the santa cruz conference on finite groups

The Santa Cruz Conference on

FINITE GROUPS

Volume 37

PROCEEDINGS OFSYMPOSIA INPURE MATHEMATICS

AMERICAN MATHEMATICAL SOCIETY

PROCEEDINGS OF SYMPOSIAIN PURE MATHEMATICS

Volume 37

THE SANTA CRUZ CONFERENCEON FINITE GROUPS

AMERICAN MATHEMATICAL SOCIETY

PROVIDENCE, RHODE ISLAND

1980

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS

OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT THE UNIVERSITY OF CALIFORNIASANTA CRUZ, CALIFORNIA

JUNE 25-JULY 20, 1979

EDITED BY

BRUCE COOPERSTEIN

GEOFFREY MASON

Prepared by the American Mathematical Societywith partial support from National Science Foundation grant MCS 78-24165

Library of Congress Cataloging in Publication Data

Santa Cruz Conference on Finite Groups, 1979.The Santa Cruz Conference on Finite Groups.

(Proceeding of symposia in pure mathematics; v. 37)Includes bibliographies.

1. Finite groups-Congresses. 1. Cooperstein, Bruce, 1950- II. Mason,Geoffrey. 1948- III. American Mathematical Society. IV. Series.

QA1'1.S26 1979 512'.2 80-26879 ISBN 0-8218-1440-0

1980 Mathematics Subject Classification. Primary 00A10, 20-02.Copyright © 1980 by the American Mathematical Society

Printed in the United States of AmericaAll rights reserved except those granted to the United States Government.

This book may not be reproduced in any form without the permission of the publishers.

TABLE OF CONTENTSPreface xiii

List of Participants xv

Part I: Classification theory of finite simple groupsAn outline of the classification of finite simple groups 3

DANIEL GORENSTEIN

Groups of characteristic 2-type 29

MICHAEL ASCHBACHER

Aschbacher blocks 37

RICHARD FOOTE

Some results on standard blocks 43

RONALD SOLOMON

Signalizer functors in groups of characteristic 2 type 47

RICHARD LYONS

The B-conjecture: 2-components in finite simple groups 57

JOHN H. WALTER

The maximal 2-component approach to the B(G) conjecture 67

RONALD SOLOMON

Finite groups having an involution centralizer with a 2-component ofdihedral type 71

MORTON E. HARRIS

On Chevalley groups over fields of odd order, the unbalanced groupconjecture and the B(G)-conjecture 75

MORTON E. HARRIS

Remarks on certain standard component problems and the unbalancedgroup conjecture 81

KENSAKU GOMIV

TABLE OF CONTENTS

Odd standard components 85

ROBERT GILMAN

Standard subgroups of Chevalley type of rank 2 and characteristic 2 91

1zuMI MIYAMOTO

Standard subgroups of type G2(3) 95HIROMICHI YAMADA

Open standard form problems 99LARRY FINKELSIE.,IN

Groups generated by a conjugacy class of involutions 103FRANZ TI.MMESFELD

The classification of finite groups with large extraspecial2-subgroups

STEPHEN D. SMITH

1]]

Some characterization theorems 121

SERGEI A. SYSKIN

On finite groups whose Sylow 2-subgroups are contained in uniquemaximal subgroups

BERND STELLMACHER

Groups having a selfcentralizing elementary abelian subgroup oforder 16

GERNOF STROTH

123

127

p-local subgroups 131

GEORGE GLAUBERMAN

Local analysis in the odd order paper 137

GEORGE GLAUBERMAN

Finite groups with a split BN-pair of rank one 139MICHIO SUZUKI

Finite groups of low 2-rank, revisited 149

KOICHIRO HARADA

Quasithin groups 1»GEOFFREY MASON

TABLE OF CONTENTS

Part II: General theory of groupsZusammengesetzte Gruppen: Holders Programm heute

HELMUT WIELANDT

Some consequences of the classification of finite simple groupsWALTER FEIT

Graphs, singularities, and finite groupsJOHN MCKAY

Works on finite group theory by some Chinese mathematiciansHsio-Fu TUAN

The prime graph components of finite groupsJ. S. WILLIAMS

7T-solvability and nilpotent Hall subgroupsZvi ARAD AND DAVID CHILLAG

On maximal subgroups with a nilpotent subgroup of index 2ZVI ARAD, MARCEL HERZOG AND AHIEZER SHAKI

Automorphisms of nilpotent groups and supersolvable ordersANTHONY HUGHES

A short survey of Fitting classesALAN R. CAMINA

Transfer theoremsTOMOYUKI YOSHIDA

Problem areas in infinite group theory for finite group theoristsGILBERT BAUMSLAG

Classification theorems for torsionfree groupsLASZLO G. KOVACS

Part III: Properties of the known groupsProperties of the known simple groups

GARY M. SEITZ

The root groups of a maximal torusGARY M. SEITZ

Geometry of long root subgroups in groups of Lie typeBRUCE COOPERSTEIN

Vii

161

175

183

187

195

197

201

205

209

213

217

225

231

239

243

viii TABLE OF CONTENTS

S and F-pairs for groups of Lie type in characteristic two 249BRUCE COOPERSTEIN

Geometric questions arising in the study of unipotent elements 255TONNY A. SPRINGER

Kleinian singularities and unipotent elements 265ROBERT STEINBERG

The construction of J4 271SIMON NORTON

Schur multipliers of the known finite simple groups. II 279ROBERT L. GRIESS, JR.

2-local geometries for some sporadic groups 283MARK A. RONAN AND STEPHEN D. SMITH

Part IV: Representation theory of groups of Lie-typeProblems concerning characters of finite groups of Lie type

CHARLES W. CURTIS

The relation between characteristic 0 representations and characteris-tic p representations of finite groups of Lie type

ROGER W. CARTER

Some problems in the representation theory of finiteChevalley groups

GEORGE LUSZTIG

293

301

313

Representations in characteristic p 319LEONARD L. SCOTT

Characters of finite groups of Lie type. 11 333BHAMA SRINIVASAN

Principal series representations of finite groups with split(B N)-pairs

ROBERT W. KILMOYER341

Cartan invariants and decomposition numbers of Chevalley groups 347JAMES E. HUMPHREYS

Duality in the character ring of a finite Chevalley group 353DEAN ALVIS

TABLE OF CONTENTS ix

Characters of projective indecomposable modules for finite Chevalleygroups

LEONARD CHASTKOFSKY

359

Some indecomposable modules of groups with split (B, N)-pairs 363NALSEY B. TINBERG

Part V: Character theory of finite groupsLocal representation theory

JON L. ALPERIN

369

Characters of solvable groups 3771. MARTIN ISAACS

Local block theory in p-solvable groups 385LLUIS PUIG

Characters of finite ¶r-separable groups 389DILIP GAJENDRAGADKAR

On characters of height zero 393MICHEL BROUE

Brauer trees and character degrees 397HARVEY 1. BLAU

A correspondence of characters 401EVERETT C. DADE

Irreducible modules forp-solvable groups 405WALTER FEIT

Finite complex linear groups of degree less than (2q + 1)/3 413PAMELA A. FERGUSON

A criterion for cyclicity 419PETER LANDROCK AND GERHARD O. MICHLER

A characterization of generalized permutation characters 423DAVID GLUCK

Character tables, trivial intersections and number of involutions 425MARCEL HERZOG

Representation theory and solvable groups: Length type problems 431T. R. BERGER

X TABLE OF CONTENTS

Part VI: CombinatoricsGroup problems arising from combinatorics

MARSHALL HALL, JR.445

Group-related geometries 457ERNEST SHULT

Near n-gons 461SAEED SHAD AND ERNEST SHULT

Orthogonal polynomials, algebraic combinatorics and sphericalt-designs

EIICHI BANNAI465

Finite translation planes and group representation 469THEODORE G. OSTROM

Finite collineation groups of projective planes containing nontrivialperspectivities

CHRISTOPH HERING473

Further problems concerning finite geometries and finite groups 479WILLIAM M. KANTOR

Part VII: Computer applicationsEffective procedures for the recognition of primitive groups

JOHN J. CANNON487

Software tools for group theory 495JOHN J. CANNON

The computation of a counterexample to the class-breadth conjectureforp-groups

VOLKMAR FELSCH503

A computer-based atlas of finite simple groups 507DAVID C. HUNT

Finding the order of a permutation group 511JEFFREY S. LEON

Part VIII: Connections with number theory and other fieldsModular functions 521

ANDREW P. OGG

TABLE OF CONTENTS

A finiteness theorem for subgroups of PSL(2, R) which arecomensurable with PSL(2, Z)

JOHN G. THOMPSON

Characters arising in the monster-modular connectionPAUL FONG

Modular functions and finite simple groupsLARISSA QUEEN

Euclidean Lie algebras and the modular function jJAMES LEPOWSKY

Exposition on an arithmetic-group theoretic connection via Riemann'sexistence theorem

MICHAEL FRIED

Burnside ring of a Galois group and the relations between zetafunctions of intermediate fields

DALE HUSEMOLLER

Finite automorphism groups of algebraic varietiesDALE HUSEMOLLER

Transformation groups and representation theoryTED PETRIE

Lie algebras with nilpotent centralizers1. MARTIN ISAACS

xi

533

557

561

567

571

603

611

621

633

PREFACE

In the last year or so there have been widespread rumors that group theory isfinished, that there is nothing more to be done. It is not so.

While it is true that we are tantalizingly close to that pinnacle representing theclassification of finite simple groups, one should remember that only by reach-ing the top can one properly look back and survey the neighboring territory. Itwas the task of the Santa Cruz conference not only to describe the tortuousroute which brings us so close to the summit of classification, but also to chartout more accessible paths-ones which might someday be open to the generalmathematical public.

A third concern was the elucidation of topics in related fields, and it is to oneof these three areas that the papers in this volume are devoted.

Just a quick glance at the table of contents will reveal a wide variety of topicswith which the modern group theorist must contend. Some of these, for example,the connections with the theory of modular functions, have very recent origins,but they leave us with the clear impression that, far from being dead, grouptheory has only just come of age.

Geoffrey MasonChicago, June 1980

xiii

LIST OF PARTICIPANTSNAMEJonathan AlperinHerbert AlwardBruno AndriamanalimananaZvi AradMichael AschbacherGeorge AvruninEiichi BannaiBernd Baumann

Dave Benson

Thomas BergerHarvey BlauMichel BroueFrancis BuekenhoutAlan Robert CaminaNeville CampbellJohn CannonAndrew ChermakDavid ChillagArjeh M. Cohen

Michael Collins

John M. ConwayBruce CoopersteinCharles W. CurtisEverett C. DadeStephen DavisAlberto DelgadoClifton Ealy, Jr.Yoshimi EgawaPaul FanWalter FeitPamela FergusonLarry Finkelstein

AFFILIATIONUniversity of ChicagoUniversity of OregonLehigh UniversityBar-Ilan University, IsraelCalifornia Institute of TechnologyUniversity of Massachusetts, AmherstOhio State UniversityUniversity of Bielefeld,

Federal Republic of GermanyTrinity College, University of Cambridge,

EnglandUniversity of Minnesota, MinneapolisNorthern Illinois UniversityCNRS, University of Paris VII, FranceFree University of Brussels, BelgiumUniversity of East Anglia, EnglandUniversity of California, Santa CruzUniversity of Sydney, AustraliaUniversity of MinnesotaTechnion-Israel Institute of Technology, IsraelMathematisch Centrum, Amsterdam,

The NetherlandsUniversity of Oxford, England, and

California Institute of TechnologyUniversity of Cambridge, EnglandUniversity of California, Santa CruzUniversity of OregonUniversity of Illinois, Urbana-ChampaignOhio State UniversityUniversity of California, BerkeleyNorthern Michigan UniversityOhio State UniversityUniversity of California, BerkeleyYale UniversityUniversity of MiamiWayne State University

xv

xvi

Bernd Fischer

Paul FongRichard FooteMike FriedDaniel FrohardtStephen M. Gagola, Jr.Dilip Gajendragadkar

Robert GilmanGeorge GlaubermanAndrew M. GleasonDavid GoldschmidtKensaku GomiDaniel GorensteinRobert GriessFletcher GrossRobert GuralnickMartin GutermanJonathan HallMarshall HallKoichiro HaradaMorton HarrisJohn HaydenMarcel HerzogJames HumphreysDavid Hunt1. Martin IsaacsDavid JacksonPeter JohnsonWilliam KantorOtto Kegel

Hiroshi KimuraLaszlo KovacsBurkhard Kuelshammer

Peter LandrockJeffrey S. LeonHenry LeonardJames LepowskyRobert LieblerRichard LyonsAvinoam MannNarendra Lal MariaGeoffrey MasonVictor Mazurov

LIST OF PARTICIPANTS

University of Bielefeld,Federal Republic of Germany

University of Illinois at Chicago CircleUniversity of Cambridge, EnglandUniversity of California, IrvineWayne State UniversityTexas A & M UniversityTata Institute of Fundamental Research,

Bombay, IndiaStevens Institute of TechnologyUniversity of ChicagoHarvard UniversityUniversity of California, BerkeleyUniversity of TokyoRutgers University, New BrunswickUniversity of Michigan, Ann ArborUniversity of UtahCalifornia Institute of TechnologyTufts UniversityMichigan State UniversityCalifornia Institute of TechnologyOhio State UniversityUniversity of Minnesota, MinneapolisBowling Green State UniversityTel-Aviv University, IsraelUniversity of Massachusetts, AmherstUniversity of New South Wales, AustraliaUniversity of Wisconsin, MadisonUniversity of Cambridge, EnglandKansas State UniversityUniversity of OregonMathematics Institute, Freiburg,

Federal Republic of GermanyHokkaido University, JapanAustralian National University, AustraliaUniversity of Dortmund,

Federal Republic of GermanyAarhus University, DenmarkUniversity of Illinois at Chicago CircleNorthern Illinois UniversityRutgers University, New BrunswickColorado State UniversityRutgers University, New BrunswickHebrew University, IsraelCalifornia State College, StanislausUniversity of California, Santa CruzInstitute of Mathematics, Novosibirsk, USSR

LIST OF PARTICIPANTS Xvii

Patrick McBrideGerald McCollumJohn McKayGerhard Michler

Izumi MiyamotoPaul MurphyMichael NewmanAnni Neumann

Volker Neumann

Jorn B. Olsson

Michael O'NanDavid ParrottMartin PettetKok W. PhanHarriet PollatsekJerry PovseUdo Preiser

Alan PrinceLluis PuigLarissa QueenMark RonanPeter RowleyBetty SalzbergUlrich Schoenwaelder

William ScottGary SeitzErnest ShultDavid SibleyCharles SimsJustine SkalbaStephen SmithRonald SolomonTonny A. SpringerBhama SrinivasanRobert SteinbergGernot Stroth

Hiroshi SuzukiMichio SuzukiSergei A. SyskinTsunj-to Tai

University of Michigan, Ann ArborHarvard UniversityConcordia UniversityUniversity of Essen,

Federal Republic of GermanyYamanashi University, JapanCalifornia Polytechnic State UniversityAustralian National University, AustraliaChristian-Albrechts-University of Kiel,

Federal Republic of GermanyEberhard-Karls-University of TObingen,

Federal Republic of GermanyUniversity of Dortmund,

Federal Republic of GermanyRutgers University, New BrunswickUniversity of Adelaide, AustraliaTexas A & M UniversityUniversity of Notre DameMt. Holyoke CollegeUniversity of California, BerkeleyUniversity of Bielefeld,

Federal Republic of GermanyHeriot-Watt University, ScotlandCNRS, University of Paris VII, FranceUniversity of Cambridge, EnglandUniversity of Illinois at Chicago CircleUniversity of Birmingham, EnglandNortheastern UniversityUniversity of Aachen,

Federal Republic of GermanyUniversity of UtahUniversity of OregonKansas State UniversityPennsylvania State University, University ParkRutgers UniversityKansas State UniversityUniversity of Illinois at Chicago CircleOhio State UniversityUniversity of Utrecht, The NetherlandsClark UniversityUniversity of California, Los AngelesRuprecht-Karl-University of Heidelberg,

Federal Republic of GermanyOhio State UniversityUniversity of Illinois, Urbana-ChampaignInstitute of Mathematics, Novosibirsk, USSRUniversity of California, Berkeley

Xvul

Olga Taussky-ToddAlvin I. ThalerFranz G. Timmesfeld

Nalsey TinbergHsio-Fu TuanDavid WalesJohn WalterMichael WardHelmut Wielandt

Bette WarrenRobert WilsonSia K. WongHiromichi YamadaHiroyoshi YamakiTomoyuki Yoshida

LIST OF PARTICIPANTS

California Institute of TechnologyNational Science FoundationUniversity of Cologne,

Federal Republic of GermanySouthern Illinois University, CarbondalePeking University, People's Republic of ChinaCalifornia Institute of TechnologyUniversity of Illinois, Urbana-ChampaignUniversity of UtahEberhard-Karls-University of TUbingen,

Federal Republic of GermanyState University of New York, BinghamtonRutgers University, New BrunswickOhio State UniversityUniversity of Tokyo, JapanOhio State UniversityHokkaido University, Japan

PART I

Classification theory of finite simple groups

Proceedings of Symposia in Pure MathematicsVolume 37, 1980

AN OUTLINE OF THE CLASSIFICATIONOF FINITE SIMPLE GROUPS

DANIEL GORENSTEIN

1. Introduction. My aim here is to present a brief outline of the classificationof the finite simple groups, now rapidly nearing completion. The major steps inthe classification will be discussed in greater detail by many authors within thesePROCEEDINGS and it is hoped that this outline will help to provide a cohesiveoverview of their individual articles as well as the subject of finite simple groupsitself.

For expository purposes, I shall divide the outline into four parts:(A) special classification theorems,(B) general classification theorems,(C) underlying techniques,(D) the remaining open problems.A classification theorem is considered to be general or special according as its

hypothesis does or does not carry over to all subgroups and homomorphicimages. This distinction is not to be taken too literally, for a special classificationtheorem often becomes general by a slight rewording of its hypothesis (theproperty of a group having dihedral Sylow 2-subgroup is not strictly speakinggeneral, but that of a group having dihedral or cyclic Sylow 2-subgroups is). Inmaking the division, I have been concerned primarily with providing what I feltwould be the clearest picture of the global classification theorem.

Likewise the distinction between a classification theorem and an underlyingtechnique is often blurred, for today's classification theorem becomestomorrow's basic tool. Bender's classification of groups with a strongly em-bedded subgroup or Timmesfeld's root involution theorem are good illustrationsof this point. However, such theorems clearly have a different flavor fromGlauberman's ZJ- or Z*-theorem or from the signalizer functor theorem, and Ihave tried to preserve this distinction in my division.

To keep the focus as sharp as possible, I shall follow a skeletal format,carefully stating the main results in each category, but limiting myself to very

1980 Mathematics Subject Classification Primary 20D05, 20-02.

0 American Madenatical Society t980

3

4 DANIEL GORENSTEIN

brief comments. In general, I shall make no attempt to assign credits for theresults I state. A (?) before the statement of a theorem will indicate that someaspect of its proof is incomplete at the present time. This will be amplified in §5,where I discuss the remaining open problems.'

One of the main purposes of the American Mathematical Society SummerConference on Finite Groups has been to stimulate what has come to be calleda "revisionist" approach to the classification of simple groups. The fact is thatover the years, very few individuals have paused to look back at any phase ofthe evolving proof-Helmut Bender being the notable exception; most of us havebeen too intent on rushing towards the finish line. But a work of the magnitudeof the classification, covering as it does many thousands of journal pages,deserves, indeed demands, continued attention. At the minimum, such a reex-amination is required to eliminate the substantial number of "local" errors eitherknown to exist or as yet undiscovered in the present proof. Given its inordinatelength, the existence of such local errors was inevitable; but obviously they mustbe removed before the classification theorem can be regarded as fully "proved".

There are, however, two additional significant ways in which the present proofis unsatisfactory. First, because it evolved over an approximately twenty-fiveyear period, some of the early papers, still essential to the overall argument, werewritten without benefit of the ideas and results subsequently developed. Sec-ondly, because of the excessive length of most of the major papers, prior resultswere quoted whenever possible, even if this meant the saving of only a fewpages. I would guess that perhaps as many as five hundred articles are listed inthe combined bibliographies of the papers which make up the classification, andit would be a substantial job to describe their logical interconnections in detail.

Most likely, all three tasks will be carried out simultaneously. The local errorswill be eliminated in the process of reorganizing the existing proof and bringingit up to date. This will be major undertaking, for considerable effort as well asmany new mathematical insights will be required to sort out the classificationtheorem and thereby discover its "essential" core. However, nothing less thansuch a major revisionist effort will suffice to convincingly communicate to themathematical community at large the great achievement embodied in theclassification of the finite simple groups.

In §6, I shall suggest some directions and problems related to this revisionism.Others will do the same in their articles.

2. Special classification theorems. We list here eleven basic special classifica-tion theorems together with some definitions needed for their statements. Forexpository reasons, we shall assume throughout that our group G is simple or atleast that F'(G) is simple (see below for definition). However, most of theresults have slightly modified formulations without such assumptions (in a fewinstances the more general results are needed in the applications).

It should be borne in mind that several of these special results depend uponthe classification of groups of low 2-rank, the latter being part of the general

'Several additional results have been established since this article was written, thus reducing thelist of open problems required to complete the classification. These will be described in footnotes in§5. All footnotes were added in proof.

CLASSIFICATION OF FINITE SIMPLE GROUPS 5

theory. Thus the logical linkage between the special and general theorems is notcompletely straightforward.

DEFINITION. A group G is a (B, N)-pair provided(1) G = BNB for suitable subgroups B, N of G;(2)BnN=H<N;(3) W = N/H is generated by involutions w;, 1 < i < r;(4) if u; is a representative of wi in N, 1 < i < r,

then for each v E N and each i,BvBv; C BvB U Bvv,B;

and(5) B B, I < i < r.W is called the Weyl group of G and the integer r is called the (Lie) rank of G.

W is a Coxeter group and hence a Euclidean group generated by reflections.Furthermore, G is said to be split if B = (B n N)U = HU, where U is anilpotent normal subgroup of G.

THEOREM S1. If G is a simple split (B, N)-pair, then G is a group of Lie typeover GF(q) for some q.

REMARK. Thus the groups of Lie type are characterized in terms of apresentation by suitable generators and relations. A similar well-known char-acterization exists for the alternating groups.

For the sporadic groups, we have

THEOREM S2. Apart from the Fischer-Griess sporadic group Ft and Janko'sfourth group J4, each of the remaining 24 sporadic groups has a characterization asa primitive permutation group of suitable degree and (permutation) rank withsuitable one-point stabilizer.

REMARKS. The "groups" Ft and J4 can also be described as primitive permuta-tion groups of suitable degree and rank with suitable one-point stabilizers.However, existence of groups satisfying these conditions has not yet beendemonstrated. For this reason, we speak of groups satisfying the respectiveconditions as groups of type F, and type J4.

We note that several of the characterizations of Theorem S2 have requiredcomputer calculations.

Theorems S l and S2 (together with the corresponding characterization of thealternating groups) represent the means by which we ultimately "recognize" anyparticular simple group or family of groups in a given classification theorem.

DEFINITION. For any group X, the generalized Fitting subgroup F*(X) _F(X)L(X), where F(X) is the Fitting subgroup of X, the unique largest nilpotentnormal subgroup of X, and L(X) is the layer of X (E(X) in the Bendernotation), the product of all quasisimple subnormal subgroups of X, withL(X) = 1 if no such quasisimple subnormal subgroups exist. (A group K isquasisimple if K is perfect and K/Z(K) is simple.) The quasisimple factors ofL(X) are called the components of X.

2See footnote 5.

6 DANIEL GORENSTEIN

REMARK. In certain general classification theorems, one reduces only to thecase in which P(G) is simple and hence in which G is a subgroup of theautomorphism group of a simple group. For this reason we state many of ourspecial results under this assumption.

DEFINITION. If P is a p-subgroup of a group X, p a prime, and k is a positiveinteger, we set

rp,k(X) _ <Nx(Q): Q < P, m(Q) > k>.(Here m(Q) denotes the maximum rank of an abelian subgroup of Q.)

If P is Sylow in X, I'pk(X) is called the k-generated p-core of X (for p = 2, thek-generated core). If I'p,(X) < X, then any proper subgroup of X containingI'p,(X) is said to be strongly p-embedded in X (if p = 2, strongly embedded).

We combine the Bender, Aschbacher, and Holt theorems in a single state-ment.

THEOREM S3. Let G be a group with F*(G) simple, M a maximal subgroup of G,and S a Sylow 2-subgroup of G.

(1) If Fs,,(G) < M, then L2(2"), Sz(2"), or U3(2").(2) If I'S2(G) < M, then either L2(q), q odd, M, I, or J1, or else

FS1(G) < M.(3) If S < M and z is an involution of Z(S) such that whenever z8 < M for

g E G, we have g E M, then either A" or Fs,(G) < M.

Here M,, denotes the smallest of the five sporadic groups of Mathieu M11,M12, M22, M23, M24 and JI the smallest of the four sporadic groups discoveredby Janko.

REMARKS. The proof of (2), Aschbacher's proper 2-generated core theorem,utilizes the classification of groups of sectional 2-rank at most 4.

The theorem is a basic tool for deriving contradictions in the study of simplegroups. One wishes to show that a group G under investigation has somespecified property (+). Assuming (+) to be false, one constructs a maximalsubgroup M satisfying (1), (2), or (3), so that F*(G) is necessarily isomorphic toone of the listed groups and G to one of their automorphism groups. One thenobtains a contradiction by checking directly that (+) holds in each of theseautomorphism groups. (If M is strongly embedded in G and has more than oneconjugacy class of involutions, Theorem S3(l) is not needed to reach a con-tradiction since it is an easy exercise that a strongly embedded subgroupnecessarily contains exactly one class of involutions )

DEFINITION. A subgroup A of a Sylow 2-subgroup S of G is strongly closed iffor a E A4 and g E G, we have a9 E S, then a9 E A. Moreover, A is weaklyclosed if for g E G, we have A 8 < S, then A 8 = A.

THEOREM S4. Let G be a group with simple and suppose that a Sylow2-subgroup S of G possesses a nontrivial strongly closed abelian subgroup A. Theneither F'*(G) has a strongly embedded subgroup or has abelian Sylow2-subgroups. In the latter case, F*(G) Q L2(2"), J1, L2(q), q - 3, 5 (mod 8), or2G2(3" ).

Here the groups 2G2(3") denote the Ree groups of characteristic 3, defined asthe fixed points on G2(3"), n odd, of a suitable automorphism of order 2.


REMARKS. The case JAI = 2 is Glauberman's Z*-theorem, which is used in theproof. The classification of groups with abelian Sylow 2-subgroups (the caseS = A) is a corollary of the theorem. However, the subcase that S has rank 2requires the dihedral Sylow 2-group classification theorem.

The theorem is basic for the study of the fusion of 2-elements in simplegroups.

DEFINITION. An involution t of the group G is called classical if C = CG(t)possesses a subnormal subgroup L such that L/O(L) = SL2(q), q odd, or A,with t E L. (In particular, L has generalized quaternion Sylow 2-subgroups andt is the unique involution of L.)

Here O(L) denotes the unique largest normal subgroup of L of odd order,while i7 denotes the nonsplit extension of A, by Z2.

DEFINITION. A subgroup H of G is tightly embedded in G if CHI is even andH n HgI is odd for g e G - NG(H).

(This generalizes the notion of strong embedding which can be defined by thecondition that I H I is even and I H n H9 I is odd for g E G - H.)

THEOREM S5 (THE CLASSICAL INVOLUTION THEOREM). Let G be a group withF*(G) simple such that either

(a) G contains a classical involution; or(b) G contains a tightly embedded subgroup with generalized quaternion Sylow

2-subgroups.Then either(1) F*(G) is of Lie type of odd characteristic; or(2) F*(G) = Sp6(2), D4(2), M11, or M12.

REMARK. The proof of the above theorem, due to Aschbacher, uses theclassification of groups of sectional 2-rank at most 4.

DEFINITION. If t is an involution of G, a component L of CG(t) is said to bestandard in G provided:

(a) H = CG(L) is tightly embedded in G; and(b) NG(L) - NG(H) and L commutes with none of its G-conjugates.Moreover, Cw(t) is said to be in standard form with standard component L.DEFINITION. A group G with O(G) = I is said to have the B -property if for

every involution t of G, every component of Ca(t)/O(CG(t)) is the image of acomponent of CG(t) (equivalently, L(CG(t)) covers L(CG(t)/O(CG(t)))).

DEFINITION. A group X is said to be a K-group if the composition factors of Xare among the known simple groups. (Groups of type F1 and J4 are consideredto be known simple groups.)

(?) THEOREM S6 (THE STANDARD COMPONENT THEOREM). Let G be a group withF*(G) simple such that

(a) G has a standard component L;(b) L is a K-group; and(c) G has the B -property.

Then F*(G) is a K-group.

REMARKS. The list of possibilities for F*(G) includes most of the groups of Lietype of odd characteristic, most of those groups of Lie type of characteristic 2

8 DANIEL GORENSTEIN

which possess an involutory outer automorphism, the alternating groups ofdegree at least 7, and many of the sporadic groups.

Essentially the method of proof is to use condition (a) together with theZ*-theorem to construct ultimately a subgroup Go of G which satisfies theassumptions of Theorems S I or S2 (so that Go is a known simple group). ThenTheorem S3 is used to force Go = G. (In some instances, when G has more thanone conjugacy class of involutions, Thompson's so-called order formula is usedto prove I Gol = I G 1, rather than Theorem S3.)

In the applications, one requires for certain choices of L a slightly strongerresult involving a weakened form of (c) (but only when H = CA(L) has Sylow2-subgroups of order 2).

DEFINITION. A group G is said to be of component type if for some involution tof G, Ca(t)/O(CG(t)) has a component; otherwise it is of noncomponent type.

REMARK. The classical involution and standard component theorems arecrucial for the study of simple groups of component type. The remaining resultsare needed only in the analysis of groups of noncomponent type; and it willsuffice to state them under the assumption that G is simple. (We deviate fromthis in the case of the root involution theorem to allow for Fischer's groupM(24), which is not simple, but has a simple subgroup of index 2.)

DEFINITION. Let D be a conjugacy class of involutions of the group G. D issaid to be a class of root involutions if for x, y E D, I xyI = 1, 2, 4 or odd; and,moreover, if Ixyl = 4, then (xy)2 E D. If IxyI = 4 for some x, y E D, D is saidto be nondegenerate. A degenerate class of root involutions is called a class ofodd transpositions.

THEOREM S7 (ROOT INVOLUTION THEOREM). Let G be a group with F*(G)simple which is generated by a class of root involutions. Then either

(1) G is a group of Lie type of characteristic 2;(2) F*(G) = Q'(3) or Q`:(5); or(3) G = A6, J2, M(22), M(23), or M(24).

(Here J2 denotes the Hall-Janko sporadic group and M(22), M(23), and M(24),the first three sporadic groups discovered by Fischer in his study of 3-transposi-tion groups.)

Theorem S7 is used in the proof of the following result, whose hypotheseshave a more local character.

THEOREM S8 (THE ABELIAN WEAKLY CLOSED T. 1. THEOREM). Let G be a simplegroup and A a nontrivial elementary abelian 2-subgroup of G. Assume

(a) A is a T. I. group (i.e., A is disjoint from its distinct conjugates in G); and(b) A centralizes none of its distinct G-conjugates.

(In particular, (b) holds if A is weakly closed in a Sylow 2-subgroup of G.)Then either(1) G = L2(2°), Sz(2"), or U3(2");(2)GA",6<n<9; or(3) G = M22, M24, or He.

Here He denotes Held's sporadic simple group.

DEFINITION. A 2-group T is said to be of symplectic type if T has no noncycliccharacteristic abelian subgroups. (By P. Hall's theorem, T is then a centralproduct of an extra-special group and a cyclic, dihedral, quasi-dihedral, orgeneralized quaternion group.)

DEFINITION. G is said to be of GF(2)-type if for some involution t of G,F*(CG(t)) = 02(CG(t)) and is of symplectic type.

REMARK. The definition is motivated by the fact that most of the groups ofLie type defined over the prime field GF(2) possess an involution whosecentralizer has this form.

THEOREM S9. If G is a simple group of GF(2)-type, then one of the followingholds:

(1) G = L"(2), U"(2), 52,' (2), 3D4(2), 2E6(2), E6(2), E7(2), or E8(2);(2) G = L2(q), q a Fermat or Mersenne prime or 9 with q > 5, L3(3), U3(3),

L4(3), U4(3), G2(3), or S28(3);(3) G = A9; or(4) G = M11, M12, M24, J2, J3, J4, HS, He, Suz, .1, .2, M(24)', F5, F3, F2, or G

is of type F1.

Here HS denotes the Higman-Sims sporadic group, Suz Suzuki's sporadicgroup, .1 and .2 two of the three simple groups constructed by Conway from the24-dimensional Leech lattice, F5 Harada's group, F3 Thompson's group, and F2the Fischer (3, 4) -transposition group. (A group G of type F1 possesses elementst1, t2, t3 of order 2, 3, 5, respectively, such that CG(t,)/<tE> = F2, F3, and F5,respectively.)

DEFINITION. A group G is said to be of characteristic 2 type if F*(H) = 02(H)for every 2-local subgroup H of G. (A 2-local subgroup is the normalizer in G ofa nontrivial 2-subgroup of G.)

REMARKS. The definition is motivated by the fact that every simple group ofLie type defined over GF(2") is of characteristic 2 type.

A group of characteristic 2 type is necessarily of noncomponent type. Thestudy of groups of noncomponent type quickly reduces to that of groups ofcharacteristic 2 type.

DEFINITION. If S is a Sylow 2-subgroup of G, the characteristic 2-core C(G; S)of G is defined to be

C(G; S) = <NG (So): 1 So char S>.

DEFINITION. A group X is called a block if X102(X) is quasisimple or ofprime order, and X has precisely one nontrivial composition factor U within02(X) and it is contained in S21(Z(02(X))). X is called a X-block if X/02(X)L2(2"), n > 1, or A", n odd and correspondingly U is the standard or naturalmodule for X/02(X). Furthermore, if X is subnormal in the group Y, then X isa block of Y.

We combine the following two results whose proofs are closely connected.

THEOREM S lO (THE X-BLOCK THEOREM). Let G be a simple group of characteris-tic 2 type and assume either

(a) The characteristic 2 core of G is a proper subgroup; or(b) some maximal 2-local subgroup of G possesses a X- block.

10 DANIEL GORENSTEIN

Then either(1) G L2(2"), Sz(2"), U3(2"), L3(2"), or Psp4(2"); or(2) G - L2(q), q a Fermat or Mersenne prime at least 9, L3(3), L4(2), U4(2),

M1 I, M22, M23, or J3.

REMARK. Obviously if a Sylow 2-subgroup S of a group G lies in a uniquemaximal subgroup M of G, then C(G; S) < M and so is proper. Hence by thetheorem such a simple group of characteristic 2 type is necessarily a K-group.

DEFINITION. Let x be an element of odd prime order p in G and let L be acomponent of C = CG (x). We call L standard for p provided:

(a) L is of Lie type of characteristic 2;(b) either p divides the order of a Cartan subgroup of L or p = 3 and L is

defined over GF(2);(c) H = CG(L) has cyclic Sylow p-subgroups;(d) <x> is not weakly closed in a Sylow p-subgroup of C;(e) some 2-local subgroup of C has p-rank at least 3; and(f) some further technical conditions are satisfied.REMARK. It is immediate from these conditions that IH n Hg' is a p'-group

for g E G - NG(H), that NG(L) = NG(H), and that L commutes with none ofits G-conjugates. Hence the analogues of the side conditions in the definition ofa standard component for centralizers of involutions automatically hold here.

(7) THEOREM Si I (STANDARD COMPONENT THEOREM FOR ODD PRIMES). Let G bea simple group of characteristic 2 type which possesses a standard component L forsome odd prime p. Then either

(1) G is of Lie type of characteristic 2; or(2)G=F3.

REMARK. Again the method of proof is to construct a subgroup Go of G whichsatisfies the assumptions of Theorem S1. This time one uses either Theorem S3or S9 to force Go = G. (In the case of Z3 X G2(3), one shows easily that G is ofGF(2)-type and this leads to Thompson's group F3 via Theorem S9.)

3. General classification theorems. We now list the principal general classifica-tion theorems which, when all are completely proved, will together give adetermination of the finite simple groups.

The first three relate to simple groups of low 2-rank, the first being thecelebrated Feit-Thompson theorem.

THEOREM G1. Every finite group of odd order is solvable. Equivalently, everyfinite simple group (apart from the trivial groups of prime order) has even order.

THEOREM G2. If G is a simple group of 2-rank at most 2, then one of thefollowing holds:

(1) G has dihedral Sylow 2-subgroups and G = L2(q), q odd, or A,;(2) G has quasi-dihedral or wreathed Sylow 2-subgroups and G = L3(q) or

U3(q), q odd, or or(3) G = U3(4).

(A 2-group is wreathed if it is isomorphic to the wreath product Z2. '- Z2.)

REMARK. The theorem depends, of course, upon Theorem G1. Moreover, ituses the Brauer-Suzuki theorem which asserts that there is no simple group withgeneralized quaternion Sylow 2-subgroups and hence no simple group (apartfrom Z2) of 2-rank 1.

DEFINITION. The sectional 2-rank r(X) of the group X is the maximum 2-rankof a section on X.

REMARK. Clearly the sectional 2-rank of X is an upper bound of the sectional2-rank of any homomorphic image of any subgroup of X.

THEOREM G3. If G is a simple group with r(X) < 4 and 2-rank at least 3, thenone of the following holds:

(1) G is of Lie type of odd characteristic (and low Lie rank);(2) G = L2(8), L2(16), L3(4), or Sz(8);(3)GA,,,8 <n < ll;or(4) G = M12, M22, M23, J1, J2, J3, Mc, or Ly.

Here Mc and Ly denote, respectively, McLaughlin's simple group and theLyons-Sims simple group.

REMARK. If a 2-group S possesses no elementary abelian normal subgroups oforder 8, then necessarily r(S) < 4. Thus as a corollary, the theorem classifiessimple groups whose Sylow 2-subgroups have this property.

The significance of the latter condition is explained by the following proposi-tion, which requires a preliminary definition.

DEFINITION. For any group G, define 9 to be the graph whose vertices consistof the four subgroups of G (i.e., subgroups isomorphic to Z2 X Z2), with twovertices A, B of 9 connected by an edge if and only if [A, B] = 1. We say thatthe group G is connected if and only if 9 is a connected graph.

PROPOSITION 1. If G is a nonconnected group, then one of the following holds:(1) A Sylow 2-subgroup of G possesses no elementary abelian normal subgroups

of order 8; or(2) G has a proper 2-generated core.

REMARKS. Combining the proposition with Theorem S3(2) and Theorems G2and G3, we conclude that every nonconnected simple group is a K-group. Thusthe classification of simple groups is reduced to the connected case.

The property of connectedness increases the effectiveness of the so-called"signalizer functor" method. As an illustration, one can easily establish thefollowing result.

PROPOSITION 2. If G is a connected group of noncomponent type and 2-rank atleast 3 in which O(G) = 1, then G is of characteristic 2 type.

REMARKS. Thus the classification of simple groups of low 2-rank and of simplegroups with a proper 2-generated core reduces the study of simple groups to thefollowing two major cases:

(A) Connected groups of component type of 2-rank at least 3.(B) Groups of characteristic 2 type.Several general classification theorems depend upon Theorem S6, which as we

have indicated, remains to be completed. However, in all other respects these


general results have been proved. We shall therefore write (?.) before each oftheir statements to indicate that they will be completely proved once TheoremS6 is established.

The heart of the analysis of groups of component type has been the verifica-tion of the B-property, which has been an elaborate undertaking. Since theB-property automatically holds in groups of noncomponent type, it can bephrased in the following general way.

(?.) THEOREM G4. If G is a finite group with O(G) = 1, then G has theB-property.

Theorem G4 has been proved simultaneously with another result, known asthe "unbalanced group" property.

DEFINITION. A simple group K is said to be locally balanced if for anysubgroup H of Aut(K) containing K and any involution t of H, we haveO(CH(t)) = 1. Otherwise K is said to be locally unbalanced. Thus if K is locallyunbalanced, then for some such H and t, O(CH(t)) 1.

(?.) THEOREM G5. Let G be a group in which F'(G) is simple and O(CG(t))1 for some involution t of G. Then one of the following holds:

(1) is a group of Lie type of odd characteristic;(2) I(G) = A., odd; or(3) I(G) = L3(4) or He and t induces an outer automorphism on F*(G).

REMARKS. If G is a minimal counterexample to both Theorems G4 and G5, itcan be shown that G has the following properties:

(a) G is connected of 2-rank at least 3;(b) is simple;(c) every proper section of G has the B-property;

_(d) for some pair of commuting involutions x, y of G, if C = CG(x) andC = C/O(C), then C contains a component k such that O(C(y) normalizes,but does not centralize K. _

Condition (d) implies that k is locally unbalanced; the minimality of G thenyields:

(e) K is isomorphic to one of the groups of the conclusion of Theorem G5.This is the situation which must be analyzed to establish Theorems G4 and

G5.The next result, Aschbacher's so-called "component" theorem, is not, strictly

speaking, a general classification theorem. However, conceptually it is part ofthe development of the general theory and this is the natural place for it to bestated. Our precise formulation incorporates Foote's analysis of the exceptionalcases of Aschbacher's theorem.

THEOREM G6. Let G be a group of component type with F'(G) simple which hasthe B-property. Then one of the following holds:

(1) G possesses a standard component; or(2) F*(G) = Psp4(q), q odd, L4(q), q 1 (mod 8), q > 3, or U4(q), q 7

(mod 8), q > 3.

DEFINITION. For brevity, we shall say that a group G satisfying the hypothesisand conclusion of Theorem G6 is of standard type for the prime 2.

For later purposes, we rephrase the contents of Theorems G4, G5, and G6, ina form which will make explicit one of the configurations arising in their proofs.

(?*) THEOREM G7. If G is a connected simple group of component type of 2-rankat least 3, then one of the following holds:

(1) G is of standard type for the prime 2; or(2) G has a proper 2-generated core.

REMARK. Theorem S6 and the definition of standard type show that a simplegroup of standard type is necessarily a K-group. Thus combining the variousresults of this section with Theorems S6 and S3(2), we reach the half-way pointof the classification of simple groups:

(?*) THE COMPONENT TYPE CLASSIFICATION THEOREM. If G is a minimalcounterexample to the classification of all finite simple groups, then G is ofcharacteristic 2 type and all its proper subgroups are K-groups.

REMARK. We re-emphasize that this result will be completely proved onceTheorem S6 is established.

REMARK. The general shape of the analysis of simple groups of characteristic 2type was laid out by Thompson in his classification of N-groups-nonsolvablegroups in which the normalizer of every nontrivial solvable subgroup is solvable.In particular, he saw the importance of the abelian p-subgroups for odd p whichlie in a 2-local subgroup and introduced the key notion of the odd 2-localrank e(G). Moreover, the major case subdivisions of his proof (including theGF(2)-type case) have been preserved in the study of the general simple group ofcharacteristic 2 type.

DEFINITION. The 2-local p-rank m2p(G) of the group G, p an odd prime, is themaximum p-rank of a 2-local subgroup of G.

DEFINITION. The odd 2-local rank e(G) of the group G is the maximum ofm2 ,(G) asp ranges over all odd primes.

REMARKS. By a theorem of Frobenius, G has a normal 2-complement ife(G) = 0, so e(G) is always positive in a simple group.

Just as the analysis of groups of component type depended on the priorclassification of groups of low 2-rank, so also groups of characteristic 2 type oflow odd 2-local rank must also be treated separately before the general case canbe undertaken.

DEFINITION. If e(G) = 1, the group G is called thin; while if e(G) < 2, then Gis called quasithin.

THEOREM G8. If G is a thin simple group of characteristic 2 type and 2-rank atleast 3, then one of the following holds:

(1) All 2-local subgroups of G are solvable and G L2(2"), Sz(2"), U3(2"), or2F4(2)'; or

(2) G & 3D4(2) or L3(4).

(?) THEOREM G9. If G is a quasithin simple group of characteristic 2 type and2-rank at least 3, then one of the following holds:

(1) G is thin;

(2) G is of Lie type over GF(2") of Lie rank 2; or(3) G = L4(2), L5(2), SP6(2), U5(2), PsP4(3), M22, M23, M24, J3, or G is of

type J4.

REMARKS. Thus the "typical" quasithin simple group is a group of Lie type ofcharacteristic 2 of Lie rank 1 or 2.

Theorems G8 and G9 both make use of Theorems S9 and S 10.Thus one is reduced to the study of simple groups G of characteristic 2 type

with e(G) > 3. Just as the signalizer functor method (for the prime 2) was usedto analyze the "general" simple group of component type, so likewise one canuse signalizer functors for odd primes effectively in the case e(G) > 3.

We need the following terminology.DEFINITION. 8k(G) = (p: m2,(G) > k).REMARK. Thus if e(G) > 3, then 83(G) 0.DEFINITION. If e(G) > 4, set a(G) = /34(G); while if e(G) = 3, seta(G) = (p: p E /33(G),p > 7) if /33(G) (1 (3, 5);a(G) = (5) if /33(G) c (3, 5) and 5 E /33(G); anda(G) _ (3) if /33(G) (3).REMARK. For technical reasons, it is preferable when e(G) = 3 to work with

the set a(G) rather than /33(G) itself.The next result shows that the "general" simple group of characteristic 2 type

has one of three possible structures. We need the following definition.DEFINITION. A simple group G of characteristic 2 type with e(G) > 3 will be

said to be of a-uniqueness type provided the following conditions hold for eachprimep E a(G):

(a) Some maximal 2-local subgroup M(p) of G is "essentially" stronglyp-embedded in G; and

(b) if H is any 2-local subgroup of G such that H n M(p) has noncyclicSylow p-subgroups, then H < M(p).

REMARK. The term "essentially" is the basis of the concept of almost stronglyp-embedded, which represents a slight weakening of strong p-embedding. Thedefinition is too technical to give in complete detail, but includes the followingconditions:

(a) I'P2(G) < M(p) for some P E Sylp(G); and(b) if M(p) is not strongly p-embedded, then M(p) is solvable and either

p > 5 and P is abelian or p = 3 and P = Z3 '_ Z3-

(?) THEOREM G10. If G is a simple group of characteristic 2 type with e(G) > 3and all proper subgroups K-groups, then one of the following holds:

(1) G is of standard type with respect to some prime p in a(G);(2) G is of GF(2)- type; or(3) G is of a-uniqueness type.

REMARKS. The special cases of Theorems G8, G9, and G10 in which G is anN-group were obtained by Thompson. (However, alternative (1) of TheoremG10 does not arise under the N-group hypothesis.)


Since simple groups of standard type with respect to some odd prime aredetermined by Theorem S11 and those of GF(2)-type are determined by Theo-rem S10, we are left to analyze those of a-uniqueness type.

Aschbacher has analyzed this last case. We can phrase his results as follows.

(?) THEOREM G11. If G is a simple group of characteristic 2 type with e(G) > 3and all proper subgroups K-groups, then G is not of a-uniqueness type. Moreprecisely, one of the following holds:

(1) G is of GF(2)-type;(2) G possesses a nontrivial abelian T. I. subgroup which commutes with none of

its G-conjugates; or(3) some maximal 2-local subgroup of G is a X-block.

REMARK. Theorems S8, S9, and S10 thus determine the possibilities for G. Onethen checks easily that none of these is of a-uniqueness type.

Combining all our general classification theorems, we obtain our objective:

(?) MAIN CLASSIFICATION THEOREM. Every finite simple group is either isomor-phic to one of the known simple groups or is of type F1 or 4.3

REMARK. I should point out that not every Special or General classificationtheorem used in the course of the present proof of the Main ClassificationTheorem has been listed explicitly. In particular, I have omitted the statementsof the classification of groups of each of the following types:

(a) With Sylow 2-subgroups isomorphic to those of(b) With Sylow 2-subgroups of order at most 210.(c) With Sylow 2-subgroups of nilpotency class 2.(d) Of order relatively prime to 3.(e) With so-called product fusion (i.e., whose 2-fusion is similar to that in the

direct product of two groups).I have deliberately limited myself to what I consider to be the major steps in

the overall classification proof in the hope of best illuminating its line ofdevelopment. I would anticipate that an efficient revisionist proof would incor-porate some of the unlisted results into the body of the argument, while derivingothers as corollaries of more general theorems.

4. The principal underlying techniques. As is to be expected, the theoremsstated above depend heavily upon a great many technical results concerningsimple groups and finite groups in general. Some of these represent generalmethods for analyzing configurations of subgroups, others are theorems whichapply to more specific situations. In this section, we shall briefly describe themain techniques.

Ti. The theory of exceptional characters. This theory has played an essentialrole in the study of groups of odd order, groups with dihedral Sylow 2-sub-groups, and split (B, N)-pairs of rank 1.

T2. The theory of modular characters. The primary uses of the theory haveoccurred in the classification of groups with quasi-dihedral and wreathed Sylow2-groups, the and in the study of many of the sporadic groups.

3J4 can now be deleted from statement. See footnote 5.


In general, the aplications of character theory to the classification theoremhave been limited to situations in which either a Sylow 2-subgroup has been"small" in some sense or in which there existed a T. I. subgroup whosenormalizer was "nearly" a Frobenius group.

As noted in the previous sections, the construction of so-called "uniquenesssubgroups" underlies the effort to prove the existence of proper 2-generatedp-cores. The thrust of such theorems is the assertion that subgroups A of aparticular shape in a group G lie in a unique maximal local subgroup of G.There have been two major approaches to this construction; they are fundamen-tal for carrying out what has come to be called "local group-theoretic analysis".We list them next.

T3. Bender's theory of (p, q)-uniqueness subgroups. This theory developedfrom Bender's simplication of Chapter IV of the Feit-Thompson odd orderpaper. It plays a critical role in Bender's recent simplification of the dihedralSylow 2-group classification as well as in Goldschmidt's strongly closed abeliantheorem and his proof of the solvable signalizer functor theorem.

T4. The signalizer functor theorem and the signalizer functor method. Signalizerfunctors constitute a fundamental tool for analyzing the structure of thep'-coresof the centralizers of elements of prime order p in simple groups. They havebeen used throughout the study of simple groups and, in particular, in provingthe B-property and the study of simple groups of characteristic 2 type withe(G) > 3.

T5. Glauberman's ZJ-theorem and related results on characteristic subgroups ofp-groups for odd primes p. This theorem, which is used throughout the subject, isespecially important in Bender's approach to the construction of uniquenesssubgroups.

T6. Alperin's theory of local control of fusion. Alperin's theorem and somerelated results form the basis for analyzing the fusion of p-elements in simplegroups, especially groups of low p-rank, in which case it can often be used toforce the structure of a Sylow p-subgroup. (It is most effective for the prime 2since it can then be combined with the and Thompson's fusionlemma.)

T7. Fischer's theory of geometric local analysis. The theory developed from thestudy of a group generated by a conjugacy class of involutions in whichconditions are imposed on the product of every pair of elements in the class. Notonly has it led to the discovery of five sporadic groups, but underlies the rootinvolution theorem and its consequences. Moreover, the theory had a strongimpact on Aschbacher in his approach to the classical involution theorem.

T8. The theory of doubly transitive and rank 3 permutation groups. Viewing agroup as a group of permutations on the (right) cosets of a subgroup is one ofthe oldest techniques in the study of finite groups. The fact that every doublytransitive group is a (B, N)-pair of rank 1 is a particular (and important)illustration. The permutation-theoretic point of view also underlies Bender'sstrong embedding theorem; in fact, the proof consists in showing that a minimalcounterexample to the theorem is doubly transitive and then analyzing this case.Furthermore, the Wielandt-D. Higman general theory of rank 3 permutation

groups (i.e. transitive permutation groups on a set 0 in which a one-pointstabilizer has exactly three orbits on 0) underlies the construction of severalsporadic groups. In addition, it plays a basic role in Fischer's study of 3-trans-position groups.

T9. The theory of computer construction of permutation groups of large degree.The existence and uniqueness of several sporadic groups has required the use ofa high speed computer. In most cases the construction has been carried out byrepresenting the group as a transitive permutation group on the cosets of asuitable known subgroup. Sims has developed a number of very sophisticatedand efficient algorithms for making such calculations.

T10. Thompson factorization and failure of factorization. This technique had itsorigins in the odd order paper and was the basis of Thompson's simplified proofof the Frobenius conjecture. The subject has had an elaborate development,with most of the subsequent applications occurring for the prime 2 in groups ofcharacteristic 2 type. If X is a group with F*(X) = OP(X) and Sylowp-subgroupP, one wants conditions on the composition factors of X enabling one to assertthat

X = Cx(Z(P))Nx(J(P)) (*)

(It turns out that this condition is considerably easier to analyze than theGlauberman condition: Z(J(P)) <X.) There are many important variations of(*)

In particular, factorizations of the type (*) are the basis of Thompson'sso-called "triple factorizations" and "three against two" argument, whichGlauberman subsequently generalized (for the prime 2) in his elegant classifica-tion of groups of order relatively prime to 3.

A substantial portion of the study of simple groups of characteristic 2 type inwhich all proper subgroups are K-groups involves the analysis of certain critical2-local subgroups in which factorizations of the type (*) fail.

T11. The theory of "pushing up" p-locals. It is this theory, which grew out ofthe work of Baumann, Glauberman, and Niles, which underlies Aschbacher'stheory of blocks. The principal difference between it and Thompson factoriza-tion is that one surveys all nontrivial characteristic subgroups of a Sylowp-subgroup of a p-constrained group X rather than just pairs of such subgroups.Again the primary application has been to groups of characteristic 2 type. Thesimplest situation involves a group X with 1'* (X) = 02(X) and F*(X/O2(X))L2(2"). If T E Sy12(X), one asks whether any nontrivial characteristic subgroupTo of T is normal in X. The basic theorem asserts that either such a subgroup Toexists or X is block. In that result, To is a function of both T and X. However,recently Glauberman, Niles, and Campbell have proved that there is a pair ofnontrivial characteristic subgroups T1, T2, depending only upon T, one of whichwill do for To (assuming that X is not a block).

T12. Aschbacher's theory of blocks. There are many related results here, theprimary goal of the theory being the X-block theorem (Theorem S 10).

T13. Thompson's theory of weak closure of elementary 2-groups. This is atechnically very difficult method, having its origins in the N-group paper andsystematically generalized by Aschbacher. It is the major tool which he uses in


showing that there are no simple groups of a-uniqueness type (Theorem G11). Itis also the basic technique used in the analysis of thin and quasithin groups ofcharacteristic 2 type. One can view these weak closure arguments as a replace-ment for the nonexistence of triple factorization theorems of the type mentionedin T10 above. The technical difficulties arise because of the very detailedinformation one requires concerning various forms of failure of Thompsonfactorization.

Last, but not least:

T14. The general theory of K-groups. Since general classification theorems areproved inductively, the critical proper subgroups of a minimal counterexampleare always K-groups. This fact is fundamental for the analysis, which dependsupon a wide variety of properties of K-groups. We have already mentionedquestions about failure of factorization in p-constrained groups. These reduceultimately to questions about the actions of quasisimple K-groups on finite-di-mensional vector spaces over GF(p). For the construction of effective signalizerfunctors, one requires so-called "local balance" and "generational" properties ofcertain proper simple sections. The solution of standard form problems requiresa thorough knowledge of the fusion of involutions (and more generally ofelements of odd prime order). Likewise the proof of the B-property (and itsanalogue for odd primes) requires complete knowledge of the centralizers ofautomorphisms of prime order (both inner and outer) of the known simplegroups. To obtain the full list of quasisimple K-groups, one must determine theSchur multiplier of each of the known simple groups. At various places in thearguments, one also needs rather detailed information concerning the localstructure of the known simple groups. This will give an indication of the types ofproperties of K-groups that enter into the analysis.

Verification of these properties is itself an elaborate undertaking. For theclassical groups, one can often use their associated classical geometries. For theexceptional groups of Lie type (and often for the classical groups as well), oneuses the Lie-theoretic approach, which depends ultimately on results from thetheory of algebraic groups. For the alternating groups, verification is usuallycarried out by direct calculation. The same is true for the sporadic groups, but inthose cases, this is often very difficult due to their essentially irregular internalstructure.

5. The remaining open problems. We now list the problems which remain to besolved to finish the classification of the finite simple groups. It should beemphasized that work is in progress on each of these problems, at various stagesof completion.

1. Preparation of final manuscipts of announced results. A number of majorclassification theorems have been announced during the last two years. Parts ofall of the proofs have been presented in seminars and/or conferences. In somecases, complete preprints already exist (the classification of simple groups ofGF(2)-type and the X-block theorem (Theorems S9 and S 10) are examples ofthis). This is also true of Bombieri's very recent completion of the classificationof groups of Ree type, which allowed us to state the classification of simple split(B, N)-pairs of rank 1 in the form of Theorem Si. However, in several cases


(Theorems S11, G10, and G11) final manuscripts (or at least portions thereof)are still in preparation. We anticipate that these manuscripts will certainly havebeen completed by the time these PROCEEDINGS are published."

Thus, leaving aside Theorems Sll, G10, and G11, the results in §§3 and 4whose proofs are still incomplete are the following: Theorems S6, G4, G5, G7,and G9. However, we have already pointed out that Theorems G4, G5, and G7will be completely proved once Theorem S6 is established. Hence, in fact (apartfrom the work in I above), the proof of the Main Classification Theorem isreduced to completion of Theorem S6: the involution standard form theorem,and Theorem G9: the classification of quasithin simple groups. Moreover, thestatement of the Main Classification Theorem incorporates the as yet unsettledquestion of the existence and uniqueness of the Janko sporadic group J4 and ofthe Fischer-Griess sporadic group Ft, so the full classification of the finitesimple groups requires resolution of these problems as well.5

We shall comment briefly on the present state of affairs of these openquestions.

II. Remaining involution standard form problems. Let G and L satisfy thehypotheses of Theorem S6, so that F'(G) is simple, L is standard, L is aK-group, and G has the B-property. In every unresolved case, CG(L) has Sylow2-subgroups of order 2. At this writing, there are precisely five individual choicesand one infinite family of choices for L which remain to be completed:L = 2F4(2)', F2, U6(2), k(22), U4(3), or 2F4(2"), n odd, n > 1.6 (Here thedenotes the nonsplit extension by Z2 of the corresponding simple group.) Inaddition, the manuscripts for two other choices of L (L4(3) and F) whosesolutions have been announced remain to be completed.

Moreover, only a single one of these standard form problems (L GN U4(3))remains to be settled to complete the proof of the B-property for finite groups(Theorem G4) and to obtain a classification of all locally unbalanced simplegroups (Theorem G5).' The remaining solutions are needed to assert that aminimal counterexample to the classification of simple groups is of characteristic2 type (with all proper subgroups K-groups).

The methods for analyzing such standard form problems is well understoodand the open cases for L are quite analogous to other cases which have alreadybeen solved, differing only in their technical details. Each of these open cases ispresently being investigated. A fuller discussion of involution standard formproblems appears in L. Finkelstein's article, these PROCEEDINGS.

III. Quasithin simple groups. G. Mason has been working for over two yearsnow on the classification of quasithin simple groups, attempting to generalizeAschbacher's thin group analysis. At this writing the bulk of the work is

'This is indeed the case: some preprints have been distributed, while the remaining manuscriptsare in the process of being typed.

5S. Norton and others in Cambridge, England have now proved the existence and uniqueness of agroup of type J4, using a slightly different approach from that described in IV below.

6Miyamoto and Yamada have now completed the 2F4(2"), n > 1, problem (under the assumptionthat G is simple) and Aschbacher, using prior partial results of Finkelstein, has completed the U4(3)problem.

"In view of 6, the B-conjecture and unbalanced group conjecture are now proved (modulo caveat(2) of the Final comments of this section).


completed and written up in at least preliminary form and only a few very tightconfigurations remain to be analyzed. In particular, a minimal counterexampleG to the theorem is of characteristic 2 type and the only nonsolvable composi-tion factor occurring in any of its 2-local subgroups are isomorphic to L2(4),L3(2), or L4(2). Mason's article outlines his work on the subject.8

IV. Existence and uniqueness of J4 and Ft. The possibility of the existence of anew sporadic group J4 was discovered by Janko out of his work on groups ofGF(2)-type. He considered a simple group G having an involution x such that

O2(CG(x)) is extra-special of order 213 and CG(x)/O2(CG(x)) has asubgroup of index 2 isomorphic to a nonsplit extension of M22 by Z3. Usingmodular character theory, Janko determined (uniquely) the order of such agroup G and much of its local structure. Several authors, including S. Norton,subsequently computed the full character table of a group of "type J4" (againunique). Thompson has conjectured that such a group must have a 2-modularrepresentation of degree 112. It was also conjectured that it must contain asubgroup H which is an extension U3(l 1) by an automorphism of order 2induced from the underlying field GF(112). Based on these two conjectures,Norton has nearly succeeded in constructing a group of type J4 having theadditional properties of possessing a 112-dimensional representation over GF(2)as well as a subgroup of the structure of H.9 His idea was to construct G as apermutation group on an appropriate set of vectors in the given representationspace. (This is analogous to the earlier Conway-Wales construction of thecovering group Ru of the Rudvalis group by Z2 as a group of permutations on asuitable set of pairs of vectors (v, -v) in a 28-dimensional space over Q(V--l ),where Q denotes the field of rational numbers.) Using all this information,Norton determined a pair of 112 X 112 matrices A, B with entries in GF(2),which were candidates for generators of the sought for group. The final step ofthe construction, which remains to be completed, consists in showing that thegroup generated by A and B has the same order as that of a group of type J4.Undoubtedly Norton's work would also prove the uniqueness of a group of typeJ4 satisfying the two given side conditions. However, to establish that there isonly one group of J4, it would still remain to show that every such grouppossesses a 112-dimensional GF(2)-representation and contains a subgroup ofthe structure of H.

Fischer and Griess independently studied the possibility of the existence of anew sporadic group Ft having an involution whose centralizer was a nonsplitextension of a group of type F2 by Z2. (The existence and uniqueness of a groupof type F2 by Leon and Sims did not come until later and incidentally wasobtained with the aid of a computer.) It appeared likely that any such group Gpossessed another involution y with O2(CG(y)) extra-special oforder 225 and Cr(y)/O2(CG(y)) isomorphic to Conway's group .1. Griess theninvestigated properties of a simple group G having two involutions whosecentralizers have these respective structures. Usng the Thompson order formula,

'Mason has completed the analysis of all the remaining configurations and is now preparing thefinal manuscripts of the quasi-thin classification theorem.

9See footnote 5.


he determined (uniquely) the order of G and much of its local structure. Therewas strong evidence that such a group must have an irreducible complexrepresentation of degree 196,883.10 Based on this assumption, Fischer andLivingston computed (again unique) the character table of such a group G. Veryrecently Griess has announced the existence of a new simple group of thespecified order. Although the details have not yet appeared, the construction isknown to be based on the assumed existence of such a representation of degree196,883.10 It should be emphasized that Griess' announced construction isaccomplished entirely by hand without recourse to any computer calculations.Some time earlier, Thompson had proved that there is at most one simple groupsatisfying all these conditions, but undoubtedly Griess' work will yield the sameresult. Presumably (as in the case of J4) full uniqueness will still require showingthat every group of type F, has an irreducible complex representation of degree196,883."

Final comments. (1) Obviously, until otherwise demonstrated, one or more ofthe open standard form problems may lead to new finite simple groups and thesame applies to the quasithin situation. Depending on the nature of the internalstructure of any such new group the effect on the existing proofs couldeither be minimal or substantial. The more closely the internal structure of G"resembles" that of one of the presently known simple groups, the smaller wouldbe the impact. (In other words, what is important for the analysis is the extent towhich the addition of G' to the list of simple K-groups affects the variousproperties of K-groups needed for the classfication arguments.) However, at thevery minimum, one would be forced to solve the involution standard formproblem with G (or any of its covering groups) in the role of L. Thustheoretically at least, the process could lead to an infinite number of new simplegroups. (At present, there exists no theorem which asserts that there are at mosta finite number of as yet undiscovered simple groups.) But what is possible isone thing, what is probable is another! The most likely prediction is that thepresent list of finite simple groups is complete.

(2) Several of the recent manuscripts are so new that they have not yet beenwidely studied. Such scrutiny is necessary to insure that some significantconfiguration has not been missed by the analysis. The same remark certainlyapplies to the manuscripts being prepared under I above and to the nearlycompleted quasithin classification theorem, since together they will undoubtedlycomprise well over 1,000 preprint pages.

(3) In the past year, there has emerged an as yet unexplained connectionbetween the "group" F1 and classical elliptic function theory. Several of the

1°Griess has since presented the details of his construction. His group F, acts as a group ofautomorphisms of a certain algebra of dimension 196884 (rather than 196883) which he mustconstruct as an integral part of his analysis.

"In view of the results listed in the preceding footnotes (and again subject to caveat (2) below),the Main Classification Theorem will be completely proved once the following four involutionstandard form problems are solved: L z 2F4(2)', F2, U6(2), and M(22).

There will then remain only the single additional problem of proving the uniqueness of a group oftype F]; equivalently, showing that such a group must have an irreducible representation of degree196883. We emphasize that this second problem is independent of the Main Classification Theorem.


papers in these PROCEEDINGS will discuss aspects of this intriguing interrelation-ship. Certainly one of the exciting problems of simple group theory, which maywell remain after the classification is completed, is to find a satisfactoryexplanation for this relationship.

6. Revision of the classification theorem. Ideally "revisionism" should involvenew ideas and approaches to simplify one or more portions of the classificationand thereby deepen our understanding of simple groups. At the present timethere are four such "new directions" which are at some stage of exploration.

(1) The investigations of the relationship between the group F1, classicalelliptic modular functions, and infinite dimensional Lie algebras will leadhopefully to a coherent explanation of the approximately 20 sporadic groupswhich are embedded one way or another inside Fl.

(2) Several group-theorists and geometers are attempting to extend Tits'building and apartment characterization of the groups of Lie type (of Lie rank atleast 3) in the hope of finding a natural generalized geometry which encom-passes essentially all simple groups. At this point, it is not clear whether theapproach will lead to alternate constructions of any of the sporadic groups, butat the least it should provide us with a more uniform picture of the finite simplegroups.

(3) A different kind of geometric analysis is evolving out of Goldschmidt'swork on the generation of groups by 2-local subgroups which "resemble" theminimal parabolics of a group of Lie type of characteristic 2. Its origins lie inearlier graph-theoretic work of Tutte and Sims' analysis of primitive permuta-tion groups in which a one-point stabilizer has an orbit of length 3. Theobjective of this approach is to bring a geometrical viewpoint into the localanalysis of simple groups at a much earlier stage than is now done. The existingresults have already had significant application, enabling one to eliminatecertain difficult minimal configurations occurring in the study of groups ofcharacteristic 2 type. However, it is too early to predict the extent to which theseideas can be pushed.

(4) G. Mason has suggested an approach to simple groups of characteristic 2type modeled on the methods which Aschbacher and he used in studying thinand quasithin groups. The idea is to focus immediately on the 2-local structure,the aim being to establish preliminary uniqueness theorems for certain 2-locals,maximal under a suitable ordering, which are strong enough to allow one tointroduce weak closure arguments much earlier in the analysis than at present.The underlying philosophy is to view a group of characteristic 2 type in terms ofits "maximal parabolics". In contrast, the present approach to the "general"group G of characteristic 2 type (i.e., with e(G) > 3) concentrates first on thecentralizers of suitable elements of odd prime order, analyzing them by means ofthe signalizer functor method. Thus its basic viewpoint is that such a group isbuilt up from the centralizers of its "semisimple" elements. A priori, themaximal parabolic and semisimple element perspectives are equally valid. How-ever, it is again too early to tell whether Mason's approach can be successfullycarried through.

These developments give a clear indication that the study of simple groupswill remain a vital subject long after the classification. Of course, such vitality isalready fully evident from the many basic questions concerning simple groupswhich the classification does not answer: their ordinary and modular representa-tions, their maximal subgroups, to name but a few of the more fundamentalones. Many of the articles in these PROCEEDINGS will deal with these topics.

My concern here, however, is on revisions of the existing classification proof,something I have been thinking about for the past year. The starting point hasbeen the observation that there exists a considerable similarity between the waythe signalizer functor method is used to study centralizers of involutions in theproof of the B-property and the way it is used to study the centralizers ofelements of odd prime order in groups of characteristic 2 type. Moreover, inboth instances the ultimate objective is to produce either an element whosecentralizer is in standard form or a suitable "uniqueness" subgroup (in the firstcase, a proper 2-generated core and in the second, a a-uniqueness subgroup). Iwas also struck by the fact that in the odd prime situation the analysis wasachieved without the availability of analogues of two of the major underpinningsof the proof of the B-property: namely, the classification of groups of sectional2-rank < 4 and the classical involution theorem. This suggested three naturalquestions:

(R1) Does there exist a proof of the B-property which parallels the odd primeanalysis and thus avoids use of these two major (and lengthy) classificationtheorems?

(R2) Beyond this, is it possible to consolidate the two analyses into a singleargument which would therefore be valid for all primes simultaneously?

(R3) Is it possible to treat at least some of the standard form problems forcentralizers of involutions and centralizers of elements of odd prime ordersimultaneously?

There is some evidence that a positive answer to all three questions may beattainable. For example, Harada has recently made some striking progresstowards incorporating portions of the sectional 2-rank < 4 analysis into thebroader study of centralizers of involutions in simple groups. My own pre-liminary investigations of (R1) with Lyons (part of which was discussed in oneof the special sessions of the conference) indicate not only that Harada's workcan be pushed further, but that an affirmative answer to (R1) is within reach.Furthermore, some of the early standard form proofs (due to W. Wong andPhan) for centralizers of involutions with standard component of Lie type ofodd characteristic, which were established prior to the classical involutiontheorem, follow a pattern very similar to that of the standard form proofs forcentralizers of elements of odd prime order with standard component of Lietype of characteristic 2. Hence it seems entirely reasonable to seek a uniformtreatment of these two cases.

If such a revision could be achieved, it would certainly provide a moreconceptual and somewhat more compact proof of the classification theorem.The following table describes the nine major components of such a prooftogether with some of their logical connections.

A POSSIBLE FORMAT OF A REVISED CLASSIFICATION PROOF

Underlying methods

Internal characterizations

of the known simple groups

groups of2-rank < 2

groups of characteristic2 type with e(G) < 2

The trichotomy theoremfor minimal counterexample G to classification:

(1) G is of standard type for some prime p;(2) G is of GF(2)-type; or(3) G is of uniqueness type for some set of primes rr.

1groups ofuniqueness typefour={2}

groups ofGF (2)-type

groups ofstandard typefor some prime p

groups ofuniqueness typefor rr = {a set of odd primes}

We have separated out the uniqueness case for 17 = (2) and 17 = (a set of oddprimes) because the present proofs are completely distinct and the prospects offinding a uniform argument covering both cases seems very remote.

Furthermore, the flow diagram of the proof is, in reality, somewhat morecomplicated than indicated by the table. Indeed, the classification of groups ofcharacteristic 2 type with e(G) < 2 uses the classification of both groups ofuniqueness type for 2 and groups of GF(2)-type. Likewise the classification ofboth groups of 2-rank < 2 and groups of standard type for some prime p usesthe classification of groups of uniqueness type for 2.

I would like now to suggest some directions for possible further simplifica-tions in the individual portions of the table (problems (R4)-(R11) below).

Much has already been done towards improving the original proofs of theclassification of simple groups of 2-rank < 2. The combined efforts of Bender,Glauberman, and Sibley have cut the Feit-Thompson odd order proof byperhaps a factor of 2, at the same time making it easier to read. Furthermore,Bender has made a much greater simplification in the original Gorenstein-Walter dihedral Sylow 2-group classification theorem. Likewise, Goldschmidt'sstrongly closed abelian theorem (Theorem S4) includes within its "Benderized"2-rank 2 analysis both Lyons' U3(4) and Brauer's Z2. x Z2., n > 1, Sylow2-group classification theorems, thus providing a completely noncharacter-theo-retic proof of these two basic results.

By an elementary argument of Alperin (together with the odd order theoremand the Brauer-Suzuki generalized quaternion theorem), a simple group of2-rank ' 2 necessarily has Sylow 2-groups which are either dihedral, quasi-di-hedral, wreathed, or isomorphic to either Z2. X Z2., n > 1, or a Sylow 2-sub-group of U3(4). There thus remains the Alperin-Brauer-Gorenstein classificationof groups G with quasi-dihedral and wreathed Sylow 2-subgroups, which has yetto be reexamined. One reason that this has not been previously attempted is thefact that in this problem one comes face-to-face for the first time with p-localsubgroups involving SLA(pm), p odd, and hence with p-locals which need not bep-stable. As is well understood, this possibility creates serious difficulties incarrying through the Bender method, related to the problem of "pushing up".However, we now have available the Glauberman-Niles pushing up theorems, sothat even without p-stability it may still be possible to "Benderize" the localanalysis arguments of the quasi-dihedral, wreathed classification theorem (whichpresently depend upon deep and extensive modular character-theoretic results ofBrauer concerning groups with quasi-dihedral or wreathed Sylow 2-subgroups).

Let me formulate the problem more precisely. Using the dihedral classificationtheorem, one shows rather easily that a minimal counterexample G to theproposed classification theorem has only oneconjugacy class of involutions andif x is an involution of G, C = CG(x), and C = C/O(C), then C has a normalsubgroup L = SL2(q), q odd.

The objective of the local analysis is to prove that G has the B-property;equivalently, that C has a normal subgroup L = SL2(q) which maps on L. (Thisis, of course, the case for the actual groups L3(q), U3(q), q odd, and M11 in theconclusion of the theorem.) Summarizing, we thus raise the following question:

(R4) Can the local analysis of groups with quasi-dihedral or wreathed Sylow2-subgroups be Benderized? Equivalently, must such a group have the B-prop-erty?

If (R4) can be achieved, it suggests the natural further question of consolida-tion:

(R5) Is it possible to give a uniform Benderized treatment of the local analysisportion of the classification of all groups of 2-rank < 2 (including groups of oddorder)?

There are two further phases of the quasi-dihedral, wreathed classificationproof that are worth reexamining. Once the B-property has been established, thegroup L is a standard component (assuming q > 3, in which case L is quasisim-ple; in the contrary case, L is solvable). We see then that the local analysis yieldsthat G is of standard type for the prime 2. At this point, it is not difficult toprove that O(C) = AB, where A is cyclic of order dividing q2 - 1 and eitherB = 1 or O(C) is a Frobenius group with complement A and kernel B.Moreover, A centralizes a Sylow 2-subgroup of C, B is a T. I. set in G, and B isinverted by an involution of C (whence B is abelian).

The next step of the classification proof is to show that, in fact, B = 1, thisbeing true in the actual groups L3(q), U3(q), M11. The present argument makesuse of Brauer's quasi-dihedral and wreathed modular character theory results.So the question arises whether the desired conclusion can be obtained, using

only the theory of exceptional characters (perhaps together with a little modularcharacter theory). Thus we have

(R6) Assuming that C is in standard form in the quasi-dihedral, wreathedproblem, does there exist an elementary proof that O(C) is cyclic and centralizesa Sylow 2-subgroup of C?

Once one has proved that B = 1, the goal of the analysis is to show that G is adoubly transitive permutation group of appropriate degree. The present argu-ment begins with the modular character-theoretically proved assertion that noelement of C induces a nontrivial field automorphism on L, which implies thatC has the form C = LAS, where S is a Sylow 2-subgroup of C, with [S, A] = 1.

It is thus natural to ask whether one can possibly avoid the use of charactertheory here by proving double transitivity without first eliminating field auto-morphisms. Thus we have

(R7) Can one extend the existing proof that G is doubly transitive to the casein which C/LAS is cyclic of odd order?

The preceding four problems were specific and specialized. In contrast, thenext idea concerns groups of standard type for the prime 2 and is both vaguerand more general. The present classification of such groups is very elaborate andextremely long. To give some substance to this observation, let me list the majorsubcases. It should be emphasized that many of these subcases themselves divideinto a number of further subcases (often consisting of a single choice for L).

Consider then a group G (having the B-property) with F*(G) simple whichpossesses a standard K-group component L and set H = CG(L). Here are themajor subcases:

(A) The 2-rank of H > 2.(B) H has generalized quaternion Sylow 2-subgroups (in which case G is

determined by the classical involution theorem).In the remaining cases a Sylow 2-subgroup T of H is cyclic. (In most of these

cases, one reduces rather easily to the case in which T = Z2.)(C) The 2-rank of L is small.The point of (C) is that certain general lines of argument break down for

small L's, forcing separate analyses in these cases. It will be understood thatthese exceptional possibilities for L are excluded from the remaining listed cases.

(D) L/Z(L) is of Lie type of odd characteristic (again G is determined by theclassical involution theorem).

(E) L is of Lie type of characteristic 2.(F) L/Z(L) is an alternating group.(G) L/Z(L) is a sporadic group.In (C)-(G) (with T = <x> = Z2), the major difficulties occur when x is not a

2-central involution. In the present proofs, one first builds a Sylow 2-subgroup Sof G containing x by pushing up various 2-local subgroups and/or consideringcentralizers of suitable elements of odd order. Then one determines the G-fusionin S in the hope of either "transferring x off" or invoking some prior classifica-tion theorem such as the root involution theorem. The process is long andsometimes painful.

I want to consider this portion of the standard form problem from a differentperspective. The first question we ask is the following:


(R8) Is it possible to organize the overall analysis of standard form problemsfor the prime 2 in such a way that the difficult cases under (C)-(G) need to behandled only under the following additional assumptions:

(a) If y is a 2-central involution of G, then C,(y) is 2-constrained; and(b) G is balanced (and connected of 2-rank at least 3)?

(The latter condition implies, of course, that O(N) = I for every 2-local sub-group N of G.)

For simplicity, let us say that G is of weak characteristic 2 type if (a) and (b)hold. Note that, together with the general property of L-balance for finitegroups, (a) and (b) imply

(c) F*(N) = 02(N) for every 2-local subgroup N of G which contains a Sylow2-subgroup of G.

REMARK. One may wish to strengthen condition (c) of the definition. Indeed,one may be able to reduce the case in which also F*(N) = 02(N) for all 2-localsubgroups N of G in which 02(N) is "sufficiently large" (in particular, if 02(N)has 2-rank at least 4).

This suggests the following question, which is our main point concerning thesestandard form problems:

(R9) Can the present analysis of groups of characteristic 2 type be extended(easily) to groups of weak characteristic 2 type?

An affirmative answer to (R8) and (R9) would greatly simplify the existingclassification proof by eliminating the necessity of treating any of the difficultstandard form problems under (C), (E), (F), and (G). (Note that (D) would notoccur under these assumptions since in that case it would follow by L-balancethat some 2-central involution has a non 2-constrained centralizer.)

There is some reason to be optimistic about (R9). First of all, the presentanalysis of groups of GF(2)-type requires the existence of only a single (neces-sarily 2-central) involution whose centralizer C has the property that F*(C) =02(C) is of symplectic type. Furthermore, the analysis of the uniqueness case forgroups of characteristic 2 type deals with a suitable (2-constrained) maximal2-local subgroup M and involves a study of the weak closure of elementaryabelian normal 2-subgroups V of M. The present arguments depend primarilyon the uniqueness properties of this single subgroup M (note that the uniquenessproperty of M includes the condition that 2-locals whose intersection with Mcontains noncyclic p-subgroups for suitable odd primes p necessarily lie in M).This indicates that (R9) may well reduce to establishing the trichotomy theoremfor groups of weak characteristic 2 type (with corresponding 2-constrainedmaximal 2-local in the uniqueness case).

I would like to mention two further problem areas. First, the analysis ofgroups of characteristic 2 type is so recent (and so lengthy) that all parts of itclearly require reexamination. I shall raise only one specific question on thissubject. The first portion of Mason's quasithin analysis establishes the existenceof a maximal 2-local subgroup M having uniqueness properties somewhatanalogous to those of the corresponding subgroup M in the a-uniqueness case ofthe e(G) > 3 situation. At this point, Mason and Aschbacher both use weakclosure arguments to help pin down the structure of M. The natural question toask is the following:


(R10) Is it possible to consolidate the weak closure arguments so that theyapply to all groups of characteristic 2 type simultaneously?

Aschbacher has already placed portions of his analysis in a sufficientlygeneral framework for them to be applicable to the quasithin case. This suggeststhe feasibility of an affirmative answer to (R10).

Finally the theory of K-groups cries out for a systematic development. Exceptfor the effort of a few individuals-notably Burgoyne and Seitz for the groups ofLie type and O'Nan for the sporadic groups-attention to K-groups has beenvery begrudging-a necessary penance required to reach the promised land of aclassification theorem. (Lyons has spent the past two years establishing close to300 preliminary lemmas, mostly about K-groups, which we need for our jointwork on groups of characteristic 2 type! The absurdity of this situation by itselfcalls for a reexamination of the proof.) Thus we have

(R11) Give a systematic treatment of the properties of K-groups required forcarrying out local analysis in simple groups.

I have pointed out earlier the types of properties of K-groups needed for suchanalysis.

In conclusion, I hope these suggestions will contribute in some measure tofurther research on finite simple groups, and ultimately to an improved versionof the classification theorem.

RUTGERS UNIVERSITY

GROUPS OF CHARACTERISTIC 2-TYPE

MICHAEL ASCHBACHERI

If my reading of various announcements and private communications iscorrect, the groups of characteristic 2-type, in which all proper simple sectionsare of known type, are, with the single exception of the quasithin groups, nowclassified. I would however like to emphasize that, while all steps but one in theclassification appear to be complete, in most cases it does not seem that adefinitive treatment of each step has been obtained. Moreover the study ofgroups of characteristic 2-type is so new that there has been little opportunity toexplore alternate methods to investigate these groups. Thus it seems to me thatthe investigation of the groups of characteristic 2-type remains one of the mostinteresting areas of finite group theory, and that it will continue to be of interestand provide ample opportunity for good mathematics for years to come.

In his article in these PROCEEDINGS, Professor Gorenstein has provided anoutline of the program to classify the finite simple groups, and has indicatedreferences for the major theorems in the program. I will supply a less detailedoutline of the subprogram to classify the groups of characteristic 2-type, discusssome of the history of the program, discuss some of the theory and techniqueswhich provide a foundation for the program, and speculate a little on how theprogram might be modified or simplified.

I will begin my discussion of the groups of characteristic 2-type by recallingsome basic terminology. A finite group G is of characteristic 2-type if

F'(M) = 02(M) for each 2-local subgroup M of G

or equivalently if CM(O2(M)) < 02(M). Given an odd prime p define the 2-localp-rank of G to be

mZ,p (g) = max{ mp(M): M is a 2-local subgroup of G }

where mp(M) denotes thep-rank of M. Another important parameter is

e(G) = max( m2,p (G): p odd prime).

1980 Mathematics Subject Classification. Primary 20D05.'Partial support supplied by the National Science Foundation.

m American Mathematical society 1980

29

30 MICHAEL ASCHBACHER

I will refer to e(G) as the rank of G. As we will see in a moment, most of theknown simple groups of characteristic 2-type are of Lie type and even character-istic. Conversely by a theorem of Borel and Tits, all groups of Lie type and evencharacteristic are of characteristic 2-type. In these groups, e(G) is a goodapproximation of the Lie rank of G.

Let 3C be the collection of known simple groups. A group G is said to be a3C-group if each simple section of G is in K. If I have not made a mistake, theknown finite simple groups of characteristic 2-type are

(1) groups of Lie type and even characteristic,(2) L2(2" ± 1), U4(3), G2(3), PSP4(3) = U4(2), U3(3) = G2(2)', 2G2(3)' = L2(8),(3) M11, M22, M23' M24' J3, J4, Cot, F3-A history of the study of groups of characteristic 2-type should probably

begin with the N-group paper. Recall that an N-group is a finite simple group inwhich each local subgroup is solvable, and that the N-groups have beenclassified by Thompson in the N-group paper. At first glance, the N-grouphypothesis is highly restrictive. There are relatively few N-groups and, at least onthe surface, the local structure of N-groups is not particularly representative ofthe local structure of simple groups in general. For example N-groups have nocentralizers with components. But a closer inspection reveals that the N-grouphypothesis is sufficiently general to require a confrontation with many of theimportant problems which must be overcome to obtain a classification of thefinite simple groups. Thompson was the first to confront these problems; thetechniques he developed to deal with them have become the basis of theprogram to classify the groups of characteristic 2-type. As we will see in amoment, all but one of the major steps in the program have their roots in theN-group paper.

Most of the material in the first twelve sections of the N-group paper is notparticularly relevent to the program to classify the groups of characteristic2-type. Preliminary lemmas aside, three major objectives are achieved. Only oneof these has a significant counterpart in the classification program: the establish-ment of certain "uniqueness theorems." For my purposes a uniqueness theoremis a statement of the form

X is contained in a unique maximal 2-local for suitable subgroups X of G.

A weak form of Thompson's uniqueness theorem may be stated as

Let p be an odd prime, Ep2 = X < M a maximal 2-local with mp(CM(X)) > 2.Then M is the unique maximal 2-local containing X.

Hence when the rank of G is at least 3, Thompson has essentially reduced towhat I should call the Uniqueness Case. This terminology will be definedprecisely later. Moreover we at once encounter the fundamental subdivision ofthe classification into the groups of rank at most 2 and the groups of rank atleast 3. The first class of groups have come to be called quasithin groups.Thompson deals with these groups in § 14-20. The groups of rank at least 3 areeliminated in § 13.

The first half of § 13 reduces the Uniqueness Case to two rather specialsubcases. These subcases must also be faced when G is quasithin, so Thompson

GROUPS OF CHARACTERISTIC 2-TYPE 31

deals with them uniformly for e(G) arbitrary in the last half of § 13. Thesesubcases are defined by the following hypotheses.

M is a maximal 2-local of G, V is a maximal normal elementary abelian2-subgroup of M. Either

(1) V has order 2, or(2) V has order 4.

The subcases reappear in the general characteristic 2 analysis in the followingguise.

M is a maximal 2-local subgroup of a group G of characteristic 2-type,V is a maximal normal elementary abelian 2-subgroup of M, and V is a TI-set

in G.

In this situation I will call V a large TI-subgroup of G. The two subcases ofthe large TI-subgroup problem considered by Thompson remain of most inter-est, with the case where V is of order 2 corresponding to the situation where02(M) is of symplectic type, i.e., 02(M) has no noncyclic characteristic abeliansubgroups.

While Thompson chose to partition his work on N-groups into the caseswhere G is of rank 1, rank 2, or rank at least 3, the analysis in the three cases isnot dissimilar. In each case Thompson produces uniqueness theorems and thenargues on the weak closure of normal elementary abelian 2-subgroups ofmaximal 2-locals to either obtain a contradiction or to pin down the structure ofsome 2-locals, and hence determine G. This approach was the one used tohandle the general Uniqueness Case and to classify the thin groups (the groupsof rank 1). It is also the approach by which G. Mason is attempting to classifythe quasithin groups.

In a moment I will display a flow diagram of the program to classify thegroups of characteristic 2-type. You will see Thompson's approach in theN-group paper duplicated, with the exception of one significant step. As Iunderstand it, that last step was provided by Gorenstein and Walter, whoobserved that, when G is of rank at least 3, Thompson used the ThompsonTransitivity Theorem, a forerunner of the signalizer functor method, to reduceto the Uniqueness Case, and that this reduction depended critically on thesolvability of p-locals for p odd. Gorenstein and Walter suggested in the generalanalysis that the signalizer functor method could be used to either produce theUniqueness Case, or to produce an element of odd prime order whose central-izer has a nice structure: a so-called standard form. The first subcase would leadto a contradiction and the second to the known groups.

The addition of this observation to Thompson's outline for N-groups givesus a flow diagram of a possible approach to the groups of characteristic 2-type.The letters A through E in the diagram indicate the five major steps, andcorrespond to the five major theorems of the program, listed after the flowdiagram. Only step E remains to be completed. Even there we have a significantpartial result. Namely the thin groups have been classified.

Flow Diagram of the program

to classify the groups of

characteristic 2-type

Groups of characteristic 2-type

e(G) < 3

0

YGroups with a

large TI-subgroup

Known groups

0Standard Form

e(G) > 3

Uniqueness Case j

-- -------------

/,lL/

Contradiction

THEOREM A. Let G be a finite simple group of characteristic 2-type witha large TI-subgroup. Then G E T.

THEOREM B. Let G be a finite simple group of characteristic 2-type withe(G) > 3 in which all proper subgroups are `3L-groups. Then one of thefollowing holds:(1) G possesses an element of odd prime order whose centralizer is instandard form.(2) G possesses a large TI-subgroup of order 2.

(3) G satisfies the Uniqueness Case.

THEOREM C. Let G satisfy the hypothesis and first conclusion of TheoremB. Then G E X.

THEOREM D. No group satisfies the hypothesis of the Uniqueness Case.

CONJECTURE E. Let G be a finite quasithin simple group of characteristic2-type in which all proper subgroups are `3C-groups. Then G E T.

Theorem A, and more generally large TI-subgroups, are discussed in S.Smith's article in these PROCEEDINGS. See Lyons' article for a discussion ofTheorem B, and Gilman's article for a discussion of Theorem C. G. Masondescribes his work on Conjecture E in his article in these PROCEEDINGS.

Theorem D has been discussed elsewhere, most particularly in my article inthe Proceedings of the Durham Conference on Finite Groups. Thus a discussionof Theorem D is restricted to the following definition of the Uniqueness Case.

Let G be a group with e(G) > 3. Define a nonempty set a(G) of odd primedivisors of the order of G by

a,(G) = (p>2:m24,(G)>3),a2(G) = (p > 5: m2,p(G) = 3),

a3(G) = (p > 3: m2,p(G) = 3),

a4(G) = (p > 3: m2,p(G) = 3),

io = min( i: ai(G) 0),a(G) = a;p(G).

UNIQUENESS CASE. G is a finite simple group of characteristic 2-type in which allproper subgroups are 3C-groups, and for each p E a(G), G possesses an almoststrongly p-embedded maximal 2-local subgroup.

For p E a(G), a maximal 2-local subgroup M of G is almost strongly p-em-bedded in G if mp(M) > 1, I'2.1(G) < M for P E Sylp(M), and one of thefollowing three conditions is satisfied.

(1) F,,p(G) < M. That is M is stronglyp-embedded in G.(2) p > 3, M is solvable, and there is a subgroup P0 of P of order p, weakly

closed in P with respect to G, such that L = E(C0(Po)) = L2(p"), n > 2,L 4 M, OD (M)C0(L) is a Frobenius group with kernel 02(M) and complementC0(L), and NN(X) < M for each subgroup X of order p in P distinct from P0.

(3) p = 3, M is solvable, P = Z3 wreath Z3, I',,J(p)(G) < M, if Z3 = X < Pwith X 4 J(P) and Q is a P-invariant Sylow 2-group of 034M), then CQ(X) _Qs.Z2n.

The reader should probably ignore cases (2) and (3). They are very easy tohandle.

The solution of any mathematical problem of sufficient scope and complexityrests upon a well developed supporting theory. The classification of the groupsof characteristic 2-type certainly requires such a theory. However because of therelative youth of the subject, much of this theory has been created on the spot inan ad hoc fashion, and is not as yet in a particularly satisfactory form. Inaddition, many interesting problems have not been completely explored.

Much of this theory had its beginnings in the N-group paper. But the limitednature of the N-group hypothesis did not require a particularly sophisticatedtheory of groups of characteristic 2-type. A sophisticated theory began to make

its appearance in Gorenstein and Lyons' work on groups of 2-local 3-rank 1, inwork of Timmesfeld on weakly closed TI-sets, and in the thin group paper. Mostof the basic concepts necessary to classify the groups of characteristic 2-typeappear already, at least in a rudimentary form, in one of these papers.

Somewhat outside, but of great importance to the program, is the classifica-tion of groups generated by root involutions due to Timmesfeld, and extendingearlier work of B. Fischer and the author. Timmesfeld's root involution theoremis used at various stages to identify the groups of Lie type and even characteris-tic. In addition it plays an important role in a theory of TI-sets, which is one ofthe major segments of the theory of groups of characteristic 2-type. One of thefirst results on TI-sets is Timmesfelds classification of groups with a weaklyclosed elementary abelian TI-set. While the main theorem of this paper is quiteuseful in the program, of even greater importance are two elementary observa-tions about TI-sets which I believe were first made in the paper. First, if V is anelementary abelian TI-set in G, and A and B are conjugates of V withNA(B) 1, then the subgroup generated by A and B can be described ratherexplicitly. Second, if H < G and E is an abelian normal subgroup of H, then theset of involutions contained in G-conjugates A of V with A < H and A n E 1

form a set of root involutions of H. These observations are of importance to thetheory of weak closure which is the basis of the analysis in steps D and E.

It is my impression that weak closure is understood by almost no one. Myintroduction to the subject came from § 13 of the N-group paper. The theorythere is rather primitive, as no deep theory is required to deal with solvable2-locals. The thin group paper contains a basis for a general theory of weakclosure, while a much more detailed development is contained in a later paper ofthe author on weak closure. G. Mason is presumably also generating someresults on weak closure.

Weak closure theory depends heavily upon certain information about K-groups, most particularly information about the GF(2)-representations of Yu-groups. Such questions reappear in many other areas of characteristic 2 analysis.Here are some of the questions about GF(2)-representations of interest.

Let G be a 3C-group with 02(G) = 1, V a faithful GF(2)G-module, A anontrivial elementary abelian 2-subgroup of G. Describe <A G >, its action on V,and the embedding of A in each of the following situations:

(1) JAI I Cv(A)l > I BI I Cv(B)l foreachB <A.(2) A is of order at least 4 and quadratic on V, i.e., [V, A, A] = 0.(3) A is of order 2 and I V: Cv(A)l is "small."B. Cooperstein, G. Mason, and the author have done work on this subject.Various other results about C-groups are important. Some are too com-

plicated to describe here. Most reduce to questions about groups of Lie type.Some are established in a paper of Seitz. For example Seitz determines allgroups G such that F'(G) is of Lie type, of characteristic distinct from some oddprime p, but contains some elementary abelian p -group A with

G <CG(B): IA : BI = p).

Signalizer functors also play a big role in the proof of Theorem B. Aside fromthe more familiar theory of solvable functors, one also requires a nonsolvabletheory to be found in work of Gorenstein and Lyons and McBride.

Another important tool in the analysis of groups of characteristic 2-type is thetheory of short groups and blocks. This concept is implicit in the thin grouppaper, but only later did a nontrivial theory of short groups emerge. A group Lis short if F'(L) = 02(L), L = 02(L), L/02(L) is quasisimple or of primeorder, [L, 02(L)] < Z(02(L)), and L has a unique noncentral 2-chief factor. Theblocks of a group are its short subnormal groups; they behave much likecomponents. R. Foote, J. Hall, F. Smith, R. Solomon, and the author have donework on short groups.

The theory of short groups is intimately connected to "pushing up". Theconcept of pushing up does not seem to be well defined. Essentially oneconsiders a group G, probably finite of characteristic 2-type, a subgroup H of Gwith F'(H) = 02(H), and a Sylow 2-group T of H. One then asks variousquestions about normal subgroups of H contained in T and about the normal-izer in G of such subgroups. For example: if there are no nontrivial characteris-tic subgroups of T normal in H, what restrictions are imposed upon H and G?

To my knowledge, such questions first arose in the N-group paper, whereThompson was led to consider the situation where G is an N-group of character-istic 2-type and H a maximal subgroup of G with H/02(H) = S3 and NG(T) <I H. Thompson appealed to work of Sims and W. Wong to handle this situation.Sim's theorem used some ideas of Tutte. Glauberman extended these results anddid some early work on the case H/ 02(H) L2(2"), which was very helpful tome when I extended the work of Sims and Wong to the case H/ 02(H)L2(2"). This extension was motivated by, and sufficient for, the purposes of thethin group paper. It was clear even then, however, that stronger theorems weredesirable. Baumann and Niles established one such result independently. Nilesused the Tutte-Sims approach, while Baumann developed some new techniques.These techniques have proven to be quite powerful and have been used byGlauberman and Niles and by Campbell to make successive improvements tothe Baumann-Niles theorem.

This completes my discussion of the theory of groups of characteristic 2-type.There are however other less well defined aspects of the theory which reoccur invarious manuscripts, and which deserve more attention.

I would like to close by mentioning four problem areas which I regard asparticularly worthy of exploration. Suitable results in any of these areas wouldlead to simplifications or alternative approaches to various steps in the classifi-cation program.

(1) Triple Factorizations. Let T be a 2-group and AC(T) the collection of finitegroups G such that F'(G) = 02(G), T E Sy12(G), G = <TG>, and G has anormal subgroup K such that F*(G/K) = L2(2") and T is contained in a uniquemaximal subgroup of G. Produce three characteristic subgroups Ti, 1 < i < 3, ofT and a "nice" subcollection %(T) of AC(T), such that for each G E AC(T) -%(T), two of T,, 1 < i < 3, are normal in G.

(2) Bp-conjecture. Establish the Bp-conjecture or some suitable weak version ofthe conjecture. Recall that the Bp-conjecture asserts the following: Let G be afinite group /with Op-(G) = 1 and x an element of G of order p. Then

E(CG(x))/Op'(CG(x)) = E(CG(x))Op'(CG(x))/Op'(CG(x))'Here p is a prime.

(3) Pushing up rank 2 groups of Lie type. Let G be a group with02(G), F'(G/02(G)) of Lie type and even characteristic and Lie rank 2,T E Sy12(G), and G = <TG>. Determine the possibilities for G when no non-trivial characteristic subgroup of T is normal in G.

(4) Theory of the lattice of subgroups containing a Sylow 2-group. Let G be agroup of characteristic 2-type, T E Sy12(G), S2 the set of subgroups of Gcontaining NG(T), H E S2, and

GJZH(NG(T)) = {K E H n S2: K is maximal in H}.

Set F = {H E S2: GJZ,H(NG(T))j = 1}.(i) Determine the structure of members of F.(ii) If H, K E IF, determine the structure of <H, K>.(iii) Use (i) and (ii) to develop a theory analogous to the theory of parabolic

subgroups of groups of Lie type and even characteristic.

CALIFORNIA INSTITUTE OF TECHNOLOGY

ASCHBACHER BLOCKS

RICHARD FOOTE

The theory of Aschbacher blocks or constrained components, a recent arrivalto simple group theory, has taken a small but important role in the currentclassification program while at the same time it suggests a promising directionfor revisionists to investigate.

Current interest in this topic was sparked by Michael Aschbacher's powerfulpushing up tool, the C(G, T) Theorem, and when a number of theorems aboutblocks were produced to augment the C(G, T) Theorem it was seen that theseresults could be built into a framework applying to certain groups of characteris-tic 2 type which mirrored the theory of groups of component type. Indeed, notonly did the component theorems have direct analogues but even the proofswere similar and, in many cases, easier in the characteristic 2 situations. Also,the block standard form problems which arose in the context of the C(G, T)Theorem were solved with a measure of elegance, and so an auspicious begin-ning of this theory was achieved.

In fact most of the known simple groups of characteristic 2 type do possessblocks in standard form so the scope of the theory is sufficiently wide to suggestan optimistic but not untenable position that a classification of characteristic 2type groups can be accomplished via the theory of blocks or subgroups similarto blocks rather than the present shifting of attention to odd local subgroups.The fundamental barrier to this vision is the scarcity of means of producingblocks in (maximal) 2-local subgroups, an issue which poses challenges toimprove existing methods, to discover entirely new techniques, and perhaps evento develop a more encompassing theory of "block type subgroups" of which thisis only a prototype.

The key definition begins the elaboration on these points: a subgroup J of agroup H is called a block of H if and only if

(B1)J(<H,(B2) J = 02(J),(B3) J/ 02(J) is quasisimple, and(B4) J has a unique noncentral 2-chief factor.

1980 Mathematics Subject Classification. Primary 20D05, 20D35.m American Mathematical Society 1980

37

38 RICHARD FOOTE

If H = J, simply say J is a block. For a block J set U(J) = [J, 02(J)],U(J) = U(J)/Cu(J)(J), so one easily sees that U(J) is the noncentral 2-chieffactor and hence is an irreducible FZJ/ 02(J)-module. In many instances in theliterature the definition of a block also includes the hypothesis

(B5) the noncentral 2-chief factor of J lies in S21(Z(02(J))), but for thedevelopment of the theory this assumption is not necessary; we will see,however, that the treatment of standard blocks which do not satisfy (B5) may beapproached uniformly by the theory of maximal T.I.-sets, and so in the end weare left with the more interesting objects which do satisfy (B5). It is alsounnecessary to assume O(J) = 1-this will be forced in characteristic 2 situa-tions.

Before stating Aschbacher's C(G, T) Theorem some further notation andcomments are useful: a subgroup J of a group H is called a solvable block of Hif and only if J, H satisfy (B 1), (B2), (B4), and (B5) with (B3) replaced by

(B3)' J/ 02(J) is of prime order.For a solvable block J, U(J) = [J, 02(J)] = U(J).

As with components, the relation between blocks and the normal structure ofblocks is particularly simple:

PROPOSITION 1. Let J, J, be distinct blocks or solvable blocks of H and ifJ/ 02(J) is of prime order p, assume 00(H) = 1.

(1) [J, ill = 1,(2) If A 4J, either A C Z(J) and J/A is a block, or U(J) C A and J/A is

quasisimple o-r of prime order,(3) U(J) is abelian if and only if U(J) C- S21 (Z(02(4),(4) if U(J) is nonabelian, U(J)' = p(U(J)) is elementary abelian.

PROOF. See [4] and [10].Some specific types of blocks are of particular interest: a block J is of L2(2m)

type, m > 2, if and only if J/ 02(J) me L2(2m), U(J) is abelian, and U(J) is thenatural 2-dimensional F2 L2(2m)-module for J/ 02(J) considered as a moduleover F2; a (solvable) block J is of A. type, n > 3, if and only if J/ 02(J) = A,,,U(J) is abelian, and U(J) is the nontrivial irreducible constituent of then-dimensional natural permutation module over F2.

For any group G of even order with T E Sy12(G) let

C(G. T) = <NG (S) I I S char T>.

C(G, T) THEOREM. Let G be a finite group of characteristic 2 type, T ESy12(G), and assume C(G, T) G; then either C(G, T) is strongly embedded in Gor there is a maximal 2-local subgroup M of G and a (possibly solvable) block J ofM which is of L2(2m) type, m > 2, or A. type, n odd > 3, such that M is theunique maximal 2-local subgroup of G containing J.

PROOF. See [2]; a sharper version is given in Theorem 3 of [3].This theorem tackled the bedrock case of failure of factorization; in particu-

lar, it gave much information about characteristic 2 type groups in which a

ASCHBACHER BLOCKS 39

Sylow 2-subgroup is contained in a unique maximal subgroup or 2-local sub-group, and a complete characterization of such groups would be achieved butfor the gap left by the second conclusion. Aschbacher had, in this case, observedand used in the proof the fact that the subgroup J of M behaves like an ordinarycomponent in some respects and that the uniqueness condition on J wasanalogous to the hypothesis on the 2-components in Theorem 5 of his funda-mental paper [1] on standard form. His suggestion was, in this situation, to firstprove J<M, which is equivalent to: Vg E G, [J, J8] 1, and Vg E G - M,I CG(J) n CG(J8)1 is odd, and secondly to solve the "standard form problem"for blocks of L2(2m) and A. types.

In [5] R. Gilman had already introduced the notion of a constrained compo-nent which is a special type of block and had proved a standard form theoremfor them, but although his definitions were too restricted to apply directly toAschbacher's configuration, the essence of the solution to the first part ofAschbacher's program was there. Indeed, in order to widen the applicability ofGilman's theorem K. Harada worked on strengthening one of its tools and hetogether with M. Aschbacher and the author proved the following BalanceTheorem which further illustrates the analogue between blocks and components.

THEOREM 1. Let x be an involution in a finite group G, J a block of CG(x), K ablock of G, and assume the outer automorphism group of K/ 02(K) is solvable;then one of the following holds:

(1) J C K with U(J) C U(K),(2) K KX and J = CKK.,(x)', or(3) [J, K] = 1.

PROOF. See Theorem B of [4].In the end, however, it seemed better to write out a new proof of a Standard

Form Theorem for blocks based on Theorem 5 of [1]. This was worked on by F.Smith in [8] and established for arbitrary blocks by the author in [4].

THEOREM 2. Let G be a finite group, M a maximal 2-local subgroup of G, J ablock of M, and assume M is the unique maximal 2-local subgroup of G containingJ; then either M = G or J <M.

In the latter case J is said to be in Standard Form in G. Under the samehypothesis as Theorem 2 except that J is a solvable block R. Solomon and S. K.Wong (in [10]) and independently the author has shown that either M = G orJ(M or G/O(G) -- A8 (with J = A4).

The proofs of these theorems for blocks satisfying (B5) are quite short, beingfacilitated by the following observation which was not available in the originalstandard form proof for components.

LEMMA 1. If J is a (solvable) block with U(J)' = 1 and x is an involution inAut(J), then x centralizes an involution in J - Z(J).

PROOF. Theorem A of [4].The standard form problem for blocks of A. type was solved by R. Solomon

in [9] (a number of exceptional configurations appear when n = 6 or 8-thedetails are omitted here).

40 RICHARD FOOTE

THEOREM 3. Let G be a finite simple group and assume J is a subgroup of Gwhich is a (solvable) block of A, type, n > 3, n 6, 8, assume M = NG(J) is theunique maximal 2-local subgroup of G containing J, 02(M), and forevery involution x in 02(M), F*(CG(x)) = 02(CG(x)); then either n = 3, G = A6,MI1, L3(3), L2(p), p a Fermat or Mersenne prime, or n = 5, G = U4(2).

The final step required to complete the C(G, T) Theorem was handledrecently by R. Solomon and S. K. Wong [10] and via an independent and quitedifferent method by M. Aschbacher in [3].

THEOREM 4. Let G be a finite simple group of characteristic 2 type, J a subgroupof G which is a block of L2(2m) type, m > 2, and assume NG(J) is the uniquemaximal 2-local subgroup of G containing J; then either G = J3, M22, M23 (withm = 2), or G L3(2m), S4(2m).

Combining these results gives

COROLLARY 1. If G is a finite simple group of characteristic 2 type, T ESy12(G), and C(G, T) G, then G A6, L3(3), L2(p), p a Fermat or Mersenneprime, M22, M23, L2(2m), Sz(2m), U3(2m), L3(2m), or S4(2m), for some m > 2.

Further theorems developing the theory of blocks are proven in [4]. Thearguments are based on those of Aschbacher in [2] and [3], and in someinstances the more general context enables one to simplify his methods. It isinteresting to note that a certain amount of ordinary component theory is usedin these proofs: in particular, the Standard Form Theorem (Theorem 5 of [1])and parts of solutions to standard form problems are employed; furthermore, ifone made complete use of these particular standard form solutions and used theUnbalanced Group Theorem, the proof of Theorem 7 would be shortenedconsiderably.

Rather than, as in the Balance Theorem, looking at blocks in centralizers ofinvolutions, to establish the uniqueness condition the following developmentappears to be the proper analogue to P-(G) and the component order: for anygroup G define

93 (G) = (JIJ is a block of NG(S), S E Sy12(Ca(J/O2(J)))}.

If J,, J2 are subgroups of G which are blocks, write J, -* J2 if and only ifJ1 C J2 with U(J1) = [02(J2), J1], and for some 2-subgroup T of NG(J?), J, is acomponent of Cj (T), where - denotes the natural projection of J2 ontoJ2/02(J2). Extend ---> via chains to a partial order on 63 (G) and let 63 *(G) be

the maximal elements under this order.

THEOREM 5. Let G be a finite group of characteristic 2 type and J E 63'(G)with J/Z(J) not of L2(2m) type, for any m, then J is a block of some maximal2-local subgroup M of G and M is the unique maximal 2-local subgroup of Gcontaining J.

A useful criterion for entry into 93 (G) is

THEOREM 6. If G is a finite group of characteristic 2 type and J is a block ofsome maximal 2-local subgroup of G, then J E 63 (G).

ASCHBACHER BLOCKS 41

For blocks the relation --* is far more restrictive than the usual componentordering.

THEOREM 7. If J1, J2 are distinct blocks with Jl ---)J2, one of the following holds:(1) J1 = A,,, J2 = An+2k, and U(J) is the natural module, i = 1, 2 (i.e. J,/Z(J,)

is a block of alternating type),(2) J1 = Sp2,,(q), J2 = 0zR+2(q), n > 1, q a power of 2, and U(J,) is the natural

F9J,/ 02(J,)-module considered as a module over F21 or(3) J1 = U4(2), J2 - Z3 ' U4(3), dimF, U(J1) = 8, dimFZ U(J2) = 12.

Finally, the excluded case in Theorem 5 arises because a block of L2(2m) typecould blow up to a "block" of S2,(2') type; since Q4 '(2) = L2(2m) x L2(2m)and the smaller block "embeds along the diagonal" of the larger one (i.e. J1would be on the diagonal of J2 when conclusion (2) of Theorem 7, n = I occursif we allowed J2 = 2,Q-)), we are furnished with an analogue to the excep-tional (orthogonal group) conclusion of the ordinary Standard Form Theoremfor components (Theorem 1(2) of [1]). S. K. Wong and the author have provedthat this configuration may be eliminated in characteristic 2 type groups (al-though it does occur in Harada's sporadic group) and so establish Theorem 5 forarbitrary J (see [11]).

Recently other standard form problems have been attacked: Harada in [6] hasobtained a solution for blocks J with J/02(J) = S2 ,(2), n > 4, where U(J) isabelian and U(J) is the natural module (with a weakening of the characteristic 2type hypothesis on the ambient group achieved by Parrott-see [7]), J. Hall hascompleted the case J/02(J) = Z3 A6, U(J) = E26 which arises in 9.3 of [3] andhas worked on blocks of symplectic type (defined as in Harada's situation). Asmentioned in the introduction, moreover, if J is a block in standard form in acharacteristic 2 type group G and U(J) is nonabelian, by Proposition 1(2) and(3) J centralizes every normal abelian subgroup of the maximal 2-local M =N,(J) and as CG(J) is tightly embedded in G, every normal abelian 2-subgroupof M is a T.I.-set in G, whence F*(G) can be determined by results ofTimmesfeld, S. Smith, G. Stroth, etc. (see the article in these PROCEEDINGS by S.Smith). Moreover, if J1, J2 are blocks with J1-* J2, then U(J1)' I impliesU(J2)' 1; so whenever 63 (G) contains an element J with U(J) nonabelian theproblem of identifying F*(G) falls into this completely charted realm.

This completes the description of the foundations of a theory of blocks butleaves unanswered some important questions, the most pressing of which is howto produce blocks in (maximal) 2-local subgroups. In fact most of the knowncharacteristic 2 type simple groups do possess standard blocks, the exceptionsbeing low BN-rank groups where the maximal 2-locals are solvable or when incases such as G2(q), F4(q), 2F4(q) the only candidates for blocks in maximal2-local subgroups all have more than one noncentral 2-chief factor. In theclassical groups where a natural underlying module is available blocks may beseen easily in the end node maximal parabolic subgroups which are the stabi-lizers of one dimensional or codimension one subspaces of this module, andthese blocks are already in standard form. At present, not only the C(G, T)

42 RICHARD FOOTE

Theorem but other failure of factorization results by Glauberman, Niles, Bau-mann and others (see the articles in these PROCEEDINGS, by G. Glauberman) arethe most fruitful means of producing blocks and evidently the intimate relationbetween blocks and pushing up requires much further study. But Aschbacherblocks do arise in other contexts such as control of fusion and so differentconstructions which yield blocks may yet be discovered.

Related to the existence problem for blocks is the issue as to which blocksshould be studied, say as standard form problems; the leading contenders are, ofcourse, ones which actually occur in known simple groups so it is natural to askif there is a general method for eliminating other blocks as standard subgroups.Furthermore, what could one say if all block standard form problems weresolved (even without such strong assumptions as the ambient group being ofcharacteristic 2 type), that is, what general properties do groups G which are notof block type (i.e. ,:B (G) = 0) possess?

Finally, perhaps the definition of a block is too narrow and, for example, oneshould weaken or remove althogether the assumption (B4) on 2-chief factors(possibly assuming the noncentral 2-chief factors of J lie in S21(Z(O2(J)))-in thiscase it seems likely that many of the proofs for blocks carry through when thesechief factors are nonisomorphic FZJ/ 02(J)-modules). For blocks or generalizedblocks it would be nice also to have a theory which works in arbitrary groups(which may require balance-type theorems on the interaction between blocksand components) and whose proofs are more closely related or even integratedwith proofs of component theorems. In any event, there is much room forexploration and innovation in this new region of group theory.

REFERENCES

1. M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.2. A factorization theorem for 2-constrained groups (to appear).3. , Some results on pushing up infinite groups (to appear).4. R. Foote, Component type theorems for finite groups in characteristic 2, Illinois J. Math. (to

appear).5. R. Gilman, Components of finite groups, Comm. Algebra 4 (1976),1133-1198.6. K. Harada, On finite simple groups possessing 2-local blocks of orthogonal type (to appear).7. K. Harada and D. Parrott, On finite groups having 2-local subgroups E22.0 `:(2n, 2) (to appear).8. F. Smith, On blocks as uniqueness subgroups (preprint).9. R. Solomon, On certain 2-local blocks, Proc. London Math. Soc. (to appear).

10. R. Solomon and S. K. Wong, On L2(2")-blocks, Proc. London Math. Soc. (to appear).11. R. Foote and S. K. Wong, On certain blocks of orthogonal type, Comm. Algebra. (to appear).

UNIVERSITY OF CAMBRIDGE, ENGLAND


SOME RESULTS ON STANDARD BLOCKS

RONALD SOLOMON

The subject of Aschbacher blocks was initiated, appropriately, by M. Asch-bacher who discovered that in finite groups of characteristic 2 type certainsubgroups, which he dubbed blocks, arose naturally as obstructions to "Thomp-son factorization theorems". Moreover he observed that these blocks enjoymany of the pleasant properties of components in groups of odd characteristic.For these reasons, he proposed the development of a body of results on theseblocks, one concrete goal of which would be the determination of all simplegroups satisfying the hypotheses of the C(G, T)-Theorem, stated below. Therequired body of results was developed by Richard Foote and is described in hisarticle. Concurrently with this work, S. K. Wong and the author proved thespecific results on blocks of A"-type and L2(2")-type needed to complete theC(G, T) analysis. These results and their corollaries are the subject of thisarticle. We begin with some definitions.

DEFINITION 1. A subnormal subgroup X of a finite group M is a block of M if:(1) X = 02(X) and X = X/ 02(X) is quasisimple or cyclic of prime order.(2) W = 102(X), X1 C l1(Z(O2(X ))) and X acts irreducibly on W/ CK,(X ).DEFINITION 2. A subgroup X of a finite group G is a maximal block in G if X

is a block of M, the unique maximal 2-local subgroup of G containing X.

THEOREM 3 ([4], [9]). Let G be a finite simple group of characteristic 2 type.Suppose that X is a maximal block in G. Then either X Aa and G = A8 orM = NG(X). In the latter case we say X is a standard block in G.

DEFINITION 4. A subgroup X of a finite group G is a block of alternating typein G if:

(1) X is a maximal block in G.(2) X = X/ 02(X) = A. for some n> 3.(3) W/CW(X) is isomorphic as F2X-module to one of the following:

(a) the nontrivial irreducible constituent of the n-dimensional permutationmodule,

(b) an irreducible 4-dimensional module for X = A,

1980 Mathematics Subject Classification. Primary 20E32.m American Mathematical Society 1980

43

44 RONALD SOLOMON

DEFINITION 5. A subgroup X of a finite group G is a block of L2(2") type if:(1) X is a maximal block in G.(2) X = L2(2") for some n > 2.(3) W/Cw(X) is isomorphic to the "standard" 2n-dimensional irreducible

F2X-module.

THEOREM 6 ([3], C(G, T)-THEOREM). Let G be a finite simple group of char-acteristic 2 type and let T E Sy12(G). Suppose that

C(G, T) = <NG(To): 1 To char T) G.

Then G has either a strongly embedded subgroup or a block of alternating type forsome odd n or a block of L2(2") type.

By Theorem 3, the classification of groups satisfying the hypotheses ofTheorem 6 reduces to the classification of those groups having a standard blockof alternating or L2(2") type. This is accomplished in the following two results.

THEOREM 7 [7]. Let G be a finite simple group of characteristic 2 type having astandard block X of alternating type. Then one of the following holds:

(1) X_ = A3 and G = A6, L3(3), M, i or L2(p) for p a Fermat or Mersenne prime.(2) X = A5 and G = U4(2).(3) X = A6 and G = U4(3), M22 or Sp(6, 2).(4) X = A7, 02(X) = E16 and G = M23-(5) X = A8 and G = Q'(8, 2).(6) 02(M) = E32 and M/ 02(M) = S6.

THEOREM 8 [9]. Let G be a finite simple group of characteristic 2 type having astandard block of L2(2") type. Then one of the following holds:

(1) X = L2(4) and G = M22, M23 or J3.(2) X = L2(2"), n > 2, and G = L3(2") or PSp(4, 2").

We remark that case (6) of Theorem 7 is currently being studied by J. Hall inthe broader context of blocks of symplectic type. Its elimination was not neededfor the purposes of this work.

The following corollaries are easily inferred from Theorems 3, 6, 7 and 8.

COROLLARY 9 [9]. Let G be a finite simple group of characteristic 2 type withT E Sy12(G). Suppose that C(G, T) =# G. Then G is isomorphic to one of thefollowing groups:

(1) A6, L3(3), M,I or L2(p) for p a Fermat or Mersenne prime.(2) M22 or M23.(3) L2(2"), Sz(22n-'), U3(2"), L3(2") or PSp(4, 2") for some n > 2.

COROLLARY 10 [9]. Let G be a finite simple group of characteristic 2 type withT E Sy12(G). Suppose that T is contained in a unique maximal 2-local subgroup ofG. Then G is isomorphic to one of the following groups:

(1) L3(3), Mi I or L2(p) for p a Fermat or Mersenne prime greater than 7.(2) L2(2"), Sz(22n-') or U3(2") for some n > 2.

COROLLARY 11 [9]. Let G be a finite simple group of characteristic 2 type withT E Sy12(G). Suppose that T is contained in a unique maximal subgroup of G.

SOME RESULTS ON STANDARD BLOCKS 45

Then G is isomorphic to one of the following groups:(1) LZ(p) for p a Fermat or Mersenne prime greater than 7.(2) L2(2n), Sz(22n-1) or U3(2n) for some n > 2.

We now say a few words about the proofs of Theorems 7 and 8. For theremainder of this article the hypotheses of Theorem 7 or 8 are assumedimplicitly. As in the case of components, the fact that X is a standard block in Gimplies that CG(X) is a tightly embedded subgroup of G. The main theorems of[8] and [10] together with Lemma 2.5 of [1] yield the following result.

LEMMA 12. CG(X) is 2-closed. If E E Syl2(CG(X)) then either E = Z4 orE = E2,,, for some m > 0. If E <1>, then E is not weakly closed in M = NG(X)with respect to G.

Also, the structure of X yields the next result directly.

LEMMA 13. 0 2'(CM(X )/ C(X )) is an indecomposable F2X- module.

At this point the E16A7 case and the case X = A3 die painlessly and theremainder of the argument splits sharply into two cases. We first discuss thealternating case. The cohomology of the permutation module permits one todetermine M fairly precisely.

LEMMA 14 (A CASE). (1) M/ 02(M) = An or S..(2) Q = 02(M) is elementary abelian and has one of six possible structures as

FZX-module.

The goal is to prove that CG(x) C M for all x E Q #. If this holds, then Q iseasily seen to be a weakly closed TI-subgroup of G, in which case G = M22 bythe main theorem of [10]. Now for most x E Q # it is true that O2(CM(x)) = Q,whence easily CG(x) C M. There remain two difficult classes. Thinking of Q asthe standard permutation module for X with basis (vl, ... , these classescorrespond to x = vl + v2 and x = VI + v2 + v3 + v4. The former case followseasily once the latter case is treated. In the latter case, the key step is to provethat O2(CG(x)) C O2(CM(x)). From this it follows easily that either

(a) O2(CG(x)) = Qs*Qs X E2,, with m > 0 and Z(O2(CG(x))) C Q, or(b) n = 8 and O2(CG(x)) = Q8*Q8*Q8*Q8, or(c) CG(x) C M.

Case (b) leads to G = S2+(8, 2). In case (a), prior restrictions on the centralizersof other involutions in Q force m = 0, whence n = 5 or 6 and G = U4(2) orU4(3). As remarked above, case (c) leads to G = M22. If Q = E32 and M/ QS61 the key step in this approach fails and life becomes much less pleasant.

We now discuss the L2(2") case. We choose T E Sy12(M). By a theorem ofAschbacher [2], either T E Sy12(G) or G = J3. The nature of the desiredconclusion dictates our goal of proving that T has class 2, in which caseG = L3(2") or PSp(4, 2") by [5]. For ease of exposition we shall assume thatQ = 02(M) = 02'(CM(X)). The first step is to transfer off 2-power field auto-morphisms of X. This is accomplished by means of the following transferlemma, essentially due to Yoshida.

46 RONALD SOLOMON

LEMMA 15 [9, 2.11]. Let M be a subgroup of a finite group G with G = 02(G),G: M odd and M D 02(M)M'. Suppose that A is a normal abelian subgroup ofT E Syl2(M) with A of exponent at most 4, NG(A) C M and A of Sylow type in_tf (i.e. A 9 C M implies A x= A' for some x E M). Let S be a maximalsubgroup of T with IM: 02(M)SI = 2. Then there exists g E G - M with JA 9:A g n M1 < 2 and Ag n M 02(M)S.

COROLLARY 16 (L2(2n) CASE). Either 02'(M) = XQ or n = 2, Q E16 andG-M22orM23

For the remainder of the argument we assume that 02'(M) = XQ and that Thas class at least 3. It follows from the action of X on Q that Q has class 3 and,in particular that E , since Q/E is an indecomposable F2X-module. As Eis not weakly closed in M, the Alperin-Goldschmidt conjugation theorem [6]guarantees the existence of a subgroup Q, with Q1 of class 2, E C Q1 1T and, ifN = NG(Q,)/Q,, with 02'(N/O(N)) = Z2 or L2(2m) for some m > 2. A finalcontradiction is achieved by proving the impossibility of embedding T inNG(Ql)

REFERENCES

1. M. Aschbacher, Finite groups in which the generalized Fitting group of the centralizer of someinvolution is symplectic but not extraspecial, Comm. Algebra 4 (1976), 595-616.

2. , A pushing up theorem for characteristic 2-type groups (to appear).3. , A factorization theorem for 2-constrained groups, Proc. London Math. Soc. (to appear).4. R. Foote, Component type theorems for finite groups in characteristic 2 (to appear).5. R. Gilman and D. Gorenstein, Finite groups with Sylow 2-subgroups of class two. II, Trans.

Amer. Math. Soc. 207 (1975), 103-126.6. D. Goldschmidt, A conjugation family for finite groups, J. Algebra 16 (1970), 138-142.7. R. Solomon, On certain 2-local blocks, Proc. London Math. Soc. (to appear).8. R. Solomon and F. G. Timmesfeld, A note on tightly embedded subgroups, Arch. Math. 31

(1978), 217-223.9. R. Solomon and S. K. Wong, On L2(2") blocks, Proc. London Math. Soc. (to appear).

10. F. G. Timmesfeld, Groups with weakly closed TI-subgroups, Math. Z. 143 (1975), 243-278.

OHIO STATE UNIVERSITY

SIGNALIZER FUNCTORS IN GROUPS OFCHARACTERISTIC 2 TYPE

RICHARD LYONS

Introduction. We begin with the observation that a finite simple Chevalleygroup G defined over a field of characteristic 2 has the following property.

For every x E G of odd prime order, 02(CG(x)) = 1.

This property nearly characterizes Chevalley groups of characteristic 2 amongall Chevalley groups; typically, a Chevalley group of odd characteristic does notsatisfy though there are plenty of exceptions. One might then hope thatcould be proved for all simple groups G of characteristic 2 type; this would be avaluable step toward classifying such groups by first pinning down odd localsubgroups. As it turns out, one need look no further than M22 or J4 forcounterexamples to which are of characteristic 2 type. Nevertheless, one mayconjecture that holds for any finite simple group of characteristic 2 type withp-rank mm(G) at least 3 for some odd prime p (or at least for those x E G oforder p with mP(CG(x)) > 3).

The techniques described in this note may be thought of as aiming to prove(.), though only weaker statements are actually proved: statements of the sortthat certain subgroups of OP.(CG(x)) have odd order, for certain x E G of orderp, for certain odd primes p. Also, the statements are not proved for arbitrarysimple groups of characteristic 2 type, but only for those G which are minimalunknown simple groups (unknown as of June 1979). Perhaps someone will intime be able to relieve the weight of this massive induction hypothesis. Ofcourse, to be able to discuss signalizer functors at all, we shall have to assumethat

e(G) (= sup m2,(G)) > 3.P

Here m2,,(G) = sup{mP(H)1H < G, 02(H) 1).Our working hypothesis and notation are as follows.(H) G is a finite simple group of characteristic 2 type (i.e., all 2-local

subgroups of G satisfy F*(H) = 02(H)) in which e(G) > 3 and all proper

1980 Mathematics Subiect Classification. Primary 20D05.© American Mathematical Society 1980

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48 RICHARD LYONS

simple sections are of known type (known as of June 1979), but G is not ofknown type. Moreover, a(G) is defined as the first of the following sets of oddprimes p which is not empty.

(pI m2p(G) > 4), (pI m2 (G) = 3,p > 7), (pI m2p(G) = 3,p > 5), (3).

M. Aschbacher [3] has just announced the following basic result.

THEOREM 1 (ASCHBACHER). Assume (H). Then there exists p E a(G) for whichG does not possess a strongly p-embedded 2-local subgroup. (A subgroup M of G isstrongly p-embedded if p I I M I and NG(P) < M for every p-subgroup 1 P < M.)

Note that if G does have a strongly p-embedded 2-local M, then there existsx E M of order p with Co2(M)(x) 1 (assuming n (G) > 1), and so O2(CG(x))= O2(CM(x)) 1. Thus, Theorem 1 is a weak version of (*).

To dovetail Theorem 1 with signalizer functor analysis, we introduce thefollowing terminology. Sp,k(G) is the set of elementary abelian subgroups of Gof order p k. We make Ep 2(G) into a graph by placing an edge between D and Eiff [D, E] = 1. We let p,2(G) be the subgraph obtained by deleting the isolatedpoints. For given G and p, we select a connected component 6 of Gp 2(G) andset F02(G) = NG(G). Here G acts by conjugation on Gp.2(G). It is easily seen,using Sylow's theorem, that G permutes transitively the connected componentsof p 2(G), so F 2(G) is well-defined up to G-conjugacy. An equivalent defini-tion is to select a Sylow p subgroup P of G and set

[ 2(G) = <NG(Q)IQ < P, mp(Q) > 2, m(QCP(Q)) > 2>.

THEOREM 2 (ASCHBACHER, GORENSTEIN AND LYONS [5]). If (H) holds, thenthere exists p E a(G) for which F 2(G) does not lie in a 2-local subgroup of G.

This technical improvement of Theorem 1 is the starting point for signalizerfunctor analysis. One objective is to prove theorems of the following type:

THEOREM 3 (GORENSTEIN AND LYONS [12]). Assume (H) holds and e(G) > 4.Then either

(a) there exist p E a(G) and an element x of order p in G such that CG(x) is instandard form or

(b) there exists an involution z E G such that O2(CG(z)) is of symplectic type.

Aschbacher [4] has announced a similar result for e(G) = 3. ("Standard form"is a powerful term, and in particular includes the condition that C = CG(x) hasa normal quasisimple subgroup L E Chev(2) with CG(L) of p-rank 1. See thenotes of R. Gilman and S. Smith from this conference for pursuit of conditions(a) and (b).)

We shall not discuss Theorems 1 or 2, but rather indicate how Theorem 2 andsignalizer (and other) functors enter into the proof of Theorem 3 and relatedresults.

Continuous conjugacy functors. To exploit Theorem 2 we introduce the follow-ing notation.

GROUPS OF CHARACTERISTIC 2 TYPE 49

A conjugacy functor on Fp 2(G) is a map H - O(H) from p,2(G) to the set ofsubgroups of G such that O(H8) = O(H)8 for all H E p2(G) and g c G. O isa continuous conjugacy functor (c.c.f.) if in addition O(D) = O(E) whenever[D, E] = 1.

Obviously we have

LEMMA 4. If O is a c.c.f. on Sp,2(G), then I'p2(G) < NG(O(D)) for (any)D E (Conversely, if X is any group normalized by i'p2(G), then O(D) = X forD E 6 extends to a unique c.c.f.)

For the purposes of this note, let us call c.c.f. trivial if O2(O(D)) = 1 for anyD E G0 2(G). Then Theorem 2 and Lemma 4 give

COROLLARY 5. Assume (H). Then there exists p E a(G) such that every c.c.f. onP2(G) is trivial.

Indeed, otherwise To ,2(G) < N6(02(O(D))), a 2-local.Now we are led to formulate the following problem, which may be termed the

main problem of (simple group-theoretic) functor theory.For any simple group G and various primes p, devise c.c.f.'s on 'p,2(G) which

are obstructions to the desired local structure of G.We are concerned here only with odd p and G of characteristic 2 type,

although the first successes of this approach were for p = 2, for analyzing coresof 2-local subgroups (using a different notion of "trivia]").

If 0 is a c.c.f. on G, the condition 02(0(D)) = 1 is a variation of (*). Thus thespecific variations of (*) which can be proved this way depend on the specifictypes of c.c.f.'s which can be constructed. We now describe some.

Three examples.(A) m23(G) < 1. Gorenstein and Lyons [11] first considered (H) in the test

case m2 3(G) < 1. An idea of Gorenstein and Walter [15] was used to construct ac.c.f. for any p E a(G). As it turned out later, it was preferable in the generalcase (see below) to use a completely different construction; but the rather simplec.c.f. used here may be of interest. The idea is to look for a c.c.f. of the form

'(D) = <A-ld E D*>where Ad is a subgroup of Cd = CG(d) of the form

Ad = Q a Sylowp-subgroup of Ad,

where Ad is the product of certain p-components [14] of LP (Cd). If Ad is chosenso that the following conditions hold

(conj) Ad' = (Ad)8 for all d of order p and g E G,(sep) Lt (Ad n Ce) a AQ* for all commuting d, e of order p,(gen) Ad = <LJ*,(Ad* n Ce)l e e E #> for all d of order p and E E 6p 2(G)with [d, E] = 1,

then it is a formality that 41 is a c.c.f. Here ly,*. is the extended p-layer:Lp (X) = Op (X)Lp.(X) Op,(Cx(R )), R is a Sylow p-subgroup of LP(X ).

How is one to define Ad so that these properties hold? Since OP (Cd) < Ad,separation essentially forces one to include in Ad all p-components L of LP(Cd)

50 RICHARD LYONS

with L/OO,p(L) not locally 1-balanced [15], [13] with respect top, and thus mostgroups in Chev(p'). On the other hand one must exclude, by (gen), those L withL/Op p(L) not well generated. Thus as a reasonable first guess one should try

Ad = <L1 L is ap-component of Lp.(Cd) and L/Op p(L) V Chev(p)>,

since typically groups in Chev(p) are not well generated. (In this test casealternating and sporadic groups hardly ever appear.) In fact, this definition ofAd essentially works; the only problems arise from configurations of the follow-ing kinds. Here p = 5, d and e are commuting elements of order 5, and K and Lare products of 5-components of L5.(Cd) and L5,(CQ), respectively; J =L5.(CK(e)) = L5.(CL(d)) andJ = J/05.5(J), etc.

K L L2(45) L2(55)

J L2(4) = L2(5)or

5'5(5 times)

Sz(225) Sz(25) x x Sz(25)

Sz(25)

In the first configuration K < Ad, while L < Ae, so (sep) is violated. In thesecond, K < Ad so if (sep) is to hold we must have L < A,,; but the 5-compo-nents of L may not satisfy generation. It can be shown by a tedious ad hoctransfer argument that the first configuration forces G to have a normalsubgroup of index 5, so cannot occur. The second is more troublesome. Since dcycles the 5-components of L, <d, e> can be shown not to lie in an elementaryabelian 5-subgroup of G of rank m2 5(G). By beginning with subgroups of type(5, 5) lying in a large elementary abelian subgroup of G, and moving carefullyaround the graph 6, one can circumvent such configurations and in any caseconstruct a c.c.f.

It is a curious feature of this approach that it works so smoothly apart fromthe possibility of a very small number of such unlikely-looking but troublesomeconfigurations. Nevertheless, "accidental" embeddings of Chevalley groups ofdifferent characteristics underlie the existence of some sporadic groups, soconfigurations like the first one above cannot be shrugged off.

In summary, it is possible in this case, assuming (H), to construct a c.c.f.such that *(D) > <OP,(Cd)jd E D#>.

(B) e(G) > 4. Extending the above construction to the general case hasseveral disadvantages. For one thing, the number of possible "confused char-acteristic" phenomena is considerably larger (owing to the possibility thatp maybe 3), and there are more situations of cycled p-components, as above. As eachof these configurations is likely to require a long special argument to eliminateor circumvent it, the prospects are discouraging. More telling is the fact that theabove construction hardly uses the key hypothesis that G is of characteristic 2type. Accordingly, one tries to construct a c.c.f. O directly contradictingCorollary 5: namely, so that O(D) is a nontrivial 2-group, indeed so thatO(D) = 02(M) for some maximal 2-local subgroup of G. In this way one canget leverage from the fact that F*(M) = 02(M).

In doing this one is quickly led to the following problem. Given a group Mwith F*(M) = 02(M), an odd prime p, and D E -p,2(M), "name" 02(M) interms of D and subgroups associated with D, in a way that can be extended to aconjugacy functor in any group G in which M is a 2-local.

Using factorization theorems of Aschbacher [2], Cooperstein [7], and Bau-mann, Glauberman and Niles [6], a partial solution to this problem has beenfound when the composition factors of M are of known type [12]. The construc-tion is as follows. For any group X and D E Ep.2(X) define

I*x(D) = <02(Cx(d))Id E D#),

2x(D) = 02(Y**(D))

For any 2-subgroup T of X set

Yx(T) = T n n {O2(Nx(C))11 Cchar T},Yz+'(T) = Yx(Yz(T)),

00

y (T) = n y ;(T).n=1

Then put

Ox(D) = Yz (2x(D))Clearly Ox is a conjugacy functor on Sp,2(X), and it is trivial that 02(X) <Ox(D) for any D. The good property is

PROPOSITION 6 (GORENSTEIN AND LYONS [12]). Suppose M is a group whosesimple sections are of known type, and with F*(M) = 02(M). Let D E Ep 2(M).Then either OM(D) = 02(M), or else M has a block L with L/02(L) Q(2) orA. for some n, and U(L) = either the standard module or (if p = 3 and L/ 02(L)

A7) a module of order 16.

(A block L of M is a perfect subnormal subgroup of M such that L/ 02(L) issimple,

[L, 02(L)] < 01(Z(02(L))),and

U(L) =[L, 02(L)]/[L, 02(L)] n Z(L)is a chief factor of L.)

The exceptions are annoying, and the smoothest way to avoid them would beto have a theorem classifying finite simple groups with a maximal 2-local Mhaving a block L of the indicated type. When Proposition 6 was first proved,such a result was unavailable. Most of these blocks were avoided by using thefunctor OM(D) = OM(D) n O2(HM(D)), where HM(D) = <OP(CM(d))IIdl d ED # > and nd is the product of all p-components K of CM(d) with K/ OOP(K) _Qzn(2) or A. for some n. However, the example: M = UJ, J = Ep2+21 U thenatural GF(2)-module for J, and D E Sp 2(J) with a regular orbit in the naturalpermutation representation, shows that no functor (DM concocted out of char-acteristic subgroups of CM(d) for d E D # can universally give 'M(D) _02(M).

Fortunately, recent work of Foote, Harada, Solomon, and others on thestructure of groups with blocks of certain types has shown that if G satisfies (H)and has a maximal 2-local M with a block as in Proposition 6, then G is ofknown type. See R. Foote's note from this conference.

52 RICHARD LYONS

In fact, the frequency with which blocks have cropped up in the analysis ofgroups of characteristic 2 type suggests the following problem: devise c.c.f's (ingroups G satisfying (H)) which are obstructions to the existence of a block in amaximal 2-local subgroup of G.

Now, with Proposition 6 in mind, we return to our group G satisfying (H) andsearch for a maximal 2-local subgroup M for which OG(D) = OM(D) for allD E G. Proving this condition of course involves comparing O2(CG(d)) withO2(CM(d)) for d E D #, as well as comparing O2(NG(C)) with O2(N.(C)) forcertain D-invariant 2-subgroups C of G. These comparisons can get extremelycomplicated, and the only successful way to obtain the necessary results so farhas been to use (heavy) detailed balance and generation properties of the knownsimple groups; in other words, to avoid any shyness about using the hypothesis(H) wherever necessary. To get started, it is assumed that there is a largeelementary abelian p-subgroup B of M such that the above comparisons arefavorable for all D E 6p 2(B), and then it is shown that the comparisons remainfavorable as one moves (in a careful way) around the graph G. Specifically, wehave

THEOREM 7 (GORENSTEIN AND LYONS [12]). Assume that (H) holds, p E a(G),M is a maximal 2-local subgroup of G containing an elementary abelian p-group Bwith mp(B) = m2,p(G) > 4 and satisfying further conditions stated below. Theneither OG is a c.c.f. on Sp°2(G) (with OG(D) = 02(M) for D E °2(M)), or elseM has a block L as in Proposition 6. Moreover, in any case, there is a nontrivialc.c.f. on Gp°,2(G). (This last statement is unimportant because of the work of Footeet al.)

The further conditions are(1) Whenever q E a(G) and m29(G) > m2,p(G), then there is a nontrivial c.c.f.

on 6q,2(G);(2) If p > 5, then I'B,,,,_,(G) < M, where m = mp(B). (By definition I'Bk(G)

_ <NG(A)I A < B, mp(A) > k>.)(3) If p < 5, then <1IG(B; p'), NG(B)> < M and <Ab I b E B4> < M where

Ab is as before (Example (A)) except that now Ab is defined as the product ofthose p-components L of Lp.(CG(b)) such that some element of I4GG(b)(NB(L); p')acts nontrivially on L/OO.(L).

(C) e(G) = 3. In this case, except when a(G) = (3), Aschbacher has con-structed a c.c.f., which will be discussed below.

Signalizer functors. By Corollary 5, the c.c.f. of Theorem 7 must be trivial forsome p E a(G), and so the assumptions of Theorem 7 must fail for somep E a(G). These facts can be exploited by using signalizer functors on singleelementary abelian p-subgroups of G. The basic tool is McBride's signalizerfunctor theorem, generalizing and based on earlier theorems of Goldschmidt [9](for p = 2) and Glauberman [8] (for solvable functors). Unlike the earliertheorems, McBride's makes certain light assumptions on the composition factorsof some subgroups of G; these assumptions are satisfied in the known simplegroups.

THEOREM 8 (MCBRIDE [16]). Suppose p is an odd prime and all simple sectionsof all p-local subgroups of G are of known type. If B is an elementary abelianp-subgroup of G of rank at least 3, then any B-signalizer functor on G is complete.

Together with facts about balance in the known simple groups, this can beused in a way entirely analogous to the way (established by Gorenstein andWalter [15] and others) signalizer functors for p = 2 have been used to studycentralizers of involutions.

We use the following notation: for E E Gp k(G), set

AG(E) = n Op.(CG(e))eEE*

If B E Ep,(G) for somej > k, G is k-balanced with respect to B if

AG(E) n CG(b) < OP(CG(b))

for all b E B" and E E Sp.k(B). G is weakly k-balanced with respect to B if

[AG(E) n CG(b), B] < Op,(CG(b))

(equivalently:

AG(E) n CG(b) < Op (CG(b))OP,(CG(B)))

for all b E B" and E E F9p,k(B).

The establishment of these properties reduces in a routine way to "localbalance" properties of the p-components of CG(b), and-in the case of weakbalance-also to the question of how B permutes these p-components by conju-gation. Seitz [18] has investigated the local balance properties of Chevalleygroups. On the basis of his results, we have the following. Here imax(G; p) isdefined as the set of B E Sp k(G) where k = m2,(G) and B lies in a 2-localsubgroup of G.

LEMMA 9. Assume (H). Let p E a(G). Then:(a) For all B E 'max(G; p), G is 3-balanced with respect to B.(b) If p > 5, then for all B E 'max(G; p), G is 2-balanced with respect to B.(c) For some B 63.,,.(G; p), G is weakly 2-balanced with respect to B.

As a result of (a) and (b) the standard signalizer functor machinery [15] gives

LEMMA 10. Assume (H) and let p E a(G), and B E max(G; p). Then if p > 5,and mp(B) > 4, 02(G; B) - <AG(D)ID E Ep2(B)> is a p'-group; if mp(B) > 5and p < 5, 03(G; B) = <AG(D)l D E Ep,3(B)> is a p'-group.

Accordingly, set Mo = NG(02(G; B)) or NG(03(G; B)). From the generalmachinery, 1'8i,,_ 1(G) < M0, where m = m2,(G). An argument of Thompson[19] suitably modified and using McLaughlin's classification of groups generatedby transvections [17] shows that if 02(G; B) has even order, then M0 lies in amaximal 2-local M of G. (If p = 3 one must assume that [02(G; B), B] has evenorder.) Then from I'8iri_l(G) < M0 < M one checks that B satisfies the hy-potheses of Theorem 7. Choosing p E a(G) so that m2,(G) is largest among allp E a(G) for which c.c.f.'s are trivial, we can prove:

54 RICHARD LYONS

THEOREM 11 (GORENSTEIN AND LYONS [12]). Assume (H) and e(G) > 4. Thenthere exists p E a(G) such that

(DO)p: For all B E 'max(G; p) and D E Ep,2(B), [OG(D), B] has odd order,and if p > 3, then OG(D) has odd order.

Moreover for any prime q E a(G) for which m2 q(G) > m2 ,(G), either (00)9holds or nontrivial c.c.f.'s exist on 69,2(G).

The important statement is the first one; the second condition is of technicalvalue in advancing from this theorem to Theorem 3.

Thus, instead of establishing (*), this theorem shows that [OG(D), B] has oddorder for suitable p, B, and D.

Low rank cases. The proof of Theorem 11 in the cases p 6 5 and m2 ,(G) = 4remains to be discussed, as well as the case e(G) = 3 of (H). Here the size of Bis too small (slightly) to take advantage of the degree of balance which exists.The most effective tool in these cases seems to be a signalizer functor of the sortintroduced by Goldschmidt [10] and refined by Aschbacher [1] to a c.c.f.

THEOREM 12 (GOLDSCHMIDT AND ASCHBACHER). Suppose G is a group which isweakly k-balanced and also k + 1-balanced (in the ordinary sense) with respect tothe elementary abelian p-subgroup B of G, of rank at least k + 2. Assume that foreach b E B#, all simple sections of Op(CG(b)) are of known type. DefineOk(G; B) = < G(E)E E Gp,k(B)>, and for each D E Gp,k+l(B) set 4i(D) _<[OG(E), D]JE E Epk(D)>OG(D). Then for all D, D' E Gp,k+I(B), we have4i(D) = 4i(D') > [Ok(G; B), B], and 4i(D) is a p'-group. In particular, I'B,k+t(G)6 NG((D(D)) for any D E 9p k+I(B)

This clever result thus allows one to draw the generational conclusion onewould be able to draw by the standard machinery if it were known that G werek-balanced with respect to B.

For example, when e(G) = 3 and a(G) {3}, Aschbacher [4] establishesproperties close to weak 1-balance and 2-balance and by the proof of Theorem12 deduces that forp E a(G),

a(D) = <[ 02(CG(d)),D] Id E D#>AG(D)defines a c.c.f. in which a(D) is a 2-group for all D E 6p2(G). Hence a(D) = 1by Corollary 5, and so strong information about O2(CG(d)) is obtained. Simi-larly, the cases p 6 5, m2 ,(G) = 4 of (H) can be handled effectively this way.Finally, the case e(G) = 3, a(G) = {3} is considerably more complicated,mainly because even weak 1-balance fails. The bulk of the argument in that caseis in fact devoted to dealing with configurations arising from the failure of thisbalance condition.

REFERENCES

1. M. Aschbacher, A characterization of Chevalley groups over fields of odd order, Ann. of Math.(2) 106 (1977), 353-398.

2. _A factorization theorem for 2-constrained groups (preprint).3. , The uniqueness case for finite groups (preprint).4. , Lectures at the Institute for Advanced Study, Princeton, N. J., November, 1978.


5. M. Aschbacher, D. Gorenstein and R. Lyons, The embedding of 2-locals in finite groups ofcharacteristic 2 type, Ann. of Math. (submitted).

6. B. Baumann, Uber endliche Gruppen mil einer zu L2(2") isomorphen Faktorgruppe, Proc. Amer.Math. Soc. 74 (1979), 215-222.

7. B. Cooperstein, An enemies list for factorization theorems, Comm. Algebra 6 (1978), 1239-1288.8. G. Glauberman, On solvable signalizes functors in finite groups, Proc. London Math. Soc. 33

(1976), 1-27.9. D. Goldschmidt, Solvable signalizes functors on finite groups, J. Algebra 21 (1972), 137-148;

2-signalizes functors on finite groups, ibid., 321-340.10. , Weakly embedded 2-local subgroups of finite groups, J. Algebra 21(1972), 341-351.11. D. Gorenstein and R. Lyons, Finite groups of 2-local 3-rank at most 1, manuscript.12. , The local structure of finite groups of characteristic 2 type, manuscript.13. , Nonsolvable signalizer functors on finite groups, Proc. London Math. Soc. 35 (1977),

1-33.14. D. Gorenstein and J. Walter, The 7r-layer of a finite group, Illinois J. Math. 15 (1971), 555-565.15. , Balance and generation in finite groups, J. Algebra 33 (1975), 224-287.16. P. McBride, Nonsolvable signalizes functors on finite groups (preprint).17. J. McLaughlin, Some subgroups of Illinois J. Math. 13 (1969), 108-115.18. G. Seitz, Generation of finite groups of Lie type, Trans. Amer. Math. Soc. (to appear).19. J. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable. I, Bull. Amer.

Math. Soc. 74 (1968), 383-437.

RUTGERS UNIVERSITY


THE B-CONJECTURE: 2-COMPONENTS INFINITE SIMPLE GROUPS

JOHN H. WALTER

A perfect group L such that L/Z(L) is simple is said to be quasisimple; it issaid to be 2-quasisimple if Z(L) is a 2-group. Designate by O(X) the maximalnormal subgroup of odd order in a group X. A 2-component L of a group X is aperfect subnormal subgroup such that L/ O(L) is 2-quasisimple. A 2-componentL such that L/ O(L) has isomorphism type X will be said to have type 9C. IfL/O(L)/Z(L/O(L)) has type 6X, L will be said to have simple type 6X. Let4(X) and G (X) denote the set of involutions and the set of nontrivial elemen-tary 2-subgroups of a group X, respectively. Set E(X) to be the set of 2-compo-nents of X. Set

E(4(G)) = U {L(CG(t))I t E 4(G)I, (1)

E(G(G)) = U {e(CG(E))IE E G(G)J. (2)

The following conjecture singles out an important step in the classification offinite simple groups G with 1_' (J (Aut G)) nonempty-that is, groups of componenttype.

B-CONJECTURE. Let G be a finite group. Then the elements of E(J (G/ O(G)))are quasisimple.

The object of this note is to announce and describe a proof of this conjecture.This result plays a key role in the characterization of simple finite groups ofcomponent type by providing a first step in the analysis of the structure of thecentralizer of an involution. The approach is to utilize a signalizer functor toconstruct a proper subgroup of a minimal counterexample with fusion propertieswhich lead to a contradiction. In fact, the signalizer functor was developed todeal with this problem. In dealing with groups of even type-that is, in whichL (J (G)) is empty-the functor O(CG(t)), I E 4(G), can be used to showO(CG(t)) = 1 (cf. [11]).

Actually a slightly more general result can be proved. Let L be a 2-componentof a group X and set NL = Autx(L/O(L)) = Nx(L)/Cx(L/O(L)). Then L is

1980 Mathematics Subject Classification. Primary 20D05.m American Mathematical Society 1980

57

58 J. H. WALTER

said to be locally balanced [locally 6-balanced] if O(Cj(t))= 1 for all t EJ(NG(L)) [O(CK(E)) = 1 for all E E E(NG(L))]. It is a consequence of Pro-position 2 of [12] that if the elements of L(J(G)) are locally balanced, O(CG(t))determines a signalizer functor; from Aschbacher's theorem on groups withproper 2-generated core [1], it follows that O(CG(t)) C O(G) for all t E J (G) orS (29t3(2) = 0. The results of Gorenstein and Harada [10] provide a classifica-tion of G when S (2 IA3(2) = 0.

Thus it remains to study groups with locally unbalanced 2-components. Myfirst attempt [23] at doing this was based on a set of axioms. The aim was toplace particular restrictions on the set of locally 6 -unbalanced elements and touse only consequences of the definition when dealing with locally 6 -balancedelements. At the Sapporo conference in 1974 during a conversation with MichaelAschbacher, it was realized that this approach could be made more effective bycombining the direct construction of the signalizer functor with a classificationof groups with locally unbalanced elements in L(J(G)). In particular, Asch-bacher's characterization of the Chevalley groups as groups with an intrinsic2-component of type SL(2, q), q odd, provided the stimulus for the enlargementof the scope of the problem.

Shortly after this, John Thompson introduced a similar approach to thisproblem based on the concept of an unbalanced triple. His approach wascompleted by Burgoyne [4] who showed that the 2-component in an unbalancedtriple of a finite group could be assumed by virtue of an inductive hypothesis tobe of type GJ1ta or GJ1b described below. We refer the reader to [21] for a morecomplete description of these ideas. The critical papers developing this approachare those of Solomon [19], Gilman and Solomon [6], Foote [5] and Harris [15].

On the other hand, we continued with our methods based on ideas developedin [12] and [23]. We consider a smaller class of types of nonquasisimple2-components, and we focus on a uniform set of properties of these 2-compo-nents. The reduction to standard form is done after the quasisimplicity isproved. Then it is immediate and many of the resulting characterization prob-lems become simpler.

There are three classes of types of simple groups which yield locally unbal-anced 2-components:

''Jlta = (PSL(2, q), q odd; PSL(3, 4)),

IJit,b = (A,,, n odd and n > 7; He),

IM, = (Chev*(p),p odd).

Here Chev*(p) denotes the set of Chevalley groups and their twisted analoguesdefined over a field of characteristic p with the exception of the groupsPSL(2, p"), when p' > 3 and the groups PSL(3, 3), PSU(3, 3), 2G2(3zi+1), andG2(3).

It is not possible to deal with this class exclusively because a 2-component ofone of the above types may appear also as a 2-component of the centralizer ofan involution acting on another 2-component. Thus it may appear as a 2-compo-nent in the centralizer CG(E) of an element E of & (X). Because of this it isnatural to consider locally 6 -unbalanced 2-components. Thus we are led to a

THE B-CONJECTURE 59

larger class of 2-quasisimple groups-namely, the groups of type 4:

('o = {PSL(2, q), q odd; TL(3, 4); An, n odd and n > 7; An, n oddand n > 7; He; Chev*(q), q odd; M12, M12, J2, JZ HS, HS;

Sz; ON; Co3; Ly, 2G2(32n+1)}

Here X represents a 2-fold covering of the group X; by TL(3, 4), we mean anycentral extension of PSL(3, 4) by a 2-group; the remaining groups are sporadicgroups denoted by the usual notation.

The following theorem is the principal result of [26].

THEOREM 1. Let G be a finite group with a unique component L(G). SupposeO(G) = 1 but O(CG(E)) 1 for some E E 6(G). Then L(G) has type 4.

Because this result gives a characterization of groups with locally unbalancedelements in L(J(G)), the following corollary holds.

C'o = {PSL(2, q), q odd; TL(3, 4); An, n odd and n > 7;

An, n odd and n > 7; He; Chev*(q), q odd;odd; M121 M12;

J2, J2; HS, HS; Sz; ON; Co3; Ly; 2G2(32n+1)).

The underlying idea in the proof of Theorem 1 is to force the existence ofstandard components of type 4 in the centralizer of some involution. Thecharacterization of G then follows. These standard component problems must befaced in order to obtain an inductive approach to the argument. These resultsare worked out in [25] for a wide variety of groups. Most of the attention in [25]is given to characterizing groups of type Chev(q). However, also arguments forthe characterizations of other sporadic groups are obtained. Namely, the casesM12, M12, Sz, Sz, and -2 as well as components of type He, ON, and Co3. Thelatter three cases are obtained from a general result which we mention later. Thecases mentioned above have also been worked out and published by others. Thereader is referred to the talk of L. Finklestein at this conference. The approachto these problems which we take is simplified by assuming the B-conjecturewhich is justified as these characterizations are applied to proper sections of aminimal counterexample to Theorem 1.

Special attention is paid to the classification of groups with 2-components oftype Chev*(p), p odd. An element L E L(CG(t)) is said to be intrinsic if t E L.Aschbacher [3] obtained a classification of this class of groups from thecondition that L (J (G)) contains an intrinsic element of type SL(2, q). In thepresent paper this result is extended to classify simple groups of type Chev*(p),p odd. In the following theorem, let $(X) = {j E XI j2 E 02.,2(X)}.

THEOREM II. Let G be a finite group with a unique normal component L(G).Assume that the 2-components belonging to L($ (H/ O(H))) are quasisimple foreach proper 2-local subgroup H of G. Assume that L($(G)) contains an element oftype Chev*(p), p odd. Then L(G) has type Chev*(p) or else it has type An, n oddwhen p = 5, TL(3, 4) when p = 7, or An, n even when q = 9.

We remark that certain minimal cases arise in the proof which require

60 J. H. WALTER

independent investigation. An element L of L($(H)) is said to be anchored ifZ*(L) C Z(H). We need the following results.

PROPOSITION IIA. Let H be a group with a unique normal component L(H)which is nonsimple. Suppose that the elements of 1($ (H)) are quasisimple and thatL($(H)) contains an anchored element of type SL(2, q), q odd and q > 3. ThenL(G) has type Chev*(q), or q = 5, 7, or 9 and L(G) respectively has one of thetypes A. with n odd, TL(3, 4), or A with n even.

This result is obtained as a corollary to the general analysis of groups in whichL(G) is simple and L(J (G)) has elements of type PSL(2, q). We allow theB-conjecture in obtaining this result since it is applied to a proper subgroup of aminimal counterexample to Theorem I. The anchoring of the element of typeSL(2, q) serves merely to limit the possibilities of L(G) in the application of thisproposition to the proof of Theorem II.

When q = 3, SL(2, q) is solvable. A minimal subnormal subgroup L of agroup X is said to be a 2-component of SL(2, 3)-type if L/O(L) has typeSL(2, 3). Intrinsic 2-components of SL(2, 3)-type can be managed. Aschbacheralso showed in [3] that if L(G) is a unique normal component of a group G andthe centralizer of some involution has an intrinsic 2-component of SL(2, 3)-type,then L(G) has type Chev(3). However, nonintrinsic 2-components of centralizersof involutions of SL(2, 3)-type satisfy weaker balance conditions. As a result, forgroups of type Chev*(3), we must begin by controlling a slightly different classof groups.

PROPOSITION IIB. Let H be a group with a unique normal component L(H)which is nonsimple. Assume that the elements of f ($ (H)) are quasisimple. SupposeL($ (H)) contains an anchored element of type SU(4, 3), Q-(6, 3), Sp(4, 3), orSL(4, 3). Then L(H) has type Chev*(3).

Again this result is an immediate consequence of the characterization ofgroups G with a unique normal simple component L(G) and a standard elementof L(4 (G)) of type PSU(4, 3), PSp(4, 3) and PSL(4, 3). The reduction to thiscase is obtained by using Theorem II on H, which is available in studying2-local subgroups to a minimal counterexample to Theorem 11. Again thestipulation that elements of L($ (H)) of type SU(4, 3), Q-(6, 3), Sp(4, 3) andSL(4, 3) be anchored serves to reduce the possibilities for the type of L(G)particularly in the case when the elements have type Sp(4, 3). The characteriza-tion of simple groups with standard subgroups of type PSp(4, 3) has beenobtained by Gomi [9] together with some unpublished work of R. Foote; workon the case PSL(4, 3) is being carried out by H. Suzuki. The case PSU(4, 3) hasbeen worked out by M. Aschbacher based on an argument of L. Finklestein.

PROPOSITION IIC. Let G be a finite group with a unique normal simplecomponent L(G). Assume that the elements of L(4(G)) are quasisimple. Supposethat L(J (G)) n Chev*(3) consists of elements of type P2-(6, 3), Sl(7, 3),PSl+(8, 3), and 2-(8, 3). Also suppose that no 2-local subgroup has a component oftype Chev*(5). Then if L (J (G)) has an element of type Chev*(3), L(G) has typeChev*(3).

THE B-CONJECTURE 61

The assumption which eliminates 2-local subgroups with components of typeChev*(5) is obtainable by induction in the application of Proposition IIC to theproof of Theorem II. This proposition is proved by characterizing an S2-sub-group of L(G) in order to identify intrinsic SL(2, 3)-components. The argumentis fusion theoretic and technically the most difficult in the proof of Theorem II.A result of Timmesfeld [22] plays a key role. The significance of the difficultiesat this point is that in this minimal case, we are forced to look at the solvablesubgroup structure using methods appropriate to a characteristic 2 analysis.

The key to the proof of Theorem II lies in the following definition. We setU*(4m, 3) SU(4, 3)/(Z(SU(4, 3))2). Then U*(4, 3) = Sl-(6, 3). In general,Z(U*(4, 3)) has order 2. Let I E J (G) for a group G. We define recursively anintrinsic 2-component L of type Chev(q), q odd, of CG(to) to be strongly intrinsicin CG(to) provided either

(i) L has type SL(2, q) or U*(4m, 3).(ii) Given t1 E 4(L) such that CL(tl) has an intrinsic 2-component L1 of type

SL(2, q), there exists a strongly intrinsic 2-component L2 in CL(t2) where t2 = 1011-Note that CL(tl) = CL(t2). We are requiring in (ii) that L1L2 d CL(tl) with

t1 E L1 and t2 E L2. The argument proceeds by showing that if P-(4 (G))contains an element of type Chev*(q), then it contains a strongly intrinsicelement of type Chev*(q). This is a result about Chevalley groups which we treatfrom the point of view of algebraic groups. One of the possibilities is that astrongly intrinsic 2-component of a centralizer of an involution be an intrinsic2-component of SL(2, q)-type. In this case, Aschbacher's result [3] applies. If Lis a strongly intrinsic 2-component for which (ii) applies, then it is shown thatthere exists a strongly intrinsic 2-component of type Spin(7, q). From this acontradiction is obtained. On the other hand, if there is a strongly intrinsic2-component of type U*(4m, q) with m > 1, it is shown that there exists astrongly intrinsic 2-component for which (ii) holds. This leaves us with only thepossibilities that L(J(G)) n Chev*(q) consists of elements of type Sl-(6, 3),Sl(7, 3), Sl-(8, 3) or PSl+(8, 3). The only intrinsic 2-components of type Chev*(q)obtainable in this case are those of type Sl-(6, 3). It is impossible to obtainstrongly intrinsic 2-components in L(J(G)) for which (ii) holds. Without theB-conjecture, this is as far as we can go; however, this result has significance inthe reduction of the proof of Theorem I to the result of Theorem IV whichfollows. If the remaining 2-components of type Chev*(3) in E(4(G)) are quasi-simple, Proposition IIC gives the result, which is required to complete the proofof Theorem I.

A preliminary version of this result was presented to the Sapporo conferenceon finite groups [24]; later the result was generalized to include the caseChev*(3).

PROPOSITION lIIA. Let G be a finite group with a unique normal simplecomponent L(G). Assume that P-(4 (G)) contains an element of type A,,, n odd andn > 7. Then L(G) has type Ly.

This result is due to Solomon [18] except for the case n = 7, which followsfrom Aschbacher's work [3]. Both of these works effectively use a particularsignalizer construction stemming from a paper of Goldschmidt [7]. Apparently

52 J. H. WALTER

this approach does not generalize. But to take advantage of these constructionswe set 4* to be the subset of ( obtained by deleting the types A. n > 7, Ly,and type Chev*(q) q odd other than type PSL(4, 3), PSU(4, 3), 9-(6, 3), 2-(8, 3),and PS2+(8, 3). It is not necessary to exclude PSp(4, 3) = 2(5, 3) and S2(7, 3)since components of this type are never locally 6 -unbalanced.

The following result is Theorem 3.4 of [25]. It is a result which appliesimmediately to the cases of standard components of type He, ON, and Co3 sincethe hypothesis on which the result is based is easy to verify. The result has otherapplications which are irrevelant to the present question.

HYPOTHESIS IIIB. Let C and D be quasisimple groups, and denote by C and Ofthe classes of quasisimple groups with homomorphs isomorphic to C and D,respectively. Assume the following:

(i) D is simple with Schur multiplier of odd order.(ii) There is one and only one class of involution 4 (D; (3) in D whose

centralizers have a single component L of type C. This component is intrinsic; infact, CD(L) = 0,(L).

(iii) Suppose H is a group with a unique quasisimple component L(H) andCH(L(H)) = Z(H). Set H = H/Z(H), and denote images by bars. Suppose that1 , (H)) contains an element L of type C such that L is a standard subgroup of H.Then L(H) has type 6D.

(iv) Let a E 4 (Aut D). Then CD(a) n 4 (D; (2) is nonempty.(v) Let a E 4(Aut D) and set Da = CD(a)°°. Then Da 1, and C possesses no

involutory automorphism 8 with Cc(i3)°°0(C)/O(C) = Da/Z for some Z CO(Z(Da)).

(vi) If Q E 4(Aut C), then either Cc(/3)"° 0 1 or Cc(#) has nonabelianS2-subgroups.

(vii) Let z E 4(D; C) and Cr = L(CD(z)). Then there exists u E 4(D; C) nCz - Z(C.,) such that UAW(c) C 4(D; C).

(viii) If D is a standard subgroup of a group E, then CE(D) has cyclicS2 -subgroups.

PROPOSITION IIIC. Let G be a finite group with O(G) = 1. Assume that theelements of C($(G)) are quasisimple. Assume that Hypothesis IIIB holds. SupposeC( (G)) contains an element of type C or 6D. Then the normal closure [L, G] of Lin G has the form [ L, G ] = L, MM' or M where M E C (G), M has type C or 600and either L = CMM (t) or L E C(CM(t)) for some involution t according as thesecond or third case holds.

The proof is relatively short and is based on an argument communicated tome by Michael Aschbacher, which handled the case D has type ON.

We now describe the results of [26] whose objective is to utilize a signalizerfunctor to obtain a contradiction in a minimal counterexample to Theorem 1.

THEOREM IV. Let G be a minimal counterexample to Theorem I. Then(i) G contains a unique normal simple component L(G) and 02.,2(G) = 1.(ii) For any proper subgroup Y of G such that 02.2(Y) 1, the elements of

C, (F9 ( Y)) and the elements of C, ( Y) have type CD.(iii) The elements of CI(S (G)) have type Cn* and C(3 (G)) contains no intrinsic

element of type SL(2, 7).

THE B-CONJECTURE 63

The preceding results obtain the reduction stated in (iii). Here a restrictedform of Theorem II is utilized. Theorem II as stated is obtained after B-conjec-ture is proved.

The basic properties which are required for the construction of the signalizerfunctors will be described on the basis of the following notation. Denote bym(X) the 2-rank of a group X. Set

Gk(X) = {E E G(X)lm(E) = k),

Gk(X) = {E E G(X)lm(E) > k),L(X) = { LI L is a 2-component of X ),

I L is locally 6 -unbalanced),

C2(X) = { L E L(X )I L is locally 6 -balanced).

Let L E L(X) and A E G (X). We say that L is A-balanced [A-even] ifO(Cj (a)) = 1 for all a E J (NA(L)) [O(CNL(E))] = I for all E E G (NA(L)).Otherwise we say that L is A-unbalanced [A-uneven]. Set

L,(X; A) = {L E L(X)IL is A-uneven),

Cz(X; A) = {L E L(X)I L is A-even),

L(X) = <LIL E L(X)>,L,(X) = <LIL E L;(X)>, i = 1, 2,

L.(X; A) = <LIL E L,(X; A)>.

As in [12], we consider a second kind of balance. For E E 6 (X), set

Ax(E) = n o(cx(t)),I E E¢

Ox(E; p) = n 0 (Cx(t)),tEE'

where p is an odd prime. Let A E G (X). Then X is said to be k-balanced[k-balanced over p] if Ax(E) C O(X) [Ax(E; p) C O0(X)] for all E E E k(A). A2-component L of X is said to be locally k-balanced with respect to A ifONC(E) = 1 for all E E Gk(NA(L)); it is said to be weakly locally k-balancedwith respect to A if AL-(E) = 1 for all E E Ek(NA(L)).

For T E Sk+1(X) set

AT,k(X) = <L1(CG(E)0(CG(E)))IE E Gk(T)>.For A E G (X) and T E Gk+ '(A), set

AT,k(X; A) = <Ll(CG(E); A)O(CG(E))I E E Gk(T)>We say that X is regularly generated with respect to T if AT,(X) = X, and we saythat X is A-regularly generated with respect to T if AT 1(X; A) = X. Theseconcepts are derived from the concept of core-layer generation introduced in[12].

A 2-component L of a group X is said to be core-regulated in X with respect toA E 6(X) provided O(Ck(a)) 1 for some a E Att implies that O(CL(E)) 1

64 J. H. WALTER

for some E E 6 (NA(L)) with a E E. It is said to be strongly core-regulated in Xwith respect to A provided O(CK(a))/O(CL-(a)) is abelian and F(O(Ck(a))) =O(CL(a)) for all a E J (NA (L)). Thus strong core-regulation implies core-regula-tion.

Now we can state the condition to be placed on the set L1(X; A) of A-uneven2-components of X in order to control signalizer functors when X is taken to bethe centralizer of an involution in a minimal counterexample to Theorem I. Wesay that an A-uneven 2-component L of a group X is A-odd provided thefollowing properties are satisfied:

(PI) L is A -regularly generated with respect to any T E EZ(NA(L)).(P2) L is weakly locally 2-balanced and locally 3-balanced with respect to A.(P3) L is strongly core-regulated with respect to A.(P4) The elements o f f (S (L); A) satisfy (PI), (P2), and (P3). Furthermore, either(J (L); A) = e1(4 (L); A) or L either has type PSL(2, r2) where r is a Fermat or

Mersenne prime or r = 9 and an element of A induces a field automorphism on Lor type Co3 and A contains a central involution of L.

Of course, the first step in the argument is to investigate the A-oddness ofA -uneven 2-components of type Go*. The critical step is to investigate theproperties (PI), (P2), (P3), and (P4) when L has type PSL(2, q), A., n odd,TL(3, 4), or He. Except in the case where L has type A. or He, L is A -oddwhenever it is A -uneven. There always exists A E S (NX(L)) such that L isA-odd when L has type A. or He. One may see that certain elements of L1(X)can be A -even for some A E 6 (NX(L)) even if A is maximal. For example, thisoccurs in the cases L has type PSL(2, q) where q is a Fermat or Mersenne primeor q = 9. This anomaly causes complications throughout the paper.

We begin the construction of the signalizer functor in the paper by choosingA E 64(G). Special characterization results deal with the case where such anelement does not exist. Set for i = 1, 2,

L.(4(G); A) = U {Ll(CG(a); A)ja E As).

We say that A is regular in G if all the elements of f (J(G); A) are A-odd. Notonly do we show the existence of regular elements in G, but we also show that ift E 4 (G) such that f (CG(t)) 0, we identify a subset of 6 (CG(t)) consistingof elementary subgroups which are regular in G is 3-connected in Ll(CG(t)). Bythis we mean that given elementary 2-subgroups A and B of CG(t) which areregular in G, there exists a sequence in 6 (CG(t)) of elements A = El,E2, ... , EE = B such that E; are regular in G, m(E, n Ei+1) > 3 for somey, E L1(CG(t)), i = 1 , 2, ... , n - 1.

Now given an elementary 2-subgroup A which is regular in G, we define thedegree k(A) of local balance to be the least integer k such that the elements ofL(CG(a)) are k-balanced for all a E Att. Obviously the elements of E(J (G); A)are locally 1-balanced. So by (PA), 1 < k(A) < 3. Let k = k(A). Then anapplication of the signalizer functor theorem [8] and some results in [12] givethat

WA(k) = <AG(E)IE E E3(A)>

THE B-CONJECTURE 65

has odd order. Set WA = WA(3). Then it can be shown that WA(k) = WA fork < 3. Set

MA = NG(WA).

Then it follows that if m(A) > k(A) + 2,

MA Q <NG(E)I E E 6k(A)+l(A)>.

It is a consequence of (P1) that MA L1(CG(a); A)O(CG(a)) for a E Att.It is required to obtain the above results when m(A) = 4 and k(A) = 3. For

this purpose, one uses the functor obtained from the groups AG(E; p) given in(8). In this case, we set

WA,P = <i0(E; p)I E E'52(A)>.

Because of the weak local 2-balance of the elements of L, (4 (G); A) as stipulatedin (P2), the signalizer functor theorem [8] and the results of [12] show that WA,Phas odd order. Furthermore, when m(A) > 4

NG(WA,P) ;? <NG(E)I E E 63(A)>.

Then by showing that WA P = WA, it follows that when m(A) > 4

MA ? <NG(E)JE E 6;3(A)>.Now these results are next extended by showing that WA = WB whenever A

and B are 3-connected regular elements of 6(G). Then using 3-connectednessresult, we show that when a E A for some regular element of &(G)

MA D LI(CG(a))O(CG(a))

Finally we improve on this by setting for each four subgroup T of A where Ais regular in & (G)

MT = <L1(CG(t))O(CG(t))I t E Ta>,

WT = O(MT).

Then it is shown that

WT=O(MA)=WA.

It is a consequence of (21) that WA = WB when A and B are 2-connectedregular elements of 6 (G) and WA 1 for all regular elements of F,, (G). Also itfollows that NG(T) C MA for all T E 6;2(A).

When WA 1 for some regular element A in &(G), we are in a position touse the above results to obtain a fusion theoretic contradiction based on work ofAschbacher [1] and [2]. We show that WA 1 on the basis of the existence of anonquasisimple element in e1(l (G)). This is a technical argument in which thetreatment of certain minimal cases is quite complicated. The control of thefusion required to employ Aschbacher's results rests on the 3-connectedness inL1(CG(t)) of the set of regular elements of 6; (G) which contain t where t is aninvolution of G such that L1(CG(t)) is nonempty. This is particularly useful whenthere are components of type A, n odd, or He. Otherwise, all elements of f 4(G)are regular. Then (3) can be used effectively. In some cases, we use it to showthat G has a proper 2-generated core (cf. [1]).

66 J. H. WALTER

It is worthwhile although not surprising to note that many of the argumentsare much simpler under the assumption that we are dealing with regularelements of rank at least 5. Special techniques are needed for the case m(A) = 4.When m(A) < 3, direct characterization is used.

REFERENCES

1. M. Aschbacher, Finite groups with a proper 2-generated core, Trans. Amer. Math. Soc. 197(1974), 87-112.

2. , On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.3. , A characterization of Chevalley groups over fields of odd order, Ann. of Math. 106

(1977), 353-468.4. N. Burgoyne, Finite groups with Chevalley type components, Pacific J. Math. 72 (1977), 341-350.5. R. Foote, Finite groups with maximal 2-components of type Lz(q), q odd, Proc. London Math.

Soc. 37 (1978), 422-458.6. R. Gilman and R. Solomon, Finite groups with small unbalancing 2-components, Pacific J.

Math. 83 (1979), 55-107.7. D. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972),

341-351.8. , 2-signalizer functors on finite groups, J. Algebra 21 (1972), 321-340.9. K. Gomi, Finite groups with a standard subgroup isomorphic to PSU(4, 2), Pacific J. Math. 79

(1978), 399-462.10. D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most 4

elements, Mem. Amer. Math. Soc. No. 147 (1974).11. D. Gorenstein and J. H. Walter, Centralizers of involution in balanced groups, J. Algebra 20

(1972), 284-319.12. , Balance and generation in finite groups, J. Algebra 33 (1975), 224-287.13. R. Griess and R. Solomon, Finite groups with unbalancing 2-components of (L3(4), He}-type, J.

Algebra 60 (1979), 96-126.14. M. Harris, Finite groups having an involution centralizer with a 2-component of dihedral type. 11,

Illinois J. Math. 21 (1977), 621-647.15. , PS1(2, q)-type 2-components and the unbalanced group conjecture (to appear).16. M. Hams and R. Solomon, Finite groups having an involution centralizer with a 2-component of

dihedral type. I, Illinois J. Math. 21 (1977), 575-620.17. G. Seitz, Chevalky groups as standard subgroups. I, II, Illinois J. Math. 23 (1979), 36-57; 23

(1979), 36-56, 516-553, 554-578.18. R. Solomon, Finite groups with intrinsic 2-components of type A,,, J. Algebra 33 (1975), 498-522.19. , Maximal 2-components in finite groups, Comm. Algebra 4 (1976), 561-594.20. , Standard components of alternating type. I, II, J. Algebra 41 (1976), 496-514; 47 (1977),

168-179.21. , The B-conjecture and unbalanced groups, Proc. Durham Conf. on Finite Groups, 1980.22. F. G. Timmesfeld, Groups with weakly closed TI-subgroups, Math. Z. 143 (1975), 243-278.23. J. H. Walter, Centralizers of involution in finite groups and the classification problem, Proc.

Gainesville Conf., North-Holland, Amsterdam, 1973.24. , Characterization of Chevalley groups, Proc. Taniguichi Internat. Sympos., Div. Math.,

1976, Tokyo, pp. 117-441.25. , Characterization of Chevalley groups (to appear).26. , B-conjecture; 2-components in finite simple groups (to appear).

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN


THE MAXIMAL 2-COMPONENT APPROACHTO THE B(G) CONJECTURE

RONALD SOLOMON

The maximal 2-component approach to the B(G) Conjecture adheres to thesame general philosophy as that originally enunciated, and still pursued, by JohnH. Walter and Daniel Gorenstein in their approaches to this problem. Thisphilosophy is to embed the B(G) problem in the more general context of theUnbalanced Group problem and to reduce the latter to the solution of certainstandard form problems. We begin by stating the Unbalanced Group Conjec-ture (or U-Conjecture).

U-CONJECTURE. Let G be a finite group with F*(G) simple. Suppose that forsome involution t of G, O(CG(t)) 1. Then F'(G) is isomorphic to one of thefollowing groups.

(1) A group of Lie type over a field of odd order.(2) An alternating group of odd degree.(3) PSL(3, 4) or Held's group, He.The point of departure for all successful attacks on this conjecture thus far is

the following deep result, which combines work of Aschbacher, Gorenstein,Harada and Walter (see [10, (2.6)]).

THEOREM 1. Let G be a minimal counterexample to the U-Conjecture. Then Gcontains involutions a and x and a 2-component J of CG(a) such that if C =CG(a)/O(CG(a)) and D = 0(Ca(x)) n CG(a), then the following hold:

(1) [J, x] = J = [J, D].(2) J/Z(J) is isomorphic to a group of Lie type over a field of odd order, an

alternating group of odd degree, PSL(3, 4) or He.

Roughly speaking the maximal 2-component approach to the problem entailsthe following steps:

(I) Among all J arising as in Theorem 1, choose one with J "maximal", insome sense.

(II) If O(J) g Z(J), define a nontrivial signalizer functor 0 on a family 6 ofelementary 2-subgroups of G and form the odd order completion, W, of 0.

1980 Mathematics Subject Classification. Primary 20D06.0 American Mathematical Society 1980

67

68 RONALD SOLOMON

(III) If O(J) Z(J), prove that N = NG(W) is a proper strongly embeddedsubgroup of G, giving a contradiction.

(IV) If O(J) C Z(J), prove that J is standard in G and then use "standardcomponent" techniques to identify G.

By the time work began on this problem, a good library of signalizer functorshad been amassed by Goldschmidt, Gorenstein and Walter, making the choiceof a good functor 0 fairly easy. Of course, the Signalizer Functor Theorem ofGorenstein and Goldschmidt guaranteed that the completion, W, of 0 had oddorder. Thus Step II of the procedure was in quite good shape.

On the other hand, Steps III and IV presented interrelated problems. Webegin with some comments on III. If 0 were defined "functorially" for allelementary 2-subgroups of G, then Step III would be an immediate consequenceof the Proper 2-Generated Core Theorem of Gorenstein, Harada and Asch-bacher [1]. However, in practice, it was only possible to define 0 on elementary2-subgroups "close to" N,(J). Because of this, it would have been impossible toprove that N = NG(W) was strongly embedded in G unless it could be estab-lished that O(N)J was subnormal in N. Now it was almost always possible(more on this later) to choose large enough to guarantee that <a, x, J > wascontained in N. The L-Balance Theorem of Gorenstein-Walter [6, (4.2)] thenassured that there was a 2-component K of N with either

(a) J 1 L(CK(a)) and K = K', or(b) K K°, J = L(CKK.(a)) and K = KX.

From this it could be inferred that either _(a) N = N/O(N) with F*(N) = K and O(CN(z)) 0 1 (whence N was known

by induction); or(/3) For some involution b of N, (b, x, K) had the properties listed in

Theorem 1 for (a, x, J).In case (a), additional information always yielded a contradiction. In case

(/3), one could infer that the 2-component K should have been studied insteadof J. This led to the formulation of the notion of a maximal unbalancing triple(a, x, J) by Gilman and Solomon. The precise definition is rather technical butthe important point is that if (a, x, J) is a maximal unbalancing triple and case(/3) holds, then O(N)J = O()V)K is subnormal in N. This is precisely what wasneeded to show that N was strongly embedded in G. Thus Step III could beaccomplished in almost all cases. This work appears in [7], [9] and [12].

Assuming Step III is done, we have O(J) C Z(J). Step IV is next: to provethat J is standard in G and then to identify G. In order to prove that J wasstandard in G, the key step was to prove that J was maximal in a suitableordering on all 2-components in G, first defined in [10]. The difficulty was thatwhile J was chosen maximal among "unbalancing 2-components" in G, J neednot have been maximal among all 2-components in G. In most cases thisproblem was easily resolved. Exploiting the properties of intrinsic SL(2, q)components and using the main theorem of [2], Burgoyne was able in [3] toeliminate the case where J/O(J) was of Lie type over a field of odd order,except for the case J/O(J) = PSL(2, q). Let A denote the set of all isomorphismclasses of quasisimple groups of the following types:

(1) Alternating groups of odd degree n and their proper covers, for n > 7,

2-COMPONENT APPROACH TO THE B(G) CONJECTURE 69

(2) He and the Lyons-Sims group, LyS.Using Burgoyne's result and the fact that the family A is closed under thesolution of standard form problems, it was possible in [7], [9], [11] and [12], atthe expense of treating the "extraneous" group LyS, to eliminate the 2-compo-nents of type A as well, proving the following result.

PROPosiTioN 2. Let G be a minimal counterexample to the U-Conjecture. ThenG has a maximal unbalancing triple (a, x, J) with PSL(3, 4) orJ/O(J) = PSL(2, q) for some odd q.

Treating the PSL(2, q) and PSL(3, 4) cases required the derivation of thefollowing sufficient condition for J to be maximal among all 2-components in G.

PROPOSITION 3 [5]. Let G be a minimal counterexample to the U-Conjecture and(a, x, J) a maximal unbalancing triple in G. Let S E Syl2(C(a) n N(J)) and letP = Cs(J/O(J)). Then J is maximal among all 2-components in G if thefollowing conditions hold:

(1) If b is an involution in P and L is a 2-component of <JL(C(b))>, thenJ/Z'(J).

(2) If b is an involution of CP(S), then S E Syh(C(b) n N(L)).

Using Proposition 3 and assuming the solution of the standard form problemsfor PSL(2, q) and PSL(3, 4) permitted the completion of the PSL(3, 4) case andthe reduction of the PSL(2, q) case to the following situation.

PROPOSITION 4 [5], [7]. Let G be a minimal counterexample to the U-Conjecture.Then G has a maximal 2-component J with J/O(J) = PSL(2, q) for some odd q.

In this case, the difficulty alluded to before in choosing 6 sufficiently largebecame critical when the 2-rank of C = CG(J/O(J)) was at most 2. This wassolved by Foote [4] using a delicate fusion argument when the 2-rank of C was2. When the 2-rank of C was 1, the problem was circumvented by M. Harris [8].Without proving that O(J) C Z(J), he reduced the L2(q) problem to thesolution of certain standard form problems in proper sections of G. Theseappear near completion by the work of J. H. Walter, C. K. Nah, H. Suzuki, M.Aschbacher and G. Seitz. We refer the reader to Harris' article for more detailson this final step in the proof of the U-Conjecture.

REFERENCES

1. M. Aschbacher, Finite groups with a proper 2-generated core, Trans. Amer. Math. Soc. 197(1974), 87-112.

2. , A characterization of the Chevalley groups over finite fields of odd order, Ann. of Math.106 (1977), 353-468.

3. N. Burgoyne, Finite groups with Chevalley-type components, Pacific J. Math. 72 (1977),341-350.

4. R. Foote, Finite groups with maximal 2-components of type L.1(q), q odd, Proc. London Math.Soc. 37 (1978), 422-458.

5. R. Gilman and R. Solomon, Finite groups with small unbalancing 2-components, Pacific J.Math. 83 (1979), 55-106.

6. D. Gorenstein and J. H. Walter, Balance and generation in finite groups, J. Algebra 33 (1975),224-287.

70 RONALD SOLOMON

7. R. Griess and R. Solomon, Finite groups with unbalancing 2-components of (L3^(4), He}-type, J.Algebra 60 (1979), 96-125.

8. M. Hams, PSL(2, q)-type 2-components and the unbalanced group conjecture (to appear).9. R. Solomon, Finite groups with intrinsic 2-components of type A,,, J. Algebra 33 (1975), 498-522.

10. , Maximal 2-components in finite groups, Comm. Algebra 4 (1976), 561-594.11. , Standard components of alternating type I, II, J. Algebra 41 (1976), 496-514; 47 (1977),

162-179.12. , 2-Signalizers in finite groups of alternating type, Comm. Algebra 6 (1978), 529-549.



FINITE GROUPS HAVING AN INVOLUTIONCENTRALIZER

WITH A 2-COMPONENT OF DIHEDRAL TYPE

MORTON E. HARRIS

Our main concern in this note is:Problem. Determine all finite groups G that satisfy:(a) is simple; and(b) G contains an involution t such that H = CG(t) possesses a 2-component

L such that CH(L/O(L)) has cyclic Sylow 2-subgroups and L is isomorphic toA7 or to PSL(2, q) for some odd prime power q > 3.

Clearly, in this problem, it suffices to determine for finite groups G thatsatisfy (a) and (b) above.

The known possibilities for in this problem are:(I) if L/O(L) ~ A7, then can be(a) sporadic: He; or(b) alternating: A9.(II) if L/O(L) PSL(2, q) for some odd prime power q > 3, then can

be(a) sporadic: J1, J2, J3, M12, HS;(b) alternating: A7, A8;(c) characteristic 2 Chevalley groups: PSL(3, 4), PSU(3, 4), PSL(2, 16),

PSp(4, 4), PSL(5, 2), PSU(5, 2); or(d) odd characteristic Chevalley groups: PSL(2, q2), PSL(3, q), PSU(3, q),

Re(32, ) for some positive integer m, PSp(4, V), PSL(4, \ ), PSU(4, \ ),PS2(8, q1/4, -1).

The existence of the exceptional isomorphisms: PSL(2, 4) = PSL(2, 5) = A5,PSL(2, 7) s PSL(3, 2) -= GL(3, 2) and PSL(2, 9) = A6 = PSp(4, 2)' suggestssuch a lengthy list.

From 11(c), it is clear that there are groups G satisfying hypotheses (a) and (b)of the problem in which HG is the only conjugacy class of non-2-constrained


71

72 M. E. HARRIS

2-local subgroups of G. Consequently, groups G satisfying hypotheses (a) and(b) of the problem that are "essentially" of characteristic 2-type must be studied.

Now we will describe the role that this problem plays in the current programof classifying all finite simple groups.

First, recent resuls of R. Gilman and R. Solomon in [7] and of R. Foote in [5]demonstrate that the Unbalanced Group Conjecture and the B(G)-Conjectureare consequences of a demonstration that if G is a finite group satisfyinghypotheses (a) and (b) of the problem, then occurs in (I) or (II) above.(Note that the B(G)-Conjecture is an easy consequence of the UnbalancedGroup Conjecture.)

Secondly, in order to classify all finite simple groups of component type thatsatisfy the B(G)-Conjecture, it suffices, by [1, Theorem 1] and [2, Corollaries IIand III] to determine all simple groups that have a proper standard subgroup ofknown type. On the other hand, suppose that G is a finite group with fl(G)simple that contains a standard subgroup A with A/O(A) isomorphic to A7 orto PSL(2, q) for some odd prime power q > 3. Then [3, Theorem] and [2,Corollary II] reduce the determination of such groups G to the problem.

Next, suppose that the finite group G satisfies (a) and (b) of the problem.Clearly by L-balance, it suffices to assume that either G = fl(G) or t (Iand G = Also, we observe that H(o°) = L and H has very fewnonsolvable proper 2-local subgroups. For, we have L = L2,(H) d H,CH(L/O(L)) = S(H) < H and H/CH(L/O(L)) is isomorphic to a subgroup ofAut(L/O(L)) containing Inn(L/O(L)). Thus H(00) = (S(H)L)(°°) = L. Set H= H/CH(L/O(L)) and let 1 A be an elementary abelian 2-subgroup of Hsuch that NH-(A) is not solvable. Then JAI = 2. Letting t be the nonidentityelement of X, we also have that if L A7, then t acts like a transposition on Land CL-(t) = A5 and if L = PSL(2, q) for some odd prime power q > 3, then tacts like a field automorphism on L, q is a square, q > 9 andCL-(t) = PGL(2, \ ). In any case, the 2-local structure of H is very meager.

Next we present the two main results of [13] and [9].

THEOREM 1. Let G be a finite group satisfying (a) and (b) of the problem andsuch that 4. Then the following three conditions hold:

(a) CH(L/O(L)) = <t> x O(H);(b) if L/O(L) A7, then H = <t> X K where K(o°) = L and K/O(K) = 17;

and(c) if L/O(L) = PSL(2, q) for some odd prime power q > 3, then q is a square,

IL12 > 23 and there is an involution u E H that acts like a field automorphism onL/O(L), so that CLIo(L)(u) = PGL(2, \ ).

THEOREM 2. Let G be a finite group satisfying (a) and (b) of the problem andsuch that 4. Suppose also that IL12 = 23. Then IF"(G)12 < 210

In view of [8, Main Theorem], the hypothesis that r2(P(G)) > 4 is just aconvenience.

Consequently, applying the characterization of all simple groups G with1G 12 < 210 described in [4] and [6], we obtain:

2-COMPONENTS OF DIHEDRAL TYPE 73

COROLLARY 3. Under the hypotheses of Theorem 2, exactly one of the followingtwo conditions holds:

(a) L/O(L) = A7, 1 O(L) = 3 and G = Aut(He); or(b) L = PSL(2, 9) and G is isomorphic to HS, Aut(PSp(4, 4)), Aut(PSL(5, 2))

or Aut(PSU(5, 2)).

The list of known possibilities for F*(G) in the problem that is given aboveindicates that Theorem 2 and Corollary 3 deal with a lengthy list of groups.

The proofs of Theorems 1 and 2 in [13] and [9] are quite technical and aredivided into the cases

(a) H/(LCH(L/O(L))) is cyclic and(/3) H/(LCH(L/O(L))) is not cyclic corresponding to [13] and [9] respec-

tively.The proof of Theorem 1 in [13] and [9] is quite technical. The proofs of

Theorem 2 under hypotheses (a) and (/3) in [13] and [9] are similar, quitetechnical and extremely long.

An outline of the proof of Theorem 2 under the additional hypothesis (a)contained in [13] is as follows. We shall assume that jP(G)j > 2" and derive acontradiction. Let S E Sy12(H) and set D = S n L. Then there are involutionsu, x, y, z E S such that D = <x, y> D8, D' = <z>, S = <t, u> X D andZ(S) = <t, u, z>. Set A = <t, u, z, y> and B = <t, u, z, x>. Then 616(S) =(A, B) and every elementary abelian subgroup of S is contained in A or in B.Also CG(A) = O(CG(A)) X A = CH(A), O(CG(A)) = O(H) n NH(A) and ifNH(A) = NH(A)/O(CG(A)), then NH(A) = <t, u) X R where D < R 14, etc.Similar results apply to CG(B) and NH(B). Also, it is easy to see that

ING(S)/ (NG(S) n CG(Z(S)))l E (2, 4).

Set M = NG(A) and for simplicity assume that O(M) = O(CG(A)) = 1. LetW = 02(M), let p be an element of order 3 in NH(A) and set V = [ W, p].

First, consider the case

I NG(S)/ (NG(S) n CG(Z(S)))l = 2.

Since a Sylow 2-subgroup of NG(S) is not a Sylow 2-subgroup of G, we concludethat to(s) _ (t, tz). Then we show that M/A = 14 and either V = E16 orV = Z4 X Z4. A short analysis eliminates the V = E16 possibility. ConsequentlyV = Z4 X Z4. We eliminate this case by studying NG(W) and concluding thatING(W)1Z = 29 and IG : NG(W)I is odd.

Next assume that ING(S)/(NG(S) n CG(Z(S)))i = 4. Then IM/A1 E{ 12, 24, 48) and the possibility I M/A 1 = 12 yields an immediate contradiction.Suppose that I M/A I = 24. Then M/A = 14 and either V = E16 or V = Z4 XZ4. A detailed study of these two possibilities for V concludes that I G 12 < 29 inthis case. Suppose that I M/A I = 48. Then M/A = Z2 x 14 and there are fivepossibilities for V. Detailed analyses of each of these five possibilities for V arepresented in [13] to complete this part of the proof of Theorem 2.

The methods and program of [13] and [9] were applied in [11] to obtain aproof of:

74 M. E. HARRIS

THEOREM 4. Let G be a finite group with F'(G) simple. Assume that G containsan involution t such that H = CG(t)possesses a 2-component L such that L/O(L)

PSL(3, 3) and CH(L/O(L)) has cyclic Sylow 2-subgroups. Then F*(G)12 <210.

This result, [4], [6], [3, Theorem] and [2, Corollary II] classify all finite simplegroups with a standard subgroup of type PSL(3, 3).

In [10, Theorem 1], the remaining part of the problem (i.e. when I LI2 > 23) isstudied in an inductive setting. A consequence of the results of [10] is that asolution to the problem, a proof of the Unbalanced Group Conjecture, theB(G)-Conjecture and a classification of groups of component type "related" tothe simple Chevalley groups over finite fields of odd order depends only on thesolution of a few "standard subgroup problems". All of these open problems arecurrently under study by some mathematician. (See [12] for a description of theresults of [10].)

Hopefully new methods of "pushing up" and new characterization theoremswill yield substantial simplifications in the proofs of the results of [4], [6], [13],[9], etc.

REFERENCES

1. M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.2. , A characterization of Chevalley groups over fields of odd order, Ann. of Math. (2) 106

(1977), 353-398.3. M. Aschbacher and G. M. Seitz, On groups with a standard component of known type, Osaka

Math. J. 13 (1976), 439-482.4. B. Beisiegel, Uber Einfache Endliche Gruppen mit Sylow 2-Gruppen der Ordnung Hochstens 210,

Comm. Algebra 5 (1977),113-170.5. R. Foote, Finite groups with maximal 2-components of type 1-2(q), q odd (preprint).6. F. J. Fritz, On centralizers of involution with components of rank two. 1, II, J. Algebra 47 (1977),

323-374, 375-399.7. R. Gilman and R. Solomon, Finite groups with small unbalancing 2-components (preprint).8. D. Gorenstein and K. Harada, Finite groups whose 2-local subgroups are generated by at most 4

elements, Mem. Amer. Math. Soc. No. 147, Amer. Math. Soc., Providence, R. I., 1974.9. M. E. Harris, Finite groups having an involution centralizer with a 2-component of dihedral type.

II, Illinois J. Math. 21 (1977), 621-647.10. , On balanced groups, the B(G)-conjecture and Chevalley groups over fields of odd order

(preprint).11. , Finite groups having an involution centralizer with a 2-component of type PSL(3, 3),

Pacific J. Math. (to appear).12. , On Chevalley groups over fields of odd order, the unbalanced group conjecture and the

B(G)-conjecture, these PROCEEDINGS, pp. 75-79.13. M. E. Harris and R. Solomon, Finite groups having an involution centralizer with a 2-component

of dihedral type. 1, Illinois J. Math. 21 (1977), 575-647.

UNIVERSITY OF MINNESOTA


ON CHEVALLEY GROUPS OVER FIELDS OFODD ORDER,

THE UNBALANCED GROUP CONJECTUREAND THE B(G)-CONJECTURE

MORTON E. HARRIS

In this note, I will state and discuss the status of several results that essentiallyappeared in [7] and [8] almost two years ago. These results prove the B(G)-con-jecture and the Unbalanced Group Conjecture and derive a classification offinite groups G with F*(G) simple and containing an involution t such thatCG(t) possesses a 2-component of type "related" to a Chevalley group over afield of odd order under the assumption that a small number of "standardcomponent problems" in which the component is a Chevalley group over a fieldof 3 elements and of Lie rank at most 4 are solved.

The proofs in [7] and [8] utilize the results of several authors and various"facts" about Chevalley groups over fields of odd order that do not seem tohave appeared in the literature. Recently I completed proofs of all of theserequired "facts". Thus complete proofs of the results to be stated below willappear in the near future (with the title of [8] changed to: PSL(2, q)-type2-components, q odd, and the Unbalanced Group Conjecture). Currently each ofthe "standard component problems" mentioned above is under investigation bysome mathematician.

All groups in this article are finite and if G is a group, we denote the Schurmultiplier of G by M(G). A subnormal 2-quasisimple subgroup of a group G iscalled a 2-component of G. Also, if t is an involution of a group G and if L is a2-component of CG(t) that contains t, then L is said to be intrinsic.

If p is a prime integer, let Chev(p) denote all simple Chevalley groups over allfinite fields of characteristic p. Also, for p = 3, Chev(3) includes all of the simplegroups of Ree type (of characteristic 3), Re(32m+1) for all positive integers in,and excludes 2G2(3)' = PSL(2, 8).


75

76 M. E. HARRIS

DEFINITION 1. Let p be an odd prime integer. A finite group H is said to be ofM(p)-type if:

(a) H is 2-quasisimple and H/Z'(H) E Chev(p);(b) I Z'(H)j is even; and(c) if H/Z'(H) is isomorphic to PSL(2n, p') or PSU(2n, p') for positive

integers n and r, then IZ'(H)12 = IM(H/Z'(H))12.Also, if H/O(H) -- SL(2, 3), then H is said to be of M(3)-type.Note that if a group H satisfies (a) and (b) of Definition 1 and if H/Z'(H) is

not isomorphic to PSL(2n, p') or to PSU(2n, p') for positive integers n and r,then H is of M(p)-type.

Let K be a 2-quasisimple group such that K/Z'(K) E Chev(p) for some oddprime p. Suppose also that every involution t of K is such that CK(t) does notpossess an intrinsic 2-component of M(p)-type. Then we show that eitherK/Z'(K) is isomorphic to certain Chevalley groups over a field of 3 elementsand of Lie rank at most 4 or K/O(K) -- PSL(2, p'") for some positive integer mor K/O(K) = Re(32m+1) for some positive integer m. Thus intrinsic 2-compo-nents of M(p)-type are available in such quasisimple groups K except in thesecases.

As in [2], we have:DEFINITION 2. If X is a subnormal subgroup of a group G such that O(X) _

O(G) and X/O(X) is isomorphic to SL(2, 3) or PSL(2, 3), then X is said to be asolvable 2-component of G.

Our first result is an extension of M. Aschbacher's "Classical InvolutionTheorem," [3, Corollary 1II]. Also this result is related to [9] and [10].

THEOREM 1. Let G be a group such that 02' (G) is 2-quasisimple. Assume that zis an involution of G such that H = CG(z) possesses an intrinsic 2-component ofM(p)-type for some odd prime p. Then 02'

(G)/Z'(02' (G)) E Chev(p) orGIO(G) -- M11.

The next result is a consequence of Theorem 1.

THEOREM 2. Let W be a 4-subgroup of the group G and let Wtt = {z1, z2, z3).Assume that CA(W) contains subgroups L, and L2 such that z; E Lj, L. is ofM(p1)-type for some odd prime p; and L. is a 2-component or solvable 2-componentof CA(W) for i = 1 and 2. Then O(G)02' (L) d < G or 02'(L,) is contained in aunique 2-component K1. of G such that K/O(K1.) s MII or K,/Z'(K,) E Chev(p)for i = 1 and 2.

The third and final main result of [7] is somewhat technical:

THEOREM 3. Let G be a group with F'(G) simple. Assume that W is a4-subgroup of G such that CA(W) contains a 2-component L of M(p)-type forsome odd prime p and such that L n W 1. Let w E (L n W)tt. Then K =<LL2 G(-))> is a single 2-component of CG(w) or F'(G) E Chev(p).

Note that in the first possibility in the conclusion of Theorem 3, if v E W -<w>, then L is a 2-component of CK(v) but there does not seem to be a methodof recognizing K1 0(K) in general without some sort of inductive setting athand.

CHEvALLEY GROUPS OVER FIELDS OF ODD ORDER 77

We shall now present the main results of [8].As is standard, if G is a group, then L(G) denotes the set of all 2-components

of all centralizers of all involutions of G.The results of [8] are proved in an inductive environment that emanates from

the Chevalley groups over fields of odd order. Consequently in order to makeour results as self-contained as we can, we define the following inductive setting.

DEFINITION 3. Let S be a family of finite simple groups. If K is a 2-quasisim-ple group such that K is said to be of -type.

DEFINITION 4. Let S be a family of simple groups. The group G is said to beproperly 6S-complete if whenever a proper section N of G has F'(N) simple andsome elements of L(N) is of -type then F'(N) E IT.

DEFINITION 5. A simple group X is said to be a blow-up of a simple group Y ifthere is an involution a of Aut(X) such that Cx(a) possesses a 2-component Lwith Y.

As is standard, a simple group of currently known type is said to be aK-group.

DEFINITION 6. Let S be a family of simple K-groups. Then the closure of IS,Cl(f ), is the minimal set 5 of simple groups containing S such that if Y E 5and X is a simple K-group that is a blow-up of Y, then X E S.

Clearly if S is a family of simple K-groups, then Cl(f) is closed, i.e.Cl(Cl(S )) = Cl(f ).

As we have demonstrated,

Cl(Chev(p)) = Cl((PSL(2,p")jn > 1))for all primes p > 5 and (since PSL(2, 3) is solvable)

Cl((PSL(2, 3")In > 2)) < Cl(Chev(3)).Also we determine the sets Cl(Chev(p)) for all primes p > 5, Cl(Chev(3)) andCl({PSL(2, 3")In > 2)).

Setting F(p) = Cl(Chev(p)) for primes p > 5 and

5(3) = Cl((PSL(2, 3")jn > 2)),we state the main result of [8]:

THEOREM 4. Let G be a group and let p be an odd prime such that:(a) F'(G) is simple;(b) G is properly F(p)-complete; and(c) there is an involution t E G such that H = CG(t) possesses a 2-component L

with L/O(L) a PSL(2,p") for some integer n > 1 and such that CH(L/O(L))has cyclic Sylow 2-subgroups. Then E F(p).

Our proof of Theorem 4 proceeeds as follows. By L-balance, we haveL < F'(G) and consequently, by induction, we may assume that either G =F'(G) or t V F'(G) and G = Also [4, Main Theorem], [5, Theorems1 and 2] and [6, Theorems 1 and 2] imply that CH(L/O(L)) _ <t> x O(H),ILI2 > 23, p" is a square, p" > 9 and there is an involution u E H such thatJ = CL(u)(°°) is a 2-component of CG(t, u) with J/O(J) - PSL(2, ). SetM = CG(u) and K = <JL='(°')). Then K is either a product of one or two2-components of M and the possibilities for the structure of K/O(K) are

78 M. E. HARRIS

determined by our (inductive) hypothesis (b). Suppose that t (4 F*(G). Then<t, u> n F'(G) 1 and we may assume that u E F'(G) < G. But then F*(G)is determined by our (inductive) hypothesis (b). Thus we may assume that G issimple. Each of the possibilities for K/O(K) are then studied and shown toeither determine G or force G' G.

If G is a group, then B(G) denotes the subgroup of G generated by all2-components of G that are not quasisimple.

The B(G)-Conjecture. If T is any 2-subgroup of any group G, then B(NG(T))< B(G).

It is easy to see that the Unbalanced Group Conjecture implies the B(G)-Con-jecture. Also, as we have shown, [3, Corollary III] and [1, Theorem 1] readilyyield:

PROPOSITION 5. Letsatisfies:

(a) F'(G) is simple;

IS be a family of simple groups. Assume that the group G

(b) all elements of L(G) are quasisimple;(c) G is properly IT-complete; and(d) C (G) contains an element of -type.

Then G contains a standard subgroup of 6S- type or F*(G) E Chev(p) for some oddprime p.

In order to state the "standard component problems" whose solutions arerequired in our applications of Theorem 4, we set

X(3) = ( PSL(4, 3), PSU(4, 3), PSp(4, 3), PS2(7, 3), PS2(8, 3, 1), PS2(8, 3, -1))

and define:

Hypothesis B. Let G be a group such that:

(a) F'(G) is simple;(b) all sections of G (including G) satisfy the Unbalanced Group Conjecture;(c) G is properly 6S(3)-complete;(d) G contains a standard subgroup of X(3)-type; and(e) if K E L(G) and K is of 6S(3)-type, then either K is of X(3)-type or

K/O(K) PSL(2, 3") for some n E (2, 4). Under these conditions, F'(G) E6S(3).

Several mathematicians are currently working on a proof of Hypothesis B.Finally, we demonstrate that the following two results are consequences of

Theorem 4 and the work of several authors:

THEOREM 6. Hypothesis B implies the Unbalanced Group Conjecture and theB(G)- Conjecture.

THEOREM 7. Assume Hypothesis B and let p be an odd prime integer. Supposethat G is a group with F'(G) simple and such that L(G) contains an element ofF(p)-type. Then F'(G) E gy(p).

CHEVALLEY GROUPS OVER FIELDS OF ODD ORDER 79

Finally, we remark that these last two results can also be formulated forChev(3) instead of 6S(3) by replacing X(3) by

X(3)' = X(3) U {PSL(3, 3), PSU(3, 3), G2(3)}

in the above.

REFERENCES

1. M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.2. , 2-components infinite groups, Comm. Algebra 3 (1975), 901-911.3. A characterization of Chevalley groups over fields of odd order, Ann. of Math. (2) 106


elements, Mem. Amer. Math. Soc., No. 147, Amer. Math. Soc., Providence, R. I., 1974.5. M. E. Harris and R. Solomon, Finite groups having an involution centralizer with a 2-component

of dihedral type. I, Illinois J. Math. 21 (1977), 575-647.6. M. E. Harris, Finite groups having an involution centralizer with a 2-component of dihedral type.

II, Illinois J. Math. 21 (1977), 621-647.7. , Finite groups containing an intrinsic 2-component of Chevalley type over a field of odd

order (unpublished).8. , On balanced groups, the B(G)-conjecture and Chevalley groups over fields of odd order

(unpublished).9. J. G. Thompson, Notes on the B-conjecture, (unpublished).

10. J. H. Walter, Characterization of Chevalley groups. I, Finite Groups (Sapporo and Kyoto, 1974),Japan Society for the Promotion of Science, Tokyo, 1976.


REMARKS ON CERTAIN STANDARDCOMPONENT PROBLEMS

AND THE UNBALANCED GROUPCONJECTURE

KENSAKU GOMI

By definition, a group L is quasisimple if L is its own commutator subgroupand the central factor group of L is simple. A quasisimple subgroup L of a finitegroup G is called a standard component if the centralizer of L in G has evenorder, L is normal in the centralizer of every involution centralizing L, and Lcommutes with none of its conjugates. The reader is referred to the expositoryarticle by L. Finkelstein in these PROCEEDINGS for an account of "StandardComponent Problems". In this note I shall make a remark on standard compo-nents isomorphic to PSU(4, 2) s PSp(4, 3) or PSp(6, 2) and on relationshipsbetween these standard component problems and the unbalanced group conjec-ture.

THEOREM 1 [5]. Let G be a finite group, and suppose L is a standard componentof G with L = PSU(4, 2). Furthermore, assume that CG(L) has cyclic Sylow2-subgroups and that LO(G) is not normal in G. Then one of the following holds:

(1) <LG)/O(<LG>) is isomorphic to a Chevalley type group of characteristic 2or 3.

(2) NG(L)/ CG(L) = Aut(L), and for each central involution z of L, CG(z) has aquasisimple subgroup K that satisfies the following conditions:

(2.1) z E K and W = 02(K) is cyclic of order 4;(2.2) K/<z> is a standard component of CG(z)/<z>, and W is a Sylow

2-subgroup of CG(K/ <z) );(2.3) either K1 0(K) = SU(4, 3) or K/ Z(K) has a Sylow 2-subgroup of type

PSL(6, q), where q = 3 mod 4;(2.4) [K, O(CG(z))] = 1.

1980 Mathematics Subject Classification. Primary 20D05.0 American Mathematical society 1980

81

82 KENSAKU GOMI

REMARK 1. During this summer institute, the Schur multiplier of the Mathieugroup M22 was amended: it is of type (4, 3). Since I used the Schur multiplier ofM22 in the proof of Theorem 1, 1 will take this opportunity to make thenecessary corrections. Fortunately, Theorem 1 as well as the lemmas used in theproof of Theorem 1 remain true without changes. The Schur multiplier of M22 isrelevant to the condition (2.3) in Theorem 1 and used only in the last paragraphof the proof of Lemma (1R) [5, p. 413] to show that if G is a group satisfying thehypothesis of the lemma, then is not isomorphic to M22. We canavoid the use of the Schur multiplier in the following way. Let bars denote theimages in G/O(G), and assume that H = F*(G) is isomorphic to M22. If A_is anE16-subgroup _of G satisfying the hypothesis of the lemma, then A C H andNH-(A)/CH-(A) = 16 or A6 (see the second paragraph of the proof). Further-more, since G = H or Aut(H) and CG-(A) = A X O(Ca(A)), it follows thatCG-(A) = A. So by conditions (5) and (6) of the lemma, an involution t acts on Gin such a way that A C CG-(t) C NG-(A) and Cat)/A = Y-3 wr Z2. However,Ca(t)I is not divisible by 9 [3, Table 1]. This contradiction shows that H is not

isomorphic to M22.REMARK 2. In a recent unpublished work, R. Foote has proved that PSL(6, q),

q = 3 mod 4, and PSU(6, q), q = 1 mod 4, are the only simple groups having aSylow 2-subgroup of type PSL(6, q), q = 3 mod 4. We can thus eliminate thesecond possibility in (2.3) of Theorem 1 by the use of the Schur multipliers ofPSL(6, q) and PSU(6, q).

REMARK 3. Let K be as in (2) of Theorem 1, and assume that K/O(K) =SU(4, 3). If K is normal in CG(z), then (2.1) and (2.2) of Theorem 1 show that Kis a standard component of G and that W is a Sylow 2-subgroup of CG(K). So aresult of M. Harris on intrinsic 2-components of Chevalley type (see a note ofHarris in these PROCEEDINGS) shows that <KG > = <L G> is a Chevalley typegroup defined over GF(3). Therefore, assume that K is notnormal in CG(z), andlet J = <Kcc(')>. Then, because of (2.1), (2.2), and (2.4) of Theorem 1, J is astandard component of G, and <z> is a Sylow 2-subgroup of CG(J). LetX = CG(z), and let bars denote images in X /<z>. Then k is a standardcomponent of X, and K/O(K) = 7 (6, 3). So the structure of J will be de-termined once we classify the groups containing a standard component of typeQJ6, 3). It is conjectured that J/Z(J) is a Chevalley type group defined overGF(3) and I Z(J)12 = 2 is equal to the 2-part of the Schur multiplier of J/Z(J).If so, <L G> = <J G > would be a Chevalley type group again by the Harris'theorem on intrinsic 2-components.

REMARK 4. Both K/Z(K) and J/Z(J) in Remarks 2 and 3 are proper simplesections of G. So if we assume that all proper simple sections of G are of knowntype (this is allowed in the most general classification problem) then we need notuse Foote's work or results on standard components of type S2-(6, 3).

THEOREM 2 [6]. Let G be a finite group, and suppose L is a standard componentof G with L PSp(6, 2). Furthermore, assume that CG(L) has cyclic Sylow2-subgroups and that if G G', then O(NG.(X)) C O(G') for every 2-subgroup Xof G'. Then if LO(G) is not normal in G, <LG> is isomorphic to SZ+(8, 2),Q J8, 2), PSU(6, 2), SU(6, 2), PSL(6, 2), PSU(7, 2), PSL(7, 2), PSp(6, 4), orPSp(6, 2) x PSp(6, 2).

THE UNBALANCED GROUP CONJECTURE 83

REMARK 5. The assumption that O(NG,(X)) C O(G') for every 2-subgroup Xof G' is related to the unbalanced group conjecture, which is usually stated asfollows:

Unbalanced group conjecture. Let G be a finite group with F'(G) simple and tbe an involution such that O(CG(t)) 1. Then F'(G) is of known type.

The following is a generalization of the unbalanced group conjecture.Generalized unbalanced group conjecture. Let G be a finite group with F'(G)

simple and T be a 2-subgroup of G such that O(NG(T)) 1. Then F'(G) is ofknown type.

The proof of the generalized unbalanced group conjecture is reduced to theproof of the usual unbalanced group conjecture and the solutions of standardcomponent problems (see the next remark). If O(NG,(X)) g O(G') for some2-subgroup X of G' in Theorem 2, then the generalized unbalanced groupconjecture implies that <L G) = <L G) is of known type.

REMARK 6. There seems to be considerable confusion in the present treatmentof the unbalanced group conjecture and the standard component problems. Onone hand, the proof [1] of the existence of standard components depends on theB(G)-conjecture. On the other hand, the B(G)-conjecture is generalized to theunbalanced group conjecture, and the present approach to the unbalancedgroup conjecture seems to depend on the solutions of almost all standardcomponent problems. Furthermore, in solving certain standard componentproblems, the B(G)-conjecture or the (generalized) unbalanced group conjecturehas been assumed to be true in many sections of the groups in question. Forinstance, Harris' work [7] on the unbalanced group conjecture depends onSeitz's work [8] on standard components of Chevalley type, in which Seitzassumes the B(G)-conjecture and the solutions of certain isolated standardcomponent problems including PSp(6, 2). As stated in Theorem 2, the de-termination of the groups with a standard component isomorphic to PSp(6, 2)requires the generalized unbalanced group conjecture. Everything suggests thepossibility of a neater formulation of the unbalanced group conjecture and thestandard component problems. Here is my proposal for such a formulation.

DEFINITION. Let K be an arbitrary family (of isomorphism classes) of (non-solvable) simple groups. A finite group X is said to be of K-type if everynonsolvable composition factor of X is a member of K. Let K(X) be the productof all 2-components of X of K-type.

The generalized unbalanced group conjecture can further be generalized tothe following:

Conjecture. If N is a 2-local subgroup of a finite group G and K(N)O(N) 1,

where K is the family of all known simple groups, then K(G)O(G) 1.

This conjecture also asserts that if a 2-local subgroup of a finite group G has a2-component of known type then so does G, which is what we are trying toprove for groups of component type. Now let G be a minimal counterexample tothe above conjecture. Then it can be shown that F'(G) is simple and that either

(A) O(CG(t)) 1 for some involution t, or(B) G has a standard component of K-type.

The latter assertion is essentially the "Component Theorem" of Aschbacher andFoote [1], [4]. In case (A), G is a minimal counterexample to the unbalanced

84 KENSAKU GOMI

group conjecture, and so the case has been reduced to case (B) (see notes of J.Walter, R. Solomon, and M. Harris in these PROCEEDINGS). There is an obviousadvantage of working with the minimal counterexample G to the above conjec-ture, for we can assume that the generalized unbalanced group conjecture holdstrue in every proper section of G and that every "Component Problem" is solvedin every proper section of G. Thus, Theorems 1, 2, and the remarks followingthem together with the theorems of Aschbacher and Seitz [2], [3] show that theminimal counterexample does not contain a standard component isomorphic toPSU(4, 2) or PSp(6, 2).

REFERENCES

1. M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.2. , A characterization of Chevalley groups over fields of odd order, Ann. of Math. (2) 106

(1977), 353-468.3. M. Aschbacher and G. Seitz, On groups with a standard component of known type, Osaka J.

Math. 13 (1976), 439-482.4. R. Foote, Finite groups with components of 2-rank 1. I, II, J. Algebra 41 (1976), 16-46, 47-56.5. K. Gomi, Finite groups with a standard subgroup isomorphic to PSU(4, 2), Pacific J. Math. 79

(1978), 399-462.6. , Standard subgroups of type Sp6(2). I, II, J. Fac. Sci. Univ. Tokyo, Sect. IA 27 (1980),

87-107, 108-156.7. M. Harris, On balanced groups, the B(G)-conjecture, and Chevalley groups over fields of odd order

(preprint).8. G. Seitz, Chevalley groups as standard subgroups. I, II, III, Illinois J. Math. 23 (1979), 36-57,

516-553, 554-578.

UNIVERSITY OF TOKYO


ODD STANDARD COMPONENTS

ROBERT GILMAN

1. The odd standard component problem. In the current approach to theclassification of finite simple groups odd standard components arise in the workof Michael Aschbacher [2], and Daniel Gorenstein and Richard Lyons [13] onthe classification of finite simple groups of characteristic two-type. In both thesearticles (which for brevity we will refer to as [A] and [GL]) one reaches a pointwhere the following conditions hold:

(1) G is finite, simple, and of characteristic two-type;(2) For some x E G, x has odd prime order and CG(x) has a p-component L;(3) CG(L) has cyclic Sylow p-subgroups.

By analogy with the (even) standard component or standard form problemone may call L an odd standard component (or say that CG(x) is in oddstandard form) and define the odd standard component (or odd standard form)problem to be the determination of all groups G satisfying (1)-(3) as L rangesover all known quasisimple groups. We will discuss the odd standard componentproblems arising in [A] and [GL]. For a more comprehensive survey consult thearticle by Robert Griess [14].

We pause to make some definitions: L is quasisimple if L is perfect andL/Z(L) is simple. L is a component of H if L is a subnormal quasisimplesubgroup of H. The central product of all components of H is the layer of H,denoted by L(H) (or sometimes by E(H)). For any prime, p, OP(H) and Op.(H)are the largest normal subgroups of H of order a power of p and prime to prespectively. If X/Op,(H) is a component of H/Op-(H) and X = MOO.(H) =NOO.(H) for normal subgroups M and N of X, then because X/OO.(H) isperfect, X = [M, N]Op.(H) = (M n N)Op,(H). Thus there is a unique mini-mum normal subgroup L of X such that X = LOO.(H). Such an L is ap-component of H, and the product of all p-components of H is the p-layer of H,Lp.(H). L has order divisible by p lest L lie in Op.(H). A p-component need notbe quasisimple; if it is, it is a component. Conversely a component whose orderis divisible by p is a p-component.


85

86 ROBERT GILMAN

The B(G)-Conjecture states that if t e G, t : t2 = 1, H = CG(t), and 02.(G)= 1, then every 2-component of H is a component of H. For any odd prime pthe BB(G)-Conjecture is obtained by substituting p for 2 in the B(G)-Conjecture.Both conjectures are valid whenever G is a known simple group.

H is a p-local subgroup of G if H = NG(T) for some nontrivial p-subgroupT C G. G is of characteristic two-type if CH(O2(H)) C 02(H) for every 2-localH C G. The p-rank mp(H) is the maximum of the ranks of the abelianp-subgroups of H. For odd prime p, mzp(G) is the maximum of mp(H) as Hranges over the 2-local subgroups of G, and finally e(G) is the maximum ofm2,p(G) asp ranges over all odd primes.

2. A refinement of the problem. [A] deals with the case e(G) = 3 and [GL] withe(G) > 4. In both cases the methods which yield conditions (2) and (3) give infact much more specific information and dispose immediately of most of thepossibilities for L. In this section we indicate how this is done. The principal toolis the assumption that, roughly speaking, a certain signalizer functor has oddorder. (The relevant signalizer functor methods are discussed by Richard Lyonsin these PROCEEDINGS.) This assumption enables one to assert that for certainp-components, K, of centralizers in G of p-elements, [K, Op.(K)] has odd order.For simplicity we will suppose that the BB(G)-Conjecture is valid in G for allodd primes p. It follows that the p-component K above is necessarily a compo-nent whence Op.(K) C Z(K) and [K, Op.(K)] = 1.

In [A] and [GL] one assumes that all proper simple sections of G areisomorphic to known simple groups. A consequence is the following form ofLp-Balance: If x and y are elements of order p in G and K is a p-component ofCG(x) with [K<x>, y] = 1, then K C J C CG(y) where J is a p-component ofCG(y) or a product of p p-components permuted transitively by <x>. In oursituation K is quasisimple and J is quasisimple or a central product of pquasisimple groups.

Let G satisfy (1). [A] and [GL] start by choosing an odd prime p withm2,p(G) > 3 and considering an elementary abelian p-group A realizing the2-local p-rank of G. That is, A lies in a 2-local of G and mp(A) = Byarguments similar to those used in the proof of the Klinger-Mason Theorem [17,Theorem A] one shows that the appropriate signalizer functor has even order, orthat for some x E A* (where A* is the nonidentity elements of A) CG(x) has ap-component, or (in [GL]) that G possesses a maximal 2-local subgroup H with02(H) of symplectic type. The third possibility is one of the conclusions of [GL].The identification of G in this case is a project separate from the solution of oddstandard component problems and is described by Steven Smith in thesePROCEEDINGS. It follows from the hypotheses governing this discussion that forsome x E A*, CG(x) has a quasisimple p-component L.

Now one of the differences between the odd and even standard componentproblems emerges. If A C W G, then A normalizes every p-component of W.Indeed if A does not normalize the p-component J of W, then by consideringthe possibilities for J one shows (with some exceptions, which we ignore) thatni,(A) < map(<A, J>), an obvious impossibility. In particular if y E CA(L), thenby Lp-Balance L lies in a single p-component of CG(y).

ODD STANDARD COMPONENTS 87

Another difference is that while in the even case one invokes MichaelAschbacher's Standard Component Theorem [3, Theorem 1] to show that CG(x)can be chosen in standard form, in the odd case the corresponding conclusion,which is condition (3) above, follows from the fact that G is of characteristictwo-type. Indeed we will show that if CB(L) has rank greater than one, then forsome y E A * and p-component K of CG(y), L 5 K. Thus by judicious choicewe can achieve CB(L) = <x>, which is a first approximation to (3).

Suppose, then, that x E D C CB(L) and `D = p2. Pick T to be a Sylow2-subgroup of L (we write T E Sy12(L)) and let Q = 02(NG(T)). Because G isof characteristic two type, D acts faithfully on Q; and since Q and D haverelatively prime orders, Q = <CQ(d)Id E D #). Thus

S =[CQ(d), <x>] 1 forsomed E D#.

We know that L lies in a single p-component J of CG(d) and that <x>normalizes every p-component of CG(d). Because S and <x> have relativelyprime orders, S = [S, <x)], whence S normalizes every component of CG(d). IfL J, we are done with the proof, so assume L = J. Now [L, x] = I implies[L, S] = 1, and we have LD C NG(S). Let R = 02(NG(S)); as before

U = [ CR(Y), LI 1 for somey E D #.

L lies in a single p-component K of CG(y) and normalizes every p-component ofCG(y). Because L is perfect, U = [ U, L] whence U normalizes every p-compo-nent of CG(y) and U C K. We conclude L 5 K as desired. If we were notassuming K to be quasisimple, we would be trying to show that K LOP-(K).We would suppose K = LOp(K) and conclude U C Op-(K). [K, OP.(K)] wouldthen have even order and we would eventually contradict our signalizer functorhypothesis.

The method we have illustrated yields many rewards. For example(4) m2,p(G) > 3 and x lies in an elementary abelian p-group A which realizes

the 2-local p-rank of G;(5) L is a group of Lie type defined over a field of order q = 2' for some m;(6) p divides q2 - I or q = 2 and p = 7;(7) if x E D C A with ID I = p2 and J = L(CL(D)) quasisimple, then for

some y E D - <x ), J is properly contained in a p-component K of CG(y) andK satisfies (5) and (6).

Conditions (1)-(7) are approximations to the actual results obtained in [A]and [GL]. We are allowing ourselves certain liberties in treating these papers.

Condition (4) appeared in the preceding discussion, and once we have (4)-(6),we produce K from J in (7) by an argument similar to the one which produced Kfrom L in the discussion above. Condition (7) plays the role that the Z*-Theo-rem [12, Theorem 2] plays in even standard component problems.

To illustrate the derivation of (5) suppose we have (1)-(4) and L is isomorphicto an alternating group, Ak. Let L act on k letters in the usual way. We haveA = (A n L) x <x>; and as mm(B) > 3 by (4), k > 2p. We also suppose k > 16to insure k - p > 9. Pick a p-cycle y E L and let J = L(CL(y)) = Ak_P. DefineD = <x, y ). For any d E D #, LP-Balance says J C K C CG(d) with K ap-component or product of p p-components permuted by <x>. In the latter case

88 ROBERT GILMAN

our conditions force m2 ,(K) > m2 ,(G), not the case, so K is quasisimple. SinceK is isomorphic to a known group, we have K - Ak+,p for some integer r > -1.

Now pick I ag A 5 with I C J C L and I acting on 5 letters and fixing the rest.Since k - p > 9, there is a 2-group T with 1 T C CJ(1). Further ITD CNG(T) and ID acts on R = O2(NG(T)). As before

S = [ CA (d), I] I for some d E D

and it follows from S = [S, I] that S C K. But I acts on 5 letters and fixes therest in the usual permutation representation of K, which precludes the existenceof a 2-group S C K with 1 S = [S, I]. We conclude L Ak.

Finally a word about (6). In [A] (6) is more or less forced by (4), (5) ande(G) = 3. In [GL] there is no upper-bound on e(G), but whenever one encoun-ters an odd prime r with m2.,(G) > m2 ,(G) one can invoke an inductivehypothesis (roughly that the 2-generated r-core of G lies in a 2-local subgroup)which allows one to prove that the appropriate signalizer functor has even order.One shows in this way that e(L) 4 m2.P(G), which together with (3) and (5)yields (6).

Recall that in [A] and [GL] one must consider the possibility that L is notquasisimple. One obtains the analogs of (3) and (5) with L replaced byL/OP.(L), and invokes the hypothesis on signalizer functors to force F =[L, OP.(L)] to have odd order. L will be quasisimple if F = 1. If F 1, oneconsiders the action of L on F and forces the existence of an odd prime r withm2.,(G) > m2,,(G). Now considerations similar to those in the preceding para-graph produce a contradiction.

3. Solutions. The standard component problems arising in [A] and [GL] andsatisfying (l)-(7) are solved in a fairly uniform way by first constructing theanswer as a subgroup Go C G and then showing Go = G. Most of the problemsfrom [A] are done in [A], [6] and [8]. Many of the problems from [GL] in the caseq = 2 were first solved by Larry Finkelstein and Daniel Frohardt [6]-[8] andLarry Finkelstein and Ronald Solomon [9], [10]. Other cases were treated byMartin Guterman [15], J. Chang [4] and Robert Miller [18]. A solution to all thestandard component problems from [GL] is contained in [11].

The process of constructing Go is essentially the same one used by WarrenWong [21] in his characterization of PSp(2n, q), n > 5, q odd, by the centralizerof an involution. The general procedure is discussed in [14]; we will limitourselves to an example.

Suppose L = SL(n, q), p I q - 1, and A has rank n. Choose D as in (7); thenJ = SL(n - 1, q). We suppose further that K SL(n, q). Think of K and Lacting in the usual way on vector spaces V and W over a field of q elements.Pick bases v,, . . . , v , , and w2, ... , for V and W so that A is representedby diagonal matrices in both cases. Arrange things so that J acts on thesubspaces (v2, ... , v.> and < W2 ,---, w.> and centralizes <v1> and (wn+1>

We want to fit L and K together in an obvious way to generate G, = <K, L>with G, c SL(n + 1, q). Choose J, C L and J, C L so that J, = J = SL(2, q),J, acts on <v,, v2> and centralizes <v3, ... , and J acts on <w,,, andcentralizes <w2, ... , The presentations for groups of Lie type given by

ODD STANDARD COMPONENTS 89

Charles Curtis [5] allow us to assert GI/Z(G1) ~ PSL(n + 1, q) (which is goodenough) once we show that J, centralizes J. If we know that NG(A) acts as itshould on A (that is, as the symmetric group of degree n + 1 acts on thediagonal matrices of SL(n + 1, q)), then as Ji and J2 are the layers of thecentralizers in G of certain subgroups of A, we know how NG(A) permutes theconjugates of Jl and J. We can choose w E NG(A) with Jl = J; and J,w C J sothat [J1, 1, and we have verified the necessary relation. At least we can dothis if n > 4; if n = 3, there is no hope of such an easy solution.

In general one considers all possible choices of y and K for some fixed D orperhaps for all D's as well and identifies the group generated by L and all theK's. This group is Go. In the above example if we know that NG(A) has thestructure given above, we do not need to invoke (7); we may take K to be aconjugate of L by an appropriate element of NG(A). Thus one might proceed byassuming only the weaker hypothesis (which follows immediately from (7) byexamining NK(A)) that (x) is not normal in N,(A). This hypothesis is naturalbecause it is an analogue of the information provided in the even case by theZ*-Theorem. One would then consider alll possibilities for NB(A) and identifythe group generated by all NG(A)-conjugates of L. However if L is as above andG is PSp(2n, q), one generates only an orthogonal group in this way, not the fullsymplectic group.

It is also worth noting that the generators obtained for Go are not alwaysthose in the presentations mentioned above. One may need other presentationssuch as [10].

The final step is to show Go = G. Let M = NG(Go); since G is simple, itsuffices to show M = G. First one shows CG(a) C M for enough p-elementsa E A *, and then one switches to the prime 2. For example if we can show thatfor some involution t lying in the center of a Sylow 2-subgroup, T, of M we haveg E M whenever t g E M, then a theorem of Derek Holt [16] contradictsM G. In [A] the procedure is somewhat different. One proves roughly that forevery nontrivial characteristic subgroup S C T, one has NG(S) C M and thenrelies on the classification of all simple groups of characteristic two-type inwhich C(G, T) = <NG(S)I1 S char T) G for a Sylow 2-subgroup T of G.This classification is the content of the C(G, T) Theorem; see [1], [20, Theorem1.10], and these PROCEEDINGS, article by Richard Foote.

It would be interesting to be able to rule out the situation M G withouttransferring to the prime 2. One might start with the following hypotheses:

(a) G is a finite simple group of characteristic two-type with m2p(G) > 3;(b) M C G is strongly p-embedded in G;(c) L(M) is quasisimple and L(M)/Z(M) is isomorphic to a group of Lie

type defined over a field of characteristic 2;(d) CG(L) has order prime to p;

and try to show that no finite group G satisfies (a)-(d). The general classifica-tion of groups with strongly p-embedded subgroups seems exceedingly difficult,yet the structure of G above is so different from the structure of known simplegroups with strongly p-embedded subgroups (see [19, Theorem 7]) that progressmay be possible.

90 ROBERT GILMAN

REFERENCES

1. M. Aschbacher, A factorization theorem for 2-constrained groups (preprint).2. , Finite groups of 3-rank 1, I, Invent. Math. (to appear).3. , On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.4. J. Chang, A characterization of Sp(6, 4") by its 3-structure, Dissertation, Rutgers University,

1974.5. C. Curtis, Central extensions of groups of Lie type, J. Reine Angew. Math. 220 (1965), 174-185.6. L. Finkelstein and D. Frohardt, A 3-local characterization of L7(2), Trans. Amer. Math. Soc.

250 (1979), 181-194.7. , Simple groups with a standard 3-component of type A"(2), n > 5, Proc. London Math.

Soc. (to appear).8. , Standard 3-components of type Sp(6, 2) (preprint).9. L. Finkelstein and R. Solomon, Finite simple groups with a standard 3-component of type

Sp(2", 2), n > 4, J. Algebra 59 (1979), 466-480.10. , A presentation of the symplectic and orthogonal groups, J. Algebra 60 (1979), 423-438.11. R. Gilman and R. Griess, A characterization of the finite groups of Lie type in characteristic 2,

manuscript.12. G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420.13. D. Gorenstein and R. Lyons, Odd standard form in finite groups of characteristic 2 type

(preprint).14. R. Griess, Odd standard form problems, Proc. London Math. Soc. Conf. on Finite Simple

Groups (Durham, 1978) (to appear).15. M. Guterman, A characterization of F4(2") as a group with standard 3-component B3(2"),

Comm. Algebra 7 (1979), 1079-1102.16. D. Holt, Transitive permutation groups in which an involution central in a Sylow 2-subgroup fixes

a unique point, Proc. London Math. Soc. 37 (1978), 165-192.17. K. Klinger and G. Mason, Centralizers of p-groups in groups of characteristic 2, p-type, J.

Algebra 37 (1975), 362-375.18. R. Miller, A characterization of SL(4, 4'') by its 3-structure, Dissertation, Rutgers University,

1972.19. G. Seitz, Generation of finite groups of Lie type (preprint).20. R. Solomon and S. K. Wong, On L2(2")-blocks (preprint).21. W. Wong, Characterization of the finite simple groups PSp2,,(q), J. Algebra 14 (1970), 531-551.

STEVENS INSTITUTE OF TECHNOLOGY

STANDARD SUBGROUPSOF CHEVALLEY TYPE

OF RANK 2 AND CHARACTERISTIC 2

IZUMI MIYAMOTO

In this note we would like to announce some results on standard formproblems of Chevalley groups of rank 2 and characteristic 2 but not ofGF(2)-type and to show an outline of the arguments used in the proofs. Thestandard form problem of type Sp4(2") is solved by K. Gomi in [1] and those oftype G2(2") and of type 3D4(2") are by H. Yamada in [4] and [5]. For unitarygroups the following results are obtained and the precise proofs can be found in[2] and [3].

THEOREM 1. Let t be an involution of a finite group G and let L be quasisimplenormal subgroup of H = CG(t). Assume L/Z(L) ^- U4(2") with n > 1, LO(G). G and CG(L) has cyclic Sylow 2-subgroups. Then one of the following holds:

(i) E(G)/Z(E(G)) L4(2 2n),

(ii) E(G)/Z(E(G)) U4(2") X U4(2").

THEOREM 2. Let t be an involution of a finite group G and let L be a quasisimplenormal subgroup of H = CG(t). Assume L/Z(L) ^- U5(2") with n > 1, LO(G)5C] G and CG(L) has cyclic Sylow 2-subgroups. Then one of the following holds:

(i) E(G)/Z(E(G)) ^- L5(2 2n),(ii) E(G)/Z(E(G)) U5(2") X U5(2").

As a Chevalley group of rank 2 has just two maximal parabolic subgroups, letM and N be those of L and let V = 02(M) and W = 02(N). We take a Sylow2-subgroup P contained in M and N, and set NL(P) = P I, where I is acomplement of P in NL(P). Then we have P I = M n N. The first step of theproof begins with the fusion argument. First we use Glauberman's Z*-theoremto show that t fuses to tL - {t). Then we consider the 2-local subgroupsNG(V<t>) and NG(W<t>). By a fusion argument we see {tgJ g E NG(V<t>)) Ct21Z(V<t>) and {tglg E NG(W<t>)) C t21Z(W<t)'). By the structure of L, atleast one of M and N satisfies one of the following properties, say M does.

1980 Mathematics Subject Classification Primary 20D10.m American Mathematical Society 1980

91

92 IZUMI MIYAMOTO

(a) There exists a normal subgroup Io of I such that 21(CP(I0)) = 21Z(V) CZ(P) and I acts transitively on S21Z(V).

(b) M/ V contains a normal subgroup isomorphic to SL2(2m) and as a modulefor the group, Q(Z(V)) is isomorphic to the natural module of dimension twoover F2,...

If either (a) or (b) holds, we can apply a theorem of doubly transitivepermutation groups and obtain that the permutation representation of NG(V<t>)on tQ1Z(V) has a regular normal subgroup. So we have the factorizationNG(V<t>) = 02(NG(V<t>))NH(V). Furthermore, since M acts on S21Z(V) irre-ducibly, t is contained in the second center of 02(NG(V<t>)). So the mapping xto [x, t] is a homomorphism from 02(NG(V<t>)) to S2IZ(V) with V<t> as itskernel. This homomorphism commutes with the action of NH(V) on02(NG(V<t>)) and so the following lemmas are useful to determine the structureof 02(NG(V<t>)) on which I acts.

LEMMA 1 (GoMI [1]). Let J1 and J2 be normal subgroups of a group J of p' orderacting on a p-group A. Assume A = CA(J1)CA(J2) and CA(J1) n CA(J2) = I.Then A = CA(Jl) X CA(J2).

LEMMA 2 (YAMADA [4]). Let J be a group of p' order acting on a p-group A.Assume CA(J) < A. Then A = CA(J) * [A, J].

On the other hand we can find a subgroup Il of I such that Z(P) C CP(Is)C(I) n 02(NG(V<t>)) and that CH(I1) contains a normal subgroup of

Bender type whose centralizer in CG(I1) has cyclic Sylow 2-subgroups. Then wecan apply the previous result for the rank 1 case due to R. Griess, D. Mason andG. Seitz [6] to obtain that 21([C(II) n 02(NG(V<t>)), I]) is a center of a Sylow2-subgroup of E(CG(Il)). Then factorizing [02(NG(V<t>)), I] by the centralizersof I1 and of some other appropriate subgroups of I which satisfy the hypothesesof the above lemmas, we have [ U, P ] = 1, where

U = Q1([ C(I1) n 02(NG(V<t>)),

Once such a subgroup is constructed, we can get a similar factorization of theother 2-local subgroup NG(W<t>) rather easily. For instance it is clear thatU C C(Z(W)) and that U C(Z(W)<t>). So if we consider the permutationrepresentation of NG(W<t>) n N(W) on tS21Z(W), U is a nontrivial subgroupof the semiregular normal subgroup NG(W<t>) n C(Z(W)). Then the action ofN on 21 Z(W) yields that < U9 I g E N > is a regular normal subgroup. Hence asbefore we have

NG(W<t>) n N(W) = 02(NG(W<t>) n N(W))NH(W).Let VI = [02(NG(V<t>)), I] and W1 = [02(NG(W<t>)) n N(W), I]. In the

second step we consider the subgroups NG(V1) n NG(V1<t>) and NG(W) nNG(Wl<t>). Starting from the parabolic subgroups M and N, in order toconstruct the corresponding parabolic subgroups of E(G) some more steps asabove may be required and in the further steps we use the argument in the latterhalf of the first step. Say in the second step, we consider the action ofNG(V1) n NG(Vl<t>) on S21(tZ2(V1)/Z(VI)) and find a subgroup in W1 nC(O1Z2(V1)/Z(V1)) not contained in V1 by applying the lemmas on W1. Thenwe have the factorization

STANDARD SUBGROUPS OF CHEVALLEY TYPE 93

NG(Vl) n NG(Vl<t>) = 02(NG(Vi) n NG(VICt>))NH(V)In such a way we can get the subgroups which are isomorphic to the parabolicsubgroups of E(G). After this stage the argument to complete the proof seems tobe rather standard so we shall not mention it here. Let us conclude by remarkingthat the argument mentioned here does not work well if the subgroup I is small,which implies that L is of GF(2)-type, because we have to take advantage of theaction of various subgroups of I on 2-subgroups to obtain the factorizationssuch as

N(W<t>) n N(W) = 02(N(W<t>) n N(W))NH(W).

REFERENCES

1. K. Gomi, Finite groups with a standard subgroup isomorphic to Sp(4, 2"), Japan. J. Math. 4(1978), 1-76.

2. I. Miyamoto, Finite groups with a standard subgroup isomorphic to U4(2"), Japan. J. Math. 5(1979), 209-244.

3. , Finite groups with a standard subgroup of type US(2"), n > 1, J. Algebra 64 (1980).4. H. Yamada, Finite groups with a standard subgroup isomorphic to G2(2"), J. Fac. Sci. Univ.

Tokyo 26 (1979), 1-52.5. , Finite groups with a standard subgroup isomorphic to 3D4(23' ), J. Fac. Sci. Univ. Tokyo

26 (1979), 255-278.6. R. Griess, D. Mason and G. Seitz, Bender groups as standard subgroups Trans. Amer. Math. Soc.

238 (1978), 179-202.

YAMANASHI UNIVERSITY

STANDARD SUBGROUPS OF TYPE G2(3)

HIROMICHI YAMADA

This article is an announcement of a result concerning the standard formproblem. The proof will appear elsewhere. Our result is the following:

THEOREM. Let G be a finite group which possesses a standard subgroup L suchthat L/Z(L) = G2(3). Assume that CG(L) has a cyclic Sylow 2-subgroup and thatLO(G) 4 G. Then either

(1) E(G)/Z(E(G)) = G2(9) or G2(3) x G2(3), or(2) NG(L)/CG(L) = Aut(G2(3)) and for an involution z of L, CG(z) has a

quasisimple subgroup K which satisfies the following conditions:(i) z E K, 02(K) is cyclic of order 4, and K/O(K) = SU4(3).(ii) [K, O(CG(z))] = 1.(iii) K/<z> is a standard subgroup of CG(z)/(z) and 02(K) is a Sylow

2-subgroup of CG(K/(z)).

We remark that case (2) does not occur in any known examples of G. Thus itis anticipated that once the classification of groups with a standard subgroup oftype PSU4(3) is established, case (2) will be superfluous.

Now we recall some terminology. More basic notation and terminology canbe found in Gorenstein's expository article in this volume. A group L isquasisimple if L is equal to L' = [L, L] and L/Z(L) is simple. A subgroup A ofa group G is tightly embedded in G if JA is even while JA f1 A91 is odd for eachg E G. A quasisimple subgroup L of a group G is said to be standard if CG(L) istightly embedded in G, NG(L) = NG(CG(L)), and [L, L8] 1 for each g E G.The following problem is of crucial importance.

Standard form problem. Find all groups having a standard subgroup L suchthat L/Z(L) is isomorphic to some known simple group.

Indeed, by the fundamental theorems of Aschbacher [1] and Foote [3] the studyof groups of component type is reduced to the proof of the unbalanced groupconjecture and the solution of the standard form problem.

A number of results on the standard form problem have already beenestablished by many authors and only a few cases still remain to be complete.


95

96 HIROMICHI YAMADA

The list of open cases can be found in Finkelstein's article in this volume.Let us outline the proof of the theorem. Let G, L be as in the hypothesis of the

theorem. We note that I Z(L)j = I or 3 by Griess [7] and Aut(G2(3)) = G2(3)<a>by Steinberg [10] where a denotes the graph automorphism. Also, I G2(3)1 = 26.36. 7 13. Let t be an involution of CG(L), so that L is normal in H = CG(t).Let S be a Sylow 2-subgroup of L. Then Z(S) _ <z> has order 2. Since LO(G)is not normal in G, t E Z*(G) and so by the Z*-theorem of Glauberman wehave tG n LCH(L) (t). Then it follows that <t> is a Sylow 2-subgroup ofCG(L), tG n L = 0, and No(<z, t)) is transitive on {t, zt). Put B = 12,(Z2(S)),which is a four-group. We have that NG(B<t)) is transitive on Bt. ThusjNG(B<t))12 = 29 or 210. Now by a property of G2(3) there is an elementaryabelian subgroup F of order 8 in S such that NL(F) = NL(F)' X Z(L) andNL(F)' is the nonsplit extension of E8 by GL3(2). Here we distinguish two cases;NG(F<t>) < H or NG(F<t)) is transitive on Ft. In the former case it can beshown that NG(B<t)) contains a Sylow 2-subgroup of G and by using a transferlemma we have t E 02 (G) and 102 (G)12 = 28. Moreover F is self-centralizing ina Sylow 2-subgroup of 02 (G), so the sectional 2-rank of 02 (G) is at most 4 by[8, Theorem 2]. Thus it follows from [6] and [2] that E(G) Q6 G2(9). Assume thatNG(F<t>) is transitive on Ft. Then we can show that NG(F<t>) has a normalsubgroup M of order 26 such that CM(t) = F and either M is elementary abelianor homocyclic abelian of exponent 4. If M is homocyclic abelian, we make useof [5, Lemma (1R)] and the classification of simple groups whose Sylow2-subgroups are isomorphic to a Sylow 2-subgroup of PSL6(q), q - 3 (mod 4) byFoote [4]. It can be shown that case (2) of the theorem holds. Suppose M iselementary abelian. Then M and F<t) are the only maximal elementary abeliansubgroups of M<t) and so NG(M<t)) = NG(F<t>). Since NG(F<t>) _NH(F)M, this provides the exact structure of the centralizer of the involution tin NG(M) = NG(M)/M, namely, it is isomorphic to NH(F)/F. Combining thisinformation with some additional conditions, we see that the normal closure Kof NL(F)'M in NG(M) has the property that K/M - GL3(2) x GL3(2) and tinterchanges its components. Let J/M be a component of K/M. Then we haveK = J' X J" and J' = CK(t) = NL(F)'. Let P be a Sylow 2-subgroup of Kcontaining S and put U = J' n P, so that P = U X U' and U = C,(t) = S. IfPO denotes a Sylow 2-subgroup of NG(K) containing P, then by examining thestructure of PO we have NG(PO) < NG(K). Hence PO is a Sylow 2-subgroup of G.Moreover, a transfer lemma gives that P is a Sylow 2-subgroup of 02(G).Finally, we show that U is strongly involution closed in P with respect to 02(G)and then apply Shult's product fusion theorem [9] to conclude thatE(G)/Z(E(G)) G2(3) X G2(3). Thus we do not construct directly a semisim-ple subgroup GO of G such that GO/Z(GO) = G2(3) X G2(3).

The group G2(3) seems to be "small" in two senses. First, G2(3) is ofcharacteristic 2-type, although it is a Chevalley group defined over a field of oddcharacteristic. By virtue of this fact the proof of the theorem entirely consists of2-local analysis depending heavily on the structure of 2-local subgroups of G2(3).Secondly, G2(3) is almost a N-group, that is to say, it has only one conjugacyclass of nonsolvable 2-local subgroups and the remaining local subgroups aresolvable. This property causes some technical difficulties in determining thestructure of certain 2-local subgroups of G.

STANDARD SUBGROUPS OF TYPE G2(3) 97

REFERENCES

1. M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.2. N. Burgoyne and C. Williamson, Centralizers of involution in Chevalley groups of odd

characteristic (unpublished).3. R. Foote, Finite groups with components of 2-rank 1, I, II, J. Algebra 41 (1976), 16-46, 47-57.4. , personal communication.5. K. Gomi, Finite groups with a standard subgroup isomorphic to PSU(4, 2), Pacific J. Math. 79


elements, Mem. Amer. Math. Soc. 147 (1974).7. R. Griess, Schur multipliers of the known finite simple groups, Bull. Amer. Math. Soc. 78 (1972),

68-71.8. K. Harada, On finite groups having self-centralizing 2-subgroups of small order, J. Algebra 33

(1975), 144-160.9. E. Shult, Disjoint triangular sets, Ann. of Math. 111 (1980), 67-94.

10. R. Steinberg, Automorphisms of finite linear groups, Canad. J. Math. 12 (1960), 606-615.

UNIVERSITY OF TOKYO

OPEN STANDARD FORM PROBLEMS

LARRY FINKELSTEIN

The determination of all finite groups with a standard component of knowntype is an essential part of the program to classify finite simple groups. In thisarticle, we shall report on recent progress made towards the resolution of thisproblem and list those standard form problems which remain open. No attemptwill be made to reference the results cited, since in some instances the mathe-matical analysis has been completed but preprints are not yet available. Instead,we refer the interested reader to the forthcoming article of G. Seitz [1] which willcontain an updated bibliography. The author would like to thank ProfessorDaniel Gorenstein for his assistance in compiling the list of open standard formproblems.

We begin by defining the notion of a standard component.

DEFINITION. A subgroup K of G is tightly embedded if K has even order andK n Kg has odd order for g E GUN(K). A quasisimple subgroup A is astandard component of G if C(A) is tightly embedded in G, N(A) = N(C(A))and [A, Ag] <1> for allg E G.

Aschbacher's component theorem combined with work of Foote yields thefollowing fundamental result.

COMPONENT THEOREM (ASCHBACHER-FOOTE). Let G be of component type withO(G) = <1>. Suppose the B-conjecture holds for G. Then with known exceptions,G has a standard component.

The Component Theorem provides the proper background for an attack onthe structure of groups of component type and this leads to the

Main Problem. Classify groups with a standard component of known type.For the remainder of the article, we shall assume that the following situation

olds. G is a group with O(G) = <1>, A is a standard component of G with,i_ = A/Z(A) of known type, R is a Sylow 2-subgroup of C(A) and X = <AG>.


99

100 LARRY FINKELSTEIN

LEMMA I. One of the following occurs:(i) X = A.(ii) A is simple, X = A X A, I R = 2 and RX = A-- Z2.(iii) X is simple and G C Aut(X).

The existence of case (ii) of Lemma 1 causes much of the difficulty in solvingstandard form problems.

THEOREM 1 (ASCHBACHER-SEITZ). Suppose m(R) > I. Then one of the followingoccurs:

(i)X=A.(ii) A A, X = Ai+4, n > 5.(iii) A = A5, X = HJ or M12.(iv) A = PSL(3, 4), X = Suz or He.(v) A = Sz(8), X = Ru.(vi) A = G2(4), X = C1.

Case (iv) remains open, although Aschbacher and Seitz have strong partialresults and will now attempt to complete the analysis. In light of the previousresult, it may be assumed that m2(R) = 1. Thus either R is a generalizedquaternion group or R is cyclic. In the former case, we have the followingcorollary to Aschbacher's classical involution theorem.

THEOREM 2 (ASCHBACHER). If R is a generalized quaternion group and X # A,then X E Chev(q), q odd.

From now on, we may assume that R is cyclic and set <t> = 21(R). The nextresult gives a criteria for reducing to the case I R I = 2.

THEOREM 3 (FINKELSTEIN). Suppose A has the property that whenever A C L CAut(A) and F is an involution of L, then CL-(z)/O(CL-(z)) has no Z4 or Z2 X Z4normal subgroup. Then either I R I = 2 or X = A.

The analysis of the cases when RA- is cyclic may be divided as follows. Hererank(A_) refers to the Lie rank when is a Chevalley group.

(1) A_ E Chev(2), rank(A) < 2.(2) A E Chev(2), rank(A) > 3.(3) A =A,,,n > 6.(4) A_ E Chev(q), q odd.(5) A sporadic.

It may also be assumed, although this will not always be necessary, that theB-conjecture holds in all sections of G whenever A is not an element of(Chev(q) (q odd), A2n+1, He).

THEOREM 4 (GRIEss-D. MASON-SEITZ). Suppose A E Chev(2), rank(A) = 1 andX # A. Then one of the following occurs:

(i)X=AxA.(ii) X E Chev(2), t E4 X.(iii) A = PSL(2, 4), X A, A9, PSL(2, 52), PSL(3, 5), PSU(3, 5), G2(5), 3D4(5),

J1, HJ or M12.(iv) A = PSL(2, 8), X = G2(3).

OPEN STANDARD FORM PROBLEMS 101

THEOREM 5. Suppose X A, I Z(A)l odd and one of the following holds:(i) A = L3(2°), a > 2 (Seitz).(ii) A PSp(4, 2°), a > 2 (Gomi).(iii) A = PSU(4, 2°), PSU(5, 2°), a > 2 (Miyamoto).(iv) A = PSU(5, 2), G2(2°), a > 2, 3D4(2°) (Yamada).

Then X A X A or X E Chev(2) and t E4 X.

Seitz has treated the rank 3 groups under the assumption that the followinghypothesis holds: _

Hypothesis I. Suppose A = Sp(6, 2), PSU(6, 2) or W:(8, 2), X = A and I Z(A )lodd. Then either

(i)X=AxA,(ii) X E Chev(2), t E4 X, or(iii) A = 0 +(8, 2), G Aut(M(22)).

THEOREM 6 (SEITZ). Assume A E Chev(2), rank(A) > 3, 1 Z(A )l odd, HypothesisI holds and X # A. Further assume A at Sp(6, 2), PSU(6, 2) or W: (8, 2). Theneither X= A X A or X E Chev(2) and t E4 X.

We indicate briefly the method of proof used in Theorems 5 and 6. In the casewhen rank(A) = 2, let Ai C A, i = 1, 2, be distinct maximal parabolic sub-groups. By "pushing up", one constructs 2-local subgroups P1 and P2 of G suchthat A; C Pi, i = 1, 2, 02(A1) C 02(P1) and Pi contains a Sylow 2-subgroup of G.Set P; = Pi/02(P;). Usually, E(A;) is standard in P; and Theorem 4 applies. SetGo = <P11 P2>. It is then shown that either Go is simple or else Go = A X A. InTheorem 6, Seitz analyzes the structure of C(J°) where J. is a root SL2 subgroupof A with a a long root. He shows that D = CA(J°) is standard in C(J°). Usinginduction and Curtis' theorem, Seitz constructs Go = A X A or Go E Chev(2).The final step is to prove that Go = G. At several key points in the argument,use is made of the assumption that the B-conjecture holds in all sections of G.

We now list the remaining known results when A E Chev(2).

THEOREM 7. The following cases have been completed:(i) A = Sp(6, 2) (Gomi), subject to the solution of the PSU(4, 3) problem.(ii) A = 0 +(8, 2) (Egawa).(iii) A = 0 -(8, 2) (Alward).(iv) A = 2E26 ) (Strothl_(v) A = Sp (6, 2) or F4 (2) (Seitz).

We now discuss the case A A, n > 6.

THEOREM 8. Suppose X A, the U-conjecture holds in all sections of G and oneof the following occurs:

(i) A = A6 or A7 (Harris-Solomon).(ii) A A, n > 8 (Solomon).(iii) A = A. (Janko-S. K. Wong, Lyons, F. Smith, Solomon).

Then either X - A X A or(1) X A+2,

<X, t> = 5n+2(2) A = A,1, X = LyS.(3) A = A10, <X, t> Aut(F5).

102 LARRY FINKELSTEIN

(4) A = A8, X = L4(4), HiS, t (4 X.(5) A = A8, X McL.(6) A_ 3 A (X, t) = Aut(He).7) A = A6, X = PSL(2, 81), PSL(3, 9), L3(9), PSU(3, 9), PSp(4, 3), PSL(4, 3),(7),4-

PSL(5, 2), PSU(5, 2), Sp(4, 4) or HiS.

Solomon's argument in the case A = A. (n ) 11) is similar to the one used bySeitz in the proof of Theorem 6.

We now define the class of groups C(3) = Chev(3) U {A2n: n > 4) U(G2(2"), Sp(4, 2), PSL(4, 2"), PSU(4, 2"): n = 2' > 2) U (PSU(5, 2"): n = 2'> 1) u {Mr, HS, C1, F5).

Hypothesis II. Let F*(G) be simple and assume the following hold:(i) The B-conjecture holds in all sections of G. _(ii) If G contains a standard subgroup A with A = PSU(3, 3), G2(3), PSL(4, 3)

or PSU(4, 3), then F*(G) E C(3).THEOREM 9 (WALTER, HARRIS). Suppose Hypothesis II holds and A E Chev(q),

q odd. Then either X= A x A or X is known.In conjunction with Theorem 8, we have

THEOREM 10. The following cases have been completed.(i) A_= PSU(3, 3) (Harris).(ii) A_- G2(3) (Yamada).(iii) A = PSL(4, 3) (H. Suzuki).

Finally, we discuss the case when A is sporadic. Finkelstein and Solomonhave worked systematically on this problem. Many other authors have treatedindividual cases as well (see Seitz [1, Table 2]). Solomon's work assumes that thefollowing hypothesis holds.

Hypothesis III. G is a group with F*(G) quasisimple. The U-conjecture holdsin all sections of G. Suppose G contains a subgroup A such that I Z(A )I even, Ais standard in d = G/Z(G) and A = PSU(6, 2), M(22), 2E6(2) or has type F2.Then F*((;) = M(22), M(23) or has type M(24)', F. or Ft.

Thus subject to the conditions of Hypothesis III being verified, the solution ofthis problem is essentially complete.

We close by listing the structure of A for which the corresponding standardform problem remains open and indicate the name of the person working on itssolution:

A = 2F4(2)' (Yamada).A = 2F4(22n+1) n > 1 (Miyamoto).A = 0 +(8, 2) (Egawa).A = F2 (Griess).A = PSU(6, 2) (Hunt).A = M"(22) (Hunt).A = PSU(4, 3) (H. Suzuki).

REFERENCE

1. G. Seitz, Standard subgroups in finite groups, Proc. London Math. Soc. Conf. on Finite SimpleGroups (Durham, 1978) (to appear).

WAYNE STATE I;NIVERsrry

GROUPS GENERATED BY A CONJUGACYCLASS OF INVOLUTIONS

F. TIMMESFELD

1. Introduction. Instead of classifying finite groups by local properties B.Fischer had the idea that it might be useful to consider global properties of aconjugacy class of involutions. Namely let G = S, the symmetric group on nletters and D the class of transpositions of G, then if d = E D and e = (kl)E D we have

o(de) = 2 iff (i,j) n (k, 1) = 0and

o(de) = 3 if qJ (i,j) n (k, 1) = 1.Taking this property of Sr, Fischer called a normal set D of involutions which

generate the group G a set of 3-transpositions of G, if o(de) < 3 for all d, e E D.In the course of classifying all "nearly simple" groups generated by a set D of3-transpositions in [6], he found 3 new simple groups, which he called M(22),M(23) and M(24).

In [1], M. Aschbacher considered the most suitable generalization of the3-transposition classification. He called a normal set D of involutions generatingthe group G a set of odd-transpositions of G, if it satisfies o(de) = 2 or odd for alld, e E D. He then classified all nearly simple groups generated by a set D ofodd-transpositions. Indeed many of Fischer's methods generalize to this case.

I tried in my thesis to generalize Fischer's work by admitting the next number4 as an order of such a product. Namely let D be a normal set of involutionsgenerating the group G such that o(de) < 4 for all d, e E D, then D is called aset of {3, 4) -transpositions of G. If the number 3 does not occur as the order ofsuch a product a theorem of R. Baer shows that G is a 2-group. If 4 does notoccur we are in the case of 3-transpositions, which has already been treated byFischer. So the interesting case is when all numbers from 1 to 4 occur as theorder of such a product. In this case call D nondegenerate. Soon I discoveredthat the problem of classifying groups generated by {3, 4) -transpositions was, atleast to this time, too difficult for me. Actually the problem is still not solved,

1980 Mathematics Subject Classification. Primary 20D05; Secondary 20B30.m American Mathematical Society 1980

103

104 F. TIMMESFELD

although B. Fischer spent some time on it, thereby finding a new simple groupgenerated by (3, 4)-transpositions, the Baby-monster.

I then introduced the + condition by calling a nondegenerate set D of{3, 4) -transpositions of G a set of (3, 4)+-transpositions of G, if

o(de) = 4 implies (de)2 E D for d, e E D.This condition is taylor made to the Chevalley-groups over GF(2), since the rootelements corresponding to long roots of the root system form a class of 3 resp.(3, 4}+-transpositions of these Chevalley groups. Groups generated by a set of(3, 4) '-transpositions were classified in [11].

The next step was the generalization of both the (3, 4)+-transposition and theodd-transposition work, to get a joint classification of all the Chevalley-groupsin characteristic 2. For this I called a normal set D generating the group G a setof "root-involutions" of G, if the following holds:

(1) o(de) = 2, 4 or odd for all d, e E D.(2) If o(de) = 4 then (de)2 E D for d, e E D.

This definition was inspired by the properties of the root-elements in Chevalley-groups in characteristic 2. Further I called D nondegenerate, if 4 occurs as theorder of such a product. Otherwise D is degenerate. So a degenerate set ofroot-involutions is a set of odd-transpositions. In [12] groups generated byroot-involutions were classified. The theorem which contains all the former is:

THEOREM. Let G be a finite group generated by a set D of root-involutions.Suppose G has no solvable normal subgroups and G' = G". Then G = X '-I Gi,G; = <D1>, Di = D n G; is a class of root involutions of G; and each pair (G,, D)satisfies one of the following:

(1) G; is a simple group of Lie-type in characteristic 2 different from 2F4(2m) andD, is the class of root-elements corresponding to long roots of the underlyingroot-system if G; C"(2') or F4(2m). If G; C"(2'") or F4(2 ") then Di is eitherthe class of root-elements corresponding to the long or the short roots.

(2) G, 0,'(2') and D, is the class of transvections.(3) G, p = 3 or 5, and D. is a class of reflections.(4) G. S. or L2(2m) Wr S. and Di fuses to the transpositions in S.(5) G A6, J2, M(22), M(23) or M(24) and in each case Di is a uniquely

determined class of involutions of G,.

In the following sections I will try to explain the fundamental ideas behind theproofs of the theorems. Furthermore I will explain the major applications andfinally describe my ideas of possible revisionism.

Since the combined length of 3, (3, 4)+, odd-transposition and root-involu-tion classification is about 350 journal-pages, the outline of the proof has to bevery brief. Further it explains the need of revisionism, although the proof isshorter than the proof of other classification theorems.

2. Odd transpositions and Fischer's Dd-theorem. For each normal set D ofinvolutions of some group G introduce the following notation:

6D (D) is the graph with vertex set D and edges (e, d) with ed = de 1.

A subset L C D is a TI-subset, if L n L9 = 0 or L for each g E G. It is aweak TI-subset, if L n Ld = 0 or L for each d E D.

GROUPS GENERATED BY A CONJUGACY CLASS 105

If d E D set

Ed := {f E DICD(e) = CD(f)} and Dd := CD(d)\Ed.

Then Ed is always aTI-subset of D. If G is a Chevalley-group in character-istic 2 and D a class of root-elements of G, then Ed is the set of elements of Dwhich are contained in the root-subgroup containing d.

Now one may express the Dd-theorem as follows:

(2.1) THEOREM. Suppose D is a class of odd-transpositions of G and there existsno TI-subset T c D satisfying:

(1) {abla, b E T and ab = ba 1) 0.(2) T g / Ef for some f E T.

Then Dd is a class of odd-transpositions of <Dd>.

The Dd-theorem is perhaps the most effective tool for the classification ofgroups generated by odd-transpositions. It is excellent for induction purposes.The proof is essentially the same as Fischer's proof in [6]. If D is a class of3-transpositions and G' simple then the hypothesis of (2.1) is always satisfied asthe following lemma shows:

(2.2) LEMMA. Suppose D is a set of 3-transpositions of G and T C D is aTI-subset of D. Then <abla, b E T> is in the kernel K of the permutationrepresentation of G on {TgIg E G}.

PROOF. Let a, b E T and x E Tg T. If o(ax) = 2, then a E T n T"whence x E N(T). But then o(bx) = 2, since if o(bx) = 3 then b - x in <b, x>,contradicting <b, x> < N(T). Hence to show ab E K we may assume o(ax) = 3= o(bx). This is equivalent to az = x° and bz = xb. Since T is a TI-set thisimplies (Tg)° = Tz = (Tg)b. But then ab E N(T9) for each g E G.

In the general odd-transposition case it is more difficult to obtain thehypothesis of (2.1). It can be shown that this amounts to the determination ofthose groups, for which 6D(D) is disconnected. This was done in the first ofAschbacher's papers. Now, with the help of the Dd-theorem, one may proceedby induction. Suppose we have for C = <Dd> and C_= C/S(C) that C' = C".Then the hypothesis of the main-theorem applies to C, whence by minimality ofG one may assume that C is one of the listed groups. With this information onetries to determine G. Of course one has to handle S(C). Here it is essentiallyshown that the following 2 cases occur:

(1) S(C) = 02(C) Z(C),(2) S(C) < Z(C).

The first case corresponds to the symplectic and unitary groups. All the othergroups generated by odd-transposition occur in case (2).

3. Maximal sets of commuting involutions and the groups M(22), M(23), andM(24). Suppose D is a conjugacy class of involutions of G such that <L> =<D n S> is abelian for S E Sy12(G). Then Burnside's Fusion-theorem showsthat N(L) is transitive on L. If D is a class of 3-transpositions of G and G'simple (2.2) shows Ed = d for d E D. Hence by (2.1) Dd = CD(d)\d is aconjugacy class in <Dd>. This implies N(L) n C(d) is transitive on L\d, whence

106 F. TIMMESFELD

N(L) is doubly transitive on L and is a transitive extension of N(L) n C(d).If now G - PSU(2n, 2) and D is the class of transvections of G, then

N(L)/C(L) ^_ L,,(4). Especially in case G 2: PSU(6, 2) we have N(L)/C(L) ^L3(4) and the latter group is known to have a transitive extension namely M22. Ifthere would exist a group generated by 3-transpositions for which N(L)/C(L) ^-M22, then <Dd> must be a 2-fold covering of PSU(6, 2). Now PSU(6, 2) isknown to have such an extension. Starting with this information Fischer con-structed a new simple group, which he called M(22). This process can berepeated twice, thereby getting the groups M(23) and M(24). Since for M(24) wehave N(L)/C(L) ^- M24, which does not have a transitive extension, the processis finished.

In his paper Fischer actually constructs the groups M(22)-M(24) by con-structing their graphs 6D (D) and showing that there is an additional automor-phism of 6 (D) besides the automorphism group induced by <Dd) .

4. The root-involution classification. For the classification of groups generatedby root-involutions one tries to copy the situations in the Chevalley-groups incharacteristic 2. If v, s are both long positive roots of the root-system of such agroup we get by the Chevalley-commutator-relations the following possibilities:

(1) ACS] = 1 iff v + s is not a root.(2) 'Xs] = iff v + s is a root.(3) (%,% ,) - SL2(q), where GF(q) is the ground-field.

Now one tries to copy this situation in the abstract group G generated byroot-involutions. That is one has to identify subgroups, which play the role ofroot-subgroups, which means, each pair of conjugates of these groups satisfy oneof the possibilities (1)-(3). Apart from technical difficulties in the identification,this is actually the major difficulty in the classification. Here the followingproblems arise:

(a) Show that the graph 6 (D) is connected. Actually one needs a somewhatstronger statement. Let F(D) be the graph with vertex-set D and edges (e, d),where f e, d, ed } C D. If D is a nondegenerate set of root-involutions of G thenone needs to show that F(D) is connected. This is needed to show that certainsubsets of D generate already the whole group. Fortunately it is not too difficultto reduce the problem of connectedness of F(D) to the connectedness of 6 (D).

(/3) Show that E(d) = Ed U 1 is a group for d E D. This is done by showingthat elements of <Ed> Ed already lie in Z(G). At this point one needsminimality of G, namely that the subgroup centralizing such an element isalready of known type.

(y) Show that for each pair E(d), E(e); e, d E D one of the possibilities(1)-(3) hold.

(y) is actually false in the groups J2 and A6, where E(d) = <d>. Unfor-tunately I could not prove (y) in general, even for I E(d)J > 2. I proved that oneeither has possibilities (1)-(3) for each pair E(d), E(e) or there exists a weakTI-subset T C D satisfying:

(i) T = <T> and I<T>I = q2, where q = E(d)J,(ii) Ed C T if d E T.

The case where these weak TI-sets T occur corresponds to the groups of type A2,

G2 and 3D4. If one calls the conjugates of T lines and the subgroups E(d) pointsand says that E(d) lies on the line T, if and only if Ed C T, one gets ageneralized hexagon, which is of course the generalized hexagon correspondingto these groups. In the course of characterizing these groups I actually showedthat one gets a generalized hexagon in this way.

Part I of [12] contains the solution of problems (a)-(y) and the treatment ofthe groups of type A2, G2, 3D4 and the groups A6, J2. In part II one can assumethat (y) holds. It contains the classification of all the other groups. One proceedsroughly as follows: Let C = CG(E(d)), x E D such that <E(d), E(x)) L2(q)and E = CD(E(d)) n CD(E(x), X = <E>. Then one shows that E is a set ofroot-involutions of X and X is a complement of 02(C) in C. By minimality of Gone may assume that X is of known type. Now one tries, as in the case ofodd-transpositions, to determine G from the structure of X.

5. Applications to characteristic 2-type groups. Apart from applications of the"root-involution" classification for the identification of certain groups, thereseem to be 2 major applications, which I will describe here.

(1) Involutions of type a2. Let V be a nondegenerate symplectic space over afield of characteristic 2 and t an involution acting on V. Then t is of type a2 if

(i) dim[ V, t] = 2,(ii) V = (vl(v, v') = 0).

Such an involution of type a2 fuses to the class of nondegenerate root-involu-tions in Sp(V). Hence if G is any group acting on V and containing t, then tG isa set of root-involutions of <tG>, whence <tG> "known" by [12]. This fact wasused extensively in [3] and [14] for the treatment of the "large extraspecialproblem", since an extraspecial 2-group Q is equipped naturally with thestructure of an orthogonal, whence symplectic space over GF(2). Actually in thiscase the classification of groups generated by 3 resp. {3, 4) '-transpositions issufficient. But the full "root-involution theorem" has been used in the generali-zation of the large extraspecial problem, the classification of groups with a"large TI-set".

(2) TI-sets. An elementary abelian 2 subgroup A of a group G is a TI-set in G,if A fl A 9 = A or I for all g E G. These TI-sets are connected withuniqueness-theorems. Namely if M is a maximal 2-local subgroup of G, H auniqueness subgroup of M and A ' C(H) an elementary abelian normal sub-group of M, then A is a TI-set in G. The connection between TI-sets androot-involutions is given by the following lemma, which is contained in [13].

(4.1) LEMMA. Suppose B is a TI-set in K such that B n Z(02(K)) # 1. LetV = <(B fl 02(K))K>. Then V is elementary abelian and one of the followingholds :

(1) B s V. _ _ _(2) Let K = K/V and D = (t E K`t - B5 in K). Then D is a set of

root-involutions of <D>.

In [4] M. Aschbacher went so far to determine all possibilities for <D) in case(2). Actually most of the root-involution groups do not occur, since they cannotact on V appropriately.

108 F. TIMMESFELD

Now using (4.1) in the above situation one tries to show that either A isweakly closed and so <AG> known by [13] or A < 02(H) for each A < H < Gsuch that A n Z(02(H)) I.1.

6. Simplifications of the proofs. There are several major simplifications of theexisting proofs, which seem to be possible at the moment.

(1) Connectedness of °D (D). It should now be possible to reduce the problemof determining those groups generated by root-involutions, for which LT(D) isdisconnected, either to Aschbacher's 2-generated core paper [2] or to thetheorem of Holt [9] and F. Smith [10]. But it would be more desirable to have acomplete classification of groups generated by a class D of involutions, forwhich D(D) is disconnected.

(2) Identification of symplectic, unitary and special orthogonal groups in char-acteristic 2. The identification of the unitary groups in [1] and [6] is relativelylong and complicated. Here the use of the Bukenhout-Shult-theorem [5], whichgives axioms for polar spaces only in terms of isotropic points and lines, wouldsave some work. The same theorem could be used for the classification oforthogonal groups in [12]. In both cases the Bukenhout-Shult-axioms are ob-tained in a relatively early stage of the proof.

(3) Identification of the exceptional Chevalley groups. I identified the groups oftype G2 and 3D4 using theorems of M. Harris [7], [8] and the groups of type F4,2E6, E6, E7, E8 by showing that certain graphs associated with E(d) are uniquelydetermined and determine already the group G as subgroup of their automor-phism group. It might be possible to find a common and shorter approach bygenerators and relations using Curtis' theorem.

(4) Of course joining the papers [1] and [6] resp. [1l.] and [12] will save manypages, since they overlap in large parts.

REFERENCES

1. M. Aschbacher, Groups generated by odd transpositions. I, Math. Z. 127 (1972), 45-56; II, IIIand IV, J. Algebra 26 (1973), 451-491.

2. , Finite groups with proper 2-generated cores, Trans. Amer. Math. Soc. 197 (1974),87-112.

3. , On finite groups in which the generalized Fitting group of the centralizer of someinvolution is extraspecial, Illinois J. Math. 83 (1977), 347-364.

4. , Weak closure in finite groups of even characteristic (preprint).S. F. Bukenhout and E. Shult, On the foundations of polar geometry, Geometriae Dedicata 3

(1974), 155-170.6. G. Fischer, Groups generated by 3-transpositions, Univ, of Warwick, 1969.7. M. Harris, A characterization of odd order extensions of G2(2") by the centralizer of one 2-central

involution, J. Algebra 23 (1972), 291-309.8. , A characterization of odd order extensions of 3D4(2") by the centralizer of one 2-central

involution, J. Algebra 24 (1973), 226-244.9. D. Holt, Transitive permutation groups in which an involution central in a 2-Sylow subgroup fixes

a unique point, Proc. London Math. Soc. 37 (1978), pp. 165-192.


10. F. Smith, On transitive permutation groups in which a 2-central involution fixes a unique point,Comm. Algebra (to appear).

11. F. Timmesfeld, A characterization of the Chevalley- and Steinberg-groups over F2, GeometriaeDedicata 1 (1973), 269-323.

12. , Groups generated by root-involution. I, J. Algebra 33 (1975), 75-135; II, 35 (1975),367-441.

13. , Groups with weakly closed TI-subgroups Math. Z. 143 (1975), 243 - 278.14.

,

, Finite simple groups in which the generalized Fitting-group of the centralizer of some

involution is extraspecial, Ann. of Math. 107 (1978), 297-369.

UNIVERSITY OF GtESSEN, FEDERAL REPUBLIC OF GERMANY


THE CLASSIFICATION OF FINITE GROUPSWITH LARGE EXTRASPECIAL

2-SUBGROUPS

STEPHEN D. SMITH

1. In the earlier days of the classification. The modem literature on largeextraspecial subgroups is firmly rooted in the classics. The idea appeared fullydeveloped, nearly a decade ago, in § 13 of Thompson's N-group paper [28]. Thesituation:

S is a maximal solvable subgroup of the N-group G, with e(G) at least3. O(S) = 1, so that F*(S) = 02(S).

Thompson eventually reduces to the case where all abelian normal subgroups of02(S) must be cyclic. In particular, this forces:

All characteristic abelian subgroups of 02(S) must be cyclic.

The group 02(S) is then said to be of symplectic type; it can be shown that02(S) is at most the central product of an extraspecial group and a cyclic group.

Thompson goes on to show that this condition is strong enough to lead tocertain explicit configurations, whose analysis disposes of the case e(G) > 3.

Much of the more recent work on classification of simple groups of character-istic 2-type follows the lines set out in the N-group paper, including verycrucially the reduction to a 2-local subgroup M with F*(M) a 2-group ofsymplectic type.

2. In the earlier days of sporadic groups. Thompson's minimal situation couldbe seen to hold "naturally" in many Chevalley groups defined over the field oftwo elements, as well as "exceptionally" in many of the sporadic simple groupsdiscovered during that era: He, Sz, Co.2, Co.l, F24. Indeed Janko's work [7]begins explicitly with such a situation: an extraspecial 2-subgroup Q is termedlarge in G if Q = F*(N(Z(Q))). Janko showed that a simple group G with alarge extraspecial 2-subgroup Q of order 25, and N(Q)/Q = A5, must beisomorphic either to J. (the Hall-Janko group) or J3.

1980 Mathematics Subject Classification. Primary 20D05-Z American Mathematical Society 1980

111

112 S. D. SMITH

Janko posed the problem: Classify all simple groups with a large extraspecial2-subgroup. Many of the early contributions to the solution were made by Jankoand his school. And the final solution of the problem was obtained within adeadline attributed to Janko.

Large extraspecial 2-subgroups can also be observed in many of the morerecently discovered sporadic groups: J4, F5 (Harada's group), F3 (Thompson'sgroup), F2 (the Baby Monster), Fi (the Fischer-Griess Monster). We remark thatthe approaches of Janko in [8] and Griess in [5] began with such configurations,while Fischer was led to the Monster by considering a large extraspecial3-subgroup.

3. The examples. From now on, we will use the following conventions.Notation.G: some finite simple group.Q: an extraspecial 2-subgroup large in G.n: the width of Q (so Q has order 22I )

M: the normalizer N(Q).z: the involution of Z(Q).

We will also use the following bar conventions for quotients:M for M/ Q.Q for Q/Z(Q). _

It is easily shown that Q is an orthogonal GF(2)-module for M.3A. Chevalley groups. As a small but still fairly typical example, we can take

the simple group L4(2). Here the configuration of interest is provided by thecentralizer of a 2-central involution. Using matrices, we can take z to be:

1

0 1

0 0 1

1 0 0 1

Then the other features are:

1 li i

Q = M = C(z) _0 1 . i .

* * * 1 *+We see that n = 2, and M = L2(2). The example may be even clearer if weconsider instead the Chevalley theory of L4(2) = A3(2), with associated Dynkindiagram:

From this point of view, we may take z to be the involution of the root subgroupfor the highest root (1, 1, 1). Then M is the parabolic subgroup obtained by"suppressing" the first and third nodes in the diagram. The unipotent radical Qis provided by the subgroup corresponding to the roots:

1 0 0 0 1 1

1 1 0 0 0 1

1 1 1

CLASSIFICATION OF FINITE GROUPS 113

We see that the highest root 111 is the sum of the pairs of roots in the first tworows; while no other possible pair, taken from these five, adds up to a root. Thisexplains, by use of the Chevalley commutator formula, why Q is extraspecial.The structure of M, on the other hand, is provided by a Levi complement in theparabolic; here it is L2(2), as described by the single remaining node.

This partition of root subgroups into noncommuting pairs, for a suitableparabolic, occurs in each of the Dynkin diagrams containing only single bonds.We indicate below which nodes are to be suppressed in the diagrams, in order todescribe the parabolic whose unipotent radical is a large extraspecial subgroup.

Ln+ 2 (2)

t' n+2(2)

IKX 4_)CK2n+4(2)

T2E6(2) 0 0 T a

E7 (2) S S S

The width of each extraspecial subgroup can be easily computed using knowl-edge of the root system. The Levi-factor structure of M should be readily visiblefrom the diagram.

114 S. D. SMITH

A number of questions should be at least partially addressed at this point. Forinstance, it might be noticed that the pairing of roots we described also occursfor type G2. However, G2(2) is not simple; so we have left it out of our table.Furthermore, for the diagrams with double bonds, a similar pairing can beshown to arise for a suitable parabolic. But in these cases, the unipotent radicalturns out to be the direct product of an extraspecial group with an elementarygroup. Of course, we can also replace 2 by an odd prime p, and our remarks giverise to large extraspecial p -subgroups. For odd primes, however, this concept hasnot (so far) had much importance. Finally if the prime p is replaced by a primepower, the corresponding unipotent radical is a "semi-extraspecial" p-group. Thecenter of this p-group is in fact a T.I. set in the whole group (a remark which istrivial in the prime case). We will return to consider some of these ideas in moredetail in the later sections.

3B. "Exceptions" from other families. A number of smaller simple groups notdefined over GF(2) also exhibit large extraspecial 2-subgroups. In this case, wecan interpret "smaller" to mean that the width of Q never exceeds 4.

Alternating: A6, A8, A9.Over GF(3): S24(3), L4(3), U4(3), G2(3), Q'(3)-3C. Sporadic examples.M11, M12, M24.Co.2, Co.l.Sz, He, F24-

J2, J3, J4.F5, F3, F2, Fl.

We mention also that large symplectic-type subgroups are found in U3(3) andHS; their automorphism groups G2(2) and HS.2 in fact have large extraspecial2-subgroups.

Work of many authors culminated in the proof (1978) that the list of examplesgiven above is complete. We will survey this work in a moment.

Another obvious question that arises here is: Why do so many of the sporadicgroups exhibit this configuration, which seems more properly to belong togroups defined over GF(2)? It appears that some of these groups can beconsidered to be "near GF(2)" in the sense that they possess a lattice of"parabolic" 2-local subgroups, which are the stabilizers of certain subconfigura-tions in a "natural" geometry over GF(2). Some investigations along these lineshave been described by the author and Mark Ronan, these PROCEEDINGS.

4. The chase is on. Beginning about 1975, work of a number of authorsbrought the idea of a general classification of simple groups with large extraspe-cial subgroups into sharper focus. I will try to list here, with fuller references inthe bibliography, these contributions.

4A. Beginnings. The small-width cases n = 1, 2 lead to a Sylow 2-group ofsectional 2-rank at most 4; and so were accounted for as special cases of themore general work of Gorenstein and Harada [4]. So we will assume n > 3 inthe statement of further results.

In the background also was Timmesfeld's fundamental work classifyinggroups generated by a class of (3, 4) +-transpositions [31]. The groups areessentially the Chevalley groups defined over GF(2); and this result would prove


to be the principal tool for the final identification of the Chevalley and twistedtypes in the large extraspecial problem.

4B. A telling blow. The most important result of this period was obtained byAschbacher, who made a study of the arguments of Thompson in the N-grouppaper, and extended them to the general extraspecial problem. In [1], he firstdealt with the case of a large symplectic (but not extraspecial) subgroup. Thenfor Q extraspecial he showed:

If zG n Q = (z), then G = or Co.2.Actually, Aschbacher's original result assumed G to be of characteristic 2-type,and F. Smith in [18] extended this to general G.

To see why a unitary group like U4(2) arises at this point, consider the unitarytransvection z given by:

0 1

0 0 1

1 0 0 1

with the form of Q and M being given as in the L4(2) example earlier:

1 li i

1*t *-*

Q + 0 1M

s 1 s s 1

s s s 1 - t----s s s 1

We see that the other unitary involutions in Q are not transvections. By contrast,recall that the Q for L4(2) contains plenty of transvections aside from z.

The effect of this result was to allow other workers to assume there is somea = z9 # z in Q. This turns out to be very helpful; it is particularly relevant tostudy the elementary quotient L = Q9 n M and its embedding in M.

4C. Other partial results. In several cases for n = 1, 2 it can be observed thatthe quotient M has the structure of the orthogonal group S22,,(2), nearly the fullautomorphism group of the orthogonal module Q. For an example, consider J2or J3 with n = 2 and M = A5 = S24(2). In [17], F. Smith showed that thissituation could further arise only in case n = 3 or 4. These cases correspond justto the sporadic groups Sz and Co.l, respectively, as is shown in Patterson [12]and Patterson and S. K. Wong [13].

The nature of the action of )Wand Q was also studied in some detail. F. Smithin [19] showed that M can contain no involution that acts as a transvection on

must act indecomposably on Q. Thisthe module Q. In [18], he showed that M-approach culminated in the result of Dempwolff and S. K. Wong [3]:

If M acts reducibly on Q, then G = L,,+2(2) or M24 or He.To see why the linear groups arise here, recall from the example of L4(2) that Qcontains two very different subgroups consisting of transvections:

and

1

0 1

0 0 1

* * * 1

which cannot be interchanged in M.

116 S. D. SMITH

It can also be seen that O(M) # 1 occurs basically in the orthogonal groupsthese groups, M = L2(2) X Elsewhere this arises only when

n = 6 and G = F24 or J4 (where M = 3 U4(3)2 or 3M222). Thompson consideredthis situation in [30]; and in the cases leading to the orthogonal groups, he wasable to establish that the conjugates of z in M form a set of (3, 4) +-transposi-tions. This essentially determines the structure of M.

The case M solvable was considered by Lundgren and S. K. Wong [9]. Also,some particular sporadic groups were characterized in work of F. Smith [20] andParrott [10], [11].

5. The problem succumbs. Timmesfeld considered the problem from thefollowing point of views: if a E zG n Q =(z), and Qa is O2(CG(a)), setL = Qa n M. What can be said about <E" >? The work of [32] used thisapproach to determine (essentially) all the possibilities for M. (In particular, theideas of Thompson in [30] were further developed for the case O(M) # 1.) Thisreduced the general problem to a small set of specific configurations, where thestructure of the centralizer M of z was partially or even completely determined.So the completion of the large extraspecial problem was brought within reach.

The remaining case leading to an infinite family, the orthogonal groups, wastreated by S. Smith in [21]. The result was in fact needed to complete the casescorresponding to E7(2) and E8(2).

The cases leading to E6(2) and 2E6(2) were first treated by Reifart in [14],using the method of Patterson and S. K. Wong [13]. Subsequently S. Smithshowed in [22] that the method of [21] could be adapted to yield an essentiallysimultaneous proof for these two groups, as well as E7(2) and E8(2).

The cases with n < 4 were considered by Bierbrauer (unpublished). Theparticular case leading to 3D4(2) was treated by Reifart in [15]. A completeanalysis of the cases with n < 6 appears in S. Smith [23].

The case leading to F2 was dealt with by a remark of Stroth and Reifart(compare also Bierbrauer [2]). This approach applied equally well to F1. Detailscan be found in [23].

Thus the classification of simple groups with large extraspecial 2-subgroupswas complete.

6. Generalizations. In §3, we noted that Dynkin diagrams with double bondsdo not quite give rise to the large extraspecial configuration. Since the orthogo-nal groups of odd dimension in characteristic 2 are isomorphic to symplecticgroups, we consider now only the diagrams:

0SPz,,(2)

F4(2)

In these cases, the unipotent radical Q of the indicated parabolic M is the directproduct of an extraspecial group and an elementary group. In [26], Strothobserved that many of the methods for the large extraspecial problem could beextended to this situation. His main result, like that of [32], determined the


possibilities for M = M1 Q. The remaining cases were subsequently analyzed byWester [35] and S. Smith [24]. The characterization of symplectic groups in thelatter is much like the treatment of the orthogonal groups in [21].

We also noted in §3 that if 2 is replaced by a power of 2, then the unipotentradical Q of the parabolic M becomes a semi-extraspecial group, whose center Zis a T.I. set. Consequently the Chevalley groups we listed, now over arbitraryfields of characteristic 2, constitute the standard examples of a more generalsituation considered by Timmesfeld:

M is a maximal 2-local subgroup, with F*(M) = 02(M), in a group Gwith 02(G) = 1. Z is a maximal elementary normal subgroup of M,and is a T.I. set in G.

In [33] and subsequent work, Timmesfeld was able to show that one of thefollowing situations must arise under this assumption:

(1) F*(G) = U3(4).(2) A is weakly closed in Q = F*(M) = 02(M).(3) Q is semi-extraspecial (possibly extraspecial) with Z = Z(Q).Now other work of Timmesfeld shows that case (2) leads to linear and unitary

groups in characteristic 2. To complete the determination of the groups in (3), itwas necessary to generalize the methods of the large extraspecial problem to thecase I Z I > 2. The analogue of the work of Timmesfeld in [32]-the determina-tion of M-was obtained by Stroth in [27]. The final analysis of the casesremaining was carried out by S. Smith in [25]. We remark that no sporadicgroups can arise in case (3) above when I Z I 2.

The natural question we now return to is: can results be obtained for oddprimes and prime powers? This problem seems less tractable for a number ofreasons. For instance, some crucial fusion results like the Z*-theorem are notavailable for odd p. And more generally, the technology of centralizers ofelements of odd prime order is not so well developed as for involutions.Nonetheless, the question has some interest. The natural examples are again theChevalley groups in characteristic p, but some sporadic groups can still arise.Not surprisingly, it seems that as p increases, it becomes more difficult for asporadic group to involve a large extraspecial p-subgroup. The most interestinggroup in this regard is the Monster F1, which contains the following normalizersof large extraspecialp-subgroups:

21+24Co.1, 31+122Sz, 51+62J2 71+42A7 13'+22S4.

The 5 primes here are exactly those for which the subgroup Fo(p) of the modulargroup SL2(Z) has genus 0. Phenomena like these will be receiving plenty ofattention in other lectures at this meeting, particularly those of Conway, Thomp-son (these PROCEEDINGS), Fong (these PROCEEDINGS), and Atkin.

7. Possible revision of existing proofs. One striking aspect of the F*-extraspe-cial work is that it is very largely self-contained; that is, it does not depend onthe rest of the simple-group classification effort, by assuming knowledge ofproper sections. In fact, the large extraspecial condition is sufficiently inductivethat suitable sub-configurations can often be identified by means of prior work.

118 S. D. SMITH

This feature suggests a number of ways in which the present rather lengthyproof might be unified and simplified.

The remarks to follow are my own speculations, and do not necessarily reflectthe views of the many other authors who have contributed to the present proof.

The assertions above about independence cannot, naturally, hold entirelywithout qualification. For the low-width cases, the present proof depends onsectional 2-rank at most 4. Indeed, it seems reasonable that any future version ofthe work would need to depend on this or some other basic minimal analysis;although it is possible that large extraspecial subgroups of width at most 2 couldbe handled independently with reasonable brevity. As regards other parts of thegeneral proof, the dependence of certain arguments in [23] on sectional 2-rank atmost 4 is due only to the indolence of the author, and could almost certainly bereleased by the application of a little energy.

The inductive possibilities in the F*-extraspecial hypothesis provide a particu-larly appealing avenue for future exploration of the proof. The various contri-butions to the present work mostly proceeded by reduction from the mostrecently available result, so by their very nature were limited in scope. Myfeeling is that the time has come to consider the proof in the light of thecomplete problem, with full use made of inductive situations; in particular in thedetermination of M, now provided by Timmesfeld's [32], as well as centralizersof suitable 3-elements that crop up in other parts of the general proof. Further-more, I suspect that the best way to achieve this would be to involve new handsin the task, rather than depending completely on the original workers to try torefine their old ideas.

This inductive direction now seems rather more natural in the light of therecent work for the groups over GF(2m) just described. Although this work againexhibited the classic division of labor [33], [27], [25], the proofs ended up beingshorter and generally easier, in the wake of the pioneering methods alreadyestablished for the extraspecial case. Maybe the best idea would be to attack thewhole work as a single unit; making all generic arguments for GF(2'") wheneverpossible, and only isolating the extraspecial case for GF(2) when absolutelynecessary.

The suggestions mentioned so far have deftly avoided dealing with a numberof the messier details of the present proof, which could not be introduced in anexpository lecture of this nature. For example, each sporadic group to beidentified in the work depends on a different recognition theorem, and theremay be no way to avoid this dependence. Of course, it is natural to ask for someuniform axiomatization of the exceptional cases, to explain in what way they areall really analogous to the groups defined over GF(2). The lecture of Ronan andSmith previously mentioned is one possible approach; I hope that others will becoming forward.

Finally I would like to reiterate my appeal to the present audience-to thosewho might be attracted by this work, to consider devoting some time to theimprovement of the existing proof. It is an elegant problem, with a fairly


tractable list in the conclusion; and one which richly repays any study by abetter understanding of the Chevalley groups over GF(2) and the sporadicgroups that come up. I hope to see a treatment, in one volume of reasonablesize, of the entire problem from start to finish.

REFERENCES

1. M. Aschbacher, Finite groups in which the generalized Fitting group of the centralizer of someinvolution is symplectic but not extraspecial, Comm. Algebra 4 (1976), 595-616; also On finite groupsin which the generalized Fitting group of the centralizer of some involution is extraspecial. Illinois J.Math. 21 (1977), 347-364.

2. J. Bierbrauer, A characterization of the "Baby Monster" F2, including a note on 2E6(2), J.Algebra 56 (1979), 384-395; also The finite simple groups containing large extraspecial subgroups ofwidth at most 4, J. Algebra (to appear).

3. U. Dempwolff and S. K. Wong, On finite groups whose centralizer of an involution has normalextraspecial and abelian subgroups. 1, J. Algebra 45 (1977), 247-253; II, J. Algebra 52 (1978),210-217.

4. D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most 4elements, Mem. Amer. Math. Soc. 147 (1974).

5. R. L. Griess, The structure of the Monster simple group, Proceedings of the Conference onFinite Groups (Utah, 1975), Academic Press, New York, 1976.

6. K. Harada, On the finite simple group F of order 21036567 11 19, Proceedings of theConference on Finite Groups (Utah, 1975), Academic Press, New York, 1976.

7. Z. Janko, Some new simple groups of finite order, Ist. Naz. Alta Math., Symposia Mathematica1 (1968), 25-64.

8. , A new finite simple group of the order 86, 775, 571, 046, 077, 562, 880 which possessesM24 and the full cover of M22 as subgroups, J. Algebra 42 (1976), 564-596.

9. R. Lundgren and S. K. Wong, On finite simple groups in which the centralizer M of an involutionis solvable and O2(M) is extraspecial, J. Algebra 41 (1976), 1-15.

10. D. Parrott, On Thompson's simple group, J. Algebra 46 (1977), 389-404.11. , Characterizations of the Fischer groups, II, Trans. Amer. Math. Soc. (to appear).12. N. Patterson, Thesis, Univ. of Cambridge, 1972.13. N. Patterson and S. K. Wong, A characterization of the sporadic Suzuki group of order 448, 345,

497, 600, J. Algebra 39 (1976), 277-286.14. A. Reifart, On finite groups with large extraspecial subgroups, I, J. Algebra 53 (1978), 452-470;

II, J. Algebra 54 (1978), 273-289.15. , A characterization of the simple group D42(2), J. Algebra 50 (1978), 63-68.16. , Eine Kennzeichnung einiger einfachen Gruppen vom Charakteristik 2- Typ (to appear).17. F. Smith, On groups with an involution z such that the generalized Fitting subgroup E of C(z) is

extraspecial and C(z)/E 3 Out(E)', Comm. Algebra 5 (1977), 207-277.18. , On finite groups with large extraspecial subgroups, J. Algebra 44 (1977), 477-487.19. , On the centralizer of an involution in fusion-simple groups, J. Algebra 38 (1976),

268-273.20. , A characterization of the Conway simple group. 2, J. Algebra 31 (1974), 477-487.21. S. Smith, A characterization of orthogonal groups over GF(2), J. Algebra 62 (1980), 39-60.22. , A characterization of finite Chevalley and twisted groups of type E over GF(2), J.

Algebra 62 (1980), 101-117.23. , Large extraspecial subgroups of widths 4 and 6, J. Algebra 58 (1979), 251-281.24. , A characterization of symplectic groups over GF(2) (preprint).25. , A characterization of some Chevalley groups in characteristic 2 (preprint).26. G. Stroth, Endliche Gruppen, die eine maximale 2-lokale Untergruppe besitzen, so dass

Z(F'(M)) eine T.I.-Menge in G ist (preprint).27. , Einige Gruppen vom Characteristik 2-Typ, J. Algebra 51 (1978), 107-143.

120 S. D. SMITH

28. J. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable. IV, Pacific J.Math. 48 (1973), 511-592.

29. , A simple subgroup of E8(3), Finite Groups, Iwahori (ed.), Japan Soc. Promotion ofScience, Tokyo, 1976.

30. , Notes on extraspecial 2-groups (unpublished).31. F. G. Timmesfeld, A characterization of Chevalley and Steinberg groups over F2, Geometriae

Dedicata 1 (1973), 269-321.32. , Finite simple groups in which the generalized Fitting group of the centralizer of some

involution is extraspecial, Ann. of Math. 107 (1978), 297-369.33. , On the structure of 2-local subgroups in finite groups, Math. Z. 161 (1978), 119-136.34. Tran van Trung, A general characterization of 2D4(2), J. Algebra 60 (1979).35. M. Wester, Endliche Gruppen, die eine Involution z besitzen, so dass das direkte

Produkt einer extraspezielle 2-Gruppe von kleiner Weite mit einer elementar abelschen 2-Gruppe ist,Dissertation, Univ. Mainz, 1979.

UNIVERSITY OF ILLINOIS AT CHICAGO CIRCLE


SOME CHARACTERIZATION THEOREMS

SERGEI A. SYSKIN

I shall speak about some results which have been proved in the Soviet Unionin the last few years. All groups considered are (of course!) finite. Simple groupmeans non-Abelian simple group.

1. 2-local subgroups. The following result is a generalization of Suzuki'swell-known classification of CIT-groups.

THEOREM 1 [6], [7]. Let G be a simple group in which the centralizer of everyfour-subgroup is a 2-group. Then G is isomorphic to one of the following groups:

L2(q) for q > 4, Sz(22n,1) for n > 1,L3(q), U3(q), G2(q) for suitable odd q,

Jn Ml v L3(4),2F4(2)'.

Theorem 1 is an "odd classification": it classifies all simple groups in whichthe centralizer of every element of odd order 1 has 2-rank at most 1. Thistheorem is special in Gorenstein's definition (the assumptions are not inductive).

In the course of the proof of Theorem 1 we prove the following:THEOREM 2 [6]. Let G be a simple group containing an elementary 2-subgroup

E 1 such that 02'(NG(E)/O(NG(E))E) = L2(2") for some n > 2, and anelement of order 3 in this group acts fixed-point freely on E. Then G is isomorphicto L3(2") or n = 2 and G = J3.

THEOREM 3 [1]. Let G be a simple group and let E be an elementary 2-subgroupof G such that NG(E) is 2-constrained and NG(E)/O(NG(E))E is isomorphic toL2(q) for some q > 4. Then q = 2" or q E (5, 7, 9) and the group G is known.

The list of such groups G is long (especially for q = 7).

THEOREM 4 [4]. Let G be a simple group, T E Sy12(G) and let A be an Abeliannormal subgroup of T. Then

(1) T/A is not a quaternion group;(2) if T/A is cyclic, then T contains an Abelian subgroup of index 2 (and G is

isomorphic to one of the following: L2(q) for q > 4, L3(q), U3(q) for odd q,Janko-Ree type groups, A7, Mi i).

1980 Mathematics Subject Classification. Primary 20D05.® Amencan Mathematical Society 1980

121

122 S. A. SYSKIN

2. Odd characterizations.

THEOREM 5 [2]. Let G be a simple group in which the centralizer of everynonidentity element of odd order is Abelian. Then G is isomorphic to one of thefollowing: L2(q) for q > 4, Sz(22n+i) for n > 1, L3(4).

Let A7 be a 3-fold covering of the alternating group A7.

THEOREM 6 L2]. Let G be a simple group, containing an element z of order 3 suchthat CG(z) = A7 and <z> is not weakly closed in CG(z). Then G is isomorphic tothe Held group.

3. Some open problems.(3.1) Conjecture (Glauberman). Let G be a finite group, p be a prime and

x E P E Sylp(G). Suppose xG n P = (x). Then x E Z(G mod OP,(G)).This conjecture is not yet proved. The case p = 3 is very important.

THEOREM 7 [5]. Let G be a finite group containing a strongly 3-embeddedsubgroup M. If 31 IZ(M)I, then 31 IZ(G mod O3.(G))I.

COROLLARY. If O3.(CG(g)) = 1 for every element of order 3 in G, then theconjecture holds.

(3.2) DEFINITION. A proper subgroup H of the group G is called weaklyembedded in G iff the order of H is even and NG(D) = O(NG(D))NH(D) forevery nonidentity 2-subgroup D of H.

In particular, every weakly embedded subgroup contains some S2 subgroup ofG and controls fusion in it. If 02.22(H) is of even order then a Sylow 2-subgroupof G contains a strongly closed Abelian subgroup and hence G is known by D.Goldschmidt's classification.

Question. Is G known in the general case?(3.3) Let G be a finite group with Sylow 3-subgroup Q, where Q is a

non-Abelian group of order 27. Suppose that zG n Q = Z(Q)# for someelement z. For example, the Hall-Janko group J2 satisfies this assumption, aswell as U3(3) and G2(q), 3D4(q) for suitable q.

Problem. Determine all such G.

REFERENCES

1. A. V. Borovik, 2-local subgroups involving L2(q), Siberian Math. J. (to appear).2. , A 3-local characterization of the Held group, Algebra and Logic 19 (1980).3. V. M. Busarkin, Finite groups with Abelian centralizers of elements of odd order, Algebra and

Logic 16 (1977), 381-388.4. A. S. Kondratev, Finite simple groups whose Sylow 2-subgroup is an extension of Abelian group by

a group of rank 1, Algebra and Logic 14 (1975), 288-303.5. N. D. Podufalov, 3-characterizations of finite groups, Algebra and Logic 18 (1979), 442-462.6. S. A. Syskin, On centralizers of 2-subgroups in finite groups, Algebra and Logic 17 (1978),

316-354.7. , Finite groups with primary centralizers of four-subgroups, Izv. Akad. Nauk SSSR Ser.

Mat. 42 (1978), 1132-1150.

INSTITUTE OF MATHEMATICS, NOVOSIBIRSK, USSR

ON FINITE GROUPS WHOSE SYLOW2-SUBGROUPS ARE CONTAINED

IN UNIQUE MAXIMAL SUBGROUPS

BERND STELLMACHER

In [3] B. Baumann characterized all finite groups generated by any pair ofdistinct Sylow 2-subgroups. In this note I want to outline the proof of a similarresult and state a related theorem on characteristic 2-type groups.

THEOREM 1. Let G be a finite group, S a Sylow 2-subgroup of G and Q asubgroup of S. Assume:

(i) The B-conjecture holds in every section of G.(ii) G is generated by any pair of distinct conjugates of Q.Then

d-G = 02,2(G), or one of the following holds for G = G/02(G)O(G).

(a) - L2(2"), U3(2") or Sz(2")_(b) F*(G) - L3(4), G = QF*(G) and I Q/Q n F*(G)I = 2.(c) F(G) = X rr31 G;, G. ^_ L2(q) for suitable odd prime powers q. _(d) G = QO(G), and O(G) is the unique minimal normal subgroup of G.

For the definition of F*(G) see [5]. We use the B-conjecture in the followingformulation:

B-CONJECTURE. Let X be a finite group and Y a 2-local subgroup of X. Then[E, O(E)] < O(X) for every 2-component E of Y.

From the hypothesis of Theorem 1 one can easily derive that NG(Q) is theunique maximal subgroup of G containing NG(S), and that G is generated by Sand any conjugate of S not contained in NG(Q). This leads to the followingdefinition.

DEFINITION. A finite group X is said to be minimally generated with respect toa subgroup H of X, iff

(1) H contains the normalizer of a Sylow 2-subgroup of X.(2) For T E Sy12(H) and every x E X and X0 = <T, TX> we have X0 = X or

X0 = (X0 n H)O(Xo).(3) X HO(X ).

1980 Mathematics Subject Classification. Primary 20D20, 20E07.® American Mathematical Society 1980

123

124 BERND STELLMACHER

THEOREM 2. Let G be a finite group minimally generated with respect to thesubgroup H, and assume:

(i) The B-conjecture holds in every subgroup of G.(ii) H is 2-constrained, or Sylow 2-subgroups of 0222(H) are not abelian.(iii) 02(G) = O(G) = 1.Then the following holds for S E Sy12(G):(a) F*(G) = X r, _I G,, G; ^ L2(q), for suitable odd q, L3(3), L4(3), L3(2"),

Sp4(2"), L2(2"), U3(2") or Sz(2").(b) S operates transitively on (G1, ... , G,).

If Q is not abelian, or if NG(Q) is 2-constrained, Theorem 1 is a corollary ofTheorem 2. In the other case a theorem of M. Aschbacher [2, Theorem 4]together with several classification theorems finishes the proof.

Now let G be a minimal counterexample to Theorem 2. We may assumewithout loss S < H. Set Z = SZ,(Z(S n 02.2(H))).

LEMMA 1. G is minimally generated with respect to NG(Z), and S n 02.2(H) <O2'2(NG(Z))

Lemma 1 enables us to assume H = NG(Z). As G is a counterexample, atheorem of D. Goldschmidt [5] implies that Z is not strongly closed in S withrespect to G.

Let GX(G) be the set of proper subgroups M of G, which are minimallygenerated with respect to NM(M n Z).

LEMMA 2. GX(G) 0.

LEMMA 3. No element of J1Z(G) contains a Sylow 2-subgroup of G.

Lemma 2 follows from Alperin's fusion theorem, as Z is not strongly closed inS. Lemma 3 is an easy consequence of the hypothesis of Theorem 2 and thedefinition of 'X(G ).

Lemmas 2 and 3 allow us to apply pushing up methods to a suitably chosen"maximal" element U of 'X(G). Set U = U/02(U)O(U).

If U is non-2-constrained, one can show that U satisfies the hypothesis ofTheorem 2. Therefore the components of F*(U) are known by induction. Now"pushing up methods for components" apply to deal with this case.

In the 2-constrained case "maximal" means maximal with respect to I U12. Thedefinition of )fL(G) implies that the normalizers of nontrivial characteristicsubgroups of a fixed Sylow 2-subgroup of U generate a proper subgroup of U.Thus, from a theorem of M. Aschbacher [1] we get F*(U) = X 1E;, E; -L2(2") or A2'+ 1, n > 1. The A2.+ 1-case for n > 2 can be eliminated by hypothe-sis (ii) of Theorem 2. Therefore 02(Nu(Z n U)) 1, and G and U (in place ofX and M) satisfy the hypothesis of the following pushing up theorem.

THEOREM 3. Let X be a finite group and O(X) = 1. Assume that the B-conjec-ture holds in X, and that there exists a subgroup M in X and a Sylow 2-subgroup Pof M with the following properties:

(a) There exists a proper subgroup W of M with PF(M) < W and <F, P'") _Mfor all m E M \ W.

ON FINITE GROUPS 125

(b) 02(W/F(M)) 1.

(c) CM(02(M)) < F(M).(d) If M is a subgroup of X containing M, then 1 or P E(e) 02(<Nx(P), U>) = 1.Then there exist subgroups XI, ... , X, (r > 1) in X, such that for i,j E

{ 1, ... , r) the following hold:(1) [Xi, Xj] = l for i j.(2) M < Nx(jl'- IX1) and P (Z Syl2(Nx(l,- 1X1))(3) X;/ O(X1) is simple.(4) Sylow 2-subgroups of Xi have nilpotency class 2 or sectional 2-rank at most

4.

For the definition of sectional 2-rank see [6]. We apply Theorem 3 to G and Uand take GI, . . . , G, and Go to be the subgroups of G corresponding toXI, ... , X. and NXQI;_IX). Then by classification theorems of R. Gillman andD. Gorenstein [4] and D. Gorenstein and K. Harada [6] the structure ofGi / O(G1) is known, and an easy argument shows that G = Go and ll;- I Gi _

which finishes the proof of Theorem 2.Another application of Theorem 3 is the following theorem about characteris-

tic-2-type groups, since in characteristic-2-type groups the B-conjecture holdsper definitionem.

THEOREM 4. Let G be a finite group of characteristic-2-type, 02(G) = O(G) _1 and S E Sy12(G). Assume that there exists a subgroup H in G with the following

property:(+) Every 2-local subgroup of G containing NG(S) is contained in H.Then G is isomorphic to L2(2"), U3(2"), Sz(2"), L3(3) or L2(q), q ± 1 a power of

2.

In particular Theorem 4 characterizes the characteristic-2-type groups, whosenormalizers of Sylow 2-subgroups are contained in unique maximal 2-localsubgroups. Theorem 4 can also be derived from a theorem of R. Footeannounced at this conference together with classification theorems of R.Solomon and S. K. Wong.

REFERENCES

1. M. Aschbacher, A factorization theorem for 2-constrained groups Proc. London Math. Soc. (toappear).

2. , On finite groups of component type, Illinois J. Math. 19 (1975), 87-115.3. B. Baumann, Endliche Gruppen, die von je zwei verschiedenen ihrer 2- Sylowuntergruppen erzeugt

werden, Arch. Math. 28 (1977), 34-40.4. R. Gillman and D. Gorenstein, Finite groups with Sylow-2-subgroups of class 2, I, II, Trans.

Amer. Math. Soc. 207 (1975), 1-101, 103-126.5. D. Goldschmidt, 2-fusion in finite groups, Ann. Math. 99 (1974), 70-117.6. D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most 4

elements, Mem. Amer. Math. Soc. 147 (1974).

UNIVERSITAT BIELEFELD, FEDERAL REPUBLIC OF GERMANY


GROUPS HAVING A SELFCENTRALIZINGELEMENTARY ABELIAN SUBGROUP

OF ORDER 16

G. STROTH

The classification problem for the groups of the title is related to theclassification problem of finite simple groups of low 2-rank, i.e. of 2-rank atmost four. The main interest in the groups of low 2-rank occurs in the study offinite groups of odd characteristic type. The signalizer functor method does notbegin to work well unless the 2-rank of G is at least 5. In view of the unbalancedgroup problem it seems necessary, therefore, to treat the case in which G has2-rank at most 4 independently. If G is a finite simple group of 2-rank at most 2then G is known by results due to J. Alperin, R. Brauer, D. Gorenstein, R.Lyons and J. Walter. If G is of 2-rank three then G is known by results due toM. O'Nan and G. Stroth. No such general characterization exists if G is of2-rank four. Perhaps we do not need such a general theorem. In dealing withgroups of 2-rank four, groups containing an elementary abelian subgroup oforder 16 which is selfcentralizing play a crucial role. Results about this kind ofgroups are very helpful in general classification problems.

For the remainder let G be a finite simple group, T a Sylow 2-subgroup of Gand E an elementary abelian subgroup of G with JEl = 16 and CG(E) = E(some of the following theorems remain true under weaker conditions onCG(E)). Set H = NG(E).

At first suppose H to be nonsolvable. Then H/E is isomorphic to A5, S5, A6,A7, a group containing a subgroup of index at most 2 isomorphic to Z3 X A5,A8, E8L2(7), L2(7) or S6. In [2] K. Harada proves the following theorem.

THEOREM 1. Suppose H/E Q5 A7, A6, S5, A5 or a group containing a subgroup ofindex at most 2 isomorphic to Z3 X A 5. Then G is of sectional 2-rank at most 4.Thus G is known.

If H/E is isomorphic to A. the combination of the work of G. Kierman [5],K. Harada [3], [4], M. O'Nan and R. Solomon [7] and H. Yamaki [4], [11] yield


127

128 G. STROTH

THEOREM 2. Suppose HI E = A8. Then G = L5(2), M24, A 16, A 17, A 18, A 19 or

C3.

If H/E = E8L2(7) we have the same Sylow 2-subgroup in H/E as in the caseH/E = A8. Thus a careful lemma-by-lemma analysis of Harada's paper [3] andO'Nan's and Solomon's paper [7] yields [9]

THEOREM 3. Suppose H/E = E8L2(7). Then G = M24 or He.

If H/E is isomorphic to L2(7) there are two cases. If Z(H) = 1 then Gcontains a subgroup X of index two which is of sectional 2-rank at most 4.Suppose now Z(H) 1. Then CG(Z(H))/Z(H) contains a selfcentralizingelementary abelian subgroup of order 8. Thus CG(Z(H)) is known by [2]. Fusionarguments yield now how G looks.

Only the case H/E = S6 is left. If H splits over E it is easy to see that Gcontains a subgroup of index 2. Then Theorem 1 can be applied. Thus assumethat the extension is nonsplit. Let S be a Sylow 2-subgroup of H. Then

J(S) = <A C SSA' = 1, A maximal> = Z2 X Z4 X Z4.

Furthermore N1(J(S))/J(S) = S4. Now we determine the structure ofNG(J(S)). Let GX = (X IX < G, CC(J(S)) = Z2- X Z2. X Z2., for some m, nX = Cx(J(S))NH(J(S))). Choose X maximal in 9C. Then it is possible todetermine the structure of X. The maximality yields

(1) CG(Cx(J(S))) = Cx(J(S)),(ii) I G: NG(Cx(J(S)))l is odd,(iii) I NG(CX(J(S))): X 12 < 8.

Up to this point mainly the structure of N,1(J(S)) was used. Thus it seems likelythat we can reach the same conclusions in some cases in which H/E is solvable.

Now fusion arguments come into the picture. First of all we show thatNG(Cx(J(S))) controls fusion in S21(Cx(J(S))). If all involutions in Cx(J(S)) areconjugate then L2(7) is involved in NG(Cx(J(S))). Hence n = m. With similararguments as in [7] we get n = 2 and G = HiS.

After that we have a certain kind of nonfusion. Take an involution w ECx(J(S)) which is not 2-central in G. In almost all cases CG(w) contains noelementary abelian subgroup of order 16. Together with the fact that only a fewinvolutions in NG(Cx(J(S))) have this property we get very good informationabout fusion and the structure of CG(w). This information yields that there areonly two possibilities for the structure of NG(Cx(J(S))) left. Thus we have theexact structure of the Sylow 2-subgroup T of G. Now the application of resultsdue to D. Mason [6] and R. Foote [1] yields [10]

THEOREM 4. Suppose H/E = S6 then G =HiS, L4(q), q - 1 (mod 8), U4(q),q - -1 (mod 8), L6(q), q - -1 (mod 4) or U4(q), q - 1 (mod 4).

Summarizing Theorems 1-4 we get

THEOREM 5. Suppose H/E to be nonsolvable. Then G is known.

For future study there remains the case that H is solvable. A lemma which iseasy to prove reads as follows.

A SELFCENTRALIZING ELEMENTARY ABELIAN SUBGROUP 129

LEMMA 8. If H is solvable with 02(H/E) = 1 then G is of sectional 2-rank atmost four. Thus G is known.

Assume now 02(H/E) 1. This case seems to be quite hopeless. But forapplications it is enough to look at the case where H is not a 2-group. Then it isquite easy to treat the case 71 JHJ. Thus the case that H/E is a (2, 3)-groupremains. One of the most difficult cases but for applications a very importantcase is HI E = S4 or Z2 X S4. It may be that we can determine the structure ofNG(J(S)) for some Sylow 2-subgroup S of H with similar arguments as in theS6-case. Then it seems very probable that we can come to a final result.

For the future we have to look for a unified short proof of all the results aboutgroups containing a selfcentralizing elementary abelian subgroup of order 16mentioned above.

REFERENCES

1. R. Foote, Finite groups with Sylow 2-subgroups of type L6(9), 9 = 3 (mod 4) (to appear).2. K. Harada, Of finite groups having self-centralizing 2-subgroups of small order, J. Algebra 33

(1975), 144-160.3. , Finite groups having 2-local subgroups E16L4(2) (to appear).4. K. Harada and H. Yamaki, Finite groups having 2-local subgroups E16L4(2), II (to appear).5. G. R. Kierman, On finite groups with a 2-local subgroup which is a nontrivial split extension of E24

by L4(2), thesis, Rutgers Univ., 1975.6. D. Mason, Finite simple groups with Sylow 2-subgroups of type PSL(4, q), q odd, J. Algebra 26

(1973), 75-98.7. M. O'Nan and R. Solomon, Simple groups transitive on internal flags, J. Algebra 39 (1976),

375-410.8. G. Stroth, Gruppen mil kleinen 2-lokalen Untergruppen, J. Algebra 47 (1977), 441-454.9. __, Endliche einfache Gruppen mil einer zentralisatorgleichen elementar abelschen

Untergruppe von der Ordnung 16, J. Algebra 47 (1977), 480-523.10. , Endliche einfache Gruppen mit einer 2-lokalen Untergruppe E1616, J. Algebra 47 (1977),

455-479.11. H. Yamaki, Finite groups with Sylow 2-subgroup of type A16, J. Algebra 33 (1975), 523-566.

FREIE UNIVERSITAT BERLIN


p-LOCAL SUBGROUPS

GEORGE GLAUBERMAN

1. Introduction. My topic isp-local analysis, i.e., the study of p-local subgroupsof a group. Recall that, for a group G and a prime p, the p-local subgroups of Gare the normalizers NG(H) of the nonidentityp-subgroups H of G. Now,p-localanalysis has considered problems in two main areas, known as control andpushing-up:

Question 1 (Control). What do thep-local subgroups tell us about G?Question 2 (Pushing-up). What do the maximal p-local subgroups look like if

G is simple?In both cases, we are concerned with the general problem of determining

relations between the local and global properties of G, that is, between thestructure of various subgroups of G and the structure of G itself.

Now, p-local analysis has developed partly in the form of general theoremsand partly in the form of ad hoc arguments in proofs of other results, such asclassification theorems or results on signalizer functors or fixed-point-freegroups of automorphisms. Often, general results have been sought and obtainedbecause special arguments have failed, and vice versa. For brevity, I will restrictmyself mainly to general results and will usually not state them in full generality.Since I have written two surveys on this subject (Chapter 1 of [PH], [GI]), I willtouch briefly on the highlights of these surveys and then proceed to recentresults and problems.

Now let G be an arbitrary group. Take a prime p and a Sylowp-subgroup S ofG. To avoid triviality, we will generally assume that S 1. The main concept in

p-local analysis is that of the fusion of S in G; two elements of S are said to befused in G if they are conjugate in G. Fusion is of interest for its own sake.Moreover:

(1) Fusion determines transfer in G, i.e., it determines the Sylowp-subgroup ofG/ G' [G4, pp. 245-251].

(2) Fusion determines whether G has a normalp-complement [PH, p. 26].In addition, fusion yields strong information aboutp-local subgroups of G.


131

132 GEORGE GLAUBERMAN

In general, it is easy to find examples of elements x, y E S and u E G suchthat x" = y, but u does not centralize or normalize any nonidentity subgroup ofS and seems to have no relation to S. Thus, at first glance, it might appear thatfusion is a rather haphazard matter. But Alperin's Fusion Theorem [PH, p. 7]tells us that this is not the case. Indeed, two elements x, y of S are fused in G ifand only if there is a chain of elements

x = x0, x1, ... , X =Y

of S such that each element is conjugate to the next in some p-local subgroupNG(T) for 1 c T c S. Thus, fusion of S in G can be viewed as the smallestequivalence relation on S that includes fusion of Ns(T) in NA(T) for everynonidentity subgroup T of S. We describe this fact by saying

(3) (Alperin, 1967) The nonidentity subgroups of S jointly control fusion of Sin G.

2. Control by one characteristic subgroup. Alperin's result and (1) tell us thatthe nonidentity subgroups of S, taken all together, jointly control transfer.However, Thompson's work on normal p-complements [H3, p. 438] andWielandt's earlier work on transfer [W] showed that it is possible for one or twocharacteristic subgroups to control transfer (or some other consequence of fusion)even if they do not control fusion. Much of the work onp-local analysis from 1960to 1970 was devoted to the subject of control by one characteristic subgroup;this was the main theme of my survey in [PH]. The major reductions for suchresults, developed largely by Alperin and Gorenstein, show that:

(4) (Alperin-Gorenstein) Most questions of control reduce to sections G of Gfor which CG.(OP(G')) C [PH, pp. 15-20], [GI, pp. 7-18].

Among the results obtained were the following (here and later, see thereferences for the definitions of the subgroups mentioned):

(5) (GG) One characteristic subgroup controls transfer if p > 5 (e.g., K,.(S)[PH, p. 36]).

(6) (GG-Thompson) One characteristic subgroup controls normal p-comple-ments if p is odd (e.g., Z(J(S)) [G4, Theorem 8.3.1]).

(7) (GG) One characteristic subgroup controls fusion if p is odd and allsections of G arep-stable (e.g., Z(J(S)) [PH, p. 43]).

Result (7) used the Alperin-Gorenstein reduction and the following:(8) (GG) Z(J(S)) < G if p is odd, CG(OP(G)) C- OO(G), and G is p-stable

[G4, Theorem 8.2.11 ].These results have been applied by Bender in his simplification [B2], [B3], [B4]

of the papers classifying groups of odd order and groups with Abelian ordihedral Sylow 2-subgroups.

3. Factorizations in E4-free groups. The outstanding open problem raised bythe results of §2 was to find an analogue of (8) forp = 2. Now,p-stability is notdefined forp = 2; the appropriate substitute appears to be the condition that Gis E4-free, i.e., that the symmetric group E4 of degree 4 is not involved in G.Thus, we ask the following [GI, p. 57], [PH, p. 49]

Question 3. Suppose P is an arbitrary nonidentity 2-group. Does there exist anonidentity characteristic subgroup L(P) of P such that L(P) < H for every

p-LOCAL SUBGROUPS 133

1:4-free group H for which CH(02(H)) C 02(H) and H contains P as a Sylow2-subgroup?

The following special case is of particular interest:Question 3'. Restrict H to range over solvable 3'-groups in Question 3.These and other questions raised by the results of 1960-1970 seemed very

hard, and relatively little progress was made in this area for about 5 years.However, side by side with the formal, general results of the 1960s, there hadbeen many special arguments using two or three subgroups and factorizations,e.g., in the Odd Order Paper and the N-Group paper. Eventually, these ideasand some results of Thompson on 3'-groups were extended to obtain thefollowing result [G1, Chapter II]:

(9) (GG) Supposep = 2 and G is 1:4-free.

(a) If CG(02(G)) C 02(G), then there exist nonidentity characteristic sub-groups A1, A2, A3 of S, depending only on S (not on G), such that

G = NG(A1)NG(A2) = NG(A1)NG(A3) = NG(A2)NG(A3)

(b) In any case, there exists a nonidentity Abelian normal (but not necessarilycharacteristic) subgroup of S that controls fusion of S in G.

Here, (a) begins with the Thompson factorization G = CG(Z(S))NG(Je(S)).Recently, T. Hayashi has proved two remarkable new results [H1, H1'] whichstrengthen (9)(b) and seem likely to help investigation of Question 3.

4. Pushing-up. In Question 2, we asked about the properties of maximalp-local subgroups in simple groups. As an example, we may ask the following:

Question 4. Suppose G is a maximal p-local subgroup in a simple group Gand CG(Op(G)) C O,(G). Does G contain NG.(S)?

This question is of interest because, whenever an affirmative answer occurs, Sis a Sylow p-subgroup of G. Moreover, by a result of Borel and Tits [G5, p. 67],it always has an affirmative answer if G' is a group of Lie type ofcharacteristic p.

Suppose in Question 4 that S possesses a nonidentity characteristic subgroupK which is normal in G. Then NG.(K) contains both G and NG.(S). Thus, if Gdid not contain NG.(S), we could "push up" G to the strictly larger p-localsubgroup NG.(K) of G. But this would contradict our choice of G as a maximalp-local subgroup. Thus, Question 4 can be answered affirmatively in this case.This applies in particular if p is odd and G is p-stable, by (8) (with K =Z(J(S))). It would also apply when p = 2 and G is 1:4-free, if one could answerQuestion 3 affirmatively.

Unfortunately, sometimes the answer to Question 4 is "no", e.g., whenG' = PSL(2, 17), p = 2, and G E4. Here, G/ 02(G) = 1:3 me SL(2, 2). In in-vestigations of "small" simple groups, such as thin groups, Question 4 arises forvarious subgroups of this type. Therefore, we will consider the following condi-tion:

(+) CG(OP(G)) C O,(G) and G/OO(G) = SL(2, p") for some n.We ask whether there exists a nonidentity characteristic subgroup K of S such

that K < G. At this point, we may consider G as an abstract group and forgetabout Question 4.

Unfortunately, K need not exist. For every prime power p", there is acounterexample, Qd(p"), consisting of all matrices of the form

a b e

c d f , a, b, c, d,e,f EGF(p"), ad - bc = 1.0 0 1

For P" = 2, Qd(p") is isomorphic to 1:4.

The examples Qd(p") were discovered about 1965. Since S has a fairly simplestructure in each case (in fact, S' is Abelian), it seemed hopeless to try to find Kin the general case, when S can be much more complicated. This inhibitedprogress on the general case for about ten years. In the meantime, however, Sims(1967) had obtained [S] strong results forp" = 2; and these were extended [Ni]by Niles (1976), who showed (for arbitraryp that one could always find K if Swere sufficiently complicated, e.g., if S' is not Abelian. A similar result (forp = 2) was obtained [Bl] at the same time independently by B. Baumann, whoused very different methods. By extending Baumann's argument, Niles and Imanaged to show [GN] that, if S' is not Abelian, then K can always be chosen tobe one of two pre-assigned characteristic subgroups S1, S2 of S, which do notdepend on G. In other words:

(10) Whenever (.) is satisfied and S' is not Abelian, then Sl < G or S2 < G.Actually, S, S2 are closely related to the two groups Z(S), J(S) of the

Thompson factorization; Sl C Z(S) and S2 is characteristic in CS(Z(J(S))).(The group CS(Z(J(S))), which contains J(S), was discovered by Baumann andhas a number of important properties.) Recently, N. Campbell has proved [C1] avariant of (10) which places restrictions on G rather than S.

One might hope to improve (10) by obtaining one subgroup which is normalfor every G. Unfortunately, this is impossible in general. To see this, letH = PSL(k, p") for k > 5. Let S be a Sylow p-subgroup of H, e.g., an uppertriangular group with all entries on the main diagonal equal to 1. LetP1, ... , Pk_ 1 be the rank one parabolic subgroups of H that contain S, and letG. = Op (P;) for each i. Then each G. satisfies (.) (in place of G) and

H = <G1, ... , Gk- I>'

Since H is simple, no nonidentity subgroup of S can be normal in every G;.Incidentally, in this case, one can take Sl = Z(S) and S2 = CS(Z(J(S))). (Moreinformation about (10) appears in my notes in [C3].)

The results above have been applied to simple groups in two ways. In somecases, the p-local subgroups are "small" and satisfy (.). In other cases, groupssatisfying are the "building blocks" of some arbitrary p-local subgroups Hsatisfying CH(OP(H)) C OP(H) [Gl, pp. 54-55]. For example, Campbell'svariant of (10) has been applied by Aschbacher to obtain a result in one of hisexpository talks, namely, a description of groups of characteristic 2 type forwhich G <CG(Sl), NG(S2)>.

In addition, (10) has been used very recently for two results on control by twocharacteristic subgroups for Schur multipliers for p > 5 by D. Holt [H2], and fortransfer for p = 3 by myself [G2]. (Holt obtains control of Schur multipliers byone characteristic subgroup for p > 11.)

p-LOCAL SUBGROUPS 135

The results above have several possible applications to the revision program.Gorenstein has suggested in his expository talk that (10) could be used tosimplify the Alperin-Brauer-Gorenstein classification [ABG] of groups withquasi-dihedral or wreathed Sylow 2-subgroups. For p odd, such groups need nothave p-stable subgroups, and (10) might help in situations where (7) and (8) donot apply. In addition, for certain other families of simple groups, it might bepossible to use (10) to show that S1 and S2 jointly control fusion unless Gsatisfies some strong restriction. This would be especially powerful for p = 2,where the determination of fusion sometimes constitutes a substantial propor-tion of the proof. Recently, P. McBride has investigated this problem.

5. Open questions. In addition to Questions 3 and 3' and the possibleapplications to revision mentioned above, there are a large number of openproblems. Several were mentioned in expository talks by Aschbacher andGorenstein. Among them are the two following:

Question 5 (Triple Factorization). Given (*) and suitable restrictions on S orG, find nonidentity characteristic subgroups S S2, and S3 of S, depending onlyon S (not on G), such that at least two of the groups S, are normal in G.

Question 6. Extend (10) to a result for groups G such that G/OP(G) isisomorphic to a group of Lie type of Lie rank 2 and characteristic p, inparticular for p = 2.

N. Campbell has obtained a partial answer [Cl] to Question 6 for p = 2 andG/OP(G) = L3(2).

The following problem, like Question 4, occurs in the study of simple groups:Question 7. Suppose H is a simple group and Gl and G2 are maximal p-local

subgroups of H containing a Sylow p-subgroup of H. Assume that C, (OO(G;))c OP(G,) for i = 1, 2. Is G1 equal to G2?

Like Question 4, this question can be answered affirmatively in "nice" cases.All one needs is a nonidentity characteristic subgroup of S (or merely normalsubgroup of S) normal in both G1 and G2. Recently, Goldschmidt [G3] hashandled a case that is not "nice," namely, when p = 2 and G1/O2(G1) aG2/02(G2) _ 23 = SL(2, 2). He proves that the answer is affirmative ifI G1 I > 3 . 2', but may be negative otherwise. His work applies to §18 of theN-group paper. It seems likely that his methods will help in the investigation ofQuestions 5 and 6.

Recently, Chermak [C2] and Niles [N2] have considered different situationsrelated to that of Question 7, where one is given three or morep-local subgroupsG...... G, and assumes for each i j that <G;, Gj> looks like a rank 2parabolic subgroup of a group of Lie type of characteristic p. In these situations,they show that <G1, . . . , G,> looks like a rank r parabolic subgroup. Theirresults are discussed in Goldschmidt's expository talk on pushing-up "from thetop down."

One may try to get around the examples with PSL(k, p ") in §4 by consideringspecial cases of (*) as follows:

Question 8 (Thompson). Assume (*). Suppose p is odd and at least onenoncentral chief factor of G within OP(G) is not a standard (natural) module forG/OP(G). ISZ(J(S)) d G?


It would be interesting to know whether the results in §4 on control of Schurmultipliers and transfer by two characteristic subgroups could be improved tocontrol by one characteristic subgroup [Gl, p. 57]. It seems likely that this wouldfollow from an affirmative answer to Question 8, possibly with Z(J(S)) replacedby some other characteristic subgroup.

For the future, it seems to me that the best chance for progress on theseproblems lies in the extension of Baumann's methods (applied in [GN]) and ofGoldschmidt's graph-theoretic methods [G3], possibly for the discovery of newcharacteristic subgroups. Unfortunately, the present depression in mathematicsmay present more of an obstacle to progress here and elsewhere than the actualdifficulty of the problem, and may require more effort to be overcome.

REFERENCES

[ABG] J. L. Alperin, R. Brauer and D. Gorenstein, Finite groups with quasi-dihedral and wreathedSylow 2-subgroups, Trans. Amer. Math. Soc. 151 (1970), 1-261.

[BI] B. Baumann, Uber endliche Gruppen mit einer zu L2(2") isomorpher Faktorgruppe, Proc. Amer.Math. Soc. 74 (1979), 215-222.

[B2] H. Bender, On the uniqueness theorem, Illinois J. Math. 14 (1970), 376-384.[B3] , On groups with abe/ian Sylow 2-subgroups, Math. Z. 117 (1970), 164-176.[B4] , On finite groups with dihedral Sylow 2-subgroups, J. Algebra (to appear).[Cl] N. Campbell, Pushing up in finite groups, Ph. D. Thesis, California Institute of Technology,

1979.

[C2] A. Chermak, On certain groups with parabolic type subgroups over Zz (preprint).[C3] M. Collins (ed.), Finite simple groups. II, Proc. Durham Conf., Academic Press, London (to

appear).[Cl] G. Glauberman, Factorizations in local subgroups of finite groups, CBMS Regional Conf. Ser.

in Math., vol. 33, Amer. Math. Soc., Providence, R. I., 1977.[G2] , Control of transfer for p = 3 (in preparation).[G3] D. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. 111 (1980), 377-406.[G4] D. Gorenstein, Finite groups, Harper and Row, New York, 1968.[G5] , The classification of finite simple groups. I, Bull. Amer. Math. Soc. (N.S.) 1 (1979),

43-200.[GN] G. Glauberman and R. Niles, A pair of characteristic subgroups for pushing-up (in prepara-

tion).[H1] T. Hayashi, 2 factorization in finite groups, Pacific J. Math. 84 (1979), 97-142.[H1] , On the existence of a characteristic 2-subgroup of a finite special group (in prepara-

tion).[132] D. F. Holt, More on the local control of Schur multipliers (preprint).[H3] B. Huppert, Endlichen Gruppen. I, Springer-Verlag, Berlin and New York, 1967.[NI] R. Niles, Pushing-up infinite groups, J. Algebra 57 (1979), 26-63.[N2] , BN-pairs and finite groups with parabolic type subgroups (preprint).[PH] M. B. Powell and G. Higman (eds.), Finite simple groups, Academic Press, New York, 1971.[S] C. C. Sims, Graphs and finite permutation groups, Math. Z. 95 (1967), 76-86.[T] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable. VI, Pacific

J. Math. 51 (1974), 573-630.[W] H. Wielandt, p-Sylowgruppen and p-Faktorgruppen, J. Reine Angew. Math. 182 (1940),

180-193.

UNIVERSITY OF CHICAGO, CHICAGO


LOCAL ANALYSISIN THE ODD ORDER PAPER

GEORGE GLAUBERMAN

For several years, I have been working on a revision of the first half of theproof of the Odd Order Theorem, i.e., the part which uses mainly local analysisrather than character theory. Very recently, Helmut Bender has substantiallysimplified this part (beyond his earlier published simplification of the Unique-ness Theorem in Illinois J. Math. 14 (1970), 376-384). Probably an up-to-dateset of lecture notes on this part will be available soon.

David Sibley has been working on the second half of the theorem, which usescharacter theory, and has obtained substantial reductions. A summary of hiswork and mine, and of progress on revising other papers, appears in my articlein the Proceedings of the Dupham Conference (Finite Simple Groups. II, M. J.Collins (ed.), Academic Press (to appear)).



137

FINITE GROUPS WITH A SPLIT BN-PAIROF RANK ONE

MICHIO SUZUKI1

1. Introduction. This is a report on the class of finite groups with splitBN-pairs of rank 1; 1 will give a survey of the known results on this class ofgroups and touch upon the significance of these results in relation to theultimate classification of finite simple groups.

A finite group G is said to be a group with a split BN-pair of rank 1 if thefollowing conditions are satisfied:

(1) There are two subgroups B and N such that

G = <B, N >.

(2) The subgroup H defined by H = B n N is a normal subgroup of N, andthe factor group N/H is of order 2.

(3) For a generator s of N/ H, we have

G = B U BsB.(4) There is a normal subgroup P of B which satisfies the splitting conditions:

B=PH and PnH=(1).The concept of BN-pairs came from the fundamental work of Tits [35] on

geometries of simple algebraic groups. So, a BN-pair is also called a Tits system.In a general BN-pair, the group W = N/ H is generated by a distinguished setS = (s) of generators which consists of elements of order 2 and satisfies thefollowing two conditions:

For any s E S and w E W, we have

sBw c BwB U BswBand

sBs B.The number of generators IS I is called the rank of the BN-pair. The above

conditions yield the relation

1980 Mathematics Subject Classification. Primary 20-02, 20B 10, 20D05, 20F05.'The author gratefully acknowledges support of the National Science Foundation.

CC American Mathematical Society 1980

139

140 MICHIO SUZUKI

sBs c B U BsBfor any s E S. This implies that B U BsB is a subgroup. Thus, if the rank is 1,the above conditions are equivalent to conditions (1), (2), and (3) stated earlier.

Tits has shown that if the rank is at least 3, all finite BN-pairs are provided bythe simple algebraic groups over finite fields in the standard way (Tits [35, § 11]).In particular, finite simple groups with BN-pairs of rank at least 3 are preciselythose simple groups of Lie type with the usual Lie rank > 3. Thus, for a finiteBN-pair of rank > 3, the splitting condition (4) is always satisfied, and thesubgroup P is the unipotent radical of the Borel subgroup B. We remark that Pis the Sylow subgroup of B corresponding to the prime which is the characteris-tic of the ground field.

The concept of BN-pairs of rank 1 comes also from the theory of permutationgroups. Let G be a permutation group on a set Q. Choose a point a of Q. The setof elements of G which fix the point a forms a subgroup, which is called thestabilizer of a. Suppose that the permutation group G is doubly transitive on Q.Then, G contains an elements which exchanges two distinct points a and b. Forthe stabilizer B of the point b and

H = B n Bs (Bs = s-'Bs),conditions (1), (2), and (3) are satisfied. The subgroup N of (1) is the onegenerated by H and s:

N = <H, s>.In general, if B is a subgroup of a group G, then an element g of G acts on the

set of cosets of B viaxB H gxB.

In this way, G is represented by a permutation group on the cosets of B. It isclear that the stabilizer of the coset B is precisely the subgroup B.

Suppose that G is a group with a BN-pair of rank 1. Then, we have

G=BUBsBby Axiom (4). This means that the subgroup B acts transitively on the cosets xBwhich are different from B. Thus, the group with a BN-pair of rank 1 admits adoubly transitive permutation representation.

A doubly transitive permutation group need not satisfy the splitting condition(4). So, condition (4) does not follow from the other axioms of BN-pairs if therank is 1 or 2, although for finite BN-pairs of rank > 3, condition (4) is aconsequence of the other axioms, as remarked earlier. The class of finite groupswith BN-pairs of rank 1 is, then, the class of those finite groups which admitdoubly transitive representations with condition (4) on the stabilizers.

2. Brief historical comments. In the early studies of finite simple groups duringthe 1950s, we came to realize the significance of a class of doubly transitivepermutation groups in the classification problem. These were the doubly transi-tive permutation groups in which the stabilizer of any three distinct points istrivial. Such a transitive permutation group is called a Zassenhaus group, sonamed after the work of Zassenhaus [37] on the triply transitive permutationgroups with the same restriction on the stabilizers of three distinct points. If B is

FINITE GROUPS WITH A SPLIT BN-PAIR 141

the stabilizer of a point, say a, in a Zassenhaus group G acting on 0, then Bitself is a Frobenius group acting on the set SZ - (a). Hence, B contains acharacteristic subgroup P which satisfies the splitting condition (4) of the splitBN-pair of rank 1. Thus, the Zassenhaus groups are groups with split BN-pairsof rank 1.

The classification of Zassenhaus groups was accomplished in the early 1960sby Feit [8], G. Higman [17], Ito [19], and Suzuki [28]. These works, together withthe Feit-Thompson paper [10] on the solvability of finite groups of odd order,provided not only the technical tools to investigate further problems concerningfinite simple groups, but also a firm belief that the classification of all finitesimple groups would not be an impossible dream.

The concept of finite groups with split BN-pairs of rank 1 appeared for thefirst time in Suzuki [30], although it was stated entirely in terms of permutationgroups and only its special case when the subgroup P of condition (4) containeda Sylow 2-subgroup of G was considered. This special case was needed todetermine the structure of finite groups which satisfy the condition:

(TI): Any two distinct Sylow 2-subgroups have a trivial intersection.It has been proved that if T is a Sylow 2-subgroup of a finite group G with the

condition (TI), then the normalizer NG(T) = B satisfies the axioms for the splitBN-pair of rank 1. Thus, the above result gives the classification of (TI)-groups(Suzuki [31]).

This intimate connection between the study of finite simple groups and thestructure of groups with split BN-pairs of rank 1 was the key to furtherdevelopments. The real significance of the class of finite groups with splitBN-pairs of rank 1 stems from the fact that this class coincides with the class ofsimple groups of Lie rank 1 (see §3). Thus, the study of groups with splitBN-pairs of rank 1 is the simplest case of the classification of simple groups.

A major step was taken in the fundamental works of Bender [5]. He gener-alized the work on (TI)-groups and proved an elegant theorem characterizing thegroups with strongly embedded subgroups. In the earlier works of Feit [9],Suzuki [29], and Thompson [33], the importance of strongly embedded sub-groups had emerged. A subgroup H of a finite group G is said to be stronglyembedded if the following conditions are satisfied:

(1) H is a proper subgroup of even order;(2) H n Hx is of odd order for all x cZ H.The second condition is equivalent to.(3) NG(U) c H

for any subgroup U { 1) of even order in H. It is then trivial to verify that thenormalizer NG(T) of a Sylow 2-subgroup T of a (TI)-group G satisfies condition(3), so that NG(T) is a strongly embedded subgroup of G unless T is a normalsubgroup of G.

Bender [5] shows that if G has a strongly embedded subgroup, then either theSylow subgroups of G contain only one element of order 2, or else G is a groupof the type studied by Suzuki [30]. Thus, the structure of G is determined byBrauer and Suzuki [7] in the first case, and by Suzuki [30] in the second case.

This remarkable theorem was one of the major results in finite group theoryduring the 1960s. It gave a very useful characterization of the simple groups of

142 MICHIO SUZUKI

Lie rank 1, and paved the way for further development of the theory. Generali-zations of Bender's work and further developments in the study of fusion inSylow 2-subgroups done by Shult [25], Aschbacher [3], Goldschmidt [13], andGlauberman [12] were some of the highlights of the recent advances in finitegroup theory.

Meanwhile, the classification of finite groups with split BN-pairs of rank 1was done in 1972 by Hering, Kantor and Seitz [16] as a culmination of the worksof Shult [25] on fusion, Bender [4] and Kantor, O'Nan and Seitz [20] on doublytransitive permutation groups, and O'Nan [22], [23] on unitary designs. Actually,the case of odd degree was handled by Shult [26], while Hering, Kantor andSeitz [16] treated the even degree case.

There was a sticky problem in the classification. This was the problem ofgroups of Ree type (see §3 of this report and [14, pp. 117-118]). Quite recently,Bombieri [6] showed that, except possibly for a finite number of groups, Ree'sgroups are the only ones of Ree type. I understand that it has since beenchecked by computer that the assertion is valid without exception.

3. Finite groups with split BN-pairs of rank one. The following theoremprovides the classification.

THEOREM (HERING, KANTOR AND SEITZ [16]; SHULT [26]; BOMBIERI [6]). Let Gbe a group with a split BN-pair of rank 1, and let K be the maximal normalsubgroup of G which is contained in B. Then, the factor group G/K contains anormal subgroup S which is isomorphic to one of the simple groups in the followinglist, and G/K is isomorphic to a subgroup of Aut S, the group of automorphisms ofthe simple group S, which contains the group of inner automorphisms.

PSL(2, q), PSU(3, q), Sz(2"), Re(3").

Furthermore, the BN-pair of G corresponds to the natural BN-pair defined inthese groups.

The group PSL(2, q) is the projective special linear group defined over thefield GF(q) of q elements. It acts on the projective line over GF(q) as a doublytransitive permutation group of degree 1 + q, which is a Zassenhaus group. Wehave

IPSL(2, q)J = q(q2 - 1)/d (d = 1 or 2).

The group PSU(3, q) is the projective special unitary group acting on the setof singular points of the projective plane over GF(q 2) with respect to aHermitian form. In this case, there are q3 + 1 singular points. The stabilizer of asingular point has order

g3(g2 - 1)/d (d = 1 or 3),

and contains a normal subgroup of order q3. It is easy to verify that PSU(3, q)satisfies all the conditions (1)-(4) of the split BN-pair of rank 1.

The next group Sz(2") is defined by Suzuki [27]. This is a Zassenhaus group ofdegree q2 + 1 with q = 2". It is the fixed point set of an exceptional automor-phism of the symplectic group Sp(4, q), and ISz(q)I = q2(q - 1)(q2 + 1).


The last series in the above list of simple groups is the family of Ree's groupsdefined in [24]. Each group Re(q) in the family is a doubly transitive permuta-tion group of degree q3 + 1 where q = 3". The stabilizer of a point has order

q3(q - 1)and coincides with the normalizer of a Sylow 3-subgroup. The group Re(q) is thefixed point set of an exceptional automorphism of the group G2(q) of Lie typeG2.

In the original theorem [16], the list of simple groups was not as precise as theone stated above; namely, in place of Re(q), the list had the "groups of Reetype". As mentioned earlier, Bombieri [6] improved the statement removing theuncertainty in the list.

4. Brief sketch of the proof of the main theorem. The proof of the maintheorem is long and has been done in a sequence of several papers, eachreducing the theorem to the cases considered before. I can only give here a briefoutline of the proof which indicates the principal tools and main ideas involved.

From now on, we will always denote by G a finite group with a split BN-pairof rank 1. In order to prove the main theorem, we may and will identify G withthe permutation group acting on the cosets of B. Thus, G is a doubly transitivepermutation group satisfying condition (4) on the stabilizer. The degree of G isthen equal to the index I G : B 1. Frequently, we speak of the fixed point set of asubset of G in this permutation group.

(1) We consider first the case when G is a Zassenhaus group. In this case, thesubgroup B is a Frobenius group. So, by the basic theorem of Thompson [32], Pis nilpotent. The same conclusion holds whenever there is an element of primeorder in H which does not commute with any nonidentity element of P.

If P is a nilpotent subgroup, we obtain two advantages: the consideration offusions of its elements is easier to manage, and the theory of exceptionalcharacters becomes available. The even degree case of Zassenhaus groups can besettled by using these two methods (Ito [19], Glauberman [11]).

(2) In the odd degree case of Zassenhaus groups, the methods described in (1)are still available and show that P is a 2-group. But, in order to handle this case,we need another method of generators and relations. We have, by condition (4),

G = B U BsB,so every element of G - B has a unique presentation in the form usv with u E Pand v E B. We can choose the element s of order 2. For any x E P - (I), wehave

sxs = f(x)sg(x)

where f(x) E P - ( 1) and g(x) E B - (1). The significance of these functions fand g comes from the following lemma.

LEMMA. The structure of G is uniquely determined by the following data:(i) the structure of B,(ii) the action of the element s on H, and(iii) the functions f and g.

It suffices to show that these data determine the multiplication table of thegroup G. Consider, for example, two element a = usv and a' = u'sv' of G - B.

144 MICHIO SUZUKI

Set

vu' = wh (wEPand hEH).Since v and u' are given elements of B, the elements w and h are uniquelydetermined by (i) and Axiom (4). If w = 1, then

aa' = (usv)(u'sv') = uhsv'.

The right side is uniquely determined by a and a'. If w 1, then

aa' = uf(w)sg(w)hsv'

is an element of G which is again uniquely determined by the given data.If the group P contains a Sylow 2-subgroup of G, then P contains the

centralizer of any element of order 2 in P. This gives a constraint on the fusionsof elements of order 2. Applying the theory of exceptional characters, we canshow that P is a Sylow 2-subgroup of G. In this case, all involutions of P are inthe center of P, and they are conjugate in B. Moreover, if P contains exactlyq - 1 involutions, then there is an element of order q - 1 which acts transitivelyon the set of involutions. This is precisely the condition which G. Higmanstudied in his paper [17]. His result gives, then, the structure of P. In order todetermine the structure of B and the action of s on H, we must determine thestructure of H. At this stage, it is convenient to assume that G contains noproper normal subgroup containing P. Then, we can study the structure of H byusing the transfer map for each prime divisor of the order of H. It can be shownthat the group H is cyclic. There were mistakes in the original transfer argumentin Suzuki [30]. However, Kantor and Seitz [21] has corrected one of themistakes, and the other has been fixed. I express my thanks to ProfessorsKantor, Seitz, and Glauberman for calling my attention to these matters. Iunderstand that Professor Glauberman has also worked out the correct argu-ment.

Once the structure of H and its action on P are known, the functions f and gof the lemma can be determined. There are three cases depending on thestructure of P, but in each case, the structure of P and the functions f and g areunique; we have the groups PSL(2, q), PSU(3, q), and Sz(q).

(3) If the group G is not a Zassenhaus group, nonidentity elements fixing atleast three points are available. We have the following lemma (Lemma 4.2 of[16], Lemma 3.3 of [4]).

LEMMA. (i) If X is a subset of G fixing at least 3 points, then the centralizer of Xacts on the fixed point set of X as a doubly transitive permutation group satisfyingthe splitting condition (4).

(ii) If U is a maximal p-subgroup of H which is subject to the condition that Ufixes at least 3 points, then NG(U) acts on the fixed point set of U as a doublytransitive permutation group. In particular, U is also maximal among p-subgroupsof G which fixes at least 3 points.

In these lemmas, the structure of CG(X) or N6(U) is known by induction.This method is combined with the transfer argument to yield information aboutthe structure or the action on P of sufficiently many subsets of H. Thisinformation in turn gives restrictions on the fusions of elements of B.

After long, penetrating arguments, Bender [5] showed that a group containinga strongly embedded subgroup is essentially a group with a split BN-pair ofrank 1.

THEOREM (BENDER [5]). If a group X contains a strongly embedded subgroup,then either a Sylow 2-subgroup of X contains only one element of order two, or thefactor group X = X1 O(X) by the maximal normal subgroup of odd order containsa normal subgroup S such that

SCXCAutS,where S = PSL(2, q), PSU(3, q), or Sz(q) for some power q of 2.

The simple groups appearing in this theorem are sometimes called Bendergroups. Bender's theorem gives a characterization of the class of Bender groupsin terms of 2-fusion. This is generalized in a paper by Shult, which has not beenpublished, although the preprint of the paper has been available in limitedcircles.

THEOREM (SHULT [25]). Let X be a finite group and t an element of order 2.Suppose that the weak closure of tin the centralizer Cx(t) is abelian. If

Xo = <tx> (x E X)denotes the smallest normal subgroup of X containing the element t, the componentsof X0/0(X0) are Bender groups.

In this theorem, the weak closure of t in Cx(t) is the subgroup of thecentralizer generated by the conjugates of t. If the group X is simple in theabove theorem, then X is one of Bender groups.

This fusion theorem of Shult is one of the keys used to prove the odd degreecase of the main theorem (Shult [26]).

(4) The even degree case is handled in an entirely different manner. Theargument depends more on the work on the doubly transitive permutationgroups. A few years earlier, Bender [4] proved the following theorem.

THEOREM (BENDER [4]). If the stabilizer of a point in a doubly transitivepermutation group X is of odd order, then either X is solvable, or X contains anormal subgroup which is isomorphic to PSL(2, q).

Solvable doubly transitive permutation groups have been determined byHuppert [18]. Assuming X to be not solvable, Bender proceeded to prove thatSylow 2-subgroups of X are dihedral. Then, the theorem follows from theGorenstein-Walter theorem [15].

In the even degree case of the main theorem, we may assume, by the theoremquoted above, that the subgroup H is of even order. Let t be an involutioncontained in H. The method of (3) is applied again to determine the action ofCG(t) on the fixed point set of t. By induction, this action may involve either asolvable doubly transitive group or one of the simple groups in the list in itsusual doubly transitive representation. These cases are considered separately.Eventually, in all cases, enough information on the structure of CG(t) can beobtained. In particular, the structure of Sylow 2-subgroups is determined; so wecan apply the known classification theorem on finite simple groups of 2-rank

146 MICHIO SUZUKI

two (Gorenstein and Walter [15], Alperin, Brauer and Gorenstein [1] and [2]). Inthe process of this reduction, the following theorem on doubly transitive permu-tation groups is also used.

THEOREM (KANTOR, O'NAN AND SEITZ [20]). Let X be a doubly transitivepermutation group. Assume that the stabilizer of any two points is cyclic. Then, Xcontains a normal subgroup S which is one of the simple groups listed in the maintheorem, and the action of S coincides with its usual doubly transitive representa-tion.

The proof outlined here for the even degree case depends greatly on theresults of permutation groups and classification theorems of simple groups oflow rank. It is possible to lessen the dependence. Kantor and Seitz [21] gives anargument by which the use of [1] and [2] can be eliminated.

(5) Finally, I like to say a few words on the identification problem.For the groups PSL(2, q), various characterizations have been known. Zas-

senhaus [37] characterized PGL(2, q) among triply transitive permutationgroups, using geometric argument. His arguments are also applicable to thegroups PSL(2, q), although he did not state them explicitly. The method ofgenerators and relations has its origin in the works of Zassenhaus, and certainly,it applies to the groups PSL(2, q). Suzuki, in [28] and [30], uses this method toprove uniqueness theorems for Sz(q) and the unitary groups of characteristictwo. The same method is used to identify Re(q), but it is considerably moredifficult to prove the uniqueness (Thompson [34] and Bombieri [6]). In theoriginal form of the main theorem, the groups of Ree type were identified byusing the results of Ward [36].

The remaining series of the unitary groups of odd characteristic is identifiedby using the theorem of O'Nan [22] and [23], which is proved by a combinationof two methods: the geometric method and the generator-relation technique.

5. Remark. The outline of the proof shows the abundance of basic ideasnecessary for the proof of the main theorem, and perhaps the degree of difficultyinvolved. It is desirable to find a simpler way to prove the theorem. Entirely newand revolutionary ideas might be necessary. On the other hand, an improvementin the method of generators and relations, or new applications of the representa-tion theory might offer substantial simplifications of the argument.

REFERENCES

1. J. L. Alperin, R. Brauer and D. Gorenstein, Finite groups with quasi-dihedral and wreathedSylow 2-subgroups, Trans. Amer. Math. Soc. 151 (1970), 1-261.

2. , Finite simple groups of 2-rank two, Scripta Math. 29 (1973), 191-214.3. M. Aschbacher, Finite groups with a proper 2-generated core, Trans. Amer. Math. Soc. 197

(1974), 87-112.4. H. Bender, Endliche zweifach transitive Permutationsgruppen, deren Involutionen keine

Fixpunkte haben, Math. Z. 104 (1968), 175-204.5. , Transitive Gnippen gerader Ordnung, in denen jede Involution genau einen Punkt

festlasst, J. Algebra 17 (1971), 527-554.6. E. Bombieri, Thompson's problem, 1979 (preprint).7. R. Brauer and M. Suzuki, On finite groups of even order whose 2-Sylow subgroup is a quaternion

group, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757-1759.


8. W. Feit, On a class of doubly transitive permutation groups, Illinois J. Math. 4 (1960), 170-186.9. , A characterization of the simple groups SL(2, 2), Amer. J. Math. 82 (1960), 281-300;

correction, 84 (1962), 201-204.10. W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963),

775-1029.11. G. Glauberman, On a class of doubly transitive permutation groups, Illinois J. Math. 13 (1969),

394-399.12. , Factorizations in local subgroups of finite groups, CBMS Regional Conf. Ser. in Math.,

no. 33, Amer. Math. Soc., Providence, R. I., 1978.13. D. Goldschmidt, 2-Fusion in finite groups, Ann. of Math. 99 (1974), 70-117.14. D. Gorenstein, The classification of finite simple groups. I. Simple groups and local analysis, Bull.

Amer. Math. Soc. (N.S.) 1 (1979), 43-199.15. D. Gorenstein and J. H. Walter, The characterization of finite groups with dihedral Sylow

2-subgroups, J. Algebra 2 (1965), 85-151, 218-270, 354-393.16. C. Hering, W. Kantor and G. Seitz, Finite groups with a split BN-pair of rank 1, J. Algebra 20

(1972), 435-475.17. G. Higman, Suzuki 2-groups, Illinois J. Math. 7 (1963), 79-96.18. B. Huppert, Zweifach transitive, auflosbare Permulationsgruppen, Math. Z. 68 (1957), 126-150.19. N. Ito, On a class of doubly transitive permutation groups, Illinois J. Math. 6 (1962), 341-352.20. W. Kantor, M. O'Nan and G. Seitz, 2-Transitive groups in which the stabilizer of two points is

cyclic, J. Algebra 21 (1972), 17-50.21. W. Kantor and G. Seitz, Finite groups with a split BN-pair of rank 1. II, J. Algebra 20 (1972),

476-494.22. M. O'Nan, Automorphisnts of unitary block designs, J. Algebra 20 (1972), 495-511.23. , A characterization of U3(q), J. Algebra 22 (1972), 254-296.24. R. Ree, A family of simple groups associated with the simple Lie algebra of type (G2), Amer. J.

Math. 83 (1961), 432-462.25. E. Shult, On the fusion of an involution in its centralizer, 1969 (preprint).26. , On a class of doubly transitive groups, Illinois J. Math. 16 (1972), 434-455.27. M. Suzuki, A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U.S.A. 46 (1960),

868-870.28. , On a class of doubly transitive groups, Ann. of Math. 75 (1962), 105-145.29. , Two characteristic properties of (ZT)-groups, Osaka Math. J. 15 (1963), 126-150.30. , On a class of doubly transitive groups: II, Ann. of Math. 79 (1964), 514-589.31. Finite groups of even order in which Sylow 2-groups are independent, Ann. of Math. 80

(1964), 58-77.32. J. Thompson, Normal p-complements for finite groups, Math. Z. 72 (1960), 332-354.33. , Nonsolvable finite groups all of whose local subgroups are solvable. I, Bull. Amer. Math.

Soc. 74 (1968), 383-437.34. , Toward a characterization of E2(q). I, II, III, J. Algebra 7 (1967), 406-414; 20 (1972),

610-621; 49 (1977), 162-166.35. J. Tits, Buildings of spherical type and finite BN-pairs, Springer-Verlag, Berlin and New York,

1974.36. H. Ward, On Ree's series of simple groups, Trans. Amer. Math. Soc. 121 (1966), 62-89.37. H. Zassenhaus, Kennzeichnung endlicher linearer Gruppen als Permulationsgruppen, Abh. Math.

Sem. Univ. Hamburg 11 (1936), 17-40.

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN


FINITE GROUPS OF LOW 2-RANK,REVISITED

KOICHIRO HARADAI

As in any other branch of mathematics, "smallness", wherever the word isappropriate, has given group theorists some advantage and disadvantage also.Obviously, an advantage will result since "smallness" means "being restricted"and so one needs not consider many possibilities. On the other hand,"smallness" implies "being lack of liberty" and so what one can do is limitedhence a disadvantage will result.

One can discuss many aspects of smallness in group theory, but here werestrict ourselves to discuss only the smallness of groups in terms of the rank ofthe Sylow 2-subgroups.

Let 2° be the maximum of the orders of all elementary abelian subgroups of aSylow 2-subgroup of a group G, then the number a is called the 2-rank of G. Ifthe 2-rank of a group G is "small", then G is said to be a group of low 2-rank.There is no precise definition of "lowness".

For the past twenty years or so, an enormous number of research papers ongroups of low 2-rank have been written. Probably as much as half of the entireproof of the classification of all simple groups of finite order has been devotedto these small groups. If the quantity of the research papers written is anyindication, then the smallness of the 2-rank certainly has given us more disad-vantage than advantage.

If we are to revisit the entire classification theorem of simple groups in future,the part concerning the structure of groups of low 2-rank should not beneglected. It seems that the difficulties and the length of those papers written ongroups of low 2-rank are mostly due to technicality and so someday the entireproof may be reduced to 10 percent or less of the present size, which is about2000 journal pages.

1. Group of low 2-rank. As mentioned earlier, there is no precise definition ofgroups of low 2-rank. The definition depends on the result one is attempting to

1980 Mathematics Subject Classification. Primary 20D20, 20D05.'Supported in part by NSF Grant MCS-7903158.

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150 KOICHIRO HARADA

prove. However, the groups of low 2-rank seem to have been divided into thefollowing categories:

(a) groups of 2-rank 0;(b) groups of 2-rank 1;(c) groups of 2-rank 2;(d) groups of 2-rank 3;(e) groups of sectional 2-rank at most 4;(f) groups of 2-rank 4;(g) all sporadic groups.

Here the sectional 2-rank of a group G is the maximum of the 2-ranks of thehomomorphic images of the subgroups-sections-of G.

For (a), we have the following celebrated theorem.

THEOREM (FEIT-THOMPSON, 1963). All groups of odd order are solvable (255journal pages).

For the second category (b), one first shows that the Sylow 2-subgroups of Gare either cyclic or generalized quaternion. In the cyclic case, it is trivial to showthat G possesses a normal subgroup of index 2 and in the latter case, we have

THEOREM (BRAUER-SUZUKI, 1959). There exist no simple groups with gener-alized quaternion Sylow 2-subgroups.

As for category (c), a combination of the following theorems treats all suchgroups.

THEOREM (GORENSTEIN-WALTER, 1962). The simple groups with dihedral Sylow2-subgroups are A7 or PSL2(q), q odd (159 journal pages).

THEOREM (ALPERIN-BRAUER-GORENSTEIN, 1970). The simple groups with semi-dihedral or wreathed (= ZZ/' Z2) Sylow 2-subgroups are M11, PSL3(q), PSU3(q),q odd (261 journal pages).

THEOREM (LYONS, 1972). If a simple group G possesses the Sylow 2-subgroupsisomorphic to those of PSU3(4), then G PSU3(4).

THEOREM (ALPERIN). If S is a 2-group of 2-rank 2 which can occur as a Sylow2-subgroup of a simple group, then S is isomorphic to one of the 2-groups appearingin the three theorems above.

Category (d) has been treated in the following theorem.

THEOREM (STROTH, 1976). The simple groups with Sylow 2-subgroup of 2-rank 3are isomorphic to ones of known type (i.e. the Chevalley groups-normal or twisted,the alternating groups, 26 sporadic groups) (108 journal pages).

In fact, Stroth obtained the theorem as a corollary to a more general theoremwhich classifies all simple groups with the property that the intersection of anytwo Sylow 2-subgroups is of 2-rank 3.

For category (e), we have the following.

THEOREM (GORENSTEIN-HARADA, 1974). The simple groups of sectional rank atmost 4 are isomorphic to simple groups of known type (464 journal pages).

FINITE GROUPS OF LOW 2-RANK, REVISITED 151

All theorems mentioned above have been proved with a full classification ofall simple groups in mind. We shall say a few words about the reason why grouptheorists had to obtain these results.

Since the birth of group theory, we wanted to know all simple groups. Anyreal attempt to classify them all did not take place for a long time. However, duemainly to Brauer-Fowler's theorem that there are only finitely many simplegroups with the centralizer of an involution having a given structure andFeit-Thompson's theorem that all groups of odd order are solvable, an attemptto classify all simple groups began. The focus was put on the structure of thecentralizer of an involution. However, it was soon realized that determining theright structure of the centralizer of an involution was very hard due to thepresence of undetermined normal subgroups of odd order. However, for thesimple groups of 2-rank 2, Brauer's modular representation theory was availableand the combined effort of Alperin, Brauer, Gorenstein, Lyons and Walterproduced the full classification of all simple groups of 2-rank 2.

For simple groups of 2-rank 3 or more, no effective modular representationtheory was available and so some new method had to be found to pursue theclassification.

Gorenstein is the person who found the new method: the signalizer functor.The signalizer functor theorem was strengthened by Goldschmidt soon after-ward and it has become one of the most powerful tools for the ultimateclassification of all simple groups.

Signalizer functors are the tools to pack a certain set of unwanted subgroupsof odd order in the centralizers of involutions into one "big" subgroup of oddorder. If we call the "big" subgroup of odd order W, then the normalizer NE(W)contains lots of elements of G due to the way W is constructed. As G is simple,NG(W) is a proper subgroup as long as W 1. So the "properness" and the"richness" of NG(W) are expected to yield a contradiction, thereby provingW = 1, and so all unwanted subgroups do not exist.

However, here again, "lowness" of 2-rank of G gives us some trouble.Namely, one cannot prove that NE(W) is rich enough. In order to cope with thisdifficulty, the notion of connectivity2 was born, and the term "sectional 2-rank"was defined, and finally all simple groups of sectional rank at most 4 wereclassified. If the sectional rank of a group is 5 or more, one can prove thatNG(W) is in general rich enough.

2. Revisionism. All simple groups will very likely be known pretty soon. Onceall simple groups are known, there is a great advantage if one wants toreestablish some of the old theorems. All we have to prove is that there are nomore simple groups other than the known simple groups. So let G be a minimalcounterexample. Then all the composition factors of the proper subgroups are ofknown type. We must then show that the centralizer C of an involution of Gresembles very much the centralizer C* of an involution of a known simplegroup G*. Quoting a characterization of G* by C', we would conclude thatG* = G, whch would obviously be a contradiction.

2Let A be the set of all four subgroups of a group G. Connect two elements of A by an edge if theycommute elementwise. If the resulting graph A is connected, G is said to be connected.

152 KOICHIRO HARADA

Here again the normal subgroup O(C) of odd order of C will be a nuisance.Unless we find a revolutionary new idea to treat this problem, the signalizerfunctor will be the unique tool to handle the situation. Since there will be nosignalizer functors conceivable for groups of 2-rank 2, we must classify allsimple groups of 2-rank 2 prior to anything.

Let G be a simple group of 2-rank 2 and let S be a Sylow 2-subgroup of G. ByAlperin's theorem, S is dihedral, semidihedral, wreathed, or of type PSU3(4).

Bender has revised the classification of all simple groups with dihedral Sylow2-subgroups by his own method, which reduces the problem to the existence of astrongly embedded subgroup. Since no quaternion groups are involved, Benderwas able to use Glauberman's ZJ-theorem freely. This together with Bender'singenious idea to deal with various maximal subgroups seems to be the key tothe simplification of the classification. The isolated group PSU3(4) was treatedby Lyons in 17 journal pages. So groups with semidihedral or wreathed Sylow2-subgroups are the only remaining cases which need to be revisited as far as thegroups of 2-rank 2 are concerned.

Conceptually, the classification of all groups with semidihedral or wreathedSylow 2-subgroups is similar to the dihedral case, but in actuality it differsconsiderably in many important technical details. Again all one must prove isthe centralizer C of an involution (G is shown to possess only one class ofinvolutions) resembles very much the corresponding group of a known simplegroup. It is not hard to show (one needs the classification of groups withdihedral Sylow 2-subgroups though) that C/O(C) resembles the correspondinggroup of a known group. Hence again the main effort is devoted to determiningO(C). Ultimately, one must establish O(C) C Z(C) among other things. IfO(C) C Z(C) is denied, some 2-element has to act nontrivially on a Sylowp-subgroup P of O(C). One would next consider the structure of NG(P) andthus the standard procedure called "pushing up" begins. In the dihedral case,the use of the "pushing up" procedure is unrestricted, as all sections involved arep-stable. On the other hand, in the semidihedral or wreathed case, it isrestricted, since some sections are not p-stable, for a certain unique p which isdetermined as the characteristic of the normal subgroup of C/O(C) isomorphicto SL2(q). So, for the prime p, the notation of relative stability with respect to a2-subgroup is defined.

By the pushing up procedure, one can construct an r-local subgroup M suchthat

(1) M contains a Sylow 2-subgroup of G;(2) M has only one class of involutions; and(3) if x c Inv(M), then Cs = O(CX)(M n Cs), where C. = CG(x).

Such a subgroup M is often called a weakly embedded subgroup of G. Withstrong use of modular character theory for the prime 2, Brauer was able to showO(M) C O(G). Hence O(M) = 1. This kind of argument eventually establishesa resemblance between C (not C/O(C)) and C* = CG.(x*) where G* is aknown simple group and x' E Inv(G*). One has to make another effort toactually obtain the classification but at this moment one is reasonably sure thatthe result will be obtained. This is roughly how Alperin, Brauer and Gorensteintreated this problem.

FINITE GROUPS OF LOW 2-RANK, REVISITED 153

How can one redo the classification of groups with semidihedral or wreathedSylow 2-subgroups? The author cannot offer any effective idea which may cutthe size of the proof into a half or less. The most desirable simplication shouldtreat all 2-rank 2 groups in one paper.

Do we need to re-establish more theorems on groups of low 2-rank other thanthe classification of groups of 2-rank 2? No one knows for sure. We may at leastassume that the (unknown) simple group G is of 2-rank 3. So theoretically wewill be able to construct a signalizer functor with respect to an elementaryabelian 2-subgroup A of order at least 8. The completion W, will be of oddorder. Now we will have to show that N(W,) contains many elements of G. Buteven under the assumption that G is connected, we can conclude at best onlyNG(WA) I's.Z(G) where S is a Sylow 2-subgroup of G. Hence the followingtwo theorems seem to be indispensable.

THEOREM A. If G is a simple group and I's.Z(G) = <NG(T)Im(T) > 2, T C S,S E Sy12(G)> is a proper subgroup of G, then G is of known type. (m(T) denotes,by definition, the 2-rank of T.)

THEOREM B. If a Sylow 2-subgroup S of a simple group G is of 2-rank at least 3and nonconnected, then G is of known type (actually G = J2 or J3).

Needless to say, Theorem A is the "2-generated core" theorem of Aschbacher.The proof is known to be very hard. A simplification may be desirable. In thenext section, we shall describe a rough sketch of the proof of Theorem B.

3. Theorem B. We want to prove Theorem B mentioned in the previoussection. However, as long as our ultimate aim is to re-establish the classification,we need only show the following.

THEOREM B*. Let G be a simple group in which the nonsolvable compositionfactors of the proper subgroups are of known type. If a Sylow 2-subgroup S of G isof 2-rank at least 3 and nonconnected then G is isomorphic to the Hall Janko groupJZ or the Higman McKay Janko group J3.

Since S is nonconnected, S does not possess a normal elementary abeliansubgroup of order 8. A theorem of A. MacWilliams implies that the sectionalrank of S is at most 4. As S is assumed to be of 2-rank at least 3, S contains anormal four subgroup U. m(X) denotes the 2-rank of X.

It is easy to show

LEMMA. (1) U is the unique normal four subgroup of S;(2) IS: Cs(U)I = 2 and m(S) = m(Cs(U));(3) there exists an involution v E S - Cs(U) such that C8(v) = Z2 X ZZ_ or

Zz X Qz-, m > 2.

DEFINITION. Let `V be the set of all four subgroups of S not connected to U.T = Cs(U).

LEMMA. If V E `V, then V is conjugate in G to a four subgroup of T.

This can be obtained by Thompson's transfer lemma.Next we use a conjugation family and show

154 KOICHIRO HARADA

LEMMA. Each V in `V is conjugate to U in G, i.e. V - U.

Clearly, V Cs(U) and if v E V - Cs(U), then V = <v, z> where z is theunique involution of Z(S). Let U = <u, z>.

We have shown above that V - U in G. If v - z then all involutions of U areconjugate in G. We can show

PROPOSITION. If the involution of U are conjugate in G, then S is of type J2.

Thus we may assume that u - z in G. Since <v, z> <u, z>, we must havev - vz - u - uz in CG(z). Since V is an arbitrary element in `V, we conclude

PROPOSITION. A11 involution of U VEw V - (z> are conjugate in CG(z) to u.

Setting H = CG(z) and H = H/O(H), we prove the following:

LEMMA. One of the following holds.(1) H is 2-constrained and 02(H) D8 * D8;(2) H is non 2-constrained and E(H) is quasi-simple but nonsimple.

In the constrained case, we argue that H/02(H) =A5 and so S is of type J2.We also show that the nonconstrained case does not occur.

Now that S is determined, namely S is of type J2, it is not hard to obtain thesimple group itself. In order to kill the core of the centralizer of an involution, itis best to construct a so-called "covering p-local subgroup".

If G has only one conjugacy class of involutions, then CG(t) is 2-constrainedfor all t E Inv(G). So O(CG(t)) = O(t) is a signalizer functor of G and it is nothard to show O(CG(t)) = 1. We only note that since S is nonconnected, it is notan absolute triviality to show O(CG(t)) = 1.

If G has more than one class of involutions, it can be shown that G possessestwo classes, represented by z and a. Moreover, CG(z) is 2-constrained andCG(a)/O(CG(a)) = Z2 X Z2 X L2(q), q = 3, 5 (mod 8). We need to show q < 5if we are to define a "good" signalizer functor O(CG(t)). Assuming O(CG(z))1, we first show that there is a p-subgroup P 1 of G such that NG(P) coversCG(z)/O(CG(z)) and CG(a)/O(CG(a)); i.e., C, = O(C,)(C, fl N) where t = z ora, C, = CG(t), and N = NG(P). By induction, N/O(N) c J2, and so q = 5 asdesired. This will complete the proof. The complete proof takes only 20 typedpages.

Of course, Theorem B* will not classify all groups of sectional rank at most 4.For example, there are many standard component problems associated withgroups of sectional rank at most 4. However, it may be better if we treat allstandard component problems as one.

Besides Theorems A and B, the author did not see any major theorem whichneeds to be re-established as far as the group of low 2-rank are concerned.

OHIO STATE UNIVERSITY, COLUMBUS

QUASTTHIN GROUPS

GEOFFREY MASONI

1. At this late stage of the group-theoretic game everyone is familiar with theinvariant e(G) and its relevance to the classification of the finite simple groupsof characteristic 2-type; so I shall merely remind the reader that a quasithingroup G is one which satisfies e(G) < 2.

I have spent some time in attempting to classify this class of groups, and atthe time of writing this work it is very close to completion? There is, however,already a fairly detailed exposition of the main themes in [5]; so in this article Ishall simply attempt to place one of the ideas that we have found useful in asomewhat broader context.

2. One of the difficulties inherent in the classification of the quasithin groups(of characteristic 2-type, say) is that much of the theory built up in the last fewyears for the `general' group of characteristic 2-type is not applicable. Themachinery of signalizer functions for odd elementary abelian p-groups A is notapplicable unless m(A) i 3, and similarly the construction of `odd standardforms' and the associated analysis does not appear to be effective for groupswith e(G) < 2. Hence one must look elsewhere for the relevant techniques.

These problems are already apparent in Thompson's classification of N-groups [6], where the cases in which e(G) < 2 are incredibly involved. The firstwork involving a `general' small simple group of characteristic 2-type occurs inM. Aschbacher's brilliant solution of the problem of classifying the thin groups(i.e., with e(G) = 1) [1]. In that work the basic object of study is what we havecome to call a subgroup of parabolic-type. Our own work employs these objectsalso; in fact they provide so powerful a weapon that one wonders whether theycan be utilized elsewhere.

To be more specific, we fix a (finite, simple) group G of characteristic 2-type,with fixed Sylow 2-subgroup T. Let 71(X) be the set of maximal 2-localsubgroups of G containing X for any subgroup X < G. For convenience, set6)Tt, = O1t(T); these are the main objects of study.

1980 Mathematics Subject Classification. Primary 20D05.1The author thanks the National Science Foundation for its continued support.21n fact the classification of quasithin groups is now complete.

O American Mathematical Society 1980

155

156 GEOFFREY MASON

A subgroup L s G is called a subgroup of parabolic type (with respect to T) incase the following conditions hold:

(i) F*(L) = 02(L), L/02(L) is a central product of isomorphic, nonabelianquasisimple groups.

(ii) There is M E OiL such that L 4 M.(iii) (M} = O1L(L).The terminology derives from the fact, easily verified, that if P is a maximal

parabolic subgroup of a finite group of Lie-type and characteristic 2, and ifL < P covers the Levi-factor of P/02(P) and satisfies (i), then L is of para-bolic-type.

It is (iii) that is the crucial requirement: it is a uniqueness result whichreplaces the type of uniqueness theorems (involving various elementary abelianp-groups) that one aims for when e(G) > 3.

Of course (i)-(iii) assume that G has a nonsolvable 2-local; after [4] this is nota problem.

The first major objective in the classification of quasithin groups is toestablish that there is a suitable supply of subgroups of parabolic-type. Onedefines certain partial orderings on the set of subgroups satisfying (i), (ii) andthen establishes that the elements maximal in this partial order also satisfy (iii).If something similar to this procedure could be extended to arbitrary groups ofcharacteristic 2-type, it would constitute a major advance toward a classificationof such groups along lines distinct from those currently being pursued. Hence

Problem 1. Establish the existence of subgroups of parabolic-type in a generalgroup of characteristic 2-type.

One now aims for more uniqueness subgroups as follows.Problem 2. Let L be a subgroup of parabolic-type with (M) = O1L(L). Show

that (M) = 6)1(J) whenever J < LT satisfies 02(LT) E 14;4(J; 2) and I02(J) J.

This result is conjecturally true in general, but even if G is quasithin, one canonly establish various special cases. One approaches Problem 2 using thetechniques of pushing-up and Aschbacher blocks. Further advances in theseareas will presumably aid in understanding Problem 2.

Now the goal is to establish that L is a `block' in the nontechnical sense that02(L) has only a few (often just one) noncentral chief L-factors. This isaccomplished by weak-closure arguments: the foundations of this subject maybe found in [2].

So in some sense we arrive at the situation discussed by R. Foote elsewhere inthese PROCEEDINGS [3]. Indeed the groups L may be considered as goodcandidates for the `generalized blocks' discussed by Foote.

3. This is admittedly an exceptionally brief analysis of a work which, whencompleted, will be long even by the standards which group theorists are used to.We wish only to emphasize the point that a study of subgroups of parabolictype, together with generalized Aschbacher blocks, offers a potentially new pathdown which the revisionists may wish to wander.

QUASITHIN GROUPS 157

REFERENCES

1. M. Aschbacher, Thin finite simple groups, J. Algebra 54 (1978), 50-152.2. , Weak-closure in groups of characteristic 2-type (preprint).3. R. Foote, Aschbacher blocks, these PROCEEDINGS, pp. 37-42.4. D. Gorenstein and R. Lyons, Nonsolvable finite groups with solvable 2-local subgroups, J. Algebra

38 (1976), 453-522.5. G. Mason, On the classification of quasithin groups, Finite Simple Groups. II, Proc. Durham

Conf., M. Collins (ed.) (to appear).6. J. G. Thompson, Finite nonsoluble groups all of which local subgroups are soluble, Bull. Amer.

Math. Soc. 74 (1968), 383-437.

UNIVERSITY OF CALIFORNIA, SANTA CRUZ

PART II

General theory of groups

ZUSAMMENGESETZTE GRUPPEN:HOLDERS PROGRAMM HEUTE

HELMUT WIELANDT

Die Strukturtheorie der nicht einfachen Gruppen endlicher Ordnung wurdevon Otto Holder wahrend seiner Tubinger Jahre begrundet (1889-1895). Ineiner Reihe von umfangreichen Arbeiten entwickelte er das Programm, zunachstdie nicht-trivialen einfachen Gruppen, die von zusammengesetzter Ordnung, zubestimmen, urn dann die zusammengesetzten Gruppen explizit aus ihrenKompositionsfaktoren zu konstruieren. Mit Hilfe von Faktorensystemen andsemidirekten Produkten fiihrte er dies fur die Gruppen durch, die hochstens dreiKompositionsfaktoren besitzen, von denen mindestens zwei trivial sind and ggf.die dritte eine Ordnung < 200 hat [11].

Wahrend der erste Teil von Holders Programm stagnierte, machte der zweiteTeil seit den dreiBiger Jahren Fortschritte in einer speziellen Richtung. Die vomersten Programmpunkt unabhangige Theorie der auflosbaren Gruppen entwik-kelte sich, ausgehend von Arbeiten von Philip Hall and Wolfgang Gaschiitz, zueinem bliihenden Gebiet, in dem es immer noch Uberraschungen gibt [5].Gleichzeitig entwickelte sich die Theorie der zusammengesetzten Gruppen mitbeliebigen Kompositionsfaktoren in einer von Holder nicht bearbeiteten Rich-tung. Ausgehend von einer Anregung durch Robert Remak erforschte dieTubinger Schule die Eigenschaften der Glieder von Kompositionsreihen, alsoder subnormalen Untergruppen.

Wie steht es mit Holders Programm heute? Die gro[ie Aufgabe der Bestim-mung der einfachen Gruppen scheint dank dem 1960 von John Thompsoneingeleiteten Durchbruch and der Zusammenarbeit von zahlreichen Grup-pentheoretikern dem Abschlu[i nahe zu sein. Damit erreicht die allgemeineTheorie der zusammengesetzten Gruppen den Punkt, von dem die Theorie derauflosbaren Gruppen vor 50 Jahren ausging. Schwieriger ist die Lage jetztnatiirlich dadurch, daB die nichttrivialen einfachen Gruppen eine wenig durch-sichtige Struktur haben. Zum Ausgleich stehen inzwischen die Hilfsmittel derSubnormaltheorie zur Verfiigung, die gerade beim Auftreten von perfektenKompositionsfaktoren besonders wirksam sind.

1980 Mathematics Subject Classification. Primary 20D35, 20D25, 20E22.m American Mathematical society 1980

161

162 HELMUT WIELANDT

Das Ziel, die zusammengesetzten Gruppen in einer Liste zu beschreiben, wiesie fur die einfachen Gruppen in Aussicht steht and Holder vorgeschwebt habenmag, ist sicher zu weit gesteckt. Moglich and wichtig sind weitere Fortschritte inder Erweiterungstheorie, vielleicht ein Ausbau der Konstruktionsmittel [8]. Furdie systernatische Entwickelung der Theorie der zusammengesetzten Gruppenerscheint es aber noch wesentlicher, diejenigen Untergruppen zu finden and zuuntersuchen, auf die es ankommt. Als Leitfaden kann die Theorie derauflosbaren Gruppen dienen. Das Vorgehen ist leicht zu beschreiben: Jedemdort als wichtig erkannten Satz suche man eine Form zu geben, die furzusammengesetzte Gruppen mit beliebig zugelassenen Kompositionsfaktorensinnvoll ist and nicht bereits durch einfache Beispiele (meist Kranzprodukte)widerlegt wird. Die richtige Verallgemeinerung der bei auflosbaren Gruppenbewahrten Begriffe zu finden, kann allerdings schwierig sein. Urn mit deco erstensich historisch bietenden Beispiel anzufangen: Welcher Begriff kann im all-gerneinen Fall die zentrale Rolle iibernehmen, die Halls Verallgerneinerung derSylowgruppen im auflosbaren Fall spielt? Dieser Frage werden wir uns nacheinern Uberblick fiber Ergebnisse and Probleme der Subnormaltheorie zuwen-den. Gelegentlich werden wir an friihere Berichte [23]-[28] ankniipfen.

1. Subnormale Untergruppen. G bezeichnet stets eine endliche Gruppe. EineUntergruppe A von G (kurz: A < G oder A E sG) heiBt subnormal (kurz: AsnGoder A E snG), wenn es eine Kette

G=Go DG1D G, = A (1.0)

gibt. Die Gesamtheit snG der Subnormalteiler von G besteht aus den Gliedernder samtlichen Kompositionsreihen von G; sie hatte also schon seit 1870Aufinerksamkeit finden konnen. Doch hat erst 1936 Remak these Untergruppendurch einen besonderen Namen hervorgehoben ("nachinvariant") and in einernSeminar das erste Problem gestellt: Ist das Erzeugnis zweier Subnormalteilerstets subnormal? Er began damals, die Erzeugungen einer Gruppe durch nichtweiter zerlegbare Subnormalteiler zu untersuchen. Er ist nicht dazu gekommen,etwas dari ber zu veroffentlichen [18]. Seine Frage, die den AnstoB zur Ent-wicklung der Theorie der subnormalen Untergruppen gegeben hat, ist seitlangern beantwortet: fur endliche Gruppen 1939 mit Ja [30], fur manche un-endliche Gruppen 1958 mit Nein [40]; die wichtigsten Kriterien sind 1968gefunden worden [21].

Fur endliche Gruppen ist die Theorie schon weit ausgebaut, doch ist nochkeine zusammenfassende Darstellung erschienen; unsere Hinweise be-riicksichtigen daher auch die altere Literatur. Von der vollstandigsten, 1971entstandenen Vorlesungsausarbeitung [29] wird eine groBere Auflage mitErganzungen vorbereitet. An einer Gesamtdarstellung fiir beliebige Gruppenarbeiten J. C. Lennox and S. E. Stonehewer.

Der erste Teil dieses Vortrags gibt einen gedrangten Uberblick fiber diejenigenEigenschaften der Subnormalteiler endlicher Gruppen, die fiir die Struk-turtheorie zusammengesetzter Gruppen besonders niitzlich erscheinen. Vorwegsind zwei Punkte hervorzuheben, in denen sich Subnormalitat wesentlich vonder Normalitat unterscheidet. Subnormalitat ist transitiv: Aus AsnBsnC folgtAsnC. Das ermoglicht meist einfache Induktionsbeweise. Erschwerend wirkt

ZUSAMMENGESETZTE GRUPPEN 163

andererseits, dab es im allgemeinen keinen Subnormalisator analog zum Nor-malisator gibt: Aus AsnB s G and AsnC S G folgt nicht, dass A im Erzeugnisvon B and C subnormal ist (dieses bezeichnen wir mit <B, C> oder B V C).Beispiel: Die symmetrische Gruppe G des Grades 5 lasst sich durch zweiUntergruppen der Ordnung 8 erzeugen, die eine Transposition a gemeinsamhaben; <a> ist subnormal in beiden, aber nicht in G.

Wir behandeln zunachst das Verhalten der Subnormalteiler einer Gruppezueinander, dann ihr Verhalten gegenuber Sylowgruppen.

1.1 ERZEUGNIS UND DURCHSCHNITT. Seien A, B E snG. Dann gilt [30]:(a) A V B E snG, A n B E snG; snG ist also ein Teilverband von sG.(b) Die Kompositionsfaktoren von A V B sind bis auf Isomorphie and Vielfach-

heit dieselben wie die von A and B zusammengenommen.(c) Die Kompositionsfaktoren von A V B oberhalb von A sind bis auf Isomorphie

die gleichen wie die von B oberhalb von A n B, and ihre Vielfachheiten sind nichtkleiner.

(d) Genau dann ist die Vielfachheit jeder einfachen Gruppe als Kompositionsfak-tor von A V B oberhalb A dieselbe wie die von B oberhalb A n B, wennAB = BA ist. Hieraus folgt:

(e) Genau dann ist der Verband sn G modular, wenn je zwei Subnormalteiler vonG vertauschbar sind.

Es stellt sich die Frage, wann zwei subnormale Untergruppen miteinandervertauschbar sind; Kriterien dafiir folgen in 1.3. Uber Arbeiten zur Verbands-struktur von sn G berichtet Zappa [39].

Die erste von vielen angenehmen Uberraschungen in der Theorie der subnor-malen Untergruppen bestand darin, dab die nichtabelschen Kompositions-faktoren keine Schwierigkeiten im Verhalten der Subnormalteiler zueinanderverursachen; im Gegenteil. Das liegt an dem folgenden, wahrscheinlich imwesentlichen schon Remak bekannten Satz:

1.2 EINKOPFIGE PERFEKTE SUBNORMALTEILER. (a) Sei (1.0) eineKompositionsreihe von G. Dann gibt es zujedem nichtabelschen Faktor G,,_1/G,,genau eine minimale unter denjenigen subnormalen Untergruppen von G,,_I, welchenicht in G,, enthalten sind. Die minimale Gruppe P ist perfekt, and sie ist einkopfig(join irreducible) in dem Sinn, dali sie genau einen maximalen Normalteiler Mbesitzt. Es ist P/M - G,,_,/G,, [30].

(b) Die Anzahl der einkopfigen perfekten Subnormalteiler von G stimmt mit derA nzahl der nichtabelschen G,, _ , / G,, iiberein [30].

(c) Sei P E sn G einkopfig and perfekt. Dann folgt aus P A E sn G, dal Aim Normalisator NG(P) liegt; aus P < A V B, A E snG, B E snG folgt: P < Aoder P < B [37].

1.3 SUBNORMALITAT VERTAUSCHBARKEIT. Seien A, B E snG. Wenn eine dernachstehenden Bedingungen (a)-(c) erfI llt ist, so sind A and B vertauschbar.

(a) A = A' [30].(b) Die Indizes IA: A'J, JB: B'I sind teilerfremd [31].(c) Jeder Homomorphismus des Erzeugnisses A V B in eine p-Gruppe fuhrt A

and B in vertauschbare Gruppen fiber [3], [38]. Diese Bedingung ist natiirlich auchnotwendig. Doch ergibt sie nicht folgende Sdtze:

164 HELMUT WIELANDT

(d) Seien A, B E snG. Dann ist der kleinste Normalteiler von A mit nilpotenterFaktorgruppe vertauschbar mit B [31]. Das ist der einfachste Sonderfall von

(e) Seien E,, ... , Em, F ... , F. abstrakte einfache Gruppen; unter ihnenmogen alle abelschen Kompositionsfaktoren von G auftreten. Seien A, B E snG.Dann ist fl A(EN) mit f1 B(F) vertauschbar. Dabei bezeichnel allgemein X (E) denkleinsten Normalteiler von X, dessen Faktorgruppe nur Kompositionsfaktoren a5 Ebesitzt [31].

Zweifellos die niltzlichsten Satze fiber das gegenseitige Verhalten von Subnor-malteilern sind die Normalisatorsatze. Einige Beispiele:

1.4 SUBNORMALITAT N ORMALITAT. Seien A, B E sn G. Wenn eine dernachstehenden Bedingungen (a)-(d) erfullt ist, so ist A < NG(B).

(a) A ist einfach and nichtabelsch [33].(b) A 1st ein minimaler Normalteiler von G [33].(c) A ist perfekt and normalisiert jedes X E snA mit A n B< X c A n B''.

(A lie these X liegen in der durch B erzeugten Fittingklasse. BA bezeichnet dasErzeugnis der Konjugierten B°, a E A) [14], [29].

(d) Kein Kompositionsfaktor von A oberhalb A n B ist isomorph zu derFaktorgruppe eines maximalen Normalteilers von B [29].

Perfekte Subnormalteiler Bind fast in der ganzen Gruppe normal:(e) Ist A = A' E sn G, so gilt die Abschi tzung [30]

I G: NG(A)l < ya, Y := 1og6OJGI, a := loge01AI.

Unter scharferen Voraussetzungen erhalt man Zentralisatorsatze:

1.5 SUBNORMALITAT = ELEMENTWEISE VERTAUSCHBARKEIT. Seien A, B E snG.Wenn eine der nachstehenden Bedingungen (a)-(c) erfullt ist, so ist die Kommuta-lorgruppe [A, B] = 1.

(a) A and B haben keinen isomorphen Kompositionsfaktor. (Das ist z.B. dannerfiillt, wenn JAI and CBI teilerfremd sind) [30].

(b) A ist einfach, nicht abelsch and nicht in B enthalten [33].(c) A n B = 1, and [A, B] enthdlt keinen abelschen Normalteiler 1 [20], [30].

1.6 AUSDEHNUNG AUF KOSUBNORMALE PAARE. Die in den bisherigen Satzengemachte Voraussetzung A, B E snG scheint Anwendungen auf die Struktureinfacher Gruppen G auszuschlieBen. Doch trifft das nicht zu. Es kommt ja nurdarauf an, daD A and B in irgend einer Untergruppe von G beide subnormalsind. Das tritt genau dann ein, wenn A and B in ihrem eigenen Erzeugnissubnormal sind. Ein solches Paar von Untergruppen nennen wir kosubnormal(join subnormal). Fur jedes kosubnormale Paar von Untergruppen gelten alsoz.B. die Normalisatorsatze 1.4(a)-(d) and liefern dann jeweils Wirkungen von Aauf B. Man darf erwarten, dali eine Untersuchung des zugehorigen Graphenlohnt (Ecken: Untergruppen von G; Kanten: Kosubnormalitat.) Behandeltworden sind bisher nur seine vollstandigen Teilgraphen, d.h., Systeme vonpaarweise kosubnormalen Untergruppen A ... , Ak (2.6).

1.7 SUBNORMALTEILER UND SYLOWGRUPPEN. (a) Sei A eln Subnormalteller vonG. Dann gilt fur jede Sylowgruppe S von G: S n A ist eine Sylowgruppe von A.

(b) Sei S eine Sylowgruppe von G. Dann gilt fur je zwei Subnormalteiler A, Bvon G: (S n A) V (S n B) = S n (A V B). Die Abbildung

snG-*snS:A"S n Aist ein Verbandshomomorphismus [32].

(c) Das Zentrum jeder p-Sylowgruppe von G normalisiert Op (A), fur jedesA E snG [35].

(d) Die entsprechenden Aussagen gelten fur Hallgruppen [35].1.8 KEGELS PROBLEM. Ist jede Untergruppe A von G, welche die Eigenschaft

1.7(a) besitzt, subnormal? Bewiesen ist das fur auflosbare A [13].

2. Kriterien fur Subnormalitat. Kriterien fur die Subnormalitat einergegebenen Untergruppe A von G sind prazise Aussagen uber die Nichteinfach-heit von G. Lange Zeit kannte man nur hinreichende Bedingungen. Sieentstanden durch Milderung der Vertauschbarkeitsbedingung von Ore [16]:AX = XA fur jedes X E sG. Wir stellen einige solche Kriterien zusammen:

2.1 VERTAUSCI-IBARKEIT r SUBNORMALITAT. Sei A < G. Jede der beiden Be-dingungen (a), (b) hat A sn G zur Folge.

(a) AS = SA fiir jede Sylowgruppe S von G [13].(b) AA9 = AMA fur jedes g E G;(A9 := g-Ag) [41].

Allgemeiner gilt:(c) Sei A c G, B 'Z G and A'B = BA' fiir jedes g e G. Dann gilt fur jede

Untergruppe H von G, welche A and B enthdlt: A H fl B H E sn G. Fur je zweiTeilmengen X, Y C G ist die Kommutatorgruppe [A X, B Y] subnormal in G [36].

Die seit 1974 entwickelte systematische Theorie der Subnormalitats-Kriterienberuht auf dem folgenden, auf minimale Gegenbeispiele zugeschnittenen Satz,der manchmal nach dem Beweisdiagramm benannt wird:

2.2 ZIPPER LEMMA. Sei A (4 snG, aber A E snX fur jedes X mit A < X < G.Dann liegt A in einer einzigen maximalen Untergruppe von G [37].

Man kann das so interpretieren: Der "Subnormalisator" von A in G existiertimmer dann, wenn A subnormal in jeder echten Zwischengruppe ist. Wir stelleneinige Folgerungen zusammen.

2.3 LOKALE KRITERIEN FOR SL'BNORMALITAT. Sei A < G. Jede der Bedingungen(a)-(d) ist notwendig and hinreichend fur A snG.

(a) A sn<A, g) fir jedes g E G.(b) Asn(A, Ar> fur jedes g E G.(c) A enthdlt zu jedem a E A auch jedes Element b E G, das zu a im Erzeugnis

<a, b> konjugiert ist.(d) Zu jedem a E A and g E G gibt es ein n E N derart, da,6 der iterierte

Kommutator [ g, ,a] in A liegt.

Offen ist die Frage, ob zu jedem n, fur das die Bedingung (d) gleichmaBig gilt,eine Normalkette der Lange n von G nach A existiert. - Die beiden letztenKriterien sind kurzlich mit Hilfe neuer Methoden wesentlich verscharft worden(Bartels [1], [2]):

166 HELMUT WIELANDT

2.4 KRITERIEN FUR ERZFUGENDE VON A. Sei A < G and A0 eine Teilmenge vonA, deren A-Konjugierte die ganze Gruppe A erzeugen. Dann ist jede der folgendenBedingungen notwendig and hinreichend fiir AsnG.

(a) A enth&lt zu jedem a E A. auch jedes b e G, das zu a in <a, b> konjugiertist.

(b) Die Ordnungen der Elemente von A0 sind Primzahlpotenzen, and zu jedemagA0undjedemgEGgibteseinnENmit[g,,a]EA.

Ob die Voraussetzung fiber die Ordnungen der Elemente von A0 entbehrlichist, ist nicht bekannt. Als Konsequenz von 2.4(a) sei erwahnt:

2.5 INVOLUTORISCHER AUTOMORPHISMUS = SUBNORMALITAT. Sei a ein Auto-morphismus der Ordnung 2 von G. Dann erzeugen diejenigen Elemente ungeraderOrdnung von G, die durch a invertiert werden, einen Subnormalteiler von G.

Wir kommen zu den vollstandigen Teilgraphen des Kosubnormalgraphen.Satz 2.3(b) kann so formuliert werden: Wenn je zwei der Konjugierten von Akosubnormal sind, so ist jede von ihnen subnormal im Erzeugnis aller. In dieserForm lasst sich der Satz von Konjugiertheitsvoraussetzungen befreien:

2.6 PAARWEISE K.OSUBNORMALITAT = SLBNORMALITAT. Seien Ai, ... , Apaarweise kosubnormale endliche Untergruppen einer Gruppe: A,sn<A;, Ak> far i,k = 1, . . . , n. Genau dann ist jedes A, subnormal im Erzeugnis A von ihnen alien,wenn jeder Homomorphismus von A, der alle Ai auf p-Gruppen zu derselbenPrimzahl p abbildet, auch A auf eine p-Gruppe abbildet. (Diese Bedingung ist z.B.dann erfiillt, wenn je zwei nicht perfekte A;, Ak miteinander vertauschbar sind [38].)

2.7 NICHTEINFACHIIEIT KOSUBNORMAL ERZEUGTER GRUPPEN. Die Gruppe Glasse sich durch paarweise kosubnormale echte Untergruppen A. erzeugen, derenOrdnungen nicht sdmtlich Potenzen derselben Primzahl sind. Dann ist G nichteinfach [38].

Die Voraussetzung fiber die Ordnungen kann in 2.7 nicht weggelassen werden,wie das Beispiel der alternierenden Gruppe G des Grades 9 zeigt; sie laBt sichdurch die Permutationen

(12 3)(4 5 6)(7 8 9), (1 4 7), (2 5 9)

erzeugen, obwohl die drei zyklischen Gruppen nut diesen Erzeugendenpaarweise kosubnormal sind. Ahnliche Beispiele gibt es ffir jede Primzahlp.

Die beiden letzten Satze wecken den Wunsch, ffir die Kosubnormalitat zweierUntergruppen Kriterien zu finden, die nicht explizit auf ihr Erzeugnis Bezugnehmen. Sind, beispielsweise, A and B schon dann kosubnormal, wenn es zujedem a E A and b E B ein n E N mit [b,a] E A and [a, b] E B gibt?

Wir schliel3en mit einem kfirzlich durch Maier entdeckten Subnormalisator-satz:

2.8 SUBNORMALITAT IN PRODUKTEN. Die auflosbare Untergruppe A sei subnor-mal in zwei vertauschbaren Untergruppen B, C von G. Dann ist A subnormal inBC [15].

Der Beweis liegt keineswegs auf der Hand. Ob die Voraussetzung derAuflosbarkeit entbehrlich ist, bleibt offen.

3. Untergruppen mit vorgeschriebenen Projektionen. Wir wenden uns derzentralen Aufgabe des zu Beginn dargelegten eingeschrankten HolderschenProgramms zu. Wir gehen davon aus, daB wir in einer Subnormalreihe derzusammengesetzten Gruppe G zwar nicht die Glieder

G=Go p G,® (3.0)

kennen, aber fiber die Faktoren

G" := G"_1/G" (p = 1, ... , n)

and deren Untergruppen Informationen besitzen. Was konnen wir dann uberdie Untergruppen von G selbst sagen? Jedes A 6 G bestimmt fur v = 1, ... , neine Untergruppe von G", bestehend aus denjenigen Nebenklassen von G" inGr_,, in welchen Elemente von A vorkommen. Wir nennen these Gruppe dieProjektion von A in G" and bezeichnen sie mit

A .-.G := (A n G"_1)G"/G" (v = 1, ... , n). (3.1)

A bestimmt seine n Projektionen eindeutig. Umgekehrt ist A durch seineProjektionen wenigstens bis zu einem gewissen Grade bestimmt: DieKompositionsfaktoren von A sind, auch unter Beriicksichtigung der Vielfach-heit, dieselben wie die aller n Projektionen A -.o.G" zusammengenommen, andes gilt

JAI = IIJA -.wG"l, I G: Al - h G": (A- G")J. (3.2)

Hiernach ist beispielsweise A genau dann eine ir-Hallgruppe von G, wenn jedeProjektion A--a,. G" eine 7r-Hallgruppe von G" ist.

Zwei Untergruppen A, B von G mit denselben Projektionen stimmen also inOrdnung and Kompositionsfaktorgruppen uberein. Sie brauchen aber nichtisomorph zu sein; ein Gegenbeispiel findet rich leicht in G = C2 X C4. Auf dasEindeutigkeitsproblem, wie weft A durch seine Projektionen (3.1) bestimmt ist,werden wir in einem wichtigen Sonderfall spater eingehen (4.7). Hier betrachtenwir das Existenzproblem: Gegeben sei in jedem der n Faktoren G" eine Un-tergruppe A'. Wann existiert eine Untergruppe A von G mit A -0. G" = A"(p = 1, ... , n)? Dieses Problem ist eingehend von Chunikhin bearbeitet worden.Aus seinem Buch [4] erwahnen wir das Hauptergebnis (Theorem 3.7.2), etwasvereinfacht and in die hier benutzte Terminologie ubersetzt.

3.3 ExISTENZSATZ. Sei G" 4 G and A" < G" (p = 1, ... , n). Fur jedes p E(2, ... , n) moge NG(A") die Faktorgruppe GIG, _, decken. Dann gibt es (zwarvielleicht nicht ein A E sG mit den Projektionen A", aber) ein B E sG mitProjektionen B", welche die gleichen Voraussetzungen wie die A" erfullen and nurwenig grosser als die A" sind, in folgendem Sinn:

(a)A"4B",A1=B1,(b) fur p E (2, ... , n) ist jeder Primfaktor von I B": A "I ein Teller von

IA'I 1A21... IA "-11,(c) B"/A" ist auflosbar mit Sylowturm.

Hinreichend fur die Existenz eines A mit den vorgeschriebenen ProjektionenA" ist also z.B. die zusatzliche Voraussetzung, dad jedes A" mit seinem Norma-lisator in G" ubereinstimmt.

168 HELMUT WIELANDT

Man kann iibrigens statt (c) sogar erreichen, daB B"/A" ein minimalerSupplementkern (3.4) and daher nilpotent ist.

3.4 EIN PROBLEM DER ERWEITERUNGSTHEORIE. Man bestimme alle "minimalenSupplementkerne", d.h., diejenigen Gruppen N, welche sich als Normalteilerohne echtes Supplement in passende Gruppen G einbetten lassen: N < G, andfiir H < G ist stets NH < G.

Zu den minimalen Supplementkernen gehoren die Frattini-Untergruppen allerendlichen Gruppen, aber z.B. nicht die nichtabelschen Gruppen der Ordnungp3vom Exponenten p2 (2 <p E P); these enthalten namlich intravariante [24]nicht charakteristische Untergruppen.

4. Maximale IT-Untergruppen. Die systematische Theorie der auflosbarenGruppen begann 1928 mit einer Entdeckung von P. Hall. Unter Vermeidung desfiir Verallgemeinerungen zu engen Begriffs der Hall-Gruppen kann man sie soformulieren (dabei bedeutet, wie iiblich, fr ein Teilmenge der Menge P allerPrimzahlen):

4.1 SATZ voN HALL. In einer auflosbaren Gruppe G sind je zwei maximaleir-Untergruppen konjugiert [6].

Auf Grund dieses Satzes stimmen die maximalen IT-Untergruppen von G fiirjedes IT C P mit den IT-Hall-Untergruppen der Ordnung G J iiberein, falls Gauflosbar ist; aber sonst nicht, wie Hall [7] gezeigt hat. Hieran liegt es, daa dieHallgruppen zwar fiir die auflosbaren Gruppen fundamental wichtig sind, aberfiir eine allgemeine Theorie der IT-Untergruppen nichtauflosbarer Gruppen trotzeiniger interessanter Einzelergebnisse (vor allem [9]) wenig hergeben. Was manoffensicht]ich braucht, ist ein Kriterium fiir die Konjugiertheit zweier maximalerr-Untergruppen. Diese Gruppen sind in der Literatur bisher kaum behandeltworden. In der Tat sind sie schwer zuganglich. Der Grund ist ihre mangelndeVertraglichkeit mit Homomorphismen and normalen Untergruppen. Um dieLage in bezug auf Homomorphismen zu beschreiben, bezeichnen wir die Mengeder IT-Untergruppen von G mit s, G and die Menge der beziiglich Inklusionmaximalen unter ihnen mit m, G. Dann gilt:

4.2 in, UND HOMOMORPHISMEN. Wenn it mindestens zwei Primzahlen, aber nichtalle enthdlt, so gibt es zu jeder Gruppe G eine Gruppe H and einen Epimorphismus4i: H - G mit 4i(m,H) = s, G. Fur H kann man jedes Kranzprodukt EwrGnehmen, wobei E Pine einfache Gruppe bedeutet, die zwei nicht isomorphe maximaleir- Untergruppen besitzt.

Fiir beliebige Homomorphismen ist die Lage also denkbar ungiinstig: DieBilder der maximalen IT-Untergruppen zeichnen sich im allgemeinen durchiiberhaupt keine besondere Eigenschaft aus. Natiirlich zeigen einzelne, derPrimzahlmenge IT angepaBte Homomorphismen ein besseres Verhalten.

4.3 7r-SEPARABLE HOMOMORPHISMEN. Ist der Kern Pines Homomorphismus 4, vonG Pine fr-Gruppe oder Pine IT'-Gruppe (allgemeiner: fr-separabel), so gilt '(m,G)= m 4,(G)

Ahnlich ist es beim Schneiden mit Normalteilern:

4.4 7T-SEPARABLE NORMALTEILER. Sei A E m, G and N < G. Wenn N 7r-sepa-rabel ist, so ist A n N eine fr-Hallgruppe von N; insbesondere ist A n N E m, N.

Im allgerneinen ist A n N E m,N. Immerhin laLit das die Moglichkeit offen,daft die Normalschnitte der maximalen 7r-Untergruppen besondere Eigenschaf-ten haben. Eine solche Eigenschaft ist nach langerem Suchen 1964 vomVerfasser [26] and unabhangig davon etwas spater von Hartley [10] gefundenworden. Urn sie kurz zu formulieren, nennen wir eine Untergruppe X von G7r-normalisatorgleich, wenn der Index ING(X): X1 keinen Primfaktor aus 7Tenthalt. Hiermit gilt:

4.5 NORMALSCHNITTE VON m,,. Sei A E and N a G. Fur jeden einfachenAusschnitt (section) E = N1 /N2 von N sei die durch NA(E) induzierte aussereAutomorphismengruppe von E auflosbar. Dann ist A n N 7r- normalisatorgleich inN; insbesondere ist A n N > 1, wenn N 1 , > 1.

Die Voraussetzung fiber die Automorphismen ist sicher darn erfiillt, wenn furdie perfekten einfachen Ausschnitte von N die Schreiersche Vermutung zutrifft;sie wird also hoffentlich eines Tages entbehrlich werden. Unabhangig von derGiiltigkeit der Schreierschen Vermutung ist die Voraussetzung fiber die Auto-morphismen natiirlich erfiillt, wenn A auflosbar ist.

Von nun an schranken wir die betrachteten Untergruppen A in einer deco Satz4.5 angepafiten Weise ein:

4.6 SCHREIER-VORAUSSETZUNG. Fur jeden einfachen perfekten Ausschnitt E vonG sei die durch NA (E) induzierte Untergruppe der auleren Automorphismengruppevon E auflosbar.

Aus 4.3-4.5 folgt unter Benutzung der Eigenschaften der einkopfigen per-fekten Subnormalteiler:

4.7 KONJUGIERTHEIT MAXIMALER 7T-UNTERGRUPPEN. Seien A, B E m G. Esgebe eine Reihe 3.0, z.B. eine Kompositionsreihe von G, derart, daB die Pro-jektionen von A and B in die nicht auflosbaren Faktoren G`' iibereinstimmen:A --o- G' = B - .G`. Dann gilt unter der Voraussetzung 4.6: A and B sindkonjugiert in ihrem Erzeugnis.

Der Beweis dieses seit 1964 angekiindigten Satzes [26], [27] ist bisher nur ininternen Tiibinger Vorlesungsausarbeitungen dargestellt; es ist beabsichtigt, ihn1981 in einern grofieren Zusammenhang zu veroffentlichen, auf den wir spatereingehen.

Satz 4.7 verdient nicht nur als eine teilweise Losung des Eindeutigkeitspro-blems von Abschnitt 3 Interesse. Er ist, falls die Schreiersche Vermutung sich alsrichtig erweist, eine auf zusammengesetzte Gruppen unbeschrankt anwendbareFassung des grundlegenden Satzes 4.1. Denn in einer auflosbaren Gruppe Gerfiillen je zwei maximale 7r-Untergruppen die Voraussetzung von 4.7 in trivialerWeise, and 4.7 geht in 4.1 fiber.

Der Satz zeigt, daft die maximalen 7r-Untergruppen nicht nur die nachstlie-gende, sondern auch eine brauchbare Verallgemeinerung der fr-Hall-Gruppenauflosbarer Gruppen darstellen. Ganz befriedigend ist sie allerdings noch nicht,

170 HELMUT WIELANDT

aus zwei Griinden. Bei der "richtigen" Verallgemeinerung sollten moglichst vieleEigenschaften der ir-Hall-Gruppen erhalten bleiben, z.B. die Vertraglichkeit mitNormalteilern; den maximalen ir-Untergruppen fehlt sie. Und die grobe Eintei-lung der Untergruppen nach den Primteilern ihrer Ordnung sollte durch Un-terscheidung der Kompositionsfaktoren verfeinert werden.

5. Eine Verallgemeinerung der Hallgruppen. Im folgenden sei 3r eine gegenuberHomomorphismen, Erweiterungen and Bildung von Untergruppen abgeschlos-sene Klasse von endlichen Gruppen. Beispiel: Die auflosbaren ir-Gruppen, oderdie Gruppen mit Kompositionsindizes aus der Reihe 60, 2, 3, 5, 7; aber nicht:60, 3, 5, 7. Wir benutzen die Bezeichnungen

s,G 3r n sG (3r-Untergruppen von G),mmG (A E s,G IA < B E s,G A = B) (maximale 3r-Untergruppen).DEFINITION. Wir nennen A eine submaximale 3E-Untergruppe von G (kurz:

AsmmG oder A E smG), wenn es zwei Gruppen H, B derart gibt, dass GsnH,B E mH and B n G = A gilt. (Es gilt dann auch AsnB.)

H

V

1 A XG

Definition der SubmaximalitatDie Definition stellt sicher, daft sich Submaximalitat in der erwiinschten

Weise vererbt:

5.1 Aus AsmmG and G,snG folgt (A n G1)sm,G,

Vertraglichkeit mit Homomorphismen besteht natiirlich nicht (4.2). Weiterhinbezeichne fr die Menge der Primzahlen, welche die Ordnung wenigstens einerGruppe aus X teilen. Wenn I,rI = 1 oder G auflosbar, allgemeiner: "3E-separabel"ist, so findet man, was zu erwarten war:

5.2 (a) Besteht it aus einer Primzahlp, so besteht sm.jG aus den p-Sylowgruppenvon G.

(b) Ist G auflosbar, so besteht sm.,G aus den ir-Halluntergruppen von G. Dasgleiche gilt allgemein, wenn jeder Kompositionsfaktor von G in X liegt oder eineir'-Gruppe ist.

Eine bei Hartley [10, p. 225] nachlesbare Schlufiweise ergibt:

5.3 Ist G einfach undperfekt, so besteht smmG aus den Durchschnitten von G mitden maximalen 3r-Untergruppen von Aut G. Dabei sind die inneren Automorphis-men mit den Elementen von G identifiziert.

Schwieriger zu beweisen sind die folgenden, unter der Schreier-Voraussetzung4.6 zu erwartenden Aussagen.

5.4 (a) Ist A E sm,G, so ist A'r- normalisatorgleich in G.(b) Seien G", a E K, die "kritischen", d.h. nicht 3r-separablen, Faktoren einer

Subnormalreihe 3.0 von G. Sei A E sm£G and B E s, G. Wenn B fur jedes a E Kdie Projektion A" von A in G" normalisiert, so ist B im Erzeugnis <A, B> zu einerUntergruppe von A konjugiert.

(c) Seien A, B.... E sm G. Wenn fur jedes a E K die Projektionen A',B% . . . in ihrem Erzeugnis J" konjugiert sind, so sind A, B, ... in ihremErzeugnis J konjugiert. Wenn fur jedes a E K sogar A" = B" = ... gilt, so ist Jir-separabel, and A, B.... sind 'r-Hallgruppen von J.

Die Beweise sind-wie das auch fur andere auf dieser Tagung vorgetrageneErwartungen der Fall ist-noch nicht in allen Einzelheiten aufgeschrieben. Aberselbst wenn noch Korrekturen erforderlich werden sollten, diirfte die Richtungdeutlich sein, in der die Losung des Problems der Verallgemeinerung derHall-Untergruppen, and das heist im Grunde: der Sylowgruppen, zu suchen ist.Ob die submaximalen 3r-Untergruppen das letzte Wort darstellen, bleibt zuprufen. Durch geeignete Modifikationen liesen sich vielleicht noch andereEigenschaften der Sylowgruppen retten oder der Anwendungsbereichvergrosern. Wie unser beruhmter Kollege aus der Baker Street 221 B sagenwiirde: Wenn alle weitergehenden Wiinsche als unmoglich erfiillbar nachgewie-sen sind, ist die Wahrheit gefunden, die sich in Sylows Entdeckung verbirgt.

6. Weitere Fragen. Zum Schlus seien einige Stichworte and Hinweiseerwahnt, die als Anknupfungspunkte fur die Weiterentwicklung der Theoriedienen konnen.

(a) Komplemente and ausgezeichnete Supplemente fur Normalteiler [12], [22].(b) Man suche in zusammengesetzten Gruppen mit gegebenen Kompositions-

faktoren nach charakteristischen Klassen konjugierter Untergruppen (d.h.nachintravarianten Untergruppen), wie sie fur auflosbare Gruppen durch For-mationen and Fittingklassen geliefert werden.

(c) Was kann man aus der Kenntnis von Faktorisierungen G = AB fiber dieKompositionsfaktoren von G schliesen [17], [19], [34]?

(d) Welche Zusatzvoraussetzungen machen eine Gruppe, die ein p-Komple-ment enthalt,p-auflosbar?

(e) Welche Faktorisierungen der bekannten einfachen Gruppen gibt es?(f) Welche bekannten einfachen Gruppen besitzen einen auseren Auto-

morphismus, der jedes Element in ein konjugiertes iiberfiihrt (W. Jehne, Math.Inst. Univ. Koln)?

(g) Man untersuche die submaximalen ir-Untergruppen der minimalen nichtauflosbaren Gruppen: Konjugiertheit in der Automorphismengruppe, In-travarianz, Pronormalitat, usw.

(h) In welchen der bekannten einfachen Gruppen gilt der "starke ir-Sylowsatz": Zu je zwei ir-Untergruppen A, B gibt es t E A V B derart, dabA V B' eine fr-Gruppe ist? Fur Jfrl = 2 vgl. [9, Theorem A4].

(i) Ist eine Untergruppe A von G subnormal, wenn ihr Schnitt mit jedermaximalen ir-Untergruppe von G ir-normalisatorgleich in A ist fur jedes - C P?Vgl. 1.8, 5.4(a).

172 HELMUT WIELANDT

(j) Set G einfach, I G 1 1. Enthalt jede maximale ir-Untergruppe der Auto-morphismengruppe von G einen inneren Automorphismus I? Das wi rdeanstelle der Schreierschen Vermutung fur die Theorie der ir-Untergruppenzusammengesetzter Gruppen geniigen.

LITERATURE

1. D. Bartels, Zur Theorie der Subnormalitat in endlichen Gruppen: Relationen auf Konjugierten-klassen, 93 S. Diss. Univ. Tubingen, 1976.

2. , Subnormality and invariant relations on conjugacy classes infinite groups, Math. Z. 157(1977), 13-17.

3. D. C. Brewster, A criterion for the permutability of subnormal subgroups, J. Algebra 36 (1975),85-87.

4. S. A. Chunikhin, Subgroups of finite groups, translated by E. Robinson, 142 S. Groningen,Wolters-Noordhoff, 1969.

5. W. Gaschi tz, Ein allgemeiner Sylowsatz in endlichen auflosbaren Gruppen, Math. Z. 170 (1980),217-220.

6. P. Hall, A note on soluble groups, J. London Math. Soc. 3 (1928), 98-105.7. , A characteristic property of soluble groups, J. London Math. Soc. 12 (1937), 198-200.8. , The construction of soluble groups, J. Reine Angew. Math. 182 (1940), 206-214.9. , Theorems like Sylow's, Proc. London Math. Soc. (3) 6 (1956), 286-304.

10. B. Hartley, A theorem of Sylow type for finite groups, Math. Z. 122 (1971), 223-226.11. O. Holder, Bildung zusammengesetzter Gruppen, Math. Ann. 46 (1895), 321-422.12. C. E. Johnson and H. Zassenhaus, On equivalence of finite group extensions, Math. Z. 123

(1971), 191-200.13. O. H. Kegel, Sylow-Gruppen and Subnormalteiler endlicher Gruppen, Math. Z. 78 (1962),

205-221.14. , Uber den Normalisator von subnormalen and erreichbaren Untergruppen, Math. Ann.

163 (1966), 248-258.15. R. Maier, Um problema da teoria dos subgroups subnormais, Bol. Soc. Brasil. Mat. 8 (1977),

127-130.16. O. Ore, Contributions to the theory of groups of finite order, Duke Math. J. 5 (1939), 431-460.17. E. Pennington, Trifactorisable groups, Bull. Austral. Math. Soc. 8 (1973), 461-469.18. M. Pinl, Kollegen in einer dunklen Zeit, Jahresbericht DMV 71(1969), 190-193.19. U. Preiser, Produkte endlicher einfacher Gruppen, Math. Z. 167 (1979), 91-98.20. J. E. Roseblade, A note on disjoint subnormal subgroups, Bull. London Math. Soc. 1 (1969),

65-69.21. J. E. Roseblade and S. E. Stonehewer, Subjunctive and locally coalescent classes of groups, J.

Algebra 8 (1968), 423-435.22. L. A. Semetkov, On the existence of H-complements for normal subgroups of finite groups, Dokl.

Akad. Nauk. SSSR 195 (1970) = Soviet Math. Dokl. 11 (1970), 1436-1438.23. H. Wielandt and B. Huppert, Arithmetical structure and normal structure of finite groups, Proc.

Sympos. Pure Math., vol. 6, Amer. Math. Soc., Providence, R. 1., 1962, pp. 17-38.24. , Entwicklungslinien in der Strukturtheorie der endlichen Gruppen, Proc. Internat. Congr.

Math. (Edinburgh, 1958), 1960, pp. 268-278.25. , Arithmetische Struktur and Normalstruktur endlicher Gruppen, Atti Conv. Teoria dei

Gruppi Finiti (Firenze, 1960), 1960, pp. 56-65.26. , Sw la structure des groupes composes, Seminaire Dubreil-Pisot, 17e annie, Paris,

1963/64, no. 17 (1964), 10 pp.27. , On the structure of composite groups, Proc. Internat. Conf. Theory of Groups

(Canberra, 1965), 1967, pp. 379-388.28. , Topics in the theory of composite groups, Lecture Notes prepared by J. Horwath, Dept.

of Math., Univ. of Wisconsin, Madison, 124 pp. (1967; inzwischen Neuauflage).29. , Subnormale Untergruppen endlicher Gruppen, Vorlesung an der Universitat Tubingen

1971. Ausgearbeitet von Max Selinka, 99 S. (Neuauflage in Vorbereitung).30. , Eine Verallgemeinerung der invarianten Untergruppen, Math. Z. 45 (1939), 209-244.


31. , Vertauschbare nachinvariante Unlergruppen, Abh. Math. Sem Univ. Hamburg 21(1957), 55-62.

32. , Sylowgruppen and Kompositions-Struktur, Abh. Math. Sent. Univ. Hamburg 22 (1958),215-228.

33. , Uber den Normalisator der subnormalen Untergruppen, Math. Z. 69 (1958), 463-465.34. , Uber die Normalstruktur von mehrfach faktorisierten Gruppen, J. Austral. Math. Soc. 1

(1960), 143-146.35. , Der Normalisator einer subnormalen Untergruppe, Acta Sci. Math. (Szeged) 21 (1960),

324-336.

36. , Vertauschbarkeit von Untergruppen and Subnormalitat, Math. Z. 133 (1973), 275-276.37. , Kriterien fur Subnormalitat in endlichen Gruppen, Math. Z. 138 (1974), 199-204.38. , Uber das Erzeugnis paarweise kosubnormaler Untergruppen, Arch. Math. (Basel)

35 (1980), 1-7.39. G. Zappa, Recenti risultati sul reticolo dei sottogruppi subnormali di un gruppo, Sent. Ist. Naz.

Alta Mat. (Roma) 1962/63 (1964), 441-448.40. H. Zassenhaus, The theory of groups, 2nd ed., Vandenhoeck & Ruprecht, Gottingen, 1958, p.

236.41. J. Sup, Bemerkungen zu einem Satz von O. Ore, Publ. Math. (Debrecen) 3 (1953), 81-82.

UNIVERSITAT TUBINGEN, BUNDESREPUBLIK DEUTSCHLAND


SOME CONSEQUENCES OF THECLASSIFICATION

OF FINITE SIMPLE GROUPS

WALTER FEIT1

1. In this paper, which is an expanded version of my talk, I will consider howthe classification of finite simple groups affects various questions in the theoryof modular representations and related areas. More precisely, I want to exploresome consequences of the following assertion.

(*) The list of finite simple groups consists of groups of Lie type, alternatinggroups and the 26 sporadic groups which are currently known.

The assertion (*) has not yet been proved and so it may be false. However thestatements below are probably independent of small perturbations.

The proof of-a statement which consists of checking all cases is often a proofwithout understanding and so is not very satisfactory. It might be called "proofby exhaustion," where the term applies equally well to the investigator or to hissubject. Nevertheless the method can be very powerful as a device for discover-ing results and has very respectable historical precedents.

After Killing and E. Cartan had classified the complex simple Lie algebrasCartan, Weyl and others discovered and proved many theorems by this method.The later attempts by them and others to give conceptual proofs has been, andstill is, a powerful impetus in the development of the subject. A similar thinghappened after the finite reflection groups were classified. At present there is avast theory of these groups including many results which have only been provedby an analysis of cases. It is not unreasonable to expect that this same evolutionwill develop in the theory of finite groups once (*) has been proved.

It should be mentioned that (*) includes the fact that the listed groups exist.This is not necessary if one wishes to verify a statement by checking all cases butof course becomes essential for the purpose of constructing counterexamples.

Some statements follow immediately from (*) and other known results, someappear to be independent of (*) and some should become accessible by using(*). I will here discuss how (*) affects a few, rather randomly chosen and mostly

1980 Mathematics Subject Clarsificatlon. Primary 20D05, 20C20.'The work in this paper was partly supported by the NSF.


175

176 WALTER FEIT

well-known, statements. Any reader will be able to add to this list. It should benoted that after I gave this talk, suggestions from various people made memodify some of my original comments.

The assertion that (*) is true is very deep and is bound to have manyconsequences. Some, such as the proof of the Schreier conjecture, are wellknown. Let me state a result that is perhaps not so well known. We first need adefinition.

A polynomial f(x) E C[x] is indecomposable if whenever f(x) = f,(f2(x)) forpolynomials f1, f2 then deg f = 1 for i = 1 or 2.

For instance a polynomial of prime degree is indecomposable.

THEOREM I.I. Assume that (*) has been proved. Let f, g be nonconstantindecomposable polynomials in C[x] and suppose that f(x) - g(y) factors inC[x, y]. Then either g(x) = flax + b) for some a, b E C or

deg f = deg g = 7, 11, 13, 15, 21, or 31.

The first alternative is the trivial case. The result is the best possible in thesense that for each of the listed numbers there exist indecomposable polynomialsf and g of the given degree which do not satisfy the first condition but such thatf(x) - g(y) factors. John McKay informed me after the talk that some electricalengineers had been interested in factoring expressions of the formf(x) - g(y) sothat the ramifications of (*) are broader than might appear at first glance. I willreturn to Theorem 1.1 below in §4.

In the rest of the paper I will use standard notation. G will always stand for afinite group and p will always denote a prime.

2. This section contains a list of questions and statements which do not seemto be affected by the proof of (*).

2.1. The Alperin-McKay conjectures. For a statement in full generality see [1].The simplest of these can be stated as follows. Let nn(G) denote the number ofirreducible characters X of G with p } X(1). Then nn(G) = nP(NG(P)) for aSP-group P of G.

This is an assertion about all finite groups, which makes it especially tantaliz-ing that there appears to be no way of approaching it with or without (*). Thereare other assertions about all finite groups which are very difficult, but they willpresumably follow once (*) is proved. For instance the B-conjecture which is anessential ingredient in the proof of (*) or analogues of Glauberman's Z*-theo-rem for odd primes as mentioned by Syskin [19].

The assertion has been proved by Dade for p-solvable groups, see [7]. Thisgeneralizes earlier results for solvable groups [13], [20]. It does not appear asthough Dade's methods can reduce the general question to the case that G issimple. It has been proved by Alperin [1] for the groups GL (q) in case q is apower of p, and by Olsson [17] in case p } q. It follows from the work of Green,Lehrer and Lusztig [12] for groups for Lie type with connected center incharacteristic p. MacDonald [16] has verified it for the symmetric groups and allprimes.

2.2. Let k(G) denote the number of conjugate classes of G. Let P be aSP-group of G. What is the relation between k(G) and k(NG(P))?

It seems that k(NG(P)) < k(G) for many groups G. This is however notalways true. It is for instance false for G = A8 c_- L4(2) and p = 2 (though it istrue for G = E8). Is there any connection between k(G) and n,(G)?

The next five questions are old questions of Brauer.2.3. When does a group have blocks of defect 0 for a given primep?2.4. Is it true that a p-block B has no characters of positive height if and only

if B has an abelian defect group?2.5. Is the number of irreducible characters in a block bounded by the order

of a defect group?2.6. Let qq be an irreducible Brauer character of G for the prime p. Is IGI/q)(1)

always a local integer?This is true for p-solvable groups by the Fong-Swan theorem. If G is of Lie

type in characteristic p this follows from the fact that q)(1) < St(l) and St(l) isthe full power of p which divides I G 1.

2.7. Is there any bound for decomposition numbers and Cartan invariants interms of the defect group D of a block and its imbedding in G?

Conceivably d < D J. At one time it had been conjectured that c G I D J.Landrock [15] showed that Sz(8) is a counterexample. More recently Chastkof-sky and I have shown that the groups Sz(2z"+1), Sp4(2"), PSL3(2m), PSU3(2m) areall counterexamples after the first few. See [5]. In all the cases where explicitresults are known, c; < ING(P)I where P is a SP-group of G. However theevidence is very skimpy as few results are actually known.

2.8. Let P be a p-group and let F be an algebraically closed field ofcharacteristic p. Consider the class of all pairs (G, V) where P is a SP-group of Gand V is an irreducible F[G] module. Let S be the class of all F[P] modules Wsuch that VI WG for some pair (G, V). Does S contain only finitely manyisomorphism classes of F[P] modules?

After my talk L. L. Puig pointed out that by using (+) it can be shown that theanswer is affirmative if G is restricted to ranging only over p-solvable groups.The consequence of (+) that is needed is the fact mentioned below just before3.2.

3. This section contains some statements which it should be possible to answerby assuming (+).

3.1. Various theorems about solvable groups should be extendable to p-soly-able groups.

For example consider the following result.Alperin [2] has defined an F[G] module to be algebraic if there exist integers

ao, . . . , a,,, not all 0, with E a.( V)' = 0 in the Green ring, where (V) is theisomorphism class containing V. Berger [3] has shown that an irreducible F[G]module for G solvable is algebraic. I observed in [10] that by using (+) this is alsotrue forp-solvable. The proof requires the following fact.

If G is simple and p G I then a SP-group of Aut(G) is cyclic. This fact is avery simple consequence of (+) but, like the Schreier conjecture, seems to becompletely inaccessible without (+).

3.2. What sort of information about G can be derived from the character tableof G? Here are two explicit questions.

178 WALTER FELT

3.2. (a) Can one decide whether a Sp-group is abelian?After I gave this talk Camina and Herzog observed that if p = 2 this can be

done, by using the classification of simple groups with abelian S2-groups.3.2. (b) Let X be an irreducible character of G. Let m be the smallest integer so

that (x(x)I x E G) C Q( ). Does G contain an element of order m?3.3. Let ml IGI and let M = {xix' = 1). By a theorem of Frobenius IMI _

c,"m for some integer c,". No one has even been able to give a group theoreticinterpretation of the integer c,". However Frobenius conjectured that if IMI = mthen M < G.

Suppose for simplicity that m is a Hall divisor of IGI, i.e. (m, IG11m) = 1.During my talk I was reminded of unpublished work which had reduced theconjecture to the case that G is simple in this case. Majority opinion seemed toexpect an easy solution by using (*). Some years ago Rust had verified it for thesymmetric and alternating groups.

3.4. What trees can be Brauer trees of a block with a cyclic Sp-group? Thisquestion may be very difficult although I would not be surprised if relatively fewtrees can occur. At this point not one single tree has been eliminated. If any treecould be eliminated it would show the existence of symmetric algebras whichhave only finitely many indecomposable modules up to isomorphism but whichare not isomorphic to a block ideal of a group algebra. I will suggest a possiblecandidate for a tree T which should not be a Brauer tree.

Let e > 248 be a prime. Let T be a tree with e + 1 vertices. Suppose that T isconstructed so that if it were a Brauer tree the block could not contain any realcharacters. A possible example is the following

e-8

VIf T is a Brauer tree then it is a Brauer tree for a simple group G as e is a

prime. The group G cannot be an alternating group as all characters ofsymmetric groups are real. Since e > 248, G cannot be sporadic or exceptionalof Lie type. Thus by (*) it remains to check the groups of types of A. - D. andtheir Steinberg variations. So much work has recently been done on theirreducible characters of groups of Lie type that I hope that this is now feasible.

3.5. In case G is p-solvable the Fong-Swan theorem implies that every Brauertree is a star. It is not known whether Brauer trees of simple groups can be starswith an arbitrary number of edges.

4. This section contains results which are, or should be, simple consequencesof (*) and other known facts. I begin with the fact that (*) together with thework of Curtis, Kantor and Seitz [6] yields the classification of all simple groupswith a faithful doubly transitive permutation representation. Their results inparticular imply the following statement.


THEOREM 4.1. Assume that (*) holds. Let G be a nonsolvable doubly transitivegroup on n letters which contains an n-cycle. Then either G - A. or Z. or one ofthe following holds:

G PSL2(11) M11 M23 PSLk(q) C_ G c PFLk(q)

(qk - 1)(q - 1)

where I'Lk(q) is the group of all semilinear transformations on a k-dimensionalspace over a field of q elements.

It should be observed that not all the groups listed in Theorem 4.1 contain ann-cycle.

COROLLARY 4.2. Assume that (*) holds. Let G be a nonsolvable doubly transitivegroup on p letters for some prime p. Then G is one of the groups listed in Theorem4.1 with (qk - 1)/(q - 1) = p.

COROLLARY 4.3 Assume that (*) holds. Let p be a prime with p > 23. ThenPSL2(p) is a maximal subgroup of the alternating group A,+ I.

Corollary 4.3. and similar results concerning the groups PTLk(q) are conse-quences of Corollary 4.2. They are of interest for coding theory amongst otherthings.

COROLLARY 4.4. Assume that (*) holds. Let p be a prime and let 1 G k Gp - 1. Let f(x) = x° + axk + b be irreducible over Q and let G be the Galoisgroup of f(x) over Q. Then one of the following holds.

(i) G is solvable.(ii)G-A.or EP.(iii) p = 7, G PSL2(7).(iv) p = 11, G ^PSL2(11) or MII.(v) p = 1 + 2` > 5, SL2(2`) C G C FL2(2`).

Since f(x) has at most 3 real roots, G is a permutation group on p letters whichcontains an involution that fixes at most 3 letters. Inspection of all cases inCorollary 4.2 yields Corollary 4.4.

It should be mentioned that no examples are known where (iv) or (v) ofCorollary 4.4 actually occurs.

COROLLARY 4.5. Assume that (*) holds. Suppose that G has two inequivalentdoubly transitive permutation representations on n letters which afford the samecharacter. Assume furthermore that some element of G is represented by an n-cyclein one (and hence both) of these representations. Then either G = PSL2(11) withn = 11 or PSLk(q) C G C PFLk(q) with n = (qk - 1)/(q - 1) for some k > 3.

Define the length 1(.r) of a permutation it to be the smallest integer such that itis a product of 1(ir) transpositions. In [8], [9] it was shown that if G is one of thegroups listed in Corollary 4.5 which is generated by elements (7r;) withirk ... 7r, = 7r, an n-cycle, and E 1(7rk) = 1(.r) = n - 1 then G ' PSL2(11) orG - PFLk(q) with (k, q) = (3, 2), (4, 2), (5, 2), (3, 3) or (3, 4). By results of Fried

180 WALTER FEIT

which motivated this work, see e.g. [11], this together with Corollary 4.5 yields aproof of Theorem I.I. Actually (*) eliminates the need for most of [8] but doesnot affect [9].

The problem of studying unexpected factorizations of polynomialsf(x) - g(y) was first considered by Cassels [4] who related the question to oneconcerning permutation groups but it was later work of Fried which showed thatthe conditions listed in Corollary 4.5 are relevant. Actual factorizations fordegrees 7 and 11 can be found in [4], [11].

Other results similar to Theorem 1.1 also follow from (*). There are alsorelated statements which may not follow directly from (*) but lead to interestinggroup theoretic questions which should be investigated. See for instance [11] fora discussion of some of these.

One of the conditions mentioned in Corollary 4.4 arises in number theory inan apparently totally unrelated context. See e.g. [18] for the following results.

Two algebraic number fields are arithmetically equivalent if they have thesame i-function.

Let KI and K2 be arithmetically equivalent number fields. Then they have thesame Galois closure F (in some algebraic closure) and the permutation represen-tations of the Galois group G of F on the cosets of the subgroups Hl and H2corresponding to K, and K2 afford the same character. Hence Corollary 4.5implies the following.

THEOREM 4.6. Assume that (*) holds. Let KI and K2 be nonisomorphic arithmeti-cally equivalent fields with [K,: Q] = [K2: Q] = p, a prime. Then either p = 11 orp = (q' - 1)/(q - 1) for some prime power q and some k > 3.

The remaining statements in this section are purely group theoretic.The following consequence of Corollary 4.2 answers a question of Wielandt.

THEOREM 4.7. Assume that (*) holds. I f p2 { I G I then there are at most 2conjugate classes of p-complements in G.

Let P be a Si-group of G with IPI = p. In case 21 ING(P): CG(P)I Wielandtshowed that any two p-complements are conjugate. See e.g. [8] for a simpleproof. If however I NG(P): CG(P)I is odd no one has been able to bound thenumber of conjugate classes of p-complements without assuming that (*) hasbeen proved.

I will conclude with a list of assertions, which I have not actually verified butwhich should follow easily from the classification of simple groups with aprimitive rank 3 permutation representation which is apparently likely to bedone in the near future.

4.8. If G is simple and a Si-group has orderp then p2 { IAut(G)I.4.9. (Wielandt's problem). If G is a primitive but not doubly transitive group

on 2p letters thenp = 5 and G = AS or ES.4.10. (Fried's problem). Suppose that G has a faithful doubly transitive

permutation representation on n letters and an element x in G is represented byan n-cycle. Assume also that G has a primitive permutation representation whichis not doubly transitive on 2n letters and x is represented by a product of twon-cycles. Then n = 5 and G = AS or ES.

Statement 4.10 has interesting consequences concerning polynomials. See [11].


4.11. Let (Do be the principal indecomposable character corresponding to thetrivial character. If t0(1) = p then either G is of type L2(p) or G has a subgroupof index p.

As a consequence of 4.11 one can get information about lattices of Q-dimen-sion p - 1, p or p + 1 which admit an automorphism of order p.

4.12. If G has a rational faithful irreducible representation of degree p - 1then either G is of type L2(p) or G has a subgroup of index p.

In 4.12 it is essential to assume that the representation is rational, it is notenough to assume that the character is rational valued as the conclusion doesnot hold under this weaker assumption. Perhaps the simplest couterexampleoccurs for U3(3) with p = 7.

REFERENCES

1. J. L. Alperin, The main problem of block theory, Proc. Conf. on Finite Groups, Academic Press,New York, 1976, pp. 341-356.

2. , On modules for the linear fractional groups, Internat. Sympos. on the Theory of FiniteGroups (1974), Japan Soc. for Promotion of Sci., Tokyo, 1976, pp. 157-163.

3. T. R. Berger, Irreducible modules of solvable groups are algebraic, Proc. Conf. on Finite Groups,Academic Press, New York, 1976, pp. 541-553.

4. J. W. S. Cassels, Factorization of polynomials in several variables, Lecture Notes in Math., vol.118, Springer-Verlag, Berlin and New York, 1970, pp. 1-17.

5. L. Chastkofsky and W. Feit, Projective characters of groups of Lie type, C. R. Math. Rep. Acad.Sci. Canada 1 (1978), 33-36.

6. C. W. Curtis, W. M. Kantor and G. M. Seitz, The 2-transitive permutation representations of thefinite Chevalley groups, Trans. Amer. Math. Soc. 218 (1976), 1-59.

7. E. C. Dade, A correspondence of characters, these PROCEEDINGS, pp. 401-403.8. W. Feit, Automorphisms of symmetric balanced incomplete block designs, Math. Z. 118 (1970),

40-49.9. , On symmetric balanced incomplete block designs with doubly transitive automorphism

groups, J. Combinatorial Theory 14 (1973), 221-247.10. , Irreducible modules of p-solvable groups, these PROCEEDINGS, pp. 405 -411.11. M. Fried, Exposition on an arithmetic-group theoretic connection via Riemann's existence

theorem, these PROCEEDINGS, pp. 571-602.12. J. A. Green, G. I. Lehrer and G. Lusztig, On the degrees of certain group characters, Quart. J.

Math. 27 (1976), 1-4.13. 1. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95 (1973), 594-635.14. V. Landazuri and G. M. Seitz, On the minimal degrees of projective representations of the finite

Chevalley groups, J. Algebra 32 (1974), 418-443.15. P. Landrock, A counterexample to a conjecture on the Cartan invariants of a group algebra, Bull.

London Math. Soc. 5 (1973), 223-224.16. I. G. MacDonald, On the degrees of the irreducible representations of symmetric groups, Bull.

London Math. Soc. 3 (1971),189-192.17. J. B. Olsson, McKay numbers and heights of characters, Math. Scand. 38 (1976), 25-42.18. R. Perlis, On the class numbers of arithmetically equivalent fields, J. Number Theory 10 (1978),

489-509.19. S. A. Syskin, Some characterization theorems, these PROCEEDINGS, pp. 121-122.20. T. Wolf, Characters of p'-degrees in solvable groups, Pacific J. Math. 74 (1978), 267-271.

YALE UNIVERSrrY


GRAPHS, SINGULARITIES,AND FINITE GROUPS

JOHN McKAYI

Introduction. We have seen during the past few years a major assault on theproblem of determining all the finite simple groups. We are told that this assaultis nearly complete; even if this is so, the story is not an easy one to tell since it isspread over thousands of pages and, apart from being long, it is a story in whichalmost all the characters play roles only within the theory of finite groups-theimpact of developments in other areas of mathematics on the classificationproblem has been minimal. I want to suggest that there is an immense wealth ofconnections with other areas which lies ready to be discovered. If I am right, Iforesee new proofs of the classification which will owe little or nothing to thecurrent proofs. They will be much shorter and will help us to understand thefinite simple groups in a context much wider than finite group theory.

Representation graphs. Let R be a representation of a group G, havingirreducible representations (R1), such that

R ®Rj = ED mjk Rk, j, k = 1, 2, ... , t.k

The representation graph rR = rR(G) is the graph with vertex set (Ri) and mjk(directed) edges from Rj to Rk. We convene that a pair of opposing directededges be represented by a single undirected edge.

PROPOSITION i. r.(G) is connected if and only if R is faithful on G.

PROPOSITION 2. rR(G) is self-dual (invariant under reversal of edge orientation)if and only if R affords a real-valued character. rR(G) is undirected if it isself-dual and has no directed loops.

An example illuminating both propositions is G = E4, the symmetric group ofdegree 4, and R, the unique two-dimensional irreducible representation.

1980 Mathematics Subject Classification. Primary 05C25, 20C99; Secondary 17B10.'Research supported by National Research Council of Canada.


183

184 JOHN MCKAY

Finite groups of quaternions. The finite subgroups of real quaternions areabstractly the binary polyhedral groups defined by the relations:

<a, b, C>: A' = Bb = C` = ABC.These are finite for the binary dihedral group <2, 2, n>, of order 4n, the binarytetrahedral group <2, 3, 3), of order 24, the binary octahedral group <2, 3, 4), oforder 48, the binary icosahedral group <2, 3, 5), of order 120 and finally thedegenerate case of the cyclic group. These groups are described in Coxeter andMoser [5, Preface and §6.5] and in Du Va] [7]. Each of these groups with the soleexception of the cyclic group of odd order contains a centre of order two,namely (ABC). The classification of finite subgroups of division rings is foundin Amitsur [1]. There is a natural embedding of the binary polyhedral groups inSL2(C), or its compact version SU2(C), a double cover of S03(R), important forthe sequel.

Generalized Coxeter graphs. A graph with vertex set V and a weight functionw: V - Rj° is a Ck-graph if

2 w(u) = k.w(v)u E S(v)

where S(v) is the multiset of successors of v E V. Because of their importance,we shall drop the suffix and use `C-graph' to mean C2-graph.

In passing it should be remarked that the defining property of a Ck-graphimplies that its adjacency matrix (the matrix whose (i,j) entry counts thenumber of edges from vertex i to adjacent vertex j) has maximum eigenvalue k,see Seneta [11].

PROPOSITION 3. rR(G) is a Ck-graph for k = dim(R), w: R; - dim R..

Lie algebras. The connection between C-graphs and Lie algebras is given by

PROPOSITION 4. The finite, undirected, connected C-graphs are precisely theCoxeter graphs (Dynkin graphs) for the affine Lie algebras of type A, (r > 0), D,(r > 4), E6, E and E8. We shall call these the standard types.

The affine graphs are described (as `graphes de Dynkin completes') inBourbaki [4]. They are constructed by adjoining to the usual graph an extra root,being the negative of the highest root.

PROPOSITION 5. All circuit-free C-graphs satisfying a `symmetrisability' condi-tion-if vertices are joined by a directed edge, then they are also joined by anundirected edge-(see Berman, Moody, and Wonenburger [3]) are obtained from theundirected C-graphs by `folding' them. Folding is a weight- and incidence-preserv-ing operation on graphs which maps r to the quotient graph F/H, by replacingv E V by its orbit (vH) under a subgroup H of the symmetry group of the graph.

The graphs of this proposition are found in [3], Dlab and Ringel [6], and Kac[8]. They include the affine graphs for the algebras of type B C G2, and F4.

Finite groups and spectral structure. For each finite group of quaternions, G,there is a faithful representation RQ such that ]F,, (G) is a graph of standardtype. This representation is the two-dimensional one mentioned above and is

GRAPHS, SINGULARITIES, AND FINITE GROUPS 185

irreducible except for the cyclic case where it is the direct sum of a faithfulirreducible representation and its dual.

PROPOSITION 6. The eigenvalues of the adjacency matrix of an undirectedC-graph are the values of the character afforded by RQ. The eigenvectors can betaken to be the columns of the character table of the appropriate finite group ofquaternions.

Eigenspaces are all one-dimensional only for A0, A,, and E8. Any linearoperator for which the columns of the character table are eigenvectors com-mutes with the regular representation of the representation algebra mentionedabove.

The Cartan matrix of standard type is symmetric and satisfies C = 21 - Awhere A is the adjacency matrix of the graph. It follows, since RQ is faithful, thatC is positive semidefinite with a one-dimensional kernel spanned by the eigen-vector whose components are the dimensions of the irreducible representationsof G.

A Cartan matrix of type A, D, or E is the presentation matrix (that is, theentry c. gives the exponent of generator xj in the ith relator) of the finitepolyhedral group whose character table is obtained from the eigenvectors of thecorresponding Cartan matrix of affine type.

The Fischer-Griess simple sporadic group M is generated by a conjugacy classof involutions such that the product of any pair lies in one of 9 conjugacy classeswhose periods are given by the weights of the E. graph. The group M contains asubgroup 2.B (a central extension of the Baby Monster) which centralizes aninvolution, and a subgroup 3.F24 which centralizes an element of period 3; eachof the groups 2.B and3.F74 contains elements bearing a similar relation asabove to the graphs E7 and E6 respectively provided the periods are readmodulo the centre.

The singularities. We have seen that each C-graph may be interpreted in twoways: firstly as a representation graph of a finite group, and secondly as aCoxeter graph in the classical sense (as a description of a Lie algebra). Aconnection between these two interpretations has been given by Steinberg in hisarticle in these PROCEEDINGS. This connection is described by Orlik [10] in hisrecent survey article and by Slodowy [12].

Very briefly, starting with the polynomial invariants of the finite subgroup ofSLA(G), a surface is defined from the single syzygy which relates the threepolynomials in two variables. This surface has a singularity (partial derivativesvanish) at the origin; the singularity can be resolved by constructing a smoothsurface which is isomorphic to the original one except for a set of componentcurves which form the pre-image of the origin. The components form a Dynkincurve and the matrix of their intersections (the matrix, indexed by the curves,with (0, 1)-entries indicating intersections of distinct curves and diagonal 'self-intersection numbers' of -2) is the negative of the Cartan matrix for theappropriate Lie algebra. The Dynkin curve is the dual of the Dynkin graph.

There are references to the affine curves in Tate [14] and several otherreferences to them in [2].

186 JOHN McKAY

The universal property. One C-graph we have excluded throughout by ourfiniteness condition is the important universal C-graph which is the representa-tion graph for SL2(C) with R = R2, the natural representation. This groupoccurs in both guises as

A0:

and as

Ate:2 3 4

The representation theory of SL2(C) is much studied (see, for example,Kirillov [9]) and all we need here is the tensor product formula

where R1 is the irreducible representation of dimension i. All the undirectedC-graphs can be embedded in A. by restricting R2 to the appropriate subgroup.

By restriction and folding we obtain the Dynkin graphs of all finite ranksimple Lie algebras.

The Cartan matrix is again positive semidefinite but now infinite with 2's onits diagonal and - l's on both adjacent diagonals, all other entries being zero.

Conclusion. A paper will appear amplifying this note and containing proofs. Ihope that I have been able to indicate that there is much more to be discoveredabout finite groups and their relation with other areas of mathematics. If thisapproach is to be successful, its merit will lie in its unifying power and itselegance. Would not the Greeks appreciate the result that the simple Liealgebras may be derived from the Platonic solids?

REFERENCES

1. I. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80(1955),361-386.2. W. L. Baily and T. Shioda, Complex analysis and algebraic geometry, Cambridge Univ. Press,

New York, 1977.3. S. Berman, R. V. Moody and M. Wonenburger, Cartan matrices with null roots and finite

Cartan matrices, Indiana Univ. Math. J. 21 (1972), 1091-1099.4. N. Bourbaki, Groupes et algebres de Lie, Hermann, Paris, 1968.5. H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 3rd ed.,

Springer-Verlag, Berlin and New York, 1972.6. V. Dlab and C. Ringel, Indecomposable representations of graphs, Mem. Amer. Math. Soc., No.

173, 1976.7. P. Du Val, Homographies, quaternions and rotations, Clarendon Press, Oxford, 1964.8. V. G. Kac, Infinite-dimensional Lie algebras, Dedekind's s1 function, classical Mobius function,

and the very strange formula, Advances in Math. 30 (1978), 85-136.9. A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, Berlin and New York,

1975.10. P. Orlik, Singularities and group actions. Bull. Amer. Math. Soc. (N.S.) 1 (1979),703-720.11. E. Seneta, Non-negative matrices: an introduction to theory and applications, Allen and Unwin,

London, 1973.12. P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Math., Springer-

Verlag, Berlin and New York (to appear).13. R. Steinberg, Kleinian singularities and unipotent elements, these PROCEEDINGS, pp. 265-270.14. J. Tate, An algorithm for determining the type of a singular fibre in an elliptic pencil, Functions

of One Variable. IV, Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975,pp. 33-52.

CONCORDIA UNIVERSITY, MONTREAL, CANADA, H3G I M8


WORKS ON FINITE GROUP THEORY BYSOME CHINESE MATHEMATICIANS

HSIO-FU TUAN

This paper is a modified version of my talk. I shall confine myself to speakabout only two such works.

§1. On the commutator and power structures of p-groups. The study of (finite)p-groups plays an important role in the classification of finite simple groups andis conversely promoted by the development of the latter. The study of p-groupsinvolves at best three different aspects: (1) the arithmetical structure, e.g. thedetermination of numbers of subgroups with certain prescribed properties andvarious kinds of arithmetical invariants; (2) the commutator structure, or prob-lems related to commutativity, mainly about the derived series, the upper andlower central series and their interrelations; (3) the power structure, or problemsrelated to the p-powers of elements, mainly about the upper and lower powerseries and their interrelations. Besides, one can study p-groups by means ofrepresentation theory, variety theory, lattice theory, etc.

The power structure is not only important but also specific for p-groups andfinds increasing interest. However, to study the power structure by itself isdifficult and few significant results have been obtained so far, with the notableexception of the work of A. Mann [8]. Most authors investigate the powerstructure in connection with the commutator structure. The most importantcontribution in this respect is the theory of regular p-groups as introduced anddeveloped by P. Hall [5], [6]. Later, N. Blackburn, J. L. Alperin, A. Mann andothers have obtained many new results [3], [4], [1], [2], [8], cf. also [7], and havemade its study an important branch of the theory of p-groups.

Now I shall give an exposition of Ming-Yao Xu's work on semi-p-abelianp-groups [11], [12], [13] in this connection.

1. Concept of semi p-abelian p-groups. All groups considered are finite p-groups.

DEFINITION 1. We call a finite p-group G semi-p-abelian, whenever (ab) = 1iff apbp = 1 for any a, b E G, or equivalently, ap = by iff (a-lb) = 1 for any a,b E G.

1980 Mathematics Subject Classification. Primary 20D15, 20D05, 20C20.© American Mathematical Society 1980

187

188 HSIO-FU TUAN

Similarly, we haveDEFINITION 2. We call a finite p-group G strongly semi-p-abelian, whenever

for any positive integer s and for any a, b E G, (ab)' = I iff a" " = 1, orequivalently, ap' = b"' iff (a-`b)°' = 1.

A p-group G having the above property for a fixed s is called semi-p `-abelian.

PROPOSITION 1. Regular p-groups are strongly semi-p-abelian, and hence theyare also semi-p-abelian.

For the proof one can refer to textbooks on group theory. (Besides, p-abeliangroups are obviously strongly semi-p-abelian.) Hence we have for the concepts(1)p-abelian, (2) regular, (3) strongly semi-p-abelian, (4) semi-p-abelian that

(1) (2) (3) (4).

For the relation between regular and semi-p-abelian, we prove:

THEOREM 1. If all sections of G are semi-p-abelian, then G is regular.

Hence we get another necessary and sufficient condition for regular p-groups.On the other hand, to be semi-p-abelian is indeed weaker than to be regular:

the former is inherited by direct products whereas the latter is not. Hence everynonregular p-group which is a direct product of regular groups can be taken asan example to show that to be semi-p-abelian is weaker than to be regular.Therefore we can find many such examples for p > 2. But for p = 2, regular2-groups are abelian, and we also have examples of nonabelian and semi-2-abelian 2-groups.

EXAMPLE. The metacyclic 2-groups determined by the following definingrelations are semi-2-abelian:

G = <a, b>, m, n, c, s positive integers.(1) a2' = b2' = 1 , a' = a1 +2'" m, n )' 2, 1 < c < min(n, m - 1);(2)az" =1,b2-=a2mab=a1+2'" m n>2,

1 <c<min{n,m-1},max{l, m - n + 1) <s <min{c,m-c+ 1).

In fact, these give all nonabelian metacyclic 2-groups which are semi-2-abelian.

The concept of semi-p-abelian p-groups arises from the study of the powerstructure of p-groups. And this concept is closely connected with the powermaps of p-groups. The power maps are the maps 1r,: G -p G, where s is a positiveinteger,

ir,: a H ap', Va E G.

For ir,, we have

ker ir, = (x e Gix°' = 1) = A,(G),

Im ir, = ( x°'j x E G) = V,(G).

In general, A,(G) and V,(G) are not subgroups of G. If A,(G) is a subgroup, i.e.,A,(G) = E2,(G), T, still may not induce a map of G/2,(G) onto V,(G). In fact,G is semi-ps-abelian if and only if 7r, induces a bijection a,: G/E2,(G) -, V,(G),Vs, that is, we have

WORKS ON FINITE GROUP THEORY 189

LEMMA 2. A finite p-group G is semi-p-abelian if and only if

(1) Q1(G) = Al(G)(2) The map 71: a l1(G) H aP, a E G, is a bijection of G/SZ1(G) onto V,(G).

LEMMA 2'. A finite p-group G is semi-ps-abelian if and only if

(1) Qs(G) = AS(G).(2) The map 7rs: a l (G) H aP', a E G, is a bUection of G/SZS(G) onto VV(G).

Hence G is strongly semi-p-abelian if and only if for all s, the induced maps iisof 7Ts have the above properties.

By the way, the power map 77s is an endomorphism if and only if G isps-abelian, i.e., (a b)P' = aPV, Va, b E G. Hence G isp-abelian if and only if7r,: a H aP is an endomorphism of G. And we have: if 7T1 is an endomorphism,so is 77s for any s. We can even infer: if 77s is an endomorphism, so is 7T, for anyt > s.

II. A class of semi-p-abelian p-groups. Because semi-p-abelian p-groups have"better" power structure and they are less restrictive than regular p-groups, wecan make use of them in wider circumstances than regular p-groups.

We state a sufficient condition for semi-p-abelian p-groups and give someapplications.

THEOREM 2. Let G be a finite p-group, p > 2. If Z (G), n < p, then Gis strongly semi-p-abelian, that is, if S2(Gp_1) < Z(G), then G is strongly semi-p-abelian.

This theorem gives a large class of examples of semi-p-abelian p-groups; ingeneral, they are not regular.

Making use of this theorem, we can generalize some results of T. J. Laffey andsome others, [9], [10].

COROLLARY 1. Let G be a finite p-group, p > 2, Z(G), n < p. SupposeL < G, and L < Z(G), then SZ((G/L) _,) < Z(G/ L).

For n - 2, this is Laffey's lemma.

COROLLARY 2. Let G be as in Theorem 2. Then d(G) < logpjSZ(G)j.

This is also a generalization of a result of Laffey.

COROLLARY 3 (THOMPSON). Let G be a finite p-group, p > 2, l(G) < Z(G).Then d(G) < d(Z(G)).

COROLLARY 4. Let G be as in Theorem 2, a, b E G. Then the order of [a, b] isequal to the order of a modulo Z ((a, b>). Hence exp G' = exp(G/Z(G)).

This also generalizes a result of Laffey. The condition with which Laffey getsthe result is l(G) < Z(G), while ours is SZ(Gp_,) < Z(G).

COROLLARY 5. Let G be a finite p-group, p > 2, s a positive integer. Suppose Mis a maximal element of the set

d = (AAA < G, A' = 1, exp A < ps).

If x E CA(M) and xP' = 1, then x E M.

190 HSIO-FU TUAN

This is due to Feit and Thompson as generalized by Alperin [2].III. The case p = 2. For the case p = 2, we prove:

THEOREM 3. A finite 2-group is semi-2-abelian if and only if l(G) < Z(G), andG has no subgroup of the following type:

(1) Quaternion group Q;(2) a4 = 1, b2' = 1, b-gab = a-1, n > 2.

COROLLARY. Let G be a finite 2-group, S22(G) < Z(G). Then G is semi-2-abelian.

THEOREM 4. Any semi-2-abelian 2-group is strongly semi-2-abelian.

COROLLARY 1. If S22(G) < Z(G), then d(G) < log21S2(G)I.

But from Q(G) < Z(G), it does not follow that d(G) < 10921S2(G)I. Thequaternion group is a counterexample.

COROLLARY 2. If SZ2(G) < Z(G), then l[a, b]I = IaZ(<a, b>)I, hence exp G' _exp(G/ Z(G)).

These two results are due to Laffey.

2. On finite simple groups with the normalizer of a Sylow p-subgroup (p odd)nonabelian of order 2p. As Richard Brauer wrote in [10] (1976), "to find therelations between the properties of the p-blocks of characters of a finite group Gand structural properties of G" had been a problem of interest to him for a verylong time.

Indeed as early as in [5] (1942), he observed that Dickson's list of 78 simplegroups of an order less than 109 contains only one group for which there is noprime dividing the order of the group to the first power only. With this in mind,in [51], [511], he made the study of finite groups having a Sylow p-subgroup oforder p for some prime p. With the block theory of characters as initiated anddeveloped by him at that time, general results were obtained and were appliedto advantage in [6], [7], [18] by him and myself to yield results on finite simplegroups, including the characterization of simple groups G of order IGI = pgbgo(p, q distinct primes, b, go positive integers and go <p - 1); and in particular,I G I = prgb (p, q, r distinct primes, b a positive integer). Also, the determinationof all simple groups of orders up to 10,000; in particular, the uniqueness up toisomorphism of simple groups of orders 5616, 6048, 7800, 7920, and 9828respectively.

Again in [10], he pointed out the remarkable fact that the order of each of theknown sporadic groups G contains prime factors with the exact exponent 1.Noticing that the same is true for all alternating groups, he remarked that itseems that all simple groups have cyclic Sylow p-subgroups for some prime pdividing I G 1. It was natural for him to study such groups, making use of Dade'swork on p-blocks with a cyclic defect group [11], and he subsequently publisheda paper on this topic [9] (1976).

Since in [10, §§III, VI] there is a concise survey about groups with a cyclicSylow p-subgroup, I shall here restrict myself to the special case of groups withthe normalizer of a Sylow p-subgroup P nonabelian of order 2p.

Following the notations in [10, §III], let G be a group which has a cyclicSylow p-subgroup P of prime order p, let C = CG(P) be the centralizer of P in Gand N = NG(P) the normalizer of P in G. Then the order g = G I of G has thefollowing form

g = pmr(l + n*p),where ICI = pm, r= IN: CI divides p - 1 (hence r = (p - 1)1t), and m > 1and n * > 0 are both integers.

We shall now consider the case m = 1, t = (p - 1)/2.Then C = P, i.e. P is self-centralizing. In fact, P is even strongly self-centraliz-

ing, as CG(x) = CG(P) for all nonidentity elements x of P. This is certainly thecase if G coincides with its commutator subgroup G' and the principal p-blockBo(p) contains a nonprincipal irreducible character of degree z < 2p [7, Lemma1]; also if G is simple of order g = p °q br ` (p, q, r being distinct primes), and P iscyclic [8, Proposition 1].

Furthermore, INI = 2p, and N is nonabelian of order 2p. For G simple, Alex[3] calls such a group an index two simple group, which is shown to have justone conjugacy class of involutions.

It follows from [51] that in this case G has just one p-block of full defect, viz.,the principal p-block Bo(p), which then consists of the 1-character o, anotherself -p-conjugate character , of degree z1, and an exceptional family of(p - 1)/2 = t p-conjugate characters 2' ... , ,+ all of the same degree z2.The degree equation for the (irreducible) characters of Bo(p) is simply

1 +61zI+62z2=0,where 6, 1, 82 = ± 1,8 = -82, zI - 6, (mod p), z2 = 82/t = -282 (mod p).

The simple form of the degree equation in this case together with otherinformation about z, and z2 would in certain cases put enough conditions on thepossible values of z, and z2 to determine them and then to yield informationabout the structure of G.

In an unpublished manuscript, on the basis of the prgb theorem mentionedabove, Brauer and I proved the following:

THEOREM. A noncyclic simple group G of order g = G = pgbr` (p, q, r beingdistinct primes) with t = (p - 1)/2 must be one of the following types: LF(2, 5);LF(2, 7); LF(2, 8); LF(2, 9).

COROLLARY. In particular, a noncyclic simple group G of order g = G I = 3p q b(3, p, q being distinct primes) is either LF(2, 5) or LF(2, 7).

In [1]-[3], utilizing a case-by-case study of the degree equation together withother character-theoretic techniques, Alex characterizes simple groups of orders2a 3b5`7dp, and index two simple groups with x < 25, x being the minimal degreeof a nonlinear character in Bo(p). More recently, Alex and Morrow [4] studiedindex four simple groups and characterized such groups having a nonlinearcharacter of degree x < 15 in Bo(p), whose Brauer trees are of three distincttypes. Notice that index four simple groups arose in Brauer's study of simplegroups of order 5 3° 2b [8] (cf. also Lie [15]).

Chia-Wei Hung in [13] (1965) studied groups G with the normalizer of aSylow p-subgroup nonabelian of order 2p, and proved the following results:

192 HSIO-FU TUAN

(1) All the nonlinear characters of Bo(p) have the same kernel H of orderprime top such that the factor group G* = G/H is either the dihedral group oforder 2p or a nonabelian simple group.

(2) Let IJ1t be a finitely generated multiplicative semigroup in the ring ofordinary integers. Then there exist only finitely many simple groups G with thenormalizer of a Sylow p-subgroup nonabelian of order 2p and with G E01, p}.

Hung then established Theorems 1 and 2 (below), as announced much later in[14] (1973).

In the meantime, Harada in [12] (1967) introduced the more general conceptof a special subgroup and a # group as follows: By a special subgroup A of afinite group H is meant a subgroup satisfying the following condition (#):

(1) A is strongly self-centralizing in H.(2) The order of the normalizer of A in H is 21A 1.

A finite group H with a special subgroup A is called a # group. It is known thatLF(2, q) and Sz(q) have special subgroups. Harada proved the following theo-rems:

THEOREM A. If G is a solvable group with a special subgroup, then there exists anilpotent normal subgroup N of G such that G/N is isomorphic with a generalizeddihedral group.

THEOREM A'. If G is a nonsolvable group with a special subgroup, then thereexists a nilpotent normal subgroup N of G such that G/N is simple.

With character-theoretic methods and other known results Harada calculatedthe order of a # group and then characterized simple # groups as certainLF(2, q) under suitable conditions.

We shall now come to Hung's two theorems [14].

THEOREM 2. For any integer n, there exists an integer m such that for any integerk < n and any prime p > m, the simple groups of order p(kp + 1)(kp + 2) must beisomorphic to LF(2, p + 1) or LF(2, 2p + 1).

It was first shown that for sufficiently large p, groups of order p(kp + 1)(kp + 2) are # groups with the Sylow p-subgroups P as special subgroups andthen Harada's results on characters of the principal p-block Bo(p) and subdivi-sion of elements for # groups can be used. For the rest of the proof, we refer tothe paper itself.

In the course of the proof of this theorem, it is seen that the question ofdetermining all simple groups of order p(kp + 1)(kp + 2), k < n, can be settledin a finite number of steps. Generalizations might be considered.

THEOREM 1. The simple groups of order p(kp + S)(kp + 26) (6 = ± 1, k < 5)must be isomorphic to

(1) LF(2, p + 1) when k = 1, 6 = 1, p = 2e - 1;(2) LF(2, p - 1) when k = 1, 6 = -1, p = 2e + 1;(3) LF(2, 2p + 1) when k = 2, 6 = 1, p = (qe - 1)/2;(4) LF(2, 2p - 1) when k = 2, 6 = -1, p = (qe + 1)/2; or(5) LF(2, 7).

It should be remarked that case (1) and case (3) under the assumption that theSylow p-subgroups of G are self-centralizing are already known in previousworks of R. Brauer [6] and 0. Nagai [16].

As no proof of this theorem has been given, an outline of Hung's proof will besketched below. A detailed proof will be published elsewhere.

Firstly, several preparatory lemmas mainly on permutation representationsare proved. Then the group G under consideration is shown to be a # group asfollows:

For k < 5, we can assume p > 3. Let the number of Sylow p-subgroups P ofG be up + I and let the normalizer N of P in G have order INI = p(sp + t)(0 < t < p). Then t - 2 (mod p) and indeed t = 2, whence we have the inde-terminate equation

p(kp + 6)(kp + 26) = p(sp + t)(up + 1)

or

k2p+3k6=usp+2u+s.Then it can be proved in succession

(1) us <k2.Notice that the above indeterminate equation has for each given k only

finitely many systems of solutions (u, s, p) such that 0 < u < k2. Fork < 5, allthese systems of solutions can be discarded, for no simple group will arise.

(2) us=0orus=k2.(3) us k2, hence us 0.For a simple group G, u 0, hence I = 0 and IN I = 2p as asserted.Finally, the isomorphism of our simple group G with certain LF(2, q) is

achieved by making use of Harada's results on characters of the principalp-block Bo(p) together with known classification theorems of Brauer, Nagai,Reynolds and Suzuki.

REFERENCES

§1

1. J. L. Alperin, On a special class of regular p-groups, Trans. Amer. Math. Soc. 106 (1963),77-99.

2. , Centralizers of abeltan normal subgroups of p-groups, J. Algebra 1 (1964), 110-113.3. N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45-92.4. , On prime power groups with two generators, Proc. Cambridge Philos. Soc. 54 (1958),

327-337.5. P. Hall, A contribution to the theory of groups of prime-power order, Proc. London Math. Soc. 36

(1933), 29-95.6. , On a theorem of Frobenius, Proc. London Math. Soc. 40 (1936), 468-507.7. B. Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin and New York, 1967.8. A. Mann, The power structure of p-groups. I, J. Algebra 42 (1976), 121-135.9. T. J. Laffey, A lemma on finite p-groups and some consequences, Proc. Cambridge Philos. Soc.

76(1974), 133-137.10. , Centralizer of elementary abelian subgroups in finite p-groups, J. Algebra 51 (1978),

88-96.11. Ming-Yao Xu and Yen-Chang Yang, Semi-p-commutativity and regularity of p-groups, Acta

Math. Sinica 19 (1976), 281-285.

194 HSIO-FU TUAN

12. Semi-abelian p-groups and their power structure, Acta Math. Sinica 23 (1980), 78-87.13. , A class of semi-p-abelian p-groups, Bull. Sci. (Kexue Tongbuo), Academia Sinica (to

appear).

§2

1. L. J. Alex, Simple groups of order 2°3b5`7°p, Trans. Amer. Math. Soc. 173 (1972), 389-399.2. , Simple groups of order 2°367 p, J. Algebra 25 (1973), 113-124.3. Index two simple groups, J. Algebra 31 (1974), 262-275.4. L. J. Alex and D. C. Morrow, Index four simple groups, Canad. J. Math. 30 (1978), 1-21.5. R. Brauer, On groups whose orders contain a prime number to the first power. I, II, Amer. J.

Math. 64 (1942), 401-420; 421-440.6. , On permutation groups of prime degree and related classes of groups, Ann. of Math. (2)

44 (1943), 57-79.7. R. Brauer and Hsio-Fu Tuan, On simple groups of finite order. I, Bull. Amer. Math. Soc. 51

(1945), 756-766.8. , On simple groups of order 5 3° 2b, Bull. Amer. Math. Soc. 74 (1968), 900-903.9. , On finite groups with a cyclic Sylow subgroup. I, J. Algebra 40 (1976), 556-584.

10. , Blocks of characters and structure of finite groups, Bull. Amer. Math. Soc. (N.S.) 1(1979), 21-38.

11. E. C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966), 20-48.12. K. Harada, A characterization of the groups LF(2, q), Illinois J. Math. 11 (1967), 647-659.13. Chia-Wei Hung, On a class of finite groups with the normalizer of a Sylow p-subgroup nonabelian

of order 2p, Graduate Student Thesis Part I, Peking University, 1965.14. , On simple groups of order p(kp + lXkp + 2), Scientia Sinica 16 (1973), 177-188.15. Hwei-Lin Lie, On simple groups of order 5 31. 2b with selfcentralizing Sylow 5-subgroups,

Graduate Student Thesis Part II, Peking University, 1965.16. O. Nagai, On simple groups related to permutation groups of prime degree, Osaka Math. J. 8

(1956), 107-117.17. B. M. Puttaswamaiah and J. D. Dixon, Modular representations of finite groups, Academic

Press, New York, 1977.18. Hsio-Fu Tuan, On simple groups of order less than 10,000 (unpublished).

PEKING UNIVERSITY


THE PRIME GRAPH COMPONENTSOF FINITE GROUPS

J. S. WILLIAMS

Let G be a finite group and construct its prime graph as follows: the verticesare the primes dividing the order of G, and two vertices ( p), ( q) are joined byan edge if and only if G contains an element of order pq. The purpose of thispaper is to announce the prime graph components of the simple groups of Lietype in odd characteristic, of the alternating groups and the sporadic groups.The proofs of these results will appear later [1]. A result of Gruenberg and Kegel(cf. [1]) shows that the structure of a finite group with more than one primegraph component is restricted. In particular G has one of the following struc-tures: Frobenius, 2-Frobenius, simple, an extension of a nilpotent 7T1-group by a

simple group, an extension of a simple group by a 7T1-solvable group, or anextension of a 7T1-group by a simple group by a 7T1-group, where i1 denotes theprime graph component containing the prime 2. A corollary to this structuretheorem is that any solvable group has at most two prime graph components,while for nonsolvable groups the number of prime graph components isbounded by the maximum number for the simple groups.

The following theorem gives the number of prime graph components of thesimple groups previously described. More explicit results concerning the primesin each component will appear in a forthcoming paper [1].

THEOREM 1. A simple group of Lie type in odd characteristic, an alternatinggroup Alt(n), n > 5, or a sporadic group has one component except for thefollowing:

(a) two components: Alt(p) (p > 7), Alt(p + 1) (p > 7), Alt(p + 2) (p + 2not prime), Aa !(g), z'4a-1(g), Bz (q), C2 (q), 2D2 (q),

3D4(q3), G2(q) (q 0

(mod 3)), F4(q), E6(q), 2E6(g2), Ap(q) (q - II p + 1), AA(q) (q + l I p + 1), Bp(3),Cp(3),Dp(3)(p>5),Dp+,(3),2Dp(3)(p>5,p 2"+1),2D,(3)(1=2"+1,1not prime), DP(5) (p > 5), M12, J2, Ru, He, McL, Co,, Co3, Fizz, F51

(b) three components: Alt(5), Alt(6), Alt(p) (p - 2 is prime), A,(q), G2(3"),2G2(32n+1), 2DP(3) (p = 2" + 1, p > 5), E7(3), M11, M23, M24, J3, HS, Sz, Co21

Fi23, F2, F3,

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195

196 J. S. WILLIAMS

(c) four components: E8(q) (q 2, 3 (mod 5)), M22, J1, On, Ly, Fi24, F?,(d) five components: E8(q) (q 0, 1, 4 (mod 5)),(e) six components: J4.

The above result is strong evidence for the following conjecture.CONJECTURE 1. There is a bound on the number of prime graph components

of any finite group.For the presently known groups this bound is six.The prime graph components not containing the prime two appear in essen-

tially the same way. A proper subgroup H of a group G is called isolated (CCT)if H n H9 = <1> or H for all gin G and if for all h in H#, CG(h) is containedin H. If a group G contains an isolated subgroup H, then the primes dividing theorder of H form a complete proper prime graph component. The secondtheorem shows that the existence of an isolated subgroup is precisely how theodd prime graph components arise.

THEOREM 2. Let G be a finite nonsolvable group whose composition factors areK-groups and let 7ri be a prime graph component of G not containing the prime 2,then G contains a nilpotent Hall v.-subgroup which is isolated in G.

The above result includes the groups of Lie type in even characteristic, and isthe basis of the second conjecture.

CONJECTURE 2. If G is any nonsolvable group with 7ri a component notcontaining the prime 2, then G contains an isolated Hall Irt-subgroup.

The more detailed calculations and tables of the prime graph components canbe used to identify certain types of groups. In particular, in view of past interestin COO groups the following propositions may be of interest. (C7 M means thatthe centralizers of Ir-elements are IT-groups, and COO is C33.)

PROPOSITION 1. If G is a simple K-group of odd type, then G is a C55 group ifand only if G is one of the following groups: Alt(7), M11, M12, PSL2(q) (q = 5m, 25' - 1,or2.5'+ 1) or type C2(q)(g2=2.5"'- 1).PROPOSITION 2. If G is a simple K-group of odd type and contains an element of

order six, then G is a CTr17T1 group with 7T1 = (2, 3) if and only if G is one of thefollowing groups: Alt(7), C2(3), G2(3), M11, M22, U3(3), PSL3(3), or PSL2(q)

where m > 2 and n > l).

The theorems may also be applied to yield information about certain integralrepresentations of finite groups. For example the following is a combination ofresults of Gruenberg, Roggenkamp and Theorem 2.

THEOREM 3. If G is a finite nonsolvable group of K-type, then the following areequivalent :

(a) the augmentation ideal of G decomposes as a right module,(b) G contains an isolated subgroup,(c) the prime graph of G has more than one component.

REFERENCE

1. J. S. Williams, Prime graph components of finite groups, unpublished manuscript.

ADELPHI UNIVERSITY


,7-SOLVABILITY AND NILPOTENT HALLSUBGROUPS

ZVI ARAD AND DAVID CHILLAG

Sufficient conditions on a factorizable finite group G = A B (where A and Bare subgroups of G) to be solvable or IT-solvable were given by several authors.We mention the following results in which A is assumed to contain a nilpotentHall subgroup of G of even order.

THEOREM (BERKOVIC [3]). Let G = AB. Assume that A = 02(A) X O(A), B isnilpotent of odd order and (JA 1, 1B1) = 1. Then G is solvable.

THEOREM (P. ROWLEY [8]). Let G = AB. Assume that A = 02(A) X O(A), Bis metanilpotent of odd order and (JAI, JBI) = 1. Then G is 7r-solvable where

= 7r(O(A)).

Assuming the Unbalanced Group Conjecture we are able to generalize theabove results.

THEOREM A. Let G = AB. Assume that A = 02(A) X O(A), B of odd orderand (JAI, CBI) = 1. Then G is 7r-solvable where 77 = ?r(O(A)).

To prove Theorem A we use Arad and Glauberman's result [2] stating that ifX is a group of odd order, then ZJ(X) 1 and X = On (X)NG(ZJ(X )) for anyset of primes IT. HereAX) is the subgroup of X generated by all subgroups of Xwhich are abelian of maximal order. Using this result we are able to show thatO(CG(t)) 1 for some central involution, t, of G. Then using the UnbalancedGroup Conjecture (the proof of which is almost completed) we finish the proof.

The above three results raise the question whether the factorization is essentialor that the Ir-solvability follows from the existence of a nilpotent Hall subgroupof even order in G.

We suggestConjecture. If a finite group G contains a nilpotent Hall IT-subgroup of even

order, then G is 77*-solvable, where 77' = IT - (2). In particular, a finite simplegroup does not contain a non-Sylow nilpotent Hall subgroup of even order.

1980 Mathematics Subject Classification. Primary 20D10; Secondary 20D25, 20D15.m American Mathematical Society 1980

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198 ZVI ARAD AND DAVID CHILLAG

The following results give evidence to the conjecture.

THEOREM B. The conjecture is true for K-groups (K-groups are finite groupswhose composition factors are known simple groups).

Following P. Hall we say that a group G is an E,"-group if G has a nilpotentHall 77-subgroup. If G is an E,"-group with 2 E 7r and X is either a normalsubgroup of G or a homomorphic image of G then, either X is of odd order (andtherefore solvable) or X is an E,"-group with 2 E IT. Therefore, in order to proveTheorem B we only need to verify that no known simple group G is an E,"-groupwith 2 E 77 and 177 n IT(G)I > 1, since the IT-solvability property is inherited bysubgroups and homomorphic images.

EXAMPLE. Let G be a group of characteristic 2-type. We will show that Gsatisfies the conjecture. If not, let A be a nilpotent, non-Sylow, Hall subgroup ofG with JA I even. Let t be a central involution of G such that t E A. ThenC = CG(t) A. Since G is of characteristic 2-type, P(C) = 02(C). Hence,Cc(02(C)) C 02(C), contradicting the fact that 02(C) 02(A) and so[02(C), O(A)] = 1.

We note that in an E,"-group with 2 E 7r and 17r n IT(G)l > 1, every 2-sub-group centralizes some 2'-subgroup. This fact and Theorem A.4 of [5] are usedto verify the conjecture for the alternating groups and for the sporadic groups ofcomponent type, except for J1, Ly and Ru. The verification of the conjecture forthese three groups as well as for Chevalley groups of odd characteristic is doneusing the structure of centralizers of involutions in these groups.

Using Glauberman's "S°-free" theorem we can show that the conjectureholds if (2, 3) C 77.

THEOREM C. If G is an E,"-group with (2, 3) C 7r then G is 7r - (2)-solvable.

More evidence is given in the following result which is a consequence of theUnbalanced Group Conjecture.

PROPOSITION D (ASSUMING THE UNBALANCED GROUP CONJECTURE). Let G bean E,"-group with 17T n IT(G)l > 1 and 2 E IT. If G contains an involution t withCG(t) solvable, then G is not simple.

Next we consider the question whether the E,"-property is inherited bysubgroups or not. In general the answer is no. For example, let G = PSL(2, 31).Then G has a cyclic Hall subgroup of order 15 and a subgroup B ^' A5 (see[6, p. 213]) which has no Hall 77-subgroup for 77 = (3, 5). We note that theE,"-property is inherited by IT-local subgroup (see [5, Lemma 3]). The questionwhether the E,"-property is inherited by subgroups if 2 E 77 is connected to theconjecture.

PROPOSITION E. Assume that the conjecture holds. Then the E,"-property isinherited by subgroups if 2 E IT.

PROOF. Let G be an E,"-group with 2 E 77 and let H be a subgroup of G. Then,by the conjecture, G and therefore H are IT - (2)-solvable. Now, Theorem 3.5,p. 229, of [4] implies that H has a IT = (77 - (2)) U (2)-Hall subgroup, say A.Now, Wielandt's theorem [6, p. 285] implies that A is nilpotent.

7T-SOLVABILITY AND NILPOTENT HALL SUBGROUPS 199

As a corollary to Theorem C we get

COROLLARY F. The E,"-property is inherited by subgroups if (2, 3) C 7r.

As a converse to Corollary F we get the following generalization of Schmidt-Iwasawa theorem on minimal nonnilpotent groups G (see [6, p. 281]) if ( 2, 3) C7r(G). We show, in fact, that a minimal non-E,"-group with (2, 3) C 77 is aminimal nonnilpotent group.

THEOREM G. Let 7r be a set of primes with (2, 3) C IT. Let G be a finite group inwhich every proper subgroup is an E,"-group. Then, either G is an E,"-group orG I = p aq$, p, q are primes, G is not nilpotent but every proper subgroup of G is

nilpotent (see properties of such a group in [6, p. 281]). In particular, G is7T - {2)-solvable.

Finally, we note that the conjecture throws a new light on the followingtheorems of Suzuki.

THEOREMS (SUZUKI [9], [10]). Let G be a finite group. Assume that either(a) G is simple of even order and the centralizer of every nonidentity element of

G is nilpotent (a CN-group), or(b) G has a proper subgroup M of even order with CG(x) C M for all x E M -

{ 1 ) and G is not a Frobenius group.Then the centralizer of every 2-subgroup of G is a 2-group (a CIT-group).

PROOF USING THE CONJECTURE. If (a) holds, the result follows from theconjecture and Theorem 1.7, p. 403, of [4]. Suppose that (b) holds. Then M is aHall subgroup of G. If NG(M) = M the result follows from the conjecture andProposition (1.3) of [1]. If NG(M) M, M is nilpotent T.I. subgroup of G. If Gis simple the result follows from the conjecture. If not, let N be a minimalnormal subgroup of G. Since I M I is even, N n M 1 (see [7, Theorem 2.3(h)]).In fact M n A = 1 for every 1 A < G. Thus, Theorem 2.3(e) of [7] impliesthat M C N and by the same argument we get that N is simple. Now the resultfollows from the conjecture.

REFERENCES

1. Z. Arad and D. Chillag, Finite groups with conditions on the centralizers of +r-elements, Comm.Algebra 7 (14) (1979), 1447-1468.

2. Z. Arad and G. Glauberman, A characteristic subgroup of a group of odd order, Pacific J. Math.56 (1975), 305-319.

3. Ja. G. Berkovic, Generalization of the theorems of Carter and Wielandt, Soviet Math. Dokl. 7(1%6), 1525-1529.

4. D. Gorenstein, Finite groups, Harper and Row, New York, 1%8.5. P. Hall, Theorem like Sylow's, Proc. London Math. Soc. 6 (1956), 286-304.6. B. Huppert, Endliche Gruppen I, Springer-Verlag, New York, 1%8.7. M. Herzog, On finite groups which contain a Frobenius subgroup, J. Algebra 6 (1%7), 192-221.8. P. J. Rowley, The +r-separability of certain factorizable groups, Math. Z. 153 (1977), 219-228.9. M. Suzuki, Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99 (1%1),

425-470.10. , Two characteristic properties of (ZT)-groups, Osaka Math. J. 15 (1%3), 143-150.

BAR-II.AN UNIVERSITY, ISRAEL

TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY

ON MAXIMAL SUBGROUPS WITH ANILPOTENT SUBGROUP OF INDEX 2

ZVI ARAD, MARCEL HERZOG AND AHIEZER SHAKI

The main aim of our study was to investigate nonsolvable finite groups G 1

with a maximal subgroup A, containing a nilpotent subgroup H of index[A:H]<2.

Thompson's celebrated result [6] asserts that G is solvable if A is nilpotent ofodd order.

Baumann [1] and Rose [5] determined the structure of a nonsolvable G,containing a maximal nilpotent subgroup A of even order. In particular, Roseshowed that if G is nonsolvable, Z(G) = 1 and A is a nilpotent maximalsubgroup of G, then A is an S2-subgroup of G.

If [A : H] < 2, we proved the following theorem (assuming the UnbalancedGroup Conjecture).

THEOREM 1. Let G 1 be a finite group with a maximal subgroup A. Supposethat F(G) = 1 and A contains a nilpotent subgroup H of index [A : H] < 2. Thenone of the following statements holds.

(1) A is nilpotent, A E Sy12(G);(2) A is nonnilpotent, and one of the following holds:(a) A is Hall and either

N - PSL(2, q) 4 G a PSL*(2, q),where q is odd, q 2" ± 1, q 81, PSL*(2, q) is a maximal subgroup ofPI'L(2, q) such that PSL*(2, q)/PSL(2, q) is an elementary abelian 2-group (oforder 2 or 4), or

N - PSL(2, 81) < G < PI'L(2, 81).Moreover, G = AN and A n N = C,(t), a nonnilpotent centralizer of an involu-tion t E N.

(b) A is non-Hall, H is of even order and G = PGL(2, q)with q ) 5 odd.Moreover A = Ca(t) for an involution T E G and 8 { A

(c) A is non-Hall and H is of odd order.

1980 Mathematics Subject Classification. Primary 20D05.© American Mathematical Society 1980

201

202 ZVI ARAD, MARCEL HERZOG AND AHIEZER SHAKI

REMARKS. (1) If F(G) ' 1 and G is a nonsolvable group satisfying the otherassumptions of Theorem 1, then it is easy to check that F3(G) < A andG = G/F3(G) satisfies all the assumptions of Theorem 1, including F(G) = 1.

(2) If A is nilpotent, then A E Sy12(G) by Theorem 1 in [5]. The structure of Gwas determined by Baumann [1] and Rose [5].

(3) If 2(a) or 2(b) holds, then I G: PSL(2, q)l is a divisor of 4, unless q = 81, inwhich case it divides 8.

(4) The open case 2(c) of Theorem I is difficult. In [4] G. Higman posed theproblem of classifying simple groups with a maximal dihedral subgroup A oforder 2p, p a prime, p > 5. We are not aware of any recent developments in thatdirection. The cases JAI = 6 and 10 were solved by Feit and Thompson [2] andby Higman [4], respectively.

(5) The proof of Theorem 1 relies heavily on the Unbalanced Group Theorem,the proof of which is essentially complete.

Theorem 1 is applied for the investigation of factorizable groups satisfying thefollowing hypotheses.

Hl. There exist subgroups A and B of G such that G = A B, JA n BI is oddand A contains a normal subgroup H of order r and index [A : H] = 2.

H2. There exist subgroups A, H, B and K of G such that G > A > H,G > B > K, G = AB, H and K are nilpotent and [A : H] < 2, [B : K] < 2.

Let G satisfy H1 and suppose that A max G and H is nilpotent. We provedthat if IHI is odd then G is solvable. On the other hand, if IHI is even andF(G) = 1, then G is known by Theorem 1.

W. R. Scott has conjectured that under Hypothesis 2 G is solvable. Scott'sconjecture generalizes the well-known theorem of Kegel and Wielandt, statingthat the conjecture holds if A = H and B = K.

There is a long list of partial results in the direction of Scott's conjecture.However, even the case when one of the factors is nilpotent is still open.

The most general result, obtained by Finkel and Lundgren [3], asserts that ifH is of odd order, then G is solvable. These authors also assume the UnbalancedGroup Conjecture.

Our main result in this direction is the following corollary of Theorem 1.

COROLLARY 1. Let G satisfy H2 and suppose that A max G. Then G is solvable.

PROOF. Let G be a minimal counterexample. Then F(G) = 1 and by [3] H isof even order. Moreover, A is not Hall, since otherwise G = AB = AB2., and [3]yields a contradiction. Thus, by Theorem 1, G = PGL(2, q) with q odd, q > 5.Moreover, A = CG(t), where t is an involution and 81 IAA. This is a contradic-tion, since PGL(2, q) AB with q, A and B as required.

Another application of Theorem I is a new proof of a result to which severalindependent papers have been dedicated (by Monahov, Finkel and Walls).

COROLLARY 2. Let G satisfy H2 and suppose that B is Dedekind. Then G issolvable.

PROOF. Let G be a minimal counterexample. By Corollary 1 A is not maximalin G. Let A < M max G. Then M = AB*, with 1 < B* = M n B < B, whence

N = (B*G> = (B*M> M, 1 <N < G.

SUBGROUPS WITH NILPOTENT SUBGROUP OF INDEX 2 203

By induction M is solvable, hence so is N. Moreover, by induction G/N issolvable. Thus G is solvable, a contradiction.

Recently Arad, Herzog and Chillag proved the following theorem:

THEOREM 2. Let G 1 be a finite group with a maximal subgroup A. Supposethat F(G) = 1 and A contains an abelian subgroup H of index [A : H] < 4. Thenone of the following holds:

(1) A is nilpotent, A E Sy12(G);(2) A is nonnilpotent and PSL(2, q) < G < PI'L(2, q), q odd;(3) H is of odd order.

The proof of Theorem 2, as well as the proofs of the above applications, isbased on Theorem 1.

Added in proof. Recently, Kazarin proved the Scott's conjecture in Math.Sbornik 110(152) (1979), 51-65 (Russian).

REFERENCES

1. B. Baumann, Endliche nichauflosbare Gruppen mit einer nilpotenten maximalen Untergruppe, J.Algebra 38 (1976), 119-135.

2. W. Feit and J. Thompson, Finite groups which contain a self-centralizing subgroup of order 3,Nagoya Math. J. 21 (1962), 185-197.

3. D. Finkel and J. R. Lundgren, Solvability of factorizable groups. II, J. Algebra 60 (1979), 43-50.4. G. Higman, Odd characterizations of finite simple groups, Lecture Notes, University of Michigan,

1968.5. J. S. Rose, On finite insoluble groups with nilpotent maximal subgroups, J. Algebra 48 (1977),

182-196.6. J. G. Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat.

Acad. Sci. U.S.A. 45 (1959), 578-581.

BAR-ILAN UNIVERSITY, ISRAEL (Z. Arad and A. Shaki)

TEL-Aviv UNrvERSITY, ISRAEL (M. Herzog)


AUTOMORPHISMS OF NILPOTENTGROUPS AND SUPERSOLVABLE ORDERS

ANTHONY HUGHES

1. Introduction. This paper is divided into two parts. The first part considersthe analogue of the result obtained by Heineken and Liebeck for the prime 2. Inthe second part we shall characterize all positive integers n such that for anyfinite group G of order n, G is supersolvable. The author would like to thank A.Mann for some helpful correspondence concerning the first part of this paper.

2. Automorphisms of nilpotent groups. Let G be a nilpotent group ofnilpotency class 2. We denote by Autc G the group of all central automorphismsof G, that is, of all automorphisms of G which act trivially on G/Z(G). In [2],Heineken and Liebeck establish the following result.

(2.1) Let K be a finite group, and let p be an odd prime. Then there exists ap-group G of nilpotency class 2 and exponentp2 such that Aut G/Autc G and Kare isomorphic.

In this section we shall discuss the analogue of this result for the prime 2:(2.2) Let K be any finite group. Then there exists a 2-group G of nilpotency

class 2 and exponent 4 such that Aut G/Autc G and K are isomorphic.To establish (2.2), we first associate with K a connected graph D(K) which

satisfies the following properties:(1) each vertex has degree at least 2,(2) every cycle contains at least 4 vertices (girth D(K) > 4), and(3) Aut(D(K)) = K.We remark that we can construct a connected graph D(K) which satisfies

(1)-(3) in a number of ways. For example, let D0(K) be the connected cubicalgraph with Aut D0(K) = K constructed by Frucht [1]. Then we can take D(K)to be the graph derived from D0(K) by replacing each line by a new linecontaining one new vertex.

With a given (connected) graph D, we may construct a 2-group G(D) in thefollowing fashion: let P1, . . . , P. be the vertices of D. Associate with eachvertex Pi a generator Xi of the free group F on the n free generators X1, ... , X,,.

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205

206 ANTHONY HUGHES

Let R be the normal subgroup of F generated by the elements X.2, [X, Xj] ifP,Pj E E(D), and [[X;, Xj], Xk]. Then G(D) = FIR. We call x; = XR thecanonical generator of G(D) associated with the vertex F. We note that G(D)satisfies the following:

(A)G=<xj,..x,.>,(B) x,z = 1, 1 < 1 < n,(C) [x xj] E Z(G), 1 < i, j < n,(D) [x;, xj] = 1 if P;Pj E E(D), 1 < i, j < n, and(E) G' is freely generated by the [x,, xj] with P,Pj a E(D), 1 < i, j < n.

From (A)-(E) we may conclude that G(D) is a 2-group of nilpotency class 2 andexponent 4. Moreover, we have

(2.3) If (z,, . . . , z,.) C Z(G) and 7r is a permutation of (1, . . . , n), then themap x; x;,,zr, 1 < i < n, is an automorphism of G(D) if, and only if, the mapP; - P,,,, 1 < i < n, is an automorphism of D.

Now let G = G(D), where D = D(K). Then by (2.3), G/Autc G possesses asubgroup isomorphic to K. From conditions (2) and (3) we have

(2.4) If x E G is an involution, then either x = x,z or x = x;xjz, where x;, xjare canonical generators of G and z E Z(G).

(2.5) Let x;, xj, xk be three canonical generators of G. Then CG(x;) andCG(xjxk) are nonisomorphic subgroups of G.

From (2.4) and (2.5) we have(2.6) If 0 E Aut G, then 0: x; - x,,z;, 1 < i < n, where (zl, . . . , z,.) C Z(G)

and 7r is a permutation of (1, . . . , n).Clearly (2.2) now follows from (2.3) and (2.6).REMARKS. Let D be a connected graph satisfying (1)-(3). Then Auto G(D) is

an elementary abelian 2-group. In particular, if D is asymmetric, then Aut G(D)= Auto G(D) is an elementary abelian 2-group.

3. Supersolvable orders.

(3.1) DEFINITION. We say that the positive integer n is a supersolvable order if,for every finite group G of order n, G is supersolvable.

In this section we shall give a result which characterizes all supersolvableorders and give an application of this characterization.

Let n be a positive integer, and let n = pr p" ... p be the canonicalexpression of n as a product of distinct primes. For all positive integers k, let

7rk(n) = (p,l a, > k).

Thus 7r1(n) consists of the prime divisors of n.(3.2) The positive integer n is a supersolvable order if, and only if, the

following are satisfied by 7r,(n):(1) If p E 7rd(n), q E 7r,(n) andPdI(q' - 1), then p' (q - 1).(2) If p E 7r3(n), q E 7r,(n) and p3J(q - 1), then q (4 Iro(n).(3) If p, q, r E 7r,(n), pl(q - 1) and pql(r - 1), then r (4 Iro(n).The proof of (3.2) is given in [4]. Before giving an application of (3.2), we

require the following definition.(3.3) DEFINITION. Let x be a positive integer. Then Fi(x) = number of

supersolvable orders n such that n 4 x.

AUTOMORPHISMS OF NILPOTENT GROUPS 207

In [6], M. Ram Murty and V. Kumar Murty obtain a number of results on theasymptotic behavior of F,(x), where T is a certain subset of the set of allisomorphism classes of finite groups. In particular, using (3.2) together with aresult of Erdos, they show

(3.4) There is a constant cl > 6/772 such that as x - oo,

F,.(x) - c1x.Finally, we shall mention some of the other results contained in [6]. If T is

taken to be the subset consisting of all isomorphism classes of either nilpotentgroups (N) or solvable groups (S), then

(3.5) As x - oo, FN(x) xe-''/log log log x, where y is Euler's constant.(3.6) There is a constant c2 such that as x - oo, FS(x) - c2x (where the value

of c2 is close to but less than 1).We note that (3.5) includes a result of Erdos [3] where T consists of all

isomorphism classes of cyclic groups. Finally, Mays also obtained (3.5) in [5].

REFERENCES

1. R. Frucht, Graphs of degree three with a given abstract group, Canad. J. Math 1 (1949), 365-378.2. H. Heineken and H. Liebeck, The occurrence of finite groups in the automorphism group of

nilpotent groups of class 2, Arch. Math. 25 (1974), 8-16.3. P. Erdos, Some asymptotic formulas in number theory, J. Indian Math. Soc. 12 (1948), 75-78.4. A. Hughes, On supersolvable orders (to appear).5. M. E. Mays, Counting abelian, nilpotent, solvable and supersolvable groups, Arch. Math. 31

(1978), 536-538.6. M. Ram Murty and V. Kumar Murty, On the density of various classes of groups (to appear).

HARVARD UNIVERSITY

A SHORT SURVEY OF FITTING CLASSES

A. R. CAMINA

This note is a short survey of some known results and open questions in thestudy of Fitting classes. Since it is based on a 45 minute talk there are manygaps and flaws. A class of finite groups X will always include the trivial groupand if X contains any given group then X contains all its isomorphic copies.

1. A class X is said to be a Fitting class if(i)G EX=>N EXforallN<Gand(ii) if G = MN, M, N < G with M, N E X then G C X2. For a class of groups X and a group G an X-injector is a subgroup U such

that U n N is a maximal X-subgroup of N for all normal subgroups N of G.

THEOREM. Let X be a class of finite soluble groups. Then X is a Fitting class ifand only if X-injectors exist in all soluble groups.

Fitting classes are defined in terms of closure operations and the relationbetween various closure operators has been much studied by R. Bryce and J.Cossey. We restrict ourselves to soluble groups.

Question 1. Is a subgroup closed Fitting class quotient closed? This has beenshown to be true for Fitting classes contained in the class of groups of Fittingheight at most 3 (Bryce and Cossey, 1978).

It is clear that the intersections of Fitting classes is again a Fitting class and sowe define X V Y = (smallest Fitting class containing both X and Y).

Question 2. Is it true that if X and Y are Fitting classes then X V Y = { G I G issubnormal in H where H = HXHY}?

Notation. GX = (NON E X and N < < G) and is called the X-radical of G.When X is a Fitting class G. is the maximal normal X subgroup of G. E. Cusackhas some partial results in this direction. F. P. Lockett introduced the followingimportant idea: Let X be a Fitting class, define X* = (GI(G X G)X is subdirectin G X G). Then

(i) X* is a Fitting class and (G X H)X. = G. X HX..(ii) (X*)* = X", (X n Y)* = X` n Y.

1980 Mathematics Subject Classification. Primary 20D10.

m American Mathematical Society 1980

209

210 A. R. CAMINA

(iii) There exists a unique smallest Fitting class X. such that X*.

(iv) If Y is a Fitting class X" = Y" X* = Y*, X* < Y < X.(v) X < Y = X* < Y* and X* < Y*.(vi) for all groups G, [G.., Aut G] < G...

Of particular interest is the set of classes Y such that X* < Y < X" and such aset is called a Lockett section. For some time there has been considerable studyof normal Fitting classes i.e. those in which every injector is normal.

THEOREM. Let X be a Fitting class then the following are equivalent:(i) X is normal.(ii) X* is the class of all soluble groups, S.(iii) If V is an X- injector of G then G' < V,(iv) If G E X and p is a prime then there is a natural number n such that for all

m, G (m) Wr Zp E X.

For some time it was an open question as to whether the Lockett section for aclass was obtained by intersecting its upper star with the section of all normalFitting classes. More precisely, given any Fitting class F such that X* < F <X", Lockett asked if there was a normal Fitting class Y such that F = X" n Y.This was shown to be false (Berger and Cossey). There has also been some effortin analyzing the smallest normal Fitting class S. T. Berger has recently shownthat this can be recognized internally, and a combination of works, by Laue,Lausch and Pain and Bryant and Kovacs has shown

THEOREM. Given any two primes p q there exists a group H E S* such thatOP(H) = H' and H/H' is cyclic of order q.

Unconnected but still an open question.Question 3. What is the Fitting class generated by the symmetric group on 3

letters?It has been shown that it does not contain the dihedral group of order 18. This

was done by constructing certain normal Fitting classes using a technique ofBlessenohl and Gaschiitz. This led Lausch and Bryce and Cossey to thefollowing important concept:

A Fitting pair for a Fitting class X such that F < X < F is an Abelian groupA and a map d which assigns to each group G in F a homomorphism dG from Ginto A such that

(i) If f is a normal embedding of a group N into some group G and G E Fthen ndN = (nf)dG, Vn E N.

(ii) If G E F then G. = ker dG.(iii)A = (ydGI g E G, G E F).(iv) If Y is a Fitting class with X < Y < F then A(Y) = (gd Ig E G E Y)

sets us a lattice isomorphism between the Fitting classes lying between X and Fand the subgroup lattice of A.

Finally G E F if and only if G can be normally embedded in a group R,where R can be generated by is normal subgroups Ni, ... , N, where eachN, E F and G < [N;, Aut N,]. This lattice isomorphism should lead to a muchbetter understanding of Fitting classes than presently exists and I feel itspotential has not been fully exploited.

SHORT SURVEY OF FITTING CLASSES 211

Relating to the lattice structure of Fitting classes, Bryce and Cossey (1974)discussed the problem of determining when one Fitting class is maximal inanother. They found a necessary condition and a sufficient one but they werenot the same.

Question 4. Determine a necessary and sufficient condition for maximality ofone Fitting class in another.

The idea of transfer has played an important role recently in some of the workin Fitting class theory. As an example let me quote the work of Laue, Lauschand Pain (1977).

Let X be a group and let K satisfy (Aut X)'<ala E Aut X; [X, a] < X> < K< Aut X. Given any group G let 0 = {HI, . . . , Hr} be the set of subgroups ofG isomorphic to X. For each i fix an isomorphism 0i from Hi to X, also for eachg E G define Hi8 = g-IHig. Set

r

fG (g) = Ki=1

Let Y be a Fitting class then (Y, X) _ (GIG E Y and f f (g) = 1 Vg.- G )isa Fitting class and further if X E Y then (Y, X) is nontrivial if K Aut X.

Apologies. To all the ideas, problems and people it is impossible to mention ina 45 minute talk, and to the references omitted.

REFERENCES

1. J. C. Beidleman, On products and normal Fitting classes, Arch. Math. 28 (1977), 347-356.2. J. C. Beidleman and B. Brewster, Strict normality in Fitting classes, I, II, J. Algebra 51 (1978),

211-217, 218-227.3. J. C. Beidleman and P. Hauck, Uber Fittingklassen and die Lockett Vermutung, Math. Z. 167

(1979), 161-167.4. T. R. Berger, More normal Fitting classes of finite soluble groups, Math. Z. 151 (1976), 1-3.5. , Normal Fitting pairs and Lockett's conjecture, Math. Z. 163 (1978), 125-132.6. , unpublished.7. T. R. Berger and J. Cossey, An example on the theory of normal Fitting classes, Math. Z. 154

(1977), 287-293.8. D. Blessenohl and W. Gaschutz, Uber normale Schunck- and Fittingklassen, Math. Z. 118

(1970), 1-8.9. D. Blessenohl and H. Laue, Fittingklassen endlicher Gruppen in denen gewisse Hauptfaktoren

einfach sind, J. Algebra 56 (1978), 516-532.10. , Vorzeichen von Automorphismen and Beispiele normaler Fittingklassen, Math. Z. 148

(1976), 119-126.11. D. Blessenohl, Uber ordentliche Fittingklassen, Habilitationschrift, Kiel, 1977.12. O. Brison, On the theory of Fitting classes of finite groups, Ph.D. thesis, University of Warwick,

1978.

13. R. M. Bryant and L. G. Kovacs, Lie representations and groups of prime power order, J. LondonMath. Soc. 17 (1978), 415-421.

14. R. A. Bryce and J. Cossey, Maximal Fitting classes of finite soluble groups, Bull. Austral. Math.Soc. 10 (1974), 169-175.

15. , A problem in the theory of normal Fitting classes, Math. Z. 141 (1975), 99-110.16. , Fitting formations of finite soluble groups, Math. Z. 127 (1972), 217-223.17. , Strong containment of Fitting classes, Proc. Miniconf. (Canberra, 1975), Lecture Notes

in Math., vol. 573, Springer-Verlag, Berlin and New York, 1977.18. , Subdirect product closed Fitting classes, Proc. 2nd Internat. Conf., Lecture Notes in

Math., vol. 372, Springer-Verlag, Berlin and New York, 1974.

212 A. R. CAMINA

19. , Metanilpotent Fitting classes, J. Austral. Math. Soc. 17 (1974), 285-304.20. , Subgroup closed Fitting classes, Math. Proc. Cambridge Philos. Soc. 83 (1978),

195-204.21. A. R. Camina, A note on Fitting classes, Math. Z. 136 (1974), 351-352.22. C. Charnes, Some results concerning Fitting classes defined by parity, Arch. Math. 32 (1979),

209-212.23. J. Cossey, Classes of finite soluble groups, Proc. 2nd Internat. Conf., Lecture Notes in Math.,

vol. 372, Springer-Verlag, Berlin and New York, 1974.24. , Products of Fitting classes, Math. Z. 141 (1975), 289-295.25. E. Cusack, On the theory of Fitting classes of finite soluble groups, Ph.D. thesis, University of

East Anglia, 1979.26. , The join of two Fitting classes, Math. Z. 167 (1979), 34-47.27. , Strong containment of Fitting classes, J. Algebra (to appear).28. K. Doerk, Uber den Rand einer Fittingklasse endlicher auflosbarer Gruppen, J. Algebra 51

(1978), 619-630.29. B. Fisher, Klassen konjugierter Untergruppen in endlichen auflosbaren Gruppen, Habilitation-

schrift, Frankfurt am Main.30. B. Fischer, W. Gaschutz and B. Hartley, /njektoren endlicher auflosbarer Gruppen, Math. Z. 102

(1967), 337-339.31. W. Frantz, Spezielle Fittingklassen and Ihre /njektoren, Diplomarbeit, Kiel, 1970.32. B. Hartley, On Fischer's dualization of formation theory, Proc. London Math. Soc. 19 (1969),

193-207.33. P. Hauck, Zur Theorie der Fittingklassen endlicher auflosbarer Gruppen, Dissertation

Universitat Mainz, 1978.34. , Eine Bemerkung zur kleinsten normalen Fittingklasse, J. Algebra 53 (1978), 395-401.35. , On products of Fitting classes, J. London Math. Soc. 20 (1979), 423-424.36. , Endliche auflosbare Gruppen mit normalen F-injektoren, Arch. Math. 28 (1977),

117-129.37. , Endliche auflosbare Gruppen mit normalen F-inkektoren, Arch, Math. 31 (1979),

529-535.38. , Fittingklassen and Krantzproducte, J. Algebra 59 (1979), 313-329.39. T. 0. Hawkes, On Fitting formations, Math. Z. 117 (1970), 177-182.40. H. Laue, Uber nicht auflosbare normale Fittingklassen, J. Algebra 45 (1977), 273-284.41. H. Laue, H. Lausch and G. R. Pain, Verlagerung and normale Fittingklassen endlicher Gruppen,

Math. Z. 154 (1977), 257-260.42. H. Lausch, on normal Fitting classes, Math. Z. 130 (1973), 67-72.43. F. P. Lockett, On the theory of Fitting classes of finite soluble groups, Ph.D. thesis, University of

Warwick, 1971.44. , The Fitting class Math. Z. 137 (1974), 131-136.45. , On the theory of Fitting classes of finite soluble groups, Math. Z. 131 (1973), 103-115.46. A. R. Makan, Fitting classes with wreath product property are normal, J. London Math. Soc. 8

(1974), 245-246.47. , Normal Fitting classes and the Lockett ordering, Math. Z. 142 (1975), 221-228.

UNIVERSITY OF EAST ANGLIA, ENGLAND

TRANSFER THEOREMS

TOMOYUKI YOSHIDA

We consider first certain transfer theorems of finite groups. Though transfertheorems are one of the easiest ways to get nontrivial local subgroups in simplegroups, they are not so powerful in studying finite groups in general. Thereremain many open problems about transfer of finite groups as is well known.But the purpose of this article is not directly related to finite group theory. Wefirst go back to D. G. Higman's classical result which follows directly from the"Mackey decomposition" for transfer.

THEOREM A. Let P be a Sylow p-subgroup of G. Then

P n G' _ <x-yJx,y E P, x c y).The concept of G-functors introduced by J. A. Green provides us with a

mechanism by which we can develop "transfer theory".DEFINITION. Let G be a finite group and k a commutative ring with unit. A

G-functor (a, T, p, a) consists of k-modules a(H) (H 5 G) and k-maps

TH a(H) - a(K) : a aK

PH : a(K) - a(H) PH,

aH : a(H) - a(Hg) : a ag,

for all H < K < G and g E G. These families must satisfy the following. (Inthese axioms, D, H, K, L < G; g, g' E G; a E a(H), 8 E a(K).)

(G.1) a11 = a, (/aK)L = if H S K < L;(G.2) $K #, RD if D < H 5 K;(G.3) (ag)8 = a%8, ah = a if h E H;(G.4) (a K)g = (ag)Ks, ($H)g = (fig)H-;(G.5) (Mackey axiom) if H, K 5 L, then

L g Ka K = a H"nK .gEH\L/K

1980 Mathematics Subject Classification. Primary 18B25, 18G25; Secondary 20D20, 20J06, 20C99,55R20, 57S17.


213

214 TOMOYUKI YOSHIDA

DEFINITION. A G-functor (a, T, p, a) is cohomological if(C) $HK = I K : H 1,0 for all H < K < G and P E a(K).EXAMPLES. H H-, R(H): the character ring, gives a G-functor, where T are

inductions. Let V be a kG-module. Then H N H*(H, V) gives a cohomologicalG-functor. Hr H/H' gives a cohomological G-functor in which p are group-theoretic transfers. Some applications to representation theory and many exam-ples of G-functors are found in Green's paper.

Theorem A is generalized as follows.

THEOREM B. Let (a, T, p, a) be a cohomological G functor over k and H < G.Assume that I G : H I-' E k. Then

(a) pH is mono, TH is epi, and a(H) = Ker TH ® Im pH;(b) Im pH = (a E a(H )' a 8H n Hr = aH n H, for all g e G) ;(c) Ker TH = <#8H - 6"1$ E a(H n gHg-1), g E G).

This theorem yields some well-known results; for example, Cartan-Eilenberg'sstable element theorem, Maschke-Higman's theorem about H-projectivity, thelemma that if Visa kG-module and I G 1-' E k, then V = CV(G) ® [ V, G ], etc.It is possible to generalize Wielandt's transfer theorem to cohomological G-func-tors. From this viewpoint, Hall-Higman's Theorem B is also a kind of transfertheorem of Wielandt type. But there remain very difficult problems aboutp-groups. For example, Sasaki and I cannot yet generalize Holt's theorem aboutSchur multipliers, because the characterization of p-groups with no proper"singularities" remains open, even though it is a purely group-theoretic problemabo ut p-groups.

DEFINITION (A. W. M. DRESS). Let (i and 'l be categories, M*: Q afunctor and M*: Q .- 9i a contravariant functor such that M*(X) = M*(X)(_: M(X)) for any X, and f* :- fM*, f* := fM* for any morphism f in Q. TheM = (M*, M*): 3 is called a Mackey functor if it satisfies the followingconditions.

(M.1) If

is a pull-back, then

M(W) M(Y)

Tu' 0 Tg

M(X) Lf4 M(Z)

(M.2) M* transforms finite sums in a into finite products.The category of G-functors is equivalent to the category of Mackey functors

from the category of finite G-sets. Thus results about G-functors are rewritten bythe language of Mackey functors. Theorem B turns itself into the following.

TRANSFER THEOREMS 215

THEOREM C. Let M: cSG - YILk be a cohomological Mackey functor from finiteG-sets to k-modules. Let p: E - B be a G-map between finite G-sets such thatn = I by 'I = const. for all b E B. Assume that n-' E k. Then

0 - M(B) M(E)"_ 4M(E X B E)

is exact, where p;: E X B E - E (i = 1, 2) are projections.

In order to develop "transfer theory" for Mackey functors from & to , werealize that & and '3 must be special categories. For example, it is almostimpossible to define cohomological Mackey functors for general categories.

DEFINITION. A category & is called a topos if(T.1) & has a final object 1, products X, pull-backs, and equalizers (i.e. & is

finite complete);(T.1') & has an initial object 0, co-products +, push-outs, and co-equalizers

(i.e. & is finite co-complete);(T.2) & is Cartesian closed, i.e. for any A E , the functor X H X

X A has a right adjoint & -* & : X H XA.(T.3) & has a subobject classifier t : 1 0, i.e., for any mono A -* B, there is

a unique B 0 such that A - 1

I JtB S2

is a pull-back.Topos theory is set theory under intuitional logic and toposes are very like the

category of sets, therefore we can develop a mathematics inside each topos. Theconcept of toposes is useful in foundation and algebraic geometry: see John-stone's book. The categories of G-sets, finite G-sets, and sheaves of sets on aspace X are all toposes. We find many Mackey decompositions about toposes,for example, the fundamental theorem for toposes [J, 1.4] yields that A H & /Asatisfies (M.1).

THEOREM D. Let Sf be a Galois category and M = (M*, M.) be acohomological Mackey functor from & to an abelian category . Let X be anobject of & of cardinal n E N. Assume that n id give automorphisms of M(1) andM(X). Take p : X -* 1 and projections p; : X X X - X. Then

(a) p`: M(1) - M(X) is mono, p*: M(X) - M(1) is epi, and M(l) = Imp*Ker p *;

(b) 0 - M(1)- M(X)p1- z

M(X X X) is exact;

(c) 0 f- M(l)M(X)P1.<_

M(X X X) is exact.

We said that M is cohomological if for any simple objects A and B and any f:A - B, the following commutes.

M(B)

IM(A) FM(B)

where IA :_ JAFJ.

216 TOMOYUKI YOSHIDA

LEMMA. Let t be a connected Grothendieck topos with a point p: S - & andV E ab(& ). For each X E &tcr (the category of locally constant finite objects), seth,q,(X) H9(&, X; V) := R7(y*)(V), where y* := hom&(X, -) : ab(6)-*ab(,5) = Qb. The f is a Galois category and hv: &i,f (fib is a cohomologicalMackey functor.

When it is known that Slcf = 5f and hq(H, G) is isomorphic to thecohomology group HQ(H, V), and so Theorem D yields -Cartan-Eilenberg'stheorem. Applying this to the category of sheaves of sets, we get a result intopology.

THEOREM E. Let p: E -> B be an n-fold covering map and e a sheaf of abeliangroups (or simply an abelian group) such that n id is an automorphism ofH* (C, 1) for C = B, E. Let pi: E X B E -* E be projections. Then

0- H*(B, L) H*(E, L) H*(E X B E, E) (exact).Applying Theorem E to p: X - X/G, we have the following well-known

result ([Br, 3.2.4], and [Bo, 3.2.3]).

THEOREM F. Assume that a finite group G acts on a space X with no fixedpoints. Let L be abelian group such that ! G id is an automorphism of L. ThenH*(X, L)° H*(X/G, L).

There are many examples of G-functors. Let F be a Galois extension of Qwith Galois group G and let A(k) be the ideal (class) group, k*, idele group,K-groups, etc. Then H t-b A(FH) is a G-functor. We note that Theorems B, C, Dare generalized to general toposes, Mackey functors, G-functors by using an ideaof Dress. Representation theory of finite groups (= 0-dimensional homologicalalgebra) is perhaps useful for other fields. For example, if & is a topos withnatural number object N and A E &, then the pull-back functor ab(& /A) -ab(f) has a left adjoint, which is constructed in a way similar to that of inducedrepresentations. In terms of sheaves of abelian groups, this means that theinverse image given by any local homeomorphism has a left adjoint.

Regard finite group theory as the theory of categories of G-sets forvarious finite group G, and then remember that the category of G-sets isa Boolean Grothendieck topos which does not always satisfy the Axiom ofChoice.

REFERENCES

[A] J. F. Adams, Infinite loop spaces, Ann. Math. Studies, Princeton Univ. Press, Princeton, N. J.,1978.

[Bo] A. Bore[, Seminar on transformation groups, Ann. Math. Studies, Princeton Univ. Press,Princeton, N. J., 1960.

[Br] G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Math.,Academic Press, New York, 1972.

[D] A. W. M. Dress, Contributions to the theory of induced representations, Algebraic K-theory. II,Lecture Notes in Math., vol. 342, Springer-Verlag, Berlin and New York, 1973, pp. 183-240.

[G] J. A. Green, Axiomatic representation theory for finite groups, J. Pure Applied Algebra 1 (1971),41-77.

(J] P. T. Johnstone, Topos theory, Academic Press, New York, 1977.]Y] T. Yoshida, Character-theoretic transfer, J. Algebra 52 (1978), 1-38.

HOKKAIDO UNIVERSITY, JAPAN


PROBLEM AREASIN INFINITE GROUP THEORY

FOR FINITE GROUP THEORISTS

GILBERT BAUMSLAGI

I have been asked to talk today about parts of infinite group theory whichoffer, and I quote, "suitable problem areas for finite group theorists". In tryingto satisfy this request I have here tailored to the taste of finite group theoriststwo old problems of Max Dehn, namely the conjugacy and the isomorphismproblems. Let me remind you that a group G has a solvable conjugacy problemif there is an algorithm whereby one can determine whether or not any pair ofelements of G are conjugate (if one of these elements is always taken to be theidentity, the corresponding problem is known as the word problem). Similarly agiven class of groups is said to have solvable isomorphism problem if there is analgorithm whereby one can determine whether or not any pair of groups in theclass are isomorphic. Thus my talk, in keeping with this conference, could wellhave been entitled "Classification and conjugacy classes". I would be remiss,however, in my duties, if I did not begin this talk with a brief discussion offinitely generated infinite simple groups even though it does not fit in with eithermy program or the theory of finite simple groups in any way.

A. Finitely generated infinite simple groups. Finitely generated infinite simplegroups have not been easy to come by. Indeed it was only in 1951 that GrahamHigman [16] demonstrated that such groups exist. Oddly enough Higman didnot actually produce a concrete example at that time. It was left to Ruth Camm,a student of Higman's, to produce, shortly afterwards in [6], continuously many2-generator infinite simple groups, explicitly given as generalised free productsof two free groups. The first example of a finitely presented infinite simple groupwas discovered, almost by accident it seems, by Richard Thompson in 1969 [27].Strangely enough this group of Thompson arose as the group of automorphismsof the free algebra of rank one in a variety (in the sense of universal algebra) of

1980 Mathematics Subject Classification. Primary 20F10; Secondary 20E32, 20E26, 20E15, 20F18,20G15.

'The author thanks the National Science Foundation for their support.C American Mathematical Society 1980

217

218 GILBERT BAUMSLAG

algebras in which the free algebras of finite rank are all isomorphic! GrahamHigman then proved in 1973 [17] that T'hompson's group is only one of aninfinite family of finitely presented infinite simple groups. Somewhat earlier, in1968, Philip Hall [15] proved that every countable group can be embedded in afinitely generated simple group; indeed in a two-generator simple group accord-ing to Goryushkin, 1974 [9]. Thus it appears that finitely generated infinitesimple groups are rather varied and quite complex. The complexity of suchgroups has recently, most shockingly, been demonstrated in some unpublished,independent work of Rips [25] and Ol'shanskii [24]. In particular Ol'shanskii hasconstructed, by the methods of small cancellation theory, the following twoexamples.

(1) A nonabelian group all of whose proper subgroups are infinite cyclic andwhose maximal subgroups have pairwise trivial intersection.

(2) An infinite group whose proper subgroups have prime order and whosesubgroups of the same order are conjugate.

Thus Ol'shanskii has constructed some two-generator infinite simple groups ofthe most untouchable kind. It seems likely, although I know nothing of thedetails, that neither of the above examples is finitely presented. But this is ofsmall comfort at this stage. It is clear then that the subject of finitely generatedinfinite simple groups has hardly begun, unlike that of finite simple groupswhere the show is almost over!

B. Finitely presented residually finite groups. Most of the groups that I willdiscuss here today are residually finite. Let me recall that if C' is a class ofgroups, then RC is the class of those groups G with the property that for eachg E G, g # 1, there exists a normal subgroup Ng of G with g (4 Ng and theresidual factor group G/Ng in C. Thus the class RC is precisely the class ofsubdirect products of C'-groups. In particular it follows readily that if 9 is theclass of all finite groups, then the class R9 is simply the class of subgroups ofdirect products of finite groups. It seems appropriate to comment first on thefinitely presented residually finite groups. In general an algorithmic classifica-tion is out of the question because of the following theorem of C. F. Miller III[21].

B1. The isomorphism problem for finitely presented residually finite groups isrecursively unsolvable.

Thus there is no effectively computable set of computable invariants whichdistinguishes one finitely presented residually finite group from another. Inci-dentally the word problem for finitely presented residually finite groups issolvable (McKinsey [22]), whereas the conjugacy problem is not (Miller [21]).Notice now that it follows from Miller's Theorem B1 that there must existfinitely presented residually finite groups with the same finite images which arenot isomorphic. In fact this is by no means an uncommon phenomenon. Evenmetacyclic groups (i.e. extensions of cyclic groups by cyclic groups) can have thesame finite images without being isomorphic (G. Baumslag [2]). This brings meto my third topic.

PROBLEM AREAS IN INFINITE GROUP THEORY 219

C. Polycyclic groups. Let me recall again the terminology. A group is termedpolycyclic if it has a finite subnormal series beginning with the identity sub-group, in which successive factors are cyclic. It follows that the polycyclicgroups are simply the solvable groups which satisfy the maximum condition forsubgroups. Of course metacyclic groups are polycyclic, and according to an oldtheorem of K. A. Hirsch [18] all polycyclic groups are residually finite. Thuspolycyclic groups provide examples of finitely presented residually finite groupswhich have precisely the same finite images and yet are not isomorphic. It israther remarkable then that Grunewald, Pickel and Segal [11] have very recentlybeen able to prove the following theorem.

Cl. There are, up to isomorphism, only finitely many polycyclic groups whichhave the same finite images as any given polycyclic group.

Theorem Cl is proved by combining a theorem of Borel and Serre onalgebraic groups, an integral version of a splitting theorem of Wang, a delicatenumber-theoretical result and some machinery from the theory of nilpotentgroups. Incidentally the conjugacy problem was finally completely proved alsoonly recently, by Formanek, in 1976 [8]. This still leaves open the following

Q1. Is the isomorphism problem for polycyclic groups solvable?In a sense the structure of polycyclic groups is well understood, for according

to an old theorem of Mal'cev every polycyclic group contains a subgroup offinite index whose derived group is nilpotent [20]. Thus part of the study ofpolycyclic groups can be translated into the representation theory of finitegroups and of finitely generated abelian groups in Gl(n, Z). Clearly finitelygenerated nilpotent groups play an important role in the study of polycyclicgroups. Indeed they are an extremely important class of groups in their ownright. A major step forward in the study of such groups has just been made byGrunewald and Segal [12], who have proved

C2. The isomorphism problem for finitely generated nilpotent groups is solvable.

The proof of C2 leaves many questions about finitely generated nilpotentgroups unanswered. It does not, for example, provide invariants which char-acterise such groups. And it yields little information about the automorphismsequence of a finitely generated nilpotent group. In particular the followingproblem remains unanswered.

Q2. Let G be any given finitely generated nilpotent group and let

AI = Aut(G), A2 = Aut(A1), ... , A = Aut(A,._1)....

be the automorphism sequence of G, where here Aut(X) denotes the automor-phism group of the group X. Are there, up to isomorphism, only finitely manygroups in the sequence?

The study of finitely generated torsion-free nilpotent groups is closely relatedto the study of finite p-groups. Indeed Gruenberg [10] has proved that everyfinitely generated torsion-free nilpotent group is residually a finite p-group, forevery prime p. It would be interesting to translate some of the theory of finitep-groups into torsion-free nilpotent group theory. Conversely the connectionsbetween nilpotent groups and lie algebras in the torsion-free case might well lead


to further insights into the nature of finite p-groups. In fact, in a sense, some ofthis is already being done by G. E. Wall, among others.

Changing the subject somewhat, it was noticed many years ago that the groupring of any group with a nontrivial element of finite order is never a domain.This suggests, conversely, that the group ring of a torsion-free group is always adomain. Very little headway has been made towards resolving this situation.Now the group ring of a polycyclic group looks very much like a twistedpolynomial ring. Indeed in the past twenty years or so, beginning with the workof P. Hall in 1954 [13], a continuing effort has been made to draw out andfurther understand the group rings of polycyclic groups in the light of commuta-tive algebra. A case in point is the domain problem for torsion-free polycyclicgroups, which turned out to be surprisingly delicate, but which was ultimatelysolved by Farkas and Snider in 1976 [7] using homological techniques.

C3. The group ring of a torsion free polycyclic group is a domain.

This suggests the followingQ3. Is the group ring of a torsion-free residually finite group a domain?I would like to turn next to another class of residually finite solvable groups.

D. Finitely generated metabelian groups. As usual a group is termedmetabelian if its derived group is abelian. So the derived group of a metabeliangroup can be viewed as a module over the group ring of its factor-derived group.If the group is finitely generated to begin with, this module is always finitelygenerated and can be viewed as a module over a polynomial ring in a finitenumber of variables. By following this line of thought P. Hall proved in 1959[14] the theorem

D1. Finitely generated metabelian groups are residually finite.

As I already indicated at the outset the residual finiteness of finitely generatedmetabelian groups implies the solvability of the word problem for such groups.This has recently been reinforced by Noskov [23] who has shown

D2. The conjugacy problem for finitely generated metabelian groups is solvable.

I should point out at this stage that finitely generated metabelian groups arenot, as a rule, polycyclic. The fact is that the derived group of a finitelygenerated metabelian group need not be finitely generated; indeed it can evenbe free abelian of infinite rank. Thus many finitely generated metabelian groupsturn out not to be finitely presented. There is, however, very little structuraldifference between finitely generated metabelian groups and finitely presentedmetabelian groups, as the following theorem of G. Baumslag [1] shows.

D3. Every finitely generated metabelian group can be embedded in a finitelypresented metabelian group.

An important means of discerning which of the finitely generated metabeliangroups are actually finitely presented, of an algebraic-geometric kind, has beenfound by Bieri and Strebel [5]. In the light of all this information about finitelygenerated metabelian groups it seems appropriate to raise the following


Q4. Is the isomorphism problem for finitely generated metabelian groupssolvable?

Let me change direction a little and remind you that if G is a finitelypresented group then the second homology group HA(G, Z) of G with trivialintegral coefficients is finitely generated. In 1963 [26] Stallings constructed afinitely presented group G with the property that H3(G, Z) is not finitelygenerated. Stallings' group was however of finite cohomological dimension andso the possibility remained that at least from some point on the integralhomology groups of a finitely presented group would all be finitely generated.Recently Eldon Dyer and I [3] in an attempt to understand better the homologyof finitely generated and finitely presented metabelian groups discovered afinitely presented metabelian group G with the following property: for each oddprime p, HJ(G, Z) contains an infinite subgroup of exponent p! Despite thenegative nature of this example it suggests a question about finite groups.

Q5. Let G be a finite group. Is there a bound on the "size" of the integralhomology groups of G?

Now a group is termed acyclic if it has the integral homology of a point, i.e.its integral homology groups are trivial in all positive dimensions. Since the onlyacyclic finite groups are the trivial ones, one might ask

Q6. Can an infinite residually finite group be acyclic?I would like to turn next to

E. Linear groups. A group is here termed linear if it is (isomorphic to) asubgroup of a group of matrices over a commutative field. In the important andas yet unpublished work of Grunewald and Segal [12] that I have already citedthe conjugacy problem for arithmetic groups is completely settled.

El. Arithmetic groups have solvable conjugacy problem.

This in itself raises a number of questions, for exampleQ7. Does every finitely presented linear group have a solvable conjugacy

problem?Even the simplest looking linear groups can fail to be finitely presented. For

example, suppose that

a= (1

0 11and t=(10 3/2).

Then the subgroup G of GL(2, Q) generated by a and t is not finitely presented(Baumslag and Strebel [4]). This raises another question.

Q8. What makes a finitely generated linear group finitely presented?There is a connection between linear groups and residual finiteness that goes

back to Mal'cev [19].

E2. Finitely generated linear groups are residually finite.

Mal'cev's Theorem E2 provides us with a host of residually finite groups-Iwill discuss this further in the next section. It leaves completely unresolved,however, the question as to which groups are linear. There is a further theoremof Mal'cev which reduces this problem, at least in part, to the finitely generatedcase [19].


E3. A group is linear if and only if its finitely generated subgroups are linear ofbounded degree over a field of the same characteristic.

I would like to turn finally to

F. Discernment. It follows from E2 that free groups and certain groups with asingle defining relation, e.g. the fundamental groups of two-dimensional orienta-ble surfaces, are residually finite. Now not all one-relator groups are residuallyfinite. Oddly enough one-relator groups with nontrivial torsion seem to be muchsimpler than the torsion-free groups with a single defining relation. This suggeststhe following

Q9. Is every one-relator group with a nontrivial element of finite orderresidually finite?

One of the major difficulties in proving a group residually finite is to findsuitable finite images which "distinguish" its elements. In fact many of thesefinite images are forced to be perfect and so knowledge of perfect finite groupswhose elements enjoy a specific relation is an essential ingredient of the proof ofresidual finiteness.

The general discernment problem is the determination of all residually finitegroups. This is, in general, an impossible task. An extraordinary step forwardhas been made by Thurston in his recent work on hyperbolic manifolds whichyields in particular the theorem

F I. Knot groups are residually finite.

Here a knot is, as usual, a tame embedding of the circle in three space, and aknot group is simply the fundamental group of the complement of the knot.

REFERENCES

1. G. Baumslag, Subgroups of finitely presented metabelian groups, J. Austral. Math. Soc. 14(1973), 98-110.

2. , Residually finite groups with the same finite images, Compositio Math. 29 (1974),249-252.

3. G. Baumslag and E. Dyer, On the integral homology groups of finitely generated metabeliangroups (in preparation).

4. G. Baumslag and R. Strebel, Some finitely generated infinitely related groups with trivialmultiplicator, J. Algebra 40 (1976), 46-62.

5. R. Bieri and R. Strebel, Valuations and finitely presented metabelian groups, Proc. LondonMath. Soc. (to appear).

6. R. Camm, Simple free products, J. London Math. Soc. 28 (1953), 66-76.7. D. R. Farkas and R. L. Snider, K0 and noetherian group rings, J. Algebra 42 (1976), 192-198.8. E. Formanek, Conjugate separability in poly cyclic groups, J. Algebra 42 (1976), 1-10.9. A. P. Goryushkin, Imbedding of countable groups in 2-generator groups, Mat. Zametki 16 (1974),

231-235.10. K. W. Gruenberg, Residual properties of finite soluble groups, Proc. London Math. Soc. 7

(1957), 29-62.11. F. J. Grunewald, P. F. Pickel and D. Segal, Finiteness theorems for polycyclic groups, Bull.

Amer. Math. Soc. (N.S.) 1 (1979), 575-578.12. F. J. Grunewald and D. Segal, The solubility of certain decision problems in arithmetic and

algebra, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 915-918.13. P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. 4 (1954), 419-436.14. , On the finiteness of certain soluble groups, Proc. London Math. Soc. 9 (1959), 595-622.


15. , Embedding a group in a join of given groups, J. Austral. Math. Soc. 17 (1974), 434-495.16. G. Higman, A finitely generated infinite simple group, J. London Math. Soc. 26 (1951), 61-64.17. , Finitely presented infinite simple groups, Notes on Pure Math. 8 (1974), Inst. for

Advanced Studies, Australian National University.18. K. A. Hirsch, On infinite soluble groups. IV, J. London Math. Soc. 27 (1952), 81-85.19. A. I. Mal'cev, On isomorphic representations of infinite groups by matrices, Mat. Sb. 8 (1940),

405-422.20. , On some classes of infinite soluble groups, Mat. Sb. 28 (70) (1951), 567-588.21. C. F. Miller, III, On group-theoretic decision problems and their classification, Ann. Math.

Studies, vol. 68, Princeton Univ. Press, Princton, N. J., 1971.22. J. C. C. McKinsey, The decision problem for some classes of sentences without quantifiers, J.

Symbolic Logic 8 (1943), 61-76.23. Noskov, Cited in a report by V. N. Remeslennikov.24. A. Yu. Olshanskii, as above.25. I. A. Rips, Unpublished lectures at Warwick University, 1977.26. J. R. Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely

generated, Amer. J. Math. 85 (1963), 541-543.27. R. J. Thompson, A finitely presented infinite simple group (unpublished).

CITY COLLEGE OF CUNY

CLASSIFICATION THEOREMS FORTORSIONFREE GROUPS

L. G. KOVACS

I shall report on some little-known results, which are neither particularly newnor "mainstream" infinite group theory, but which can at least be transposed toa key not too far removed from that of this meeting.

As classifying infinite groups up to isomorphism is a hopeless goal (even inthe abelian case), let us consider a coarser equivalence relation: call two groups"similar" if each is isomorphic to some section of an unrestricted direct productof suitably many copies of the other. It is not hard to see that two finite simplegroups are similar if and only if they are isomorphic, but all nontrivialtorsionfree abelian groups are in a single similarity class: so this seems areasonable classification principle for the present occasion. To keep a long storyshort, until the concluding remarks we shall consider only torsionfree groups.

The easiest result to state extends our observation concerning abelian groups:the torsionfree metabelian groups which are nilpotent of a given nilpotency classc form a single similarity class, say, Mc. It may make more sense to write E forMo (the class of all groups of order 1) and A for Ml (the class of all othertorsionfree abelian groups). Other similarity classes of immediate interest are:the similarity class NN of F,/yc+,F, where F,. is a free group of rank r (finite orinfinite, but not smaller than c) and yc+,F, is the term of its lower central serieswhich yields a quotient of class precisely c. (It is, of course, nontrivial that NN isindependent of r.) Note No = E, Nt = A, N2 = M2, N3 = M3, but N4 M4.Before more can be said, we need to define a partial order on the set of allsimilarity classes of torsionfree groups: put X < Y whenever G E X and H EY imply that G is isomorphic to some section of an unrestricted direct productof suitably many copies of H. Then clearly

< N4E <A <Nt <N2 <N3 <M4 <N5 < ....

<M5

This poset is in fact a lattice (with 2"° elements). It is an open and, as far as Iknow, unexplored question whether this lattice is modular; the sublattice consist-ing of the classes of torsionfree nilpotent-by-abelian groups certainly is, and we

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226 L. G. KOVACS

shall not step beyond that today. We shall see from the next result that thelattice is not distributive.

Let (12 ... c) be a cyclic permutation, C the subgroup it generates in thesymmetric group S of degree c; take any faithful irreducible complex character yof C, induce to S and decompose:

75ddxxx

where the x, are the irreducible characters of S indexed as usual by thepartitions X of c. For each A, let Qx denote the lattice of all subspaces of arational vector space of dimension dd, and form the direct product lattice IIx Q,,.(If dd, = 0, interpret Qx as a singleton, redundant in this direct product.)

The sublattice (X I NN _ , < X < N,) is isomorphic to ll,, Q.

For each c in (1, 2, 3), one dd is 1 and all others are 0: thus there are nosimilarity classes strictly between NN_, and N.

For c = 4, two dX are 1, and all others 0, so in that case we have Figure 1where the unnamed point can be identified as the class of torsionfree nilpotentgroups of class precisely 4 with all 2-generator subgroups of class at most 3.

For c = 5, five of the d>, are 1 and the rest 0, so then we have a 5-dimensionalcube pictured so that one diameter is vertical. Indeed, one can show that by nowwe have seen all (39) members of the sublattice (X I X < N5), the only one notcovered by (*) being M5: this contains M4, and its join with N4 is a vertex of the5-cube adjacent to N4. "Recognition theorems" exist for each of these 39 classes.

FIGURE 1

FIGURE 2

CLASSIFICATION THEOREMS FOR TORSIONFREE GROUPS 227

For each c from 6 on, at least one dX is greater than 1, so the correspondingQ. is infinite and nondistributive. Very little is known about similarity classes oftorsionfree nilpotent groups not covered by for instance, we do not knowwhether we are missing finitely or infinitely many in (X I X < N6).

The only area where we have further conclusive information in the nilpotentcase is that of the centre-by-metabelian torsionfree nilpotent groups; the latticeof their similarity classes is shown in Figure 2.

Of the nonnilpotent story, we only know the metabelian case. Let M be theclass of the largest metabelian quotient of any noncyclic free group. If G is atorsionfree metabelian group not in M, then for some positive integer n thesubgroup of G generated by the commutators and the nth powers must benilpotent. This is a relatively elementary observation, but good enough to startus on the long road to the following conclusions. The lattice of similarity classesof torsionfree metabelian groups is distributive and satisfies the descendingchain condition, so each element of it has a unique expression as a finiteirredundant join of join-irreducibles. The join-irreducibles other than E or M areparametrized by ordered pairs (c, n) of positive integers: M, is the largestmember of the lattice which consists of groups in which the subgroup generatedby all commutators and nth powers is nilpotent of class c. The lattice can nowbe fully described by adding that Mb,,, < M, if and only if b < c and m1n.(Note our previous MM is now called MM.1.)

Does this mean anything for finite groups? The nilpotent classification de-scribed here did, in fact, start in the context of groups of prime exponent and isvalid there, mutatis mutandis, provided the prime is larger than the class.Extension to the prime-power exponent case presents no serious difficulties. Itcould be feasible to relax the small class restriction, say, to c < 2p, by exploitingthe fact that the symmetric group of degree c has cyclic Sylow p-subgroupsunder this assumption. One would need to use not only the appropriate modularrepresentation theory of the symmetric group, but also the corresponding part ofthe representation theory of GL(n, p) over GF(p), extended to representationsof the semigroup of all n X n matrices over GF(p). There is certainly scope inthis area for using some of the expertise present at this conference.

I must own up, though: there seems to be no information on similarity ofinfinite simple groups-for all we know, all simple groups of infinite exponentcould be similar to the noncyclic free groups. Also, similarity is an ad hoc termappropriated for the occasion; to contact with the literature, one has to translate"are similar" as "generate the same variety". The rest of our notation was alsoad hoc.

For a detailed exposition and references, see a recent paper of mine [5]. Thebasic ideas for the nilpotent case owe much to Magnus, but it was perhapsThrall who first glimpsed the whole picture [7]. G. Higman [2] and A. A.Kljacko [3] appear to have rediscovered it independently. Most of thetorsionfree nilpotent version was elaborated in 1967 by M. F. Newman and me,but remained unpublished until last year [4]. The nilpotent centre-by-metabeliancase is due essentially to A. G. R. Stewart [6]. The torsionfree metabelian resultswere found by M. F. Newman and me; they rely heavily on, and wereeventually incorporated into, the work of R. A. Bryce [1].

228 L. G. KOVACS

REFERENCES

1. R. A. Bryce, Metabelian groups and varieties, Philos. Trans. Roy. Soc. London Ser. A 266 (1970),281-355.

2. Graham Higman, Representations of general linear groups and varieties of p-groups, Proc.Internat. Conf. Theory of Groups (Canberra, 1965), Gordon and Breach, New York, 1967, pp.167-173.

3. A. A. Kljacko, Varieties of p-groups of a small class, Ordered Sets and Lattices No. 1, Izdat.Saratov. Univ., Saratov, 1971, pp. 31-42.

4. L. G. Kovacs, Varieties of nilpotent groups of small class, Topics in Algebra (Proc. 18th SRI,Canberra, 1978), Lecture Notes in Math., vol. 697, Springer-Verlag, Berlin and New York, 1978, pp.205-229.

5. , The thirty-nine varieties, Math. Sci. 4 (1979), 113-128.6. A. G. R. Stewart, On centre-extended-by-metabelian groups, Math. Ann. 185 (1970), 285-302.7. Robert M. Thrall, A note on a theorem by Witt, Bull. Amer. Math. Soc. 47 (1941), 303-308.

AUSTRALIAN NATIONAL UNIVERSITY, CANBERRA

PART III

Properties of the known groups

PROPERTIES OF THE KNOWN SIMPLEGROUPS

GARY M. SEITZ

In our discussion of the known simple groups we have two points of view.First, we are interested in those properties that are relevant to the classificationprogram. For the success of this program will depend, in part, on knowing agreat many properties of the known groups. Second, these groups are of interestin their own right and we are still a long way from a complete understanding ofthem.

Let `3C denote the current list of simple groups. So `3C consists of the simplegroups of Lie type, the alternating groups, and the sporadic groups. Because oftheir common theoretical framework we choose to concentrate on the groups ofLie type. We begin with a review of some of the properties of groups in `3C thathave been particularly important for work on the classification problem.

1. General properties. Let G be a simple group and suppose we know that allsimple sections of G are in X. To study G we take t E Inv(G). Set C = Ca(t)and C = Cl O(C ). Then either C has a component or C has a normal 2-sub-group, Q, such that Cc(Q) < Q. In the first case, we are immediately led toquestions about covering groups and automorphism groups of groups in X. Thisinformation is available and is the combined work of a number of authors. Inparticular, Steinberg [17], [18] gives general results for groups of Lie type,settling all but a few exceptional cases. Griess [9], [10] deals with certainexceptional coverings of groups of Lie type and with coverings of some sporadicgroups.

In the second case, C/Q may very well contain a component. So to fullyunderstand the structure of C we would need to know the action of thecomponent on chief factors of C lying in Q. This involves knowing the2-modular representations of coverings of groups in `3C. Here the availableinformation is incomplete, even for the groups of Lie type in characteristic 2.What is missing here is a description of the basic representations. One direction

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232 G. M. SEITZ

for future work would be the area of p-modular representations (irreducible,indecomposable, and projective) for groups of Lie type in characteristic p.Fortunately, the classification was able to proceed using previously knowninformation about particular groups (e.g. L2(q°)) and certain technical resultsdeveloped as needed (e.g. codimensions of fixed point sets of particular elementsacting on particular modules).

If we are presented with an explicit description of C, one uses this to build upthe structure of G. For example, if x e C we may try to find D = CG(x). Onemethod for doing this is to find CD(t) and then apply previous classificationtheorems. Of course CD(t) = Cc(x), so we are led to the problem of findingcentralizers of elements for groups in X. Much is known here, but there is stillwork to be done. Suppose, for example, x E L and L is a group of Lie type incharacteristic p. If x is a p'-element, then things are fairly well understood, butonly after the introduction of some fairly heavy machinery. Let L =O°'(Lo),where L is an appropriate algebraic group and a an endomorphism of L. Then xis contained in a a-stable maximal torus of L and CL-(x) is determined using theBruhat decomposition. Taking fixed points under a we obtain CL(x). Animportant advantage of this approach is the clear connection between CL(x) andthe root system of L. We will have further comments on this later and also oncentralizers of p-elements. There is more to do on the latter topic.

Along with centralizers of elements in C, we will need information on theconjugacy classes within C in order to study fusion. This, ultimately, reduces toquestions involving the conjugacy classes of groups in `3C (and their coveringgroups and automorphism groups). So already we require information along thefollowing lines:

(a) Covering groups;(b) Automorphism groups;(c) Modular representations;(d) Centralizers of elements; and(e) Conjugacy classes.

2. Specific properties. The previously discussed properties are all of interestquite independently of the classification program. However, a number of factsabout groups in `3C were verified mainly to meet the requirements of theclassification program. We mention a few of these that are particularly im-portant in the study of groups of characteristic two type. Let G be a simplegroup with proper simple sections in `3C and letp be an odd prime.

A notion of major importance is that of a p-strongly embedded subgroup.Recall, H < G is p-strongly embedded if p divides I H I and NG(U) < H for eachnontrivial p-subgroup of H. Of course, if the Sylow p-subgroups of G havep-rank 1, then one obtains such a subgroup by taking the normalizer of asubgroup of order p. But for groups of p-rank at least 2, such subgroups are rare.For example, if G is of Lie type, then with very few exceptions, G has Lie rank 1and is defined over a field of characteristic p (Theorem 7 of [14]). For thecomplete list of examples for the groups in `3C see p. 174 of [8].

Suppose H is p-strongly embedded in G and ZP X ZP = E < H. We mayassume that H is a maximal subgroup of G and we would like to prove that H is

PROPERTIES OF THE KNOWN SIMPLE GROUPS 233

the unique maximal subgroup of G containing E. Given E < L < G, then wehave L < H, provided L = <CL(e)Je E Es>. Therefore, we are led to theproblem of generation by centralizers for groups in `3C. Fairly complete informa-tion is available here and we refer the reader to [8] for a complete statement ofknown results. As a sample result we have: If L is a group of Lie type incharacteristic r 2, and if r gyp, then L = <CL(e)le E Ett> for each ZP X ZP

E < L (see Theorem 1 of [14]).Let x E G have order p. The group D = OP.(CG(x)) is of interest. Typically,

one is interested in restricting the structure of D. For p = 2, this is a reasonablegoal and the focus of much attention. Indeed if G E `3C, then D is cyclic. But forp > 2, very little can be said in general. For example, D may contain acomponent of CG(x). One nice case occurs when G is of Lie type in characteris-tic p. For the Borel-Tits theorem (3.12 of [4]) asserts that CG(x) is contained in aproper parabolic subgroup of G, and from here an application of the P X Qtheorem yields D = 1.

While there are many possibilities for D, there are restrictions on a relatedsubgroup. Suppose ZP X ZP = E < G and let 0G(E) = n ,1E' OP.(CG(e)). It isknown (Theorem 5 in [14]) that if G is a simple group of Lie type, then0G(E) = 1 for all possible E. However, DAut(G)(E) 1 for certain linear andunitary groups. The relevant notion is that of local 2-balance and we refer thereader to Chapter IV, §3 of [8] for a complete statement of results and asummary of applications. We simply remark that local 2-balance of the propersimple sections of G leads to a signalizer functor just as local balance yields asignalizer functor.

3. Root groups. Let G be a simple group of Lie type defined over the field Fqof characteristic p, and let X be a root subgroup of G, associated with a longroot in the root system of G. Exclude the cases G = L2(q), 2F4(q), and 2G2(q).The conjugates of X in G satisfy conditions that form the basis for some of themost important results in the classification program. Namely, for g,, g2 E G,

<Xg', Xg2> = X, X X X, SL(2, q), or a Sylowp-group of L3(q).In the latter case Z(<X91, X92>) - X. The SL(2, q) option necessarily occurs,

so let g E G satisfy J = <X, Xg> - SL(2, q). If q is even, let t E Xtt; and if q isodd, let <t> = Z(J). Then

(i) If q is even and gI, g2 E G, then tg1t921 = 1, 2, 4, or odd, and if 4, then(t91t92)2 ,,, t.

(ii) If q is odd, J < < Ca(t).So in case (i) t is a root involution in G, while in case (ii) t is a classical involutionin G. The combined work of Fischer [7], Aschbacher [1], Timmesfeld [19], andAschbacher [2] determines all finite simple groups containing such an involu-tion. These results are highly significant for the classification program, but alsouseful in the study of subgroups of known groups. For example, if X < Y =<X y> < G and if Y has no proper solvable normal subgroups, then we obtainthe abstract structure of G. Of course the list of possibilities must then benarrowed and the embeddings described. This has been carried out for theclassical groups by Kantor [11] and for the exceptional groups by Cooperstein[5]. Actually, their results are more general, only assuming x fl Y 1.

234 G. M. SEITZ

There is another way in which the root subgroups of G have been relevant tothe classification problem. This is linked with the Steinberg presentation of thecovering group of G. Let E be the root system of G and (a,, . . . , funda-mental roots for E. Choose a corresponding system of root subgroups (UQJa EY) of G, and set J , = <Ua,, U_a.>, for i = 1, ... , n.

We have G = <Jl, ... , J,,> and for each i j the group G. = <J;, Jj> is agroup with Lie rank 2. There is a result of Curtis (Theorem 1.4 of [6]) that, ineffect, asserts that if G* is a group containing subgroups Jl , ... , J,' withG* _ <J*, ... , J,> and <J,*, Jj*> naturally isomorphic to <J,, Jj> for each i, j,then G* is a central extension of G. This result has had a number of recentapplications of the following type. Given a group L, find subgroups L1, L2 of Lietype such that their root system are each part of a larger system and withsuitable labeling L1 = <JI , ... , J,'_ 1> and L2 = <J2 , . . . , J,>. Determine thestructure of <Jl , J,'> and apply Curtis' theorem to conclude that <L1, L2> is acentral extension of a group of Lie type, thereby identifying a large subgroup ofL.

4. Looking ahead. Having reviewed some of the work that has been done, wenow indicate some possible directions for future work. Let G = G(q) be a groupof Lie type in characteristic r and let E be the corresponding root system. Whenstudying various subgroups of G, things are relatively easy provided the sub-groups in question have a clear connection with the root system. But when sucha connection is not apparent, things are much more difficult. In some cases itappears almost irrelevant that G is a group of Lie type; for the special featuresof G are seemingly out of reach. For example, if x and y are semisimpleelements of G then the general theory described in §1 tells us a lot about thegroups CG(x) and CG(y). But what can we expect for the group A =<CG(x), CG(y)>? The group A is much harder to deal with, and this sort ofdifficulty is partly responsible for the long proofs required to establish some ofthe properties mentioned in §2.

A major project for the future is the systematic description of "relevant"subgroups of groups of Lie type. There are various possible interpretations ofthe word "relevant" and we will discuss some of these. As a guideline for suchwork we suggest never leaving a particular subgroup until its embedding isdescribed in terms of the root system of G or of some related group. We statethe following result as an indication of the type of theorem one might look for.Let T be a split torus of G and ( UQla E Y) the corresponding root subgroups.

THEOREM T [14]. Assume q is odd, q > 13, and T < Y < G. Then Y =YQNy(T), where Yo = <UQIUQ < Y>.

This result determines all subgroups of G that contain T, when q is suitablyrestricted. The point is that the structure of Yo is easily determined from Y,while Ny(T) is the preimage in N(T) of a subgroup of the Weyl group,W = N(T)/ T. In particular, the number of possibilities for Y is more or lessindependent of the field. The restrictions on q in Theorem T are too severe.Presumably, the restriction that q be odd is irrelevant, although there areexceptions when q < 13. But it should be possible to analyze these exceptions.

As a first problem we suggestProblem 1. Extend Theorem T to cover all q > 4.We observe that there are connections between the above theorem and

properties we have discussed so far. For let x be a semisimple element of G andE a subgroup of G with E = Z, X Z,. If x E T and y is any element of G, thenT < <CG(x), Cr(y))', while if E < T, then T < N(AG(E)) and T < N(<CG(e)leE Es>). So (for suitable q), each of these subgroups can be described in termsof 2. This could very well give a better approach to the verification of certainproperties in §2. Of course it need not be the case that x and E are contained ina split torus of G. However, if we consider arbitrary maximal ton', then one suchcontains x, and if p r, then usually E is contained in a maximal torus (in anycase it normalizes one). So it would be highly desirable to have a version ofTheorem T for arbitrary maximal tori.

Problem 2. Extend Theorem T to the case of an arbitrary maximal torus.'Our recent work has been directed toward a solution of Problem 2. The first

difficulty was how to even formulate such an extension. This led to a theory ofroot subgroups for an arbitrary maximal torus (see [16]). These root subgroupsare considerably more complicated than the usual root subgroups; the rootstructure is based on a certain orbit space of the root system of the associatedalgebraic group of G. Nonetheless, the situation is manageable and some of themost useful results can be proved in the more general context. For example, if Tis an arbitrary maximal torus of G and if J is any r-subgroup of G with jT = J,then (for q > 5) we can assert that J is a product of T-root subgroups.Consequently, the structure of J can be determined and the number of suchgroups is independent of q.

This result in the case of a split torus is a key result for the proof of TheoremT, and it is hoped that once the theory of T-root subgroups is more fullydeveloped it will be possible to give a solution to Problem 2. Partial solutionsexist at present, but there is still much to do. There are a number of additionalproblems associated with the above work on maximal tori. For example, thereare analogues of the notions of Borel subgroup and parabolic subgroup, basedon an arbitrary maximal torus, and these should be studied. These Borelsubgroups do not necessarily fall into a single G-conjugacy class of subgroups.Another topic of interest is the interplay between the lattices of subgroups thatcontain different maximal tori.

Problem 3. Study Borel subgroups and parabolic subgroups, based on anarbitrary maximal torus.

Problem 4. Study connections between the lattices of subgroups that containdifferent maximal tori.

Assuming that positive results arise from Problem 2 there are various ways ofextending the work to cover other "relevant" subgroups of G.

Problem 5. Let T be a maximal torus of G and Tt < T. Find all subgroups ofG that contain TI, when T, is one of the following:

'Added in proof. Problem 2 has now been solved (see [161). However, some of the proofs requirethe classification of finite simple groups.

236 G. M. SEITZ

(a) the fixed points of T under a field automorphism;(b) the centralizer in T of a particular r-subgroup of G; or(c) the group generated by a regular element of T or a set of such elements.Each of (a), (b), (c) represents a sizeable problem in itself, and results along

these lines would have numerous applications. For example, solutions to (a)would probably lead to new proofs that certain subgroups of G are maximalsubgroups. A solution to (c) might very well lead to a determination of allsubgroups of G that are transitive on the cosets of a parabolic subgroup of G.And a solution to (b) might be relevant to the determination of the conjugacyclasses of unipotent elements in G and provide a useful description of thecentralizers of such elements.

Problem 6. Describe all conjugacy classes of unipotent elements of G and thecentralizers of such elements.

In [3] all involutions in Aut(G) are determined along with their centralizers,for the case r = 2. Additional work has been done more recently by Mizuno[12]. For the case r = 2 and G an exceptional group, the results in [3] giverepresentations of the classes of involutions, in terms of the root system. Thecentralizers of these involutions are also given explicitly (in terms of the rootsystem). Such descriptions are useful for computations and it is hoped that ageneral solution to Problem 6 will include such precise information on central-izers of unipotent elements. A final remark here is that the method in [3] used todetermine the centralizers was to first find all maximal parabolic subgroups of Gcontaining a given centralizer. This method might be useful for the general caseand would probably give information along the lines of Problem 5(b).

Another area for future work is in the direction of possible extensions ofCurtis' result described in §3. Let T be a maximal torus of G. For each T-rootsubgroup X, of G there is an opposite T-root subgroup X,', and J, = <X, X> isof Lie type.

Problem 7. Let X1, . . . , Xk be T-root subgroups of G such that G =<J1, ... , Jk>. Let r be the graph with nodes J1, ... , Jk and J, joined to Jjprovided [J,, Jj] 0 1. Assume that r has no cycles. Let G* be the groupgenerated by isomorphic copies of J1, . . . , Jk and subject to all relations presentin the groups <J,, Jj> for i, j E (1, . . . , k). Is G* a central extension of G?

We note that the above formulation extends Curtis' result even when T is asplit torus. This is because the root subgroups involved are not assumed tocorrespond to a fundamental system of roots. Even if Problem 7 is not correct inthe above generality, there are certainly additional results in this area and theseshould be investigated.

A final problem is one that concerns subgroups of G generated by rootsubgroups for long roots. (We do not assign a number to this problem becausein large part the problem is one of reformulation of known results.) We wouldlike to see, at least for reasonably large q, a classification of those subgroupsY < G generated by long root subgroups of G and satisfying Sol(Y) = 1, thattreats the classical and exceptional groups simultaneously. Moreover, we wouldlike to see the results presented in a compact form, clearly indicating how thevarious possibilities for Y arise out of the root system of the associated algebraicgroup with certain twistings by elements of the Weyl group. Results presented in


this way would supplement the current tabular description of the groups andwould make the results easier to apply in certain contexts.

REFERENCES

1. M. Aschbacher, Finite groups generated by odd transpositions. I, II, III, IV, Math. Z. 127 (1972),45-46; J. Algebra 26 (1973), 451-459; 26 (1973), 460-478; 26 (1973), 479-491.

2. , A characterization of Chevalley groups over fields of odd characteristic, Ann. of Math.(2) 106 (1977), 353-398, 399-468.

3. M Aschbacher and G. Seitz, Involution in Chevalley groups over fields of even order, NagoyaMath. J. 63 (1976), 1-91.

4. A. Borel and J. Tits, Elements unipotents et sousgroupes paraboliques de groupes reductifs. I,Invent. Math. 12 (1971), 95-104.

5. B. Cooperstein, Subgroups of exceptional groups of Lie type generated by long root elements. I, II(to appear).

6. C. Curtis, Central extensions of groups of Lie type, Crelle J. 220 (1964), 174-185.7. B. Fischer, Finite groups generated by 3-transposition, University of Warwick, preprint.8. D. Gorenstein, The classification of finite simple groups. I. Simple groups and local analysis, Bull

Amer. Math. Soc. (N.S.) 1 (1979), 43-199.9. R. Griess, Schur multipliers of finite simple groups of Lie type, Trans. Amer. Math. Soc. 183

(1973), 355-421.10. , Schur multipliers of some sporadic simple groups, J. Algebra 20 (1972), 320-349.11. W. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math.

Soc. 248 (1979), 347-379.12. H. Mizuno, (preprint).13. M. ON an, Local properties of sporadic groups (unpublished).14. G. Seitz, Generation of finite groups of Lie type, Trans. Amer. Math. Soc. (to appear).15. , Subgroups of finite groups of Lie type, J. Algebra 61 (1979), 16-27.16. , Root subgroups for maximal Lori in finite groups of Lie type (preprint).17. R. Steinberg, Generateurs, relations, et revetements de grouper algebriques, Colloque sur la

theorie des groupes algebriques, Centre Belge de Recherches Mathematiques, Brussels, 1962, pp.113-127.

18. , Automorphisms of finite linear groups, Canad. J. Math. 12 (1960), 606-615.19. F. Timmesfeld, Groups generated by root involution. I, II, J. Algebra 33 (1975), 75-134; 35

(1975), 367-441.

UNIVERSITY OF OREGON

THE ROOT GROUPS OF A MAXIMAL TORUS

GARY M. SEITZ

Let G = G(q) be a finite simple group of Lie type in characteristic p, and let Tbe a maximal torus of G. We will associate with T a collection of "rootsubgroups". These subgroups generalize the usual root subgroups and areimportant in the analysis of subgroups of G containing T. _

Write G = O"'(G°), where G is an algebraic group defined over K = Fq and ais an endomorphism of G. Then there is a maximal torus, T of G, such that T isa-invariant and T = T n OP'(G°). Associated with T is a collection of rootsubgroups A = (UQIa E 2), where 2 is the root system of G. Then A = A° andwe write A = Al U u A, the orbit decomposition of A under <a>. Fori = 1, ... , v, let X, = <A,> and X, = O°'((X)°). The groups X1, ... , X, are theT-root subgroups of G.

LEMMA 1 . F o r i = 1, ... , v the group Xi is either a p-group or a group of Lietype defined over an extension field of F9.

Arrange notation so that X1, ... , X, are p-groups and X,+ ... , X. aregroups of Lie type. If T is a Cartan subgroup of G, then t = v and X1, ... , X,are the usual root subgroups, while if T is minisotropic we have t = 0. Thegroups Xl, ... , X, are well behaved, although they may have fairly complicatedstructure.

PROPOSITION 2. Fix 1 < i < t. Then Xi has nilpotence class at most c, where c isas follows:

c=1ifGc = 2 if G = PSp(n, q), 52,E (q), Sz(q);

c = 3 if G = E6(q), G2(q), 3D4(q), 2G2(q);c = 4 if G = E7(q), F4(q), 2E6(q);c = 5 if G = 2F4(q);c=6ifGme E8(q).

For i = 1, ... , t let V,. = X,/ b(X). Then T acts on each of the groupsV1, ... , V, and these may be regarded as Fp[ T]-modules.

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240 G. M. SEITZ

THEOREM 3. There is an integer N such that, for q > N and i j ET has inequivalent, irreducible representations on V; and V j.

Theorem 3 is a key result. The exact bound on q has not yet been completelyworked out. For classical groups, q > 5 will suffice'. There is additional infor-mation available. For if we set V, = K ® V, then V; viewed as a K[ T]-moduleis the direct sum of copies of the spaces U. for U. E A,, where we regard each ofthe U. as a 1-dimensional K[T]-module. Moreover, for q > N, the K[ T]-mod-ules UQ and UG are inequivalent whenever a f are roots in 2. Consequently,the module V; completely determines the orbit A,.

THEOREM 4. Suppose that q > N and that J is a T-invariant p-subgroup of G.Then J is the product of T-root subgroups.

THEOREM 4, for T a Cartan subgroup, is a helpful tool in the study of finitegroups of Lie-type. One consequence is a result that is quite useful for computa-tions in that it allows one to pass back and forth between the algebraic groupand the finite group.

THEOREM 5. Suppose q > N and X,,, ... , are T-root subgroups such that<X , ... , is a p-group. Then _

(i) (X,,,, . . . , <X,,,, . . . J,,k%; and

]0.

A main goal along these lines would be the following conjecture, generalizinga known result for the case of a split torus (see [1]).

CONJECTURE .2 Suppose q > N and T ' Y < G. Then(I) <Ty> = Yo T, where Yo = (X;lX; < Y>; and(ii) <X, I X, < Y> = OP'((x, I X, < Y>o)-

Statement (i) determines the possibilities for the group Y in terms of theT-root subgroups. Statement (ii) would yield the structure of the group Yo, sincethe structure of <X; I X; < Y> can be determined from 2. So, if true, theconjecture would provide an excellent description of all groups containing amaximal torus of G, at least for suitably large q. There are several difficulties toovercome in the proof of either of these statements. As a first approximation wehave the following result.

THEOREM 6. Suppose q > N, T < Y < G, and Y is solvable. Then(i) Y = OP(Y)Ny(T); and

-(ii) OP(Y) = <X,I X, < Y> = <Xi1 Xt < Y>0.

So the conjecture holds in case Y is solvable. We have established a numberof partial results that aim toward a proof of the conjecture. Theorem 6 should beuseful in the general proof; possibly reducing the problem to the case where Y isthe commuting product of quasisimple groups. But there are definite difficulties,even if Y is known to be a group of Lie type in characteristic p. In this last case

'It is now known that q > 5 suffices in all cases.2The conjecture has been established given certain field restrictions and assuming that simple

actions of G are of known type.

ROOT GROUPS OF A MAXIMAL TORUS 241

we do have a method for establishing the conjecture. Namely, first show that Tis a maximal torus of Y, and then show that a Cartan subgroup of Y, say H, isalso a maximal torus of G. At this stage we have a method for determining theembedding of Y in G, based on the torus H. The last step is to establish theembedding of Y, with respect to T.

There are a number of other properties of the T-root subgroups. For example,each X has an opposite, X,'. If i > t, then X; = X', while if X, is a p-group,X' = X for some j < t. These groups can be used as follows to give a betterunderstanding of the embedding of T-root subgroups.

THEOREM 7. Let Y = <X;, X7> and Y = O°'(Y0). Then the following conditionshold:

(i) Y is a commuting product of a <a>-orbit of semisimple groups, each generatedby T-root subgroups of G.

(ii) Y is a group of Lie type over an extension field of Fq._(iii) There is a unique T<a>-stable parabolic subgroup, P of Y, such that

(iv) If q > 4, then there is a unique parabolic subgroup, P of Y, satisfyingpT = P and X G Op(P).

In a number of cases we actually have X = OO(P). This is always the case forG = L (q) and often (but not always) the case when G is a classical group.

There are notions of Borel subgroups and parabolic subgroups for an arbi-trary maximal torus. As an example we state

PROPOSITION 8. The set (X1, ... , X,) can be partitioned as {X1, ... , X,) _{X;,...,X1) U (X;'...... X;') in such a way that each of U <X...... X;>and U' = <X7, . . . , X7> is a p group.

Given such a partition, the group B = UT would be a candidate for a T-Borelsubgroup. Such Borel groups will not necessarily fall into a single conjugacyclass. Along with further work on the T-parabolic subgroups, one can study theinterplay between the lattice of subgroups containing different parabolic sub-groups. We are also interested in replacing T by suitable subgroups of T.

REFERENCES

1. G. Seitz, Subgroups of finite groups of Lie type, J. Algebra 61(1979), 16-27.2. , Root subgroups for maximal Lori in finite groups of Lie type (in preparation).

GEOMETRY OF LONG ROOT SUBGROUPSIN GROUPS OF LIE TYPE

B. COOPERSTEIN

1. Introduction. Let G = GD(q), q = pe, be a finite group of Lie type, (D theassociated root system. Associated with the roots in (D are certain subgroups R.which are generally parametrized by Fq and elementary Abelian, and these arethe root subgroups. When a is long, D not 2B2 or 2F4, then R. is elementary oforder q unless G =2A2i,(q). In this case JRa = q3, and Z(Ra) is elementaryAbelian of order q. Let a* be the root of maximal height, Xa = Z(Ra) and set9C = HQ . We abuse language and say 9C is the conjugacy class of long rootsubgroups. Let r = U x e qcX so r is the class of long root elements of G. Fora subgroup K of G let r(K) _ {x e F: x E K) and '(K) = {x E: x <K). A subgroup K of G is an -subgroup, resp. F-subgroup, if K = <9c(K)>(K = <F(K))'). There has been a considerable amount of interest in the follow-ing two problems.

(A) Determine the-subgroups of G.(B) Determine the F-subgroups of G.In [19], J. McLaughlin states that J. Thompson posed problem (A) for

G = SL(V) and in this paper McLaughlin determines the-subgroups whenq > 2. In a second paper [11] the case q = 2 is handled. Of course this alsohandles the case of G = Sp(V). For an orthogonal group, problem (A) is doneby Betty Stark (Salzberg) [12]. Finally I have given a solution to (A) in the caseG is exceptional [1]. A number of individuals have considered problem (B) forSL(V), and it was completely solved by W. Kantor for all classical and lineargroups [8]. Recently problem (B) was completed for the exceptional group aswell [2].

Applications of these results can be made in such problems as finding therank three permutation representations of G (see cf. [9]), the subgroups ofminimal index of G (see [3]), and to the problem of showing certain subgroupsof G are maximal in G (see [4]). In this note we make explicit certain of theproperties of the Lie incidence structure on 9C.

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244 B. COOPERSTEIN

2. Preliminaries. For us a graph will consist of a pair (?X, f) with 6X. a set,f c c (2) that is 2-subsets of 9C. X. is called the vertex set and f the edges. Thusour graphs are undirected without loops or multiple edges.

An incidence structure is a pair (9C, l) consisting of a set IX whose elementsare called points and L a collection of distinguished subsets of 'A whosemembers are called lines, and each line has at least two points. (6X , p) is thick ifall lines have at least three points. The point graph of (A, L) is the graph withX as vertices and (x, y) on edge if and only if xy are collinear.

Now let G < E(X) be transitive, and f a nontrivial self-paired orbital for theaction of G on 9C, that is A. is a symmetric orbit for G acting on t X 6X andf I _ {(x, x): x E ¶ ). Such orbitals always exist if I G I is even. For x E 9C,let f(x) _ { y e 'X, : (x, y) E f) and x' = (x) a f(x). For C 9C, let ' -nyE.y y1 We now define the singular line on a pair (x, y) e f. If (x, y) E f,then xy = (x1 n y1)1 = [{x,y)1]1. Properties of such lines are well known(see [7]). Among them are

(1) If z E xy - (x), then xy = xz.(2) If z E X - xy and I f(z) n xyI > 2, then xy c f(z).(3) G yl is two-transitive on the points of xy.

Let 13 = {xy: (x, y) E f). Then ()C, L) is an incidence structure and thepoint graph of (9C, L) is (9C, f) (actually the edges are ((x, y): (x, y) E f)), butsince f is symmetric there is no loss in identifying the edges with f. By (2) thisstructure has the 0, 1, all property, that is for 1 E L, x E 9C - 1, f(x) n 11 is 0,

or 111.

Of course it may be the case that L = f, i.e. xy = (x, y). Instances whereff, L) is thick are or are not rare according to one's point of view. In generalthey seem to be a property of parabolic representations of groups of Lie type.

Let G = G,(F), 1 the root system of G, i t = (al, ... , a,) a fundamental basefor 1 and assumeI > 2. For J c{1,2,...,1),letJ'= (1,2,...,1) - J and21, = (: a > 0, spt(a) n {aj: j E J') 0>. Then NN(c21j) - P1 is a parabolicsubgroup of G and the action of G on the cosets of Pj is called a parabolicrepresentation of G. In [5] the following is proved.

THEOREM. If (G, `X,) is a parabolic representation of G, and G, --- Pj forx E XC, then there are JJ'J distinguished self-paired orbitals f for the action of G onX such that the lines with respect to f carry at least three points.

3. The Lie incidence structure on long root subgroups. Recall G. = GD(q), a,was the root of maximal height, Ra- the full root subgroup based on a*,Xa. = Z(R,,) is elementary of order q and 9C = X' . Let Q.1 = <Ra: a E -0 +B = 921H = No(Q1). Since H normalizes all root subgroups, H < N(Xa,). AlsoX, < Z(64), so B < NG(Xa.) and consequently N6(Xa,) is a parabolic sub-group, so the action (G, %) is a parabolic representation of G. We candetermine which parabolic subgroup from the extended Dynkin diagram. Recallthe extended Dynkin diagram is the Dynkin diagram together with a node for-a*, and we join this node to another in the usual manner. The extendedDynkin diagrams for groups with Weyl rank at least two (except 2F4) are

LONG ROOT SUBGROUPS IN GROUPS OF LIE TYPE

-a*

Ar

Bra2Dr+1

al a2 ar

al a2

. 0>0ar_ 1 ar

Cl, 2Ar+1 o0 0 _ O

a* al ar-1 at

Dr

-a*o----p_-al

O/a2a1_ - , a` O r

F4, 2E6 O 'a3 O-a* al a2 a4

E6

al a2 a3 a5 a6

0 0 O

a4

E7 00al a2 a3 1

E8

a5

- a* al a2 a3 a4

a6 a70-o-

a57

O

0 a6

245

Gig, 3D4 0a* al a2

246 B. COOPERSTEIN

Now the stabilizer X.. is found by deleting the nodes connected to -aVNote in all cases this is a maximal subgroup unless G = A,(q) = L,+1(q). Thusin all groups except A,(q) there is a unique distinguished orbit such that linescarry greater than two points. We describe the lines shortly.

Fundamental to the theory is the fact that there are few possibilities for<X, Y>, X z# Y E 6X. This is summed up in

THEOREM. Let X z# Y E 6JE,. Then one of the following holds

(I) [X, Y] = 1 , 1'(<X, Y>)I = q + 1 , <X, Y> = U 9((<X, Y>) Z#. In thiscase we say (X, Y) E f.

(ii) [X, Y] = 1, 6X (<X, Y>) _ {X, Y}. Here we write (X, Y) E f2.(iii) [X, Y] _ <X, Y>' = Z(<X, Y>) = Z E 6X, and (X, Z), (Y, Z) c f. Here

we say (X. Y) E f3.(iv) <X, Y> = SL2(q). We say X and Y are opposites, write (X, Y) C f, and

call ¶X,(<X, Y>) a hyperbolic line.

REMARKS. (1) In the case G = A,(q) = SL,+1(q), 1 > 3, f and f,, all occur. Inthis case f and f3 each break up into two suborbitals and f is the union of the twodistinguished orbitals. However, for application it is preferable to use the graph frather than either of the orbitals. In this case the line on (X, Y) c f is6X (<X, Y>).

(2) If G = C,(q) or A,(q) then f = 0 = f3 and f2 is the distinguished orbital.These incidence structures are the Symplectic and Unitary prepolar spaces.

(3) Suppose G = B,(q), D,(q), or 2D,(q). Then G = 7(V) for some vectorspace V. If the rank is at least four, thenf2 is a union of two orbitals. In all casesf is the distinguished orbital and for (X, Y) E f, XY = 6X,(<X, Y>).

(4) F4, E6, E7, E8 are all rank five on 6X, and each off, f,, is an orbital. f is thedistinguished orbital and the line on (X, Y) E f is 6X (<X, Y>).

(5) G2, 3D4 are rank four on 6X (f2 = 0), f is the distinguished orbital, andthe line on (X, Y) E f is again XX,(<X, Y>). Here the structure (6X,, 2) is ageneralized hexagon, dual to the standard hexagon for these groups. This is alsothe case for SL3(q), but here there are only two lines on a point.

This can all be proved by calculating in the Weyl group, W, that is if GG = Pj,determining the double cosets of Wj. It is also found that if WjwWj correspondsto f4 then WjwWj = wWj. From this it follows that

LEMMA. Op(Gx) acts regularly on f4(X) for X E 6JE,.

Now let G be any of the groups other than C,(q) or 2A,(q). Using the abovelemma the following is immediate

PROPOSITION. If 1 E 2, X E 1 and W E f4(X), then I f3(W) n l l= 1 and1 - f3(W) c f4(W)

For X E 6X,, let 2. = {l E 2 IX E 2 }, i.e. the lines on X. We say 0 C 6X, is asingular subspace of (6X,, 2) if for any X, Y E 0, (X, Y) E f and XY S A. In [5]it is shown the singular subspaces of any Lie incidence structure, together withtheir lines, have a structure as a projective space, and their stabilizers areparabolic subgroups. Now for X E 6JE, we can induce an incidence structure onex as follows: 1, m E Ex are adjacent or collinear if 6X (<I, m>) is a singular

LONG ROOT SUBGROUPS IN GROUPS OF LIE TYPE 247

plane on X. Then we can define the block on 1 and m to be (n E E:X E n C XX(<l, m>)). Let (X, W) E f4 and let A(X, W) = f(X) n f3(W), andset C(X, W) = (I E E: I C A(X, W)). Then the incidence structure induced onEx is isomorphic to (A(X, W), C(X, W)) by Z E A(X, W) goes to XZ. SetC = C(X, W) = C(<X, W>). Then C acts as an automorphism on (S(X, W),C(X, W)) and this is a Lie incidence structure for C (note that OP'(GX) =UP(GX) Q. By the results of [5] the representation (C, A(X, W)) can beidentified. Once this identification is made the following are immediate

LEMMA. If (Y1, Y2) E f3, [Y1, Y2] = X, then f(Y1) n f(Y2) = (X ).

LEMMA. Let A C IX be a singular subspace on X and Y E f(X) - A andassume A n f3(Y) 0. Then C< >(Y) is a hyperplane of <0> and Y induces on<A> the full group of elation with center X and axis C,a>(Y).

A subset d of . is said to be a subspace of 'X. if whenever 1 E C and11 n Al > 2, then I C A. For a subspace A of X, E(0) = (I E Cl 1C 0).

Now suppose (X, Y) E f2. Then clearly f(X, Y) is a subspace of X as isS(X, Y) = XE.(<X, Y,f(X, Y)>). From the identification of the induced inci-dence structure on E,r for X E `: and [6] we can deduce

THEOREM. The incidence structures (f(X, Y), C(f(X, Y))) and(S(X, Y), C(S(X, Y))) are prepolar spaces of orthogonal type and rank S(X, Y)= rank f(X, Y) + 1. Note if Z E S(X, Y) n f2(X), then S(X, Y) = S(X, Z).

The spaces that occur are given the following table

S(X, Y)

Al Al x Al

Br

Dr

2DI

F4

E6

Br-, or D3

D,_1 orD3

2Dr_ 1 or D3

B3

D4

E7 D5

Ea D7

This is especially useful.

LEMMA. If (X, W) E f4, Y E f2(X), then f2(W) n S(X, Y) contains a uniquepoint.

248 B. COOPERSTEIN

4. Subgroups generated by three long root subgroups. Crucial to the determina-tion of subgroups K of G with OP(K) = 1 and K = <X(K)> is the determina-tion of the group <X, Y, Z >, X, Y, Z E IX, with X, Y opposites. Let B =XX,(<X, Y>) and C = CG(<O>), N = NG(<O>). By exploiting the geometry offf, L) it is not difficult to find the orbits of N on X - B and to compute theisomorphism class of <9, Z > for a representative of each orbit. The possibilitiesfor <9, Z > are

op (.-(D)

(E4 in the case q = 2)

(SU3(2)' in the case q = 2)

and two other groups X with X = OP(X)<O >, and OP(X) either elementary oforder q3 or q4.

These relations are exploited continuously in the solution of problem A forgroups of Lie type.

REFERENCES

1. B. Cooperstein, The geometry of root subgroups in exceptional groups I, Geom. Dedicata 8(1979), 317-381.

2. , Subgroups of exceptional groups of Lie type generated by long root elements I. Odd

characteristic, J. Algebra. II, Characteristic two (submitted).3. , Minimal degree for a permutation representation of a classical group, Israel J. Math. 30

(1978).4. , Nearly maximal for the special linear group, Michigan Math. J. 27 (1980), 3-19.5. , Some geometries associated with parabolic representations of groups of Lie type, Canad.

J. Math. 28 (1976), 1021-1031.6. , A characterization of some Lie incidence structures, Geom. Dedicata 6 (1977), 205-258.7. P. G. Higman, Finite permutation groups of rank 3, Math. Z. 86 (1964), 145-156.8. W. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math.

Soc. 248 (1979), 347-379.9. W. Kantor and R. Liebler, Rank 3-subgroups of classical groups (to appear).

10. J. E. McLaughlin, Some groups generated by transvections, Arch. Math. 28 (1967), 364-368.11. , Some subgroups of SL (F2), Illinois J. Math. 15 (1969), 108-115.12. B. Stark, Irreducible subgroups of orthogonal groups generated by groups of root type I, Pacific J.

Math. 53 (1974).

UNIVERsrrY OF CALIFORNIA, SANTA CRUZ

S AND F-PAIRS FOR GROUPS OF LIE TYPEIN CHARACTERISTIC TWO

B. COOPERSTEIN

1. Introduction. Let G be a (simple) group of characteristic 2-type, X < G amaximal 2-local subgroup, so Q = "'M = 02(X ). Let T E Sy12(X). In prac-tice, in order to derive contradictions, by weak closure arguments, and othermethods, it is desirable that T E Sy12(G). This will be the case if we can findC char Q so C char T. Let M = S21(Z(Q)). This is one candidate for such asubgroup: if M 'Z Z(T), then M = S21(Z(T)) is characteristic in T. Thus we maysuppose M +C Z(T). Another possible candidate is the Thompson J-subgroup.

DEFINITION. For P a p-group, let m(P) be the maximal rank of an elementaryAbelian subgroup of P, ?l(P) = (E < P: I(E) = 1, m(E) = m(P)) and J(P)_ <91(P)>. Then for any R J(P) we have J(R) = J(P) char R.

Now suppose in our situation CX(M) = Q and set f = X/Q. Suppose thatJ(Q) r J(T). Then there is an E E 91(T) with E Q. Set B = E n Q andA = EQ/Q < X. Now since B s Q, M = S21(Z(Q)), BM is an elementaryAbelian subgroup of Q. Since E E 91(T), I E J > I BM I . However E l =IAI - IBIand therefore we have

IAI , IBI > IBMI = IBInIMI

=IBI IM:BnMI.Hence Al I> IM: B n MI > IM: CM(A)I since B n M = E n M < CM(A).This suggests the following

DEFINITION. A pair (Y, M) consisting of a group Y and a faithful F2 Y-moduleM is an F-pair [or is said to satisfy condition (F.F)] if there is an elementaryAbelian 2-subgroup A of Y, with Al I> IM: CM(A)I. We will call such anelementary subgroup an offending subgroup. Now set m = m2(Y) = m(T) forT E Syl2(Y) and suppose (Y, M) is an F-pair, A an offending subgroup. Lett E A". Then dimF2 M/CM(t) < dimF2 M/CM(A) S m(A) 6 m. Therefore anecessary condition that (Y, M) be an F-pair is that there exists t E Inv( Y) suchthat m = m2(Y) > dimF2 M/ CM(t). When this is satisfied we say (Y, M) is anS-pair.


249

250 B. COOPERSTEIN

2. S-pairs (Y, M) with Y E Chev(2). In joint work with G. Mason (see [2]) allS-pairs (Y, M) have been determined with Y E Chev(2), M an irreducibleF2 -module. These pairs are given in Table I of [2].

Suppose Y = G,.(2e) where V is the root system of Y and c is the untwistedroot system. Let 77 = {a1, ... , a,) be a fundamental base for c, and let A; be thefundamental weights, that is defined so 2(a,, Aj)/(a;, aj) = S,,. For each J C{1, 2, ... , 1) we have a basic weight, A, = > jEJ Aj and a basic module M(,\,)with A, as high weight. By Steinberg's tensor product theorem every irreducibleF2 Y-module is the tensor product

M(AJ)°' ® M(AA)o2 ®.. ®M(AJk)0k

where a...... o,, E Aut(F2e) and are distinct. For the most part the determina-tion of S-modules for Y reduces to the determination of the basic modules for Ywhich are S-modules. Thus in order to find the S-modules for Y, we will need

(1) m2(Y), then or at least upper and lower bounds for m2(Y),(2) representatives for the conjugacy classes in Inv(Y) and for each rep-

resentative t and basic module M = M(A), dimF2 CM(t) (or at least upper andlower bounds).

We first discuss (2): In [1] Aschbacher and Seitz determine the classes ofinvolutions in Y E Chev(2) and give representatives for each class. It is easilyverified that W = W(Y), the Weyl group of Y, contains a representative of eachclass. Also in [3] a direct, independent proof of this result is given using my Liegeometric methods. Thus we have some control over the classes in Inv(Y). Nowsuppose M = M(A). Set B(A) = AW and N(A) = F2eB(A). Then N(A) is a W-submodule of M and is a permutation module for W. Therefore for a E Inv( Y),

__ B(A)I + JFixB(A)(a)I2

Using either a concrete description of W or a formula due to P. McClurg [7] it iseasy to show that dimF2eCN(A)(a) is maximal when a is a long root element.Moreover we can get exact values for dim, 2eCN(,X)(a) for representatives a of allclasses of involutions in W. Once m2(Y) is known, we can determine the basicS-modules for Y.

3. The 2-rank of Y and some J-subgroups. In [2] several different methods areused to determine m2(Y) for Y E Chev(2), and in most instances we are able toget an exact value for m2(Y) and find the J-subgroup of a 2-Sylow. ForY = SP2-(2e) or we get upper and lower bounds which coincide. Thedetermination of m2(Y) when Y = GO(2e), c the root system for Y, where c hasonly roots of one length is considerably easier.

DEFINITION. A subset E of c+ is an Abelian set of roots if whenever a, /3 E E,then a + /3 is not a root. Let U = (Xe: a E cI > and for E C C+, E Abelian,

set U. = (X.: a E E>. By the Chevalley commutator relations, U,: is elemen-tary Abelian. By modifying a result of Mal'cev [5] on the maximal Abeliansubalgebras of a simple Lie algebra over C we are able to prove

THEOREM. Let a = max{lEI: E C (D+, E abelian). Then m2(G,(2e)) = ae.

S AND F-PAIRS FOR GROUPS OF LIE TYPE 251

In [5], Mal'cev also determines the maximal Abelian subsets of each indecom-posable root system. We give some examples and list some consequences.

EXAMPLES. (1) Suppose Y = A2m(2e) = SL2m+1(2e), ?r = (al, ... , a2m) afundamental base for the associated root system. There are two maximalAbelian subsets of +: 2m = (a > 01am E spt(a)) and Em+1 = (a > O1am+lE spt(a)). In this case, if U = <X a > 0>, then J(U) = UST U1:T. .

(2) Suppose Y = A2m_1(2e) = SL2m(2e), IT _ (al, ... , a2m_1) a fundamentalbase. Now there is a unique maximal Abelian subset of c+: 2m = (a > O1am Espt(a)). Here J(U) = UST, and in matrix notation we have

Gm- : , E MM(2e).IM

Now let Y1 = SLm(2e) x SLm(2e) and map Y, into SL2m(2e) by

(A, B) H

Under conjugation Y, acts on UST by

A

0

'J1t, )(A-'

I 0

A

0

0

B

B-1

Im A c,B -'

0IM

This action is equivalent to the action of Y, on M = M(2e) by (A, B) o 9TC =A I MB -'. A consequence of this is

LEMMA. (Yl, M) is not an F-pair.

We will make use of this in the last section.(3) Let Y = E7(2e), r = (al, ... , a7) with Dynkin diagram

5Ol0

03 0 0

Ia4

Now V has a unique maximal Abelian subset of roots 2, and 2 = (a > 01a7E spt(a)), 121 = 27. J(U) = U,: and is of course normalized by <X-q: 1 = Yo = E6(2e). As a module for Yo, Uj: is isomorphic to M(A6).

4. F-modules for groups of Lie type in characteristic 2. As remarked in § 1 anF-pair is necessarily an S-pair. Certain of these are rather obviously F-pairs.That these are the only F-pairs is our main result.

THEOREM [4]. Assume (Y, M) is an F-pair with Y E Chev(2), M an irreducibleF2 Y-module. Then we have one of the following.

(1) Y = SL(V), V an n-dimensional vector space over Fee and M is one of V,A2(V),

A"-2( V) = HomF2e(A2(V), F2e),

252 B. COOPERSTEIN

or

A" -I( V) = HomF,e( V, F2e).

[In Lie notation M = M(A,), M(A2), M(As_2), M(A"_,).](2) Y = Sp(V), V a 2m-dim vector space over Fee and M = V or m = 3, and

M is the spin module of dim 8 for Y = f27(2e). [In Lie notation M = M(A,) orwhen m = 3, M = M(A3).]

(3) Y = SU(V), Van n-dim vector space over Fee and M = V or n = 4 and Mis a 6-dimensional Fee space in A2( V) left fixed by Y. M = M(A1) or M(A2) in therespective cases. We remark that SU4(2e) 96 (2e) and this six-space over Fee isthe standard module for the latter group.

(4) Y = 2'(V), V a 2m-dimensional space over Fee, m > 3 and M = V, orm = 4 and M is one of the two spin modules of dimension eight (note V and thespin modules are conjugate under the triality automorphism), or m = 5 and M oneof the two spin modules of dimension sixteen.

(5) Y = 9-(V), V a 2m-dimensional space over Fee and M = V; or(6) Y = G2(2e) and M is the six-dimensional symplectic module for Y.

REMARKS. (1) Suppose Y = SL(V), M = A2(V), and dim V > 5. Let Vo be ahyperplane of V and X(V0) the transvections with axis V0. Then any offendingelementary subgroup of Y is conjugate to X(V0). When dim V = 4, an offendingelementary is either conjugate to X( Vo) or X(P), the transvections with center P,P a one-space of V. Also note that SL4(2e) = S26 (2e) and in this case if V is4-dimensional standard module for SL4(2e), then A2( V) is the six-dimensionalstandard module for f26 (2e)

(2) For V 5-dimensional over F2 the F-pair (SL( V), A2( V)) is especiallyinteresting since the extension of A2( V) by SL(V) occurs as a maximal 2-local inJ4, and this 2-local does not contain a 2-Sylow of J4. G. Mason characterizes J4by this property [6].

In proving this theorem the results stated in §3 are used often. In particularexample (3) eliminates the S-pairs (E6(2e), M(A;)), i = 1, 6. Example (2) can beused to eliminate most of the tensor-product S-pairs. The other S-pairs (Y, M)are eliminated by establishing certain lower bounds on the codimension in M ofCM(A) for nontrivial elementary 2-subgroups A. We illustrate this for the case ofY = A"_,(2e) 2- SL"(2e) on the adjoint module M = M(A, + A"_,). This mod-ule can be identified in the following way: Let Mo = M"(2e), M, = {71t E Mo:tr )1L = 0). Let Zo = (A I": A E F2e), Z, = M, fl Z,. Then M = M,/Z,.When n is odd, Z, = 0, Mo = M, ® Zo and M = M,. When n is even, Z, = Zo.

PROPOSITION. If 1 0 A < Y, A-elementary 2-subgroup, then IM,: CM,(A)l >qn-1 14

This is proved in a sequence of lemmas and by induction on n. Once this hasbeen established it is immediate, by our above remark, that if n > 3, n odd then(Y, M) is not an F-pair. Thus we may assume n = 2m.

DEFINITION. An involution T E SL(V), V a space over Fee has rank k ifdimF,, V/ C,i(T) = k. That is to say, in canonical form r has k Jordan blocks.

S AND F-PAIRS FOR GROUPS OF LIE TYPE 253

We next prove

LEMMA. If A is an offending elementary then A does not contain an involution ofrank k.

By straightforward linear algebra it is shown

dimF, M/ CM(T) _ 2m2 m even,2m + 1, m odd,

and since m(Y) = m2e the lemma follows.Again using linear algebra we show

LEMMA. If a E Inv(Y), rank a < m, then CM(a) = C,y (o)/ZI.

From this we immediately get CM(A) = CM (A)/ZI. This together with theproposition yields

THEOREM. (Y, M) is not an F-pair.

REMARKS. Many of these results have been simplified and extended by astudent of Geoff Mason's, Phil McClurg. In his dissertation [7] he determinesS + 1-pairs and F + 1-pairs and allows for outer involutions.

REFERENCES

1. M. Aschbacher and G. Seitz, Involutions in Chevalley groups over fields of even order, NagoyaMath. J. 63 (1976), 1-91.

2. B. Cooperstein and G. Mason, Some questions concerning representations of groups of Lie typeover fields of even order (unpublished).

3. B. Cooperstein, A remark on involutions in groups of Lie type over fields of characteristic two(unpublished).

4. , An enemies list for factorization theorems, Comm. Algebra 6 (1978), 1239-1288.5. A. I. Mal'cev, Commutative subalgebras of semisinple Lie algebras, Amer. Math. Soc. Transl.,

Ser. 1, vol. 9, Amer. Math. Soc., Providence, R. I., 1962.6. G. Mason, Some remarks on groups of type J4, Arch. Math. 29 (1977), 574-582.7. P. McClurg, Ph. D. Dissertation, University of California, Santa Cruz, 1979.

UNIVERSITY OF CALIFORNIA, SANTA CRUZ


GEOMETRIC QUESTIONS ARISINGIN THE STUDY

OF UNIPOTENT ELEMENTS

T. A. SPRINGER

1. Introduction.I.I. K is an algebraically closed field of characteristic p, either C or the

algebraic closure Fq of a finite field F4. Let G be a connected reductive linearalgebraic group over K. In the second case, G is defined over some subfield k ofK. We denote by F the corresponding Frobenius endomorphism of G. Thegroup GF of fixed points of F is then a finite group of Lie type (as in Curtis [7]).

Recall that we may imbed G as a Zariski-closed subgroup in someG is unipotent if all its eigenvalues are 1 (viewed as a matrix of

GL (K)). This is independent of the choice of the imbedding. If K = F4, anx E G is unipotent if and only if it has p-power order.

I shall discuss here some results and problems of a geometric nature, involvingunipotent elements, which one encounters, for example, in the study of the finitegroups of Lie type.

1.2. Notions and results about linear algebraic groups, mentioned withoutfurther reference, can mostly be found in [3]. If G is as before, let g be its Liealgebra. It can be viewed as a subalgebra of some Lie algebra gl (K) (alln X n-matrices). In q one has the notion of nilpotent element. Questions aboutunipotents in G often have an analogue for nilpotents in g.

In the study of unipotents exceptional situations are encountered in smallpositive characteristics. We shall say that p is good for G if either p = 0 or if pdoes not divide a highest root coefficient of one of the components of the rootsystem of G. Otherwise p is bad. In concrete terms: if G is simple, thepossibilities for bad characteristics are as follows, in the various simple types,A1:none; B,,C1,Dl:p=2;E6,E7,F4,G2:p=2,3;E8:p=2,3, 5. Quite a fewof the results to be discussed are already nontrivial and interesting in the casethat G = GL,(K), GF = where knowledge of the general theory ofreductive groups is not required.

1980 Mathematics Subject Classification. Primary 20G15, 14L35.m American Mathematical Society 1980

255

256 T. A. SPRINGER

2. Classification. In this section a brief review will be given of results about theclassification of unipotents in G and GF, known to me at the time of writing ofthis report. It has not always been possible to give precise references to resultswhich have not yet appeared in print.

2.1. If G = the theory of Jordan normal forms gives a classificationof unipotent conjugacy classes in G, and also exhibits representatives of theseclasses. The problem of doing this for general G was posed by Steinberg in histalk at the Moscow Congress [37, p. 279]. In my opinion it has not yet found asatisfactory solution. A result about the classification of unipotents is too often"in the form of a list, which, though finite, is very long, thus subject to error andinconvenient for applications" [loc. cit.]. The following general result has nowbeen established.

2.2. THEOREM. The number of unipotent conjugacy classes in G is finite.

In good characteristics this was proved, already some time ago, by Richardson(see e.g. [4, p. 185]). The general case was settled by Lusztig [20]. In his proof heexploits the Deligne-Lusztig approach to the representation theory of the finitegroups G F ([8], reported on in [36]) which uses the machinery of l-adic cohomol-ogy.

2.3. Problem. Prove the analogue for g of 2.2.Richardson's proof carries over and gives this result in good characteristics.

But Lusztig's proof does not carry over.Information about conjugacy classes of p-elements in the finite groups GF can

be deduced from information about unipotent classes in G (over Fq), via thefollowing result. We denote by ZG(x) the centralizer in G of x E G, and byZG(x)° its identity component. Put C(x) = ZG(x)/ ZG(x)°, this is a finite group.If F is a Frobenius endomorphism and x E GF, then F acts on C(x).

2.4. PROPOSITION. Let x E GF There is a bijection of the set of conjugacyclasses of GF fused in G with that of x, and the set of classes in C(x) for twistedconjugacy: c - c' if c' = dc(Fd)-I for some d E C(x). In particular: if ZG(x) isconnected then y E G F is conjugate to x in G F if and only if it is conjugate to x inG.

This is an easy application of Lang's theorem about algebraic groups overfinite fields, see [4, pp. 176-177] for details. One can also easily describe thecentralizer of x in GF (although this is not done explicitly in [loc.cit.]).

From 2.4 one sees that the groups C(x), for x unipotent, are important for theclassification of unipotent classes in GF. We shall say more about these groupsbelow.

There are three methods which have been used to obtain, for a given G,explicit information about unipotent classes (like a list of representatives).

2.5. If G is a classical group (a general linear, orthogonal or symplectic group,or a related one), the standard description of such a group enables one to findan explicit description of unipotent conjugacy classes. The simplest instance isthe description via Jordan normal forms in GL,(K), and this is the basis for thetreatment of the other cases. A thorough discussion can be found in [42]. Foranother treatment see [4, E IV], which however excludes the characteristic 2 case

THE STUDY OF UNIPOTENT ELEMENTS 257

in orthogonal and symplectic groups. This case is dealt with in [13]. In [4] onealso finds a discussion of the structure of centralizers. In particular, it can beestablished that all groups C(x) are elementary abelian 2-groups, in the case oforthogonal and symplectic groups.

2.6. Using the result stated in 3.1, one may reduce the classification ofunipotents in G to that of nilpotents in g, if the characteristic is good. The latterproblem may be treated by "linear methods". Let X be a nonzero nilpotent in g.If p = 0, or if p is sufficiently large, the theorem of Jacobson-Morozov holds. Itstates that X can be imbedded in a 3-dimensional simple subalgebra gI spannedby X and two other elements Y, H, such that [H, X] = 2X, [HY] = -2Y,[X, Y] = H. One knows that gI is unique up to conjugacy, and also that theconjugacy class of the semisimple element H determines that of X (see e.g. [4,pp. 235-241]).

Dynkin was the first to use these facts (forp = 0). In his basic paper [10] onefinds a great number of results. It contains tables of representatives of nilpotentconjugacy classes in the simple exceptional Lie algebras over C.

Dynkin's method has been elaborated upon by various authors. We mentionthe work of Bala and Carter [2], giving an approach to the classification ofnilpotents which, in principle, is independent of the use of the Jacobson-Moro-zov theorem, although this theorem is still used as a tool by them (which forcesone to make unnatural restrictions on the characteristicp). However, recently K.Pommerening has avoided the use of the Jacobson-Morozov theorem. He hasbeen able to establish that the Bala-Carter classification holds in all goodcharacteristics. In particular, the classification of nilpotents is "the same" in allgood characteristics.

A key result [2, Theorem 4.2] for the Bala-Carter classification, proved in [loc.cit.] via a lengthy case by case analysis, has recently been proved in a muchsimpler way by Kac [16]. Results about the structure of centralizers of nilpotentsin g have been given by Elashvili (see [11]).

The structure of the finite groups C(X) (defined for a nilpotent X similar tothe definition given before 2.4) has been determined for the exceptional typesover C by A. Alexeevsky [0]. It turns out that these groups are either elementaryabelian 2-groups, or symmetric groups S31 S4, S5 (the last possibility occurs inonly one case if g is simple, in type E8).

2.7. The semisimple linear algebraic groups and the corresponding finitegroups GF can be described by generators and relations, using Chevalley'sformulas. These are so explicit that they can be used, in principle, to computeconjugacy classes and centralizers. For groups of type G2 such computationshave been made by Chang and Enamoto ([5], [12]) and by Stuhler [41], also forthe Lie algebra case. For the finite groups of type F4 see the papers of Shoji andShinoda ([25], [24]).

In the case of the finite groups of exceptional types E6, E6, E8, the (verylaborious) computations have been made by K. Mizuno, in all characteristics (atthe moment of writing, only the results for E6 have been published [22]).

This review of known results could give some support to the opinion given in2.1. It should be added that quite recently Lusztig has stated a conjecture on aconnection between unipotent classes and affine Weyl groups. This connection

258 T. A. SPRINGER

might be the key to a better understanding of the classification. See Problem 5 inLusztig's contribution [21 ].

3. The unipotent variety. Let V be the set of unipotent elements of G and 8the set of nilpotent elements of g. Both V and 8 are irreducible affine algebraicvarieties, whose dimension is dim G - rank G (see [33]).

3.1. THEOREM. Let G be semisimple and simply connected, let p be good. Thereexists an isomorphism of algebraic varieties 0: V -. 8, such that O(xyx -') =Ad(x)i(y) (x E G, y E V). If G is defined over Fq then 0 may be taken tocommute with the corresponding Frobenius endomorphism.

Here Ad denotes the adjoint action of G on g. A result which is slightlyweaker than this theorem is proved in [33, p. 380]. The stronger result follows byusing the normality of 8, which is now known, and comes from a result ofDemazure [9, Theoreme, p. 287]. See also [13a, p. 147].

As was pointed out already in 2.6, this theorem can be used for the classifica-tion of unipotent classes. It is not true in bad characteristics (see [13, no. 6]).

3.2. Assume (for simplicity) that G is semisimple and simply connected. Letbe the "flag manifold" of G, i.e. the projective algebraic variety of all Borelsubgroups of G. If B0 is such a subgroup then J can be identified with thequotient variety G / Bo. The group G acts on J , by conjugation of Borelsubgroups, or equivalently, by left translations in G/Bo. Now let

V= ((x, B) E V X J3 I x E B),this is a closed subvariety of V X J3 , stable under the product action of G. Let7r: V -). V be the projection map. It is known that (V, ir) is a desingularization ofV, i.e. that V is a smooth (= nonsingular) variety, and 77 a proper, suoective andbirational morphism (see [39], see also Steinberg's contribution in these PRO-CEEDINGS [40]). It is clear that 77 is G-equivariant.

If x E V put -'x = ({x), 'x); then J?x is a closed subvariety of Jf. It canbe viewed as the variety of Borel subgroups which contain x, or as the fixedpoint set of x E G in Jf . The ffix are very interesting geometric objects, whichseem to be important in a finer study of unipotent elements.

3.3. PROPOSITION. (i) ffix is connected.(ii) All irreducible components of ffix have the same dimension, which is'(dim ZG(x) - rank G).(iii) If either p = 0 or p is sufficiently large then equality holds in the last

statement of (ii).

(i) is proved in [33, pp. 377-378]. For the first part of (ii) see [31] and for thesecond part [38, pp. 133-134] (this can also be deduced from the discussion in3.7). (iii) is proved in [39, Theorem 4.6]. This uses the Bala-Carter classification,mentioned in 2.6. The results of Pommerening, mentioned there, would imply(iii) for good characteristics.

3.4. Problem. Prove (iii) in all cases.As we saw, the good characteristics seem to have been settled. Also, if G is

simple of classical type the result is true, see [39, p. 217]. There remain the casesof the exceptional simple groups.


The case of type G2 can probably be handled without difficulty.3.5. If x is a nonsingular point of V then J?x is reduced to a point. These x are

the regular unipotent elements, whose properties are due to Steinberg (see e.g.[38, pp. 93-100)].

The next case is that of the subregular x E V, i.e. those for which dim1. In this case there are very interesting connections with the Kleinian surfacesingularities. This is discussed in Steinberg's contribution [40]. A thoroughdiscussion can be found in Slodowy's thesis [28].

Slodowy has studied the general over C, using the methods of differentialtopology. We quote one geometric result [29, 4.3, Proposition 1].

3.6. PROPOSITION. There is an irreducible affine algebraic variety, of dimensiondim ZG(x) - rank G, which has the same homotopy type as Jfx.

3.7. We can "assemble" the into a smooth variety, viz. the variety V of 3.2.A similar variety, introduced essentially by Steinberg [38, p. 133], will bediscussed now. For simplicity we assume k = C.

Fix a Borel subgroup B0, let To C B0 be a maximal torus and denote byW = Nc(T°)/T° the corresponding Weyl group. Recall that B, B' E i are saidto be in position w E W, if there are g, g' E G such that B = gB° g -', B' =g' B°(g') -', g -g' E B°wB°. Let X C V X 6B X J) be the set of triples(x, B, B') such that x c B n B'; it is a closed subvariety. If w e W, let X. C Xbe the set of those triples (x, B, B') for which B and B' are in position w. ThenX. is a locally closed subset of V X J) X J) , and one shows that the X. areirreducible algebraic varieties, all of the same dimension 2 dim Jf . It followsthat the closures Xw are the irreducible components of X.

These components can also be obtained in another way. First observe that Goperates on X, by conjugation on the three factors of V X J3 X ff, .

If x E X let S(x) denote the set of components of 1,%. The centralizer ZG(x)acts on Jfx, and the identity component ZG(x)° stabilizes each component.Hence the finite group C(x) = ZG(x)/ZG(x)° operates on S(x).

If C and C' are components of Jfx then {x} X C X C' is a subset of X; letXx,c,c, be the union of the G-conjugates of this set. Then Xx,c,c, is an irreduciblesubset of X, whose dimension is 2 dim ff, (here one uses 3.3(iii)). Consequently,the closure Xx c,c, must also be an irreducible component of X, from which oneconcludes that the sets Xw and Xx,c,c, must coincide. This implies the following.Let F be a set of representations of the unipotent conjugacy classes.

3.8. PROPOSITION. There is a bijection of W onto IIxEF (C(x) \ S(x) X S(x)).

Here C(x) \ S(x) X S(x) denotes the set of orbits of C(x), acting in S(x) XS(x).

For a further result of this kind see [32, Proposition 2.2].In the case of the classical groups the Jfx were studied thoroughly by N.

Spaltenstein [32], in all characteristics. We shall discuss some of his results forG = GL (and K arbitrary). It will appear that one finds geometric interpreta-tions of various combinatorial concepts.

3.9. Let G = GL,,, let x E G be unipotent. Take x in Jordan normal form,and write the sizes of the Jordan blocks which occur in descending order

260 T. A. SPRINGER

Al > A2 > .... Then Al + A2 + = n, hence A = (A1, AZ, ...) is a partitionof n. It follows that the unipotent conjugacy classes of G are parametrized bythe partitions of n. One knows that a partition A of n can be represented by aYoung diagram D(A), i.e. an array of n points in the plane in consecutive rows oflength Al, A2, .... all rows starting in the same column. For example,D((4, 2, 2, 1)) is

A Young standard tableau of shape D(A) is a labelling of the n points of D(A) bythe integers 1, 2, ... , n in such a manner that the labels increase in the rows(going to the right) and in the columns (going down).

The next result, proved in [30], gives a geometric interpretation of standardtableaux. For more general results see [23] (see also [14]).

3.10. PROPOSrrION. These is a bijection of the set of irreducible components ofix onto the set of standard tableaux of shape D(A).

It is of interest to know (for example in connection with the questions raisedin [18, 6.3]) when two components of J% have an intersection of dimensiondim 1. Hence the following problem.

3.11. Problem. Study the intersection pattern of the irreducible components of J?xfor G = GL,,.

Results about this problem will probably lead to interesting combinatorialfacts about standard tableaux.

We next turn to an interpretation of 3.8 in this case. In GL all centralizersare connected [4, p. 233], so all groups C(x) are trivial (x unipotent). In thatcase, using 3.10, we obtain from 3.8 a bijection of the symmetric group S. ontothe set of ordered pairs of standard tableaux of the same shape, on n points.Such a bijection has been given by Robinson and Schensted (see e.g. [6, pp.29-136], or [19, pp. 48-60]).

3.12. PROPOsrrION. The bijection of 3.8 coincides with that of Robinson andSchensted.

This has been proved by Steinberg (unpublished). The following problem isone which arises naturally here.

3.13. Problem. Describe the of 3.8 combinatorially in the case of theother classical groups.

4. Representations of Weyl groups. From now on, assume that either p = 0 orp is sufficiently large. The notations are as before.

4.1. It has become apparent that the geometry of the algebraic varieties J%x istied up with the representation theory of the Weyl group W.

We need the cohomology groups H'(J%), with coefficients in a suitable fieldE. If K = C, the X are compact complex algebraic varieties, and the cohomol-ogy groups are classical. We may take E = Q.

Over any field K, we have l-adic cohomology groups, where I is a prime,I p. Now E is any extension of the field Q1 of 1-adic numbers. If K = C, thel-adic groups are "the same" as the classical ones.


The are the finite dimensional vector spaces over E. Putting

e(x) = 2 (dim ZG(x) - rank G)

-p being so large that dim fx = e(x), cf. 3.3(iii)-we have that H'(ff, x) = 0unless 0 4 i 4 2e(x). One can construct representations of Win the H'(63x). Inthe case K =f1_1 a construction in 1-adic cohomology is given in [34]. Thisconstruction emerges in the study of certain trigonometric sums. A version ofthe construction for K = C, in classical cohomology, was given in [35]. Anotherprocedure for obtaining representations of W in H'(63x) was given by Slodowy[29, no. 4]. This is an outgrowth of a study of the singularities associated tounipotent elements, generalizing that for the case of subregular elements (whichare discussed in Steinberg's contribution [40]). It is likely that these representa-tions of W in H'(Jfx) coincide with the ones of [34] and [35] tensored with thesign representation, but this has not yet been fully established. It is true, at anyrate, if G = GL (see [29, 4.6]). We shall discuss two questions where the Weylgroup representations of [34] and [35] are useful: the description of Greenfunctions of finite Chevalley groups and the realization of irreducible represen-tations of Weyl groups.

Yet another construction of Weyl group representations is given in [18a].4.2. Assume that G is defined over Fq and take K =F9. Let F be the

Frobenius endomorphism. Choose To to be an F-stable torus. Then the G F-con-jugacy classes of Fq-tori of G are parametrized by twisted conjugacy classes inW (as in 2.4.). If T is such a torus, let w(T) E W be a representative of thecorresponding class.

If x E G F then F acts on ffix and also (linearly) in the cohomology groupsH'(fx).

Recall that the Green function Q ° is the value of the Deligne-Lusztig characterRT (B) of GF on the unipotent elements of GF, where 0 is any character of TF,with values in a suitable extension E of Q, (which is also the coefficient field forcohomology). We refer to B. Srinivasan's contribution [36] for more details.

4.3. THEOREM. If p and q are sufficiently large there exist representations r' of Won H'(Jfx) such that

QT (x) _ (- l)'Tr(Fr'(w(T)), H'(3x)).i>o

Here x is a unipotent element of G F.This follows from [17] and [34]. The theorem gives a complement to the

character formulas of Deligne and Lusztig [8] (see also [36]). The methods of [8]do not (as yet) give character formulas on unipotent elements.

In the proof of 4.3 the Lie algebra gF is heavily used.4.4. Problem. Give a proof of 4.3 avoiding the use of the Lie algebra, and

working for all p and q.4.5. Now let K = C and assume G to be simple. Let Rx = HZ`(x)(6ix, Q) be

the top cohomology group (see 4.1). Then Rx has a basis (ec) indexed by theirreducible components C of JJx.

262 T. A. SPRINGER

Now the centralizer ZG(x) acts on hence on the cohomology of 63., andthe connected subgroup ZG(x)° acts trivially. It follows that we have a represen-tation of the finite group C(x) in the and it turns out that it commuteswith the representation of W in that space which was mentioned in 4.1 (this istrue over any field K, see [34]).

In particular, we get a representation of C(x) X W in R.. Let 0 be anirreducible character of C(x). Using the fact that the corresponding representa-tion is defined over Q (see 2.6), we obtain a direct sum decomposition

R. = ®R.,O,

where Rx,, denotes the i-isotypic subspace of R., relative to the representationof C(x) in that space. For any 0 such that Rx,, 0 there is a character X,,, of Wsuch that the character of the representation of C(x) X W in the (C(x) X W-stable) subspace Rx,, is 0 ®Xx,O

4.6. THEOREM. The Xx,, are absolutely irreducible characters of W. Eachabsolutely irreducible character of W is a XY 0, the pair (x, 0) being unique up toconjugacy.

4.7. COROLLARY. The irreducible representations of W are defined over Q.

4.8. COROLLARY. The map Xx., H x defines a surjective map of irreduciblerepresentations of W onto unipotent classes of G.

These results are proved in [34] and [35]. The rationality result 4.7 was firstproved, via a case by case check in [1].

4.9. If G = PGL,,, then W = S. and all C(x) are trivial. The map of 4.6 thenturns out to be the identity map of the set of partitions of n: these parametrizeboth the irreducible characters of S. and the unipotent classes of PGL (see [15,Proposition 27]).

In the case of the other classical groups the parametrization of the irreduciblecharacters of W given by 4.6 has been described by T. Shoji [26]. He alsostudied the case of the exceptional group F4 [27]. The (easy) case of G2 ishandled using [34, pp. 205-206].

We have seen that the space R. comes with a distinguished basis {ec}. Lusztigand Hotta (unpublished) have studied the action of a simple reflection on thisbasis. An axiomatic description of representations of Weyl groups, and moregenerally of representations of Hecke algebras of Coxeter groups, in spaceshaving a distinguished basis with similar properties is given by Kazhdan andLusztig in [18].

REFERENCES

0. Alexeevsky, Component groups of centralizers of unipotent elements in semi-simple algebraicgroups, Sakharth. SSR Mecn. Akad. Math. Inst. Srom (= Akad. Nauk Gruzin SSR Trudy TbilisiMath. Inst. Razmadze - Publ. Math. Inst. Tbilisi) 52 (1979), 5-27. (Russian)

1. M. Benard, On the Schur indices of characters of the exceptional Weyl groups, Ann. of Math. (2)94 (1971), 89-107.

2. P. Bala and R. Carter, Classes of unipotent elements in simple algebraic groups. I, Math. Proc.Cambridge Philos. Soc. 79 (1976), 401-425; II, ibid. 80 (1976), 1-17.


3. A. Borel, Linear algebraic groups, Benjamin, New York, 1969.4. A. Borel et al., Seminar in algebraic groups and related finite groups, Lecture Notes in Math.,

vol. 131, Springer-Verlag, Berlin and New York, 1970.5. B. Cbang, The conjugate classes of Chevalley groups of type (G2), J. Algebra 9 (1968), 190-211.6. , Combinatoire et representation du groupe symetrique, Lecture Notes in Math., vol. 579,

Springer-Verlag, Berlin and New York, 1977.7. C. W. Curtis, Problems concerning characters of finite groups of Lie type, these PROCEEDINGS,

pp. 293-299.8. P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math.

(2) 103 (1976), 103-161.9. M. Demazure, Invariants symetriques en tiers des groupes de Weyl e1 torsion, Invent. Math. 21

(1973), 287-301.10. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. Ser.

2, vol. 6, 1957, pp. 111-245 = Mat. Sb. (N. S.) 30 (1952), 349-462.11. A. G. Elashvili, Centralizers of nilpotent elements in semi-simple Lie algebras, Sakharth. SSR

Mecn. Akad. Math. Inst. Srom (= Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze - .Publ. Math. Inst. Tbilisi) 46 (1975), 109-132. (Russian)

12. H. Enamoto, The conjugacy classes of Chevalley groups of type (G2) over finite fields ofcharacteristic 2 or 3, J. Fac. Sci. Univ. Tokyo 16 (1970), 497-512.

13. W. H. Hesselink, Nilpotency in classical groups over a field of characteristic 2, Math. Z. 166(1979), 165-181.

13a. , Desingularization of varieties of nullforms, Invent. Math. 55 (1979), 141-163.14. R. Hotta and N. Shimomura, The fixed point subvarieties of unipotent transformations on

generalized flag varieties and the green functions-combinatorial and cohomological treatments centeringGL,,, Math. Ann. 241 (1979), 193-208.

15. R. Hotta and T. A. Springer, A specialization theorem for certain Weyl group representations andan application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), 113-127.

16. V. Kac, Some remarks on nilpotent orbits, (preprint).17. D. Kazhdan, Proof of Springer's hypothesis, Israel J. Math. 28 (1977), 272-286.18. D. Kazbdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent.

Math. 53 (1979), 165-184.18a. , A topological approach to Springer's representations, Adv. in Math. (to appear).19. D. E. Knuth, The art of computer programming, vol. 3, Addison-Wesley, Reading, Mass., 1973.20. G. Lusztig, On the finiteness of the number of unipotent classes, invent. Math. 34 (1976),

201-213.21. , Some problems in the representation theory of finite Chevalley groups, these PRocEEI -

INOS, pp. 313-317.22. K. Mizuno, The conjugate classes of Chevalley groups of type E6, J. Fac. Sci. Univ. Tokyo 24

(1977), 525-563.23. N. Shimomura, A theorem on the fixed point set of a unipotent transformation on the flag

manifold, J. Math. Soc. Japan 32 (1980), 55-64.24. K. Shinoda, The conjugacy classes of Chevalley groups of ty pe (F4) over finite fields of

characteristic 2, J. Fac. Sci. Univ. Tokyo 21 (1974), 133-159.25. T. Shoji, The conjugacy classes of Chevalley groups of type (F4) over finite fields of characteristic

p 2, J. Fac. Sci. Univ. Tokyo 21 (1974), 1-19.26. , On the Springer representations of the Weyl groups of classical algebraic groups, Comm.

Algebra 7 (1979), 1713-1745 and 2027-2033.27. , On the Springer representations of Chevalley groups of type F4, ibid. 8 (1980), 409-440.28. P. Slodowy, Einfache Singularitoten and einfache algebraische Gruppen, Regensburger Math.

Schr., vol. 2; English transl., Lecture Notes in Math., Springer-Verlag, Berlin and New York (toappear).

29. , Four lectures on simple groups and singularities, Comm. Math. Inst. Univ. Utrecht, no.11, 1980.

30. N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold, Proc.Kon. Akad. Wetenscb Amsterdam 79 (1976), 452-456.

31. , On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology16 (1977), 203-204.

264 T. A. SPRINGER

32. , Sous-groupes de Borel contenant un unipotent donna, Lecture Notes in Math.,Springer-Verlag, Berlin and New York, (to appear).

33. T. A. Springer, The unipotent variety of a semi-simple group, Bombay Colloq. AlgebraicGeometry, Oxford Univ. Press, London, 1969, pp. 373-391.

34. , Trigonometric sums, Green functions of finite groups and representations of Weyl groups,Invent. Math. 36 (1976), 173-207.

35. , A construction of representations of Weyl groups, Invent. Math. 44 (1978), 279-293.36. B. Srinivasan, Characters of finite groups of Lie type. II, these PROCEEDINGS, pp. 333-339.37. R. Steinberg, Classes of elements of semisimple algebraic groups, Proc. Internat. Congr. Math.,

Moscow, 1966, pp. 277-284.38. , Conjugacy classes in algebraic groups, Lecture Notes in Math., vol. 366, Springer-

Verlag, Berlin and New York, 1974.39. , On the desingularization of the unipotent variety, Invent. Math. 36 (1976), 209-224.40. , Kleinian singularities and unipotent elements, these PROCEEDINGS, pp. 265-270.41. U. Stuhler, Unipotente and nilpotente Klassen in einfachen Gruppen and Lie-Algebren vom Typ

G2, Proc. Kon Akad. Wetensch. Amsterdam 74 (1971), 365-378.42. G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral.

Math. Soc. 3 (1963), 1-62.

RIJKSUNIVERSITEIT UTRECHT, THE NETHERLANDS


KLEINIAN SINGULARITIES ANDUNIPOTENT ELEMENTS

ROBERT STEINBERG

1. Let F be a finite nontrivial subgroup of SU2(C). Then C2/F is a surfacewith an isolated singularity at the origin. These singularities were classified andstudied by Klein in 1872 (or so) in his work on the invariant theory of theregular solids in R3. However, they also arise in many other contexts which arenot totally understood, including that of simple (Lie or algebraic) groups. Thepurpose of this talk was to discuss some of these connections, concentrating onthe connection with simple groups, and then at the end to mention some areasthat (we think) deserve further investigation. The main reference is [S] whichcontains a comprehensive treatment of the subject, including exact definitions,and an extensive bibliography.

2. The Kleinian singularities. Here is Klein's classification.

F I F1 I Degrees of invariants I Relator Type

Cyclic r+l 2, r + 1, r + 1 X'+1 + YZ Ar

Bin. dihed. 4(r-2) 4, 2(r - 2), 2(r - 1) Xr-1 + XY2 + Z2 Dr

Bin. tetra. 24 6, 8, 12 X4 + Y3 + Z2 E6

Bin. octa. 48 8, 12, 18 X 3Y + Y3 + Z2 E7

Bin. icosa. 120 12, 20, 30 X5+ Y3+ Z2 ES

Bin. = Binary, dihed. = dihedral, tetra. = tetrahedral,octa. = octahedral, icosa. = icosahedral.

Each F here is just the stabilizer in SO3 of a regular solid in R3, which may bedegenerate, lifted from SO3 to SU2 via the spin map. In each case the polynomi-als on C2 invariant under F form an algebra with three generators X, Y, Zbound by a single relation as given in the table, and this yields C2/ F as asurface in A3 with a singularity at 0. For example, in the first case F is cyclic,

1980 Mathematics Subject Classification. Primary 20G20, 14J17; Secondary 20H 15.m American Mathematical society 1980

265

266 ROBERT STEINBERG

generated, in terms of the underlying coordinates, by the map (u, v) -* (au, a-'v)with a a primitive (r + 1)th root of 1. Generating invariants are thus X = uv,Y = u'+', Z = v'+', and the relation X'+' - YZ = 0 on them implies allothers. In the second case we take r + 1 even in the first case, thus replace r + 1by 2(r - 2), and adjoin (u, v) -* (-v, u) to our group. The invariants that remainare generated by X = u2v2, Y = u2r-4 + v2r-4, Z = uv(u''-4 - V2r-4) andthese satisfy -4Xr-1 + XY2 - Z2 = 0. The remaining cases are more com-plicated. The last column will be explained presently.

Our discussion of these singularities will be local. Thus only the germs of thesurfaces, algebraic or analytic as the reader chooses, will be relevant, andsimilarly for various functions that occur along the way. As examples we notethat the singularity represented by X2 + cX 3 at 0 is isomorphic to that repre-sented by X2 under the local map X -*XV+ cX , while that represented byXY(X - Y)(X - cY) (c E C) depends essentially on c since any local isomor-phism acts linearly mod terms of degree 5 and higher and hence fixes thecross-ratio of the 4 linear factors, which is c.

3. Other incarnations. (a) Let f: C3 -* C, f(0) = 0, have a singularity at 0.Assume that in "the space of all such singularities" there is a neighborhood of fcontaining only finitely many singularities up to isomorphism (e.g. this conditionfails for the example at the end of §2). Then f is a Kleinian singularity, i.e., isgiven by one of the polynomials in the table in §2.

This result of Arnold and of Siersma shows that the Kleinian singularitiesoccupy an open part of "singularity space" and can be expected to occurfrequently, as in fact they do.

(b) Let R be a nonregular two-dimensional analytic local ring in which uniquefactorization holds. Then R is of type E8, i.e., isomorphic to the local ring ofC(X, V. Z)/(X2 + Y3 + Z5) at 0 or its completion.

Brieskorn [B1] is responsible for this remarkable result.

(c) THEOREM. For an isolated normal surface singularity (S, so) the following areequivalent: (1) it is Kleinian, (2) it is rational with multiplicity 2, (3) it is rationalwith imbedding dimension 3, (4) the minimal resolution has as singular fibre aunion of projective lines having intersection matrix minus that of the Cartan matrixof the root system of type A,, D,, E6, E7 or E8. The correspondence between (1) and(4) is as in the table of §2.

Du Val [D] and Artin [Ar] get the credit here.Explanation. A resolution of the singularity of S at so is a map ir: S' -* S with

S' a nonsingular surface and it an isomorphism from r'(S - (so)) to S - (so)-The singular fibre is ,r-1(so). "Rational" means, roughly, that S' necessarily hasthe same arithmetic genus as S.

EXAMPLE. Following Brieskorn (see [B2] for the reference) we give a resolutionof the singularity of type A,. We have S: Xr+1 = YZ, so = (0, 0, 0), all in A3.Let U1, U2, ... , U, be coordinates in (P')'. Define a surface S' in A3 X (P')'thus:

XU1 = Y, XU,+1 = U; (1 < i < r - 1), X = U,Z.

KLEINIAN SINGULARITIES AND UNIPOTENT ELEMENTS 267

Then if 7r: A3 X (P')' -* A3 is the natural projection, 7rl s.: S' -* S is a resolutionof (S, 0). The relevant points, that S' is nonsingular and 77 an isomorphism onS' - i7'(0), are easily checked (the first, e.g., since the Jacobian of the equa-tions for S' has rank r + 1 everywhere). To get a typical point(0, 0, 0; U1, U2, ... , U,) of the singular fibre, we let j (1 < j < r) be thesmallest index for which L. , 0. From the equations for S' it readily followsthat U, = 0 for i <j and U; = oo for i >j; and conversely. Thus the singularfibre is the union of r projective lines Lj (1 < j < r) and these intersect in the A,pattern as shown.

Brieskorn has also given ad hoc resolutions of the other Kleinian singularities,that of type E8 being quite complicated indeed.

4. Universal deformation (or unfolding). For a surface singularity (S, so) this is(roughly) a map

7r: (B, bo) - (T, to), 7r(bo) = to

with B nonsingular and (ir-'(to), bo) isomorphic to (S, so). The fibres representthe stages that the original singularity (in the fibre over to) goes through as it getsdeformed (unfolded). For universality it is required that every deformationfactor uniquely through this one.

so

to

EXAMPLE. For type A, the parameter space is A' with coordinates To,T1, ... , T,_1, B is the zero set of YZ + X'+' + Ti_1Xi 1 + Ti_2Xi-2+ + To, and 77 is projection onto the space of the first three coordinatesX, Y, Z. The rule here, which applies quite generally, is to add on to the originalpolynomial f(X, Y, Z) (as in the table of §2) the above linear combinationformed from a set of polynomials ((X'-', Xi-2, ... , X, 1) for type A,) whichprojects into a basis for C[X, Y, Z] modulo the ideal generated by of/aX,of/aY, of/az.


5. Resolution (desingularization) of the unipotent variety. From now on G is asimple, simply connected Lie or algebraic group over C of type A,, D,,, E6, E7 orE8. V is the variety of unipotent elements of G, B is a Borel subgroup and G/Bis the flag manifold.

THEOREM (a). Let W be the subset of V X G/B formed by all (x, gB) such thatxgB = gB. Then the natural projection 7ri: W -* V is a resolution of V.

According to this remarkable result of Springer [Sp] the singularity of a pointx in V is thus measured by the variety of flags (G/B),, that if fixes, or,equivalently, by the variety of Borel subgroups in which it is contained. Inaccordance with (a) we have:

THEOREM (b) (SEE [St,]). The elements of V each fixing a single flag form asingle conjugacy class Vreg which is dense and open in V. On 771'( V7eg) the map 7r1is an isomorphism.

DEFINITION. For each simple root a let Pa be the corresponding parabolicsubgroup so that Pa/B is a projective line in G/B. A Dynkin curve is a union oftranslates La of such lines, one for each a, so that La and L', intersect just whena and /3 are joined in the Dynkin diagram of G.

THEOREM (c). (1) In V - Vreg there exists a single dense open conjugacy classVsubreg It has codimension 2.

(2) Dynkin curves exist and all are congruent under G.(3) An element x of V is in Vsubreg if and only if the fibre (G/B),, of Theorem (a)

is a Dynkin curve.

This result is due to Tits and the author. For a proof see [St,].

6. The main theorems. In addition to the notation of §5 let T be a maximaltorus of B, W the Weyl group, and T/ W the corresponding affine space A',coordinatized by the fundamental characters. Here r is the rank of G.

THEOREM. Let x be a subregular unipotent element of G and S a transversal (ofdimension r + 2) to Vsubreg in G passing through x.

(1) x is an isolated Kleinian singularity of V n S of the type of (the Dynkindiagram of) G.

(2) The restriction of 7r1 of Theorem 5(a) to 7ri'( V n S) is a minimal resolutionof(V n S,x).

(3) The natural map (S, V n S) *' 2 (T/ W, 1), given by y (in S) -*yr (semi-simple part) -* conjugate in T, is a universal deformation of ( V n S, 1).

This result was originally conjectured by Grothendieck. The proof is due toBrieskorn (see [B2]) in the analytic case, to Slodowy [S] in the algebraic case,with a significant contribution by H. Esnault.

Thus as t changes from 1 to a nearby value the singularity in the fibre above tchanges from Kleinian of type G to Kleinian of type ZG(t) (one singular pointfor each simple component of ZG(t)), a "simpler" singularity. If the map 7rI of(2) is extended so that G takes the place of V (see the start of §5), then thecombination S n2 . . . yields in effect a simultaneous resolution ofthe singularities of the whole deformation (Grothendieck, see [B2]).

KLEINIAN SINGULARITIES AND UNIPOTENT ELEMENTS 269

A FINAL EXAMPLE. We take G = SL,,,. The standard regular element (un-ipotent) is the identity with l's filled in just above the diagonal. In the standardsubregular x the (12)-entry is then replaced by 0. SL, and hence x, acts on theunderlying space V,+ 1, hence also on the space of flags: { V1 C V2 C CV,+1, dim V; = i). As is easily seen the flags fixed by x form r projective lines Lj(1 < j < r) with Lj all flags of the form:

<e1,e2,...,e1> ifi<j,V, = <el, e2, ... , e,_ 1, ae, + beo> (a, b E C) if i

<eo,e1,...,e,> ifi>j,in terms of a basis eo, e1, ... , e, of V,+1. This is a Dynkin curve of type A, andis in close analogy with the ad hoc desingularization given in V. To exhibit auniversal deformation (following Arnold [A]) we switch to the nilpotent elementN = x - 1 in the Lie algebra s1,+1 and take as our cross-section S the set shownbelow, analogous to the general rational single Jordan block that it would be ifN were regular. For the characteristic polynomial f(t) we have in terms of that,fl(t), of the lower right hand block f(t) = (t - X1)f1(t) - YZ, and on settingt = X1, f(Xl) + YZ = 0. This is just the deformation given in §4 for thesingularity Xi + 1 + YZ = 0 of type A, with To, T1.... , T,_ 1 the coefficients off, i.e., the fundamental characters, i.e., the coordinates on t1 W as mentionedearlier.

rx1

Z

Y

0 1

-X, -X2 -`1'I

7. Remarks and problems. At several places in the known proofs of thetheorems above, the classification is used (although large parts of the develop-ment are quite satisfactory). In particular, no one seems to have found a reallydirect connection between the finite group F used at the start and the algebraicgroup G used at the end. Nor is there yet an explanation for J. McKay'sastonishing observation that the character table for F is an eigenmatrix for theextended Cartan matrix of G. Thus our first problem is:

(1) Repair this situation.(2) Explain why the Kleinian singularities and the Arnold-Siersma singulari-

ties (see 3(a)) are the same.


(3) Study the fibres (G/B), of the resolution 5(a) of V. Some nice propertiesare known: each is connected, with all irreducible components of the samedimension, which is known, as is the number of components. Further Springerhas given a realization of the Weyl group representations in these fibres and I aconnection between Weyl group elements and pairs of components. However,there is a lot about these fibres that is not known. This problem is included inthe following one which 1 first posed about 13 years ago and which has, I like tothink, led to some of the above developments.

(4) Study the unipotent variety thoroughly.

REFERENCES

[A] V. Arnold, Normal forms for functions near degenerate critical points, the Weyl groups Ak, Dkand Ek and Lagrangian singularities, Functional Anal. Appl. 6 (1972),254-272.

[A,] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136.[Bt] E. Brieskorn, Rationale Singularitaten komplexer Flachen, Invent. Math. 4 (1967/68), 336-358.

358.[B2] , Singular elements of semi-simple algebraic groups, Internat. Congr. Math. Nice (1970),

vol. 2, pp. 279-284.[D] P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction,

Proc. Cambridge Philos. Soc. 30 (1933/34), 453-465; 483-491.[S] P. Slodowy, Regensburg thesis; Lecture Notes in Math., Springer-Verlag, Berlin and New York

(to appear).[Sp] T. Springer, The unipotent variety of a semisimple group, Proc. Bombay Colloq. on Alg. Geom.,

Oxford Press, London, 1969, pp. 373-391.[Stt] R. Steinberg, Lectures on conjugacy classes, Lecture Notes in Math., vol. 366, Springer-Verlag,

Berlin and New York, 1974.[St,,] , Desingularization of the unipotent variety, Invent. Math. 36 (1976), 209-224.

QUEEN MARY COLLEGE, ENGLAND


THE CONSTRUCTION OF J4

SIMON NORTON

In this paper we outline a series of steps which should, when completed, leadto an existence proof for the Janko group J4 of order 221 33 5 7 113. 23 29

31 37 43. While studying the extra-special problem Janko was led to conjec-ture the existence of a group with involution centraliser 21 +12.3 'M22- 2, anddetermined its class list and 2-local structure. There is one other class ofinvolutions, and the 2-local subgroups also include 211. M24, 210. L5(2) and23. 212 S5 X L3(2), where in all but the last of these the extension of the02-subgroup is split. The complete p-local structure was given in [1], althoughforp = 11 this was not obtained until a later stage. It should be noted that at thetime, the covering group of M22 was believed to be 6 M22 rather than 12 'M22-

The fact that the 5- and 7-blocks of defect 1 intersect in rectangular fashionmade the determination of the character table (by several people, including theauthor, at Cambridge) relatively easy. However the least degree characters haddegree 1333 (involving the irrationality ), which is rather high for computerwork. But, while considering 2-modular characters, Thompson was led tosuspect that there was one of degree 112, over the field GF(2). To work with2-modular representations, one needs to consider subgroups of odd characteris-tic, and it turns out that the largest possibility is PE U3(11). We thereforeproceed to outline a construction for J4 based on Thompson's conjecture andthe existence of a subgroup U3(l 1). It should be said that the main impetus forthis work came from the development of R. A. Parker's methods for workingwith modular representations, as described by J. G. Thackray [2], as well as theconsiderations of the following paragraphs involving U3(l 1).

First it is necessary to obtain the relevant 2-modular representation of U3(11).The 112-space T, which is self-dual, splits 1 + 110 + 1 over U3(11), where the110-character lifts to characteristic 0. There is a unique invariant 1-space, say S.A central 11-element of U3(11) acts fixed point freely on the 110-dimensionalrepresentation, and consequently determines a unique 1-space (i.e. vector) inT/S. It therefore follows that T/S is a quotient of the permutation module of

1980 Mathematics Subject Classification. Primary 20D08.


272 SIMON NORTON

U3(11) on the normaliser of a central 11-element. As this element is a unitarytransvection, the permutation module is also that on the stabiliser of an isotropic1-space.

The next step is to construct this permutation module. For conveniencea hermitian form is used under which the norm of the vector (a, b, c) is?(ac + cd) - bb, wher° d denotes d11, and all numbers are in GF(121). Theisotropic vectors are now, up to proportionality, (1, 0, 0) and (x2 + y2 + zi,x + yi, 1) where i is a particular square root of -1, and x, y, z are in GF(11).Apart from the first these may be indexed by (x, y, z), or 121 x + l ly + z withthe least nonnegative residues of x, y and z. Now one can put generators forU3(11) into matrix form on a 1332-dimensional space over GF(2) with a basisindexed by the same index set.

We observe that as our 11-element, say the transvection corresponding to thevector (1, 0, 0), is to act fixed point freely on the hyperplane of even sums of the1332 generators (in T/S), the kernel of the homomorphism from the permuta-tion module to T/S must contain the sum of the 12 vectors corresponding to(1, 0, 0) and (zi, 0, 1), where z E GF(11), and all its U3(11)-transforms. It turnsout that these are all that are required to make the quotient space of exactly 111dimensions.

To construct the 112-space, we observe that a typical matrix for an element ofU3(11) over T/S is (o ') where a is a row-vector and 0 the zero column-vector.The method of [2] may be used to find a corresponding matrix (o b) in the dualrepresentation, for the same A. (This corresponds essentially to finding theisomorphism between the 110-dimensional representation and its dual.) Then the112-dimensional matrix will be

I a c

0 A b

0 0 1

where c = 0 or 1. As the group generated by all such matrices for any value of cwill be 2 x U3(11), there is essentially no problem in resolving the ambiguity inc, e.g. by using an odd order generating set.

To complete the group J4, we need one more generator. This is to be found inthe J4-normaliser of our 11-element, which is 111 +2. 5 X 254. When restricted tothis the 112-space T decomposes into a 2-space (on which the action of the abovegroup is that of its quotient S3) and a 110-space. When further restricted to111,2 - 5 the 110-space becomes a 55-space over GF(4). The method of [2] willdetermine an element of order 3 that normalises this in a known way, fromwhich all such elements may be found by multiplying by a scalar matrix on our55-space. If our 11-element is t, such a scalar matrix will be a power oft+t3+t4+ts+ t9.

From this we see that there are only a few possibilities for a set of generatingmatrices for J4. In fact, all but one of these were eliminated by taking randomproducts and observing matrices of orders not occurring in J4, while this onewas confirmed in the same way. Character values for the 112-representationwere also used. But it is much more difficult to prove the group generated is J4.

THE CONSTRUCTION OF J4 273

In obtaining such a proof, our first step is to obtain generators for certaininteresting subgroups. A method was devised by R. A. Parker for obtaininginvolution centralisers, as follows: if g, h are involutions whose product has evenorder, the group they generate will contain an involution centralising g. Conse-quently, we take involutions g, h of different classes, and conjugate one of themat random by our generators. We will then obtain a series of involutions in thecentraliser of the other, which will eventually generate this centraliser. We maythus obtain groups G = 21 +12 - 3 M22. 2 and 211 M22. 2. We then obtain anelement of the 211 in the latter case, other than the central element, butconjugate thereto in J4, (this was done by finding an element of order 28 andtaking its 14th power), and adjoin an element centralising it to obtain the group211. M24. We call this group M, and observe that M and G may be chosen sothat their intersection is 21 +12 3 24S6 = 211- 263 S6. This will happen if thecentral involution of G is in the 211 of M.

By manipulations within the groups M and G, sufficiently many elementshave been obtained to enable any given element to be produced easily. Further-more, using the presentations of Todd [3], it has been verified that the groups Mand G are what they have been stated to be. In addition, a "dictionary" has beenproduced enabling one to pass from the M-form of an element of D = M n Gto the G-form and vice versa.

It was suspected that the splitting of the 112-space when restricted to Mwould be 1 + 11 + 44 + 44+ 11 + 1, where the only invariant subspaces arethe left partial sums, and for g E 211 the map v -* v + vg yields the natural mapfrom the tensor product of the 11-space corresponding to the 211 with anycomposition factor of the 112-space to the composition factor immediatelypreceding. This has now been confirmed. Also a complete set of subspaces havebeen obtained for the group G. In particular we note that for each subgroupconjugate to M there is a unique fixed vector in the 112-space. We call thesestandard vectors.

We now follow with the results of some purely theoretical investigations, notall of which have been confirmed in practice. The groups M, G and D haverespectively 7, 9 and 36 orbits on standard vectors. In Table 1 we give thelengths of the 36 D-orbits, together with their fusion into M- and G-orbits, towhich we have given respectively Roman and Greek letters as names. In Table 2we give "concepts" for each M- and G-orbit. The definition of a concept is anobject belonging to a set on which M24 or M22. 2 acts, where the stabiliser is thesame as the projection of the M- or G-stabiliser corresponding to the relevantorbit into the group M24 or M22. 2. As the group D contains both the kernels ofthese projections, we may see that these concepts are all that are required todetermine the decomposition of the M- or G-orbits into D-orbits.

We now argue as follows: let us start with a standard vector (say that fixed byM), and obtain representatives of the other 35 D-orbits by the action of G andM. 56 equalities are required to show that these 36 vectors lie in 7 M-orbits and9 G-orbits. Of these we used 35 in our definition of these vectors, so there are 21that have to be confirmed experimentally. (We see below how this number canbe substantially reduced.) Now if we can determine the exact M- or G-stabiliser

274 SIMON NORTON

for each of the 7 or 9 orbits, and they correspond to our "concepts" above, wewill have shown that the union of our 36 D-orbits is the same as the union of the7 M-orbits or the 9 G-orbits, and hence that the set of vectors in these orbits isfixed by both M and G, and hence by the group <M, G>, which acts transitivelyon the set. Also we will know the cardinality of the set, which will be 173067389,the index of 211 M24 in J4.

TABLE 1

o o E r x A

I 1 0 0 0 0 0 0 0 0

T 60 720 0 2880 11520 0 0 0 0

S 16 1440 3840 0 0 23040 0 0 0

Z 0 5760 7680 23040 138240 276480 368640 1105920 1474560

F 0 0 0 18432 552960 368640 8847360737280,

176947204423680

M 0 0 30720 0 184320 11059201474560, 1474560,

353894405898240 8847360

L 0 0T 0 0 0 0 11796480 1179684023592960.

35389440

TABLE 2

I 211 M24

T 29. 26(S3 X L3(2))S 27. 263 S6

Z 24. 26(S3 X S4)F 2 26S5

M 1.26(S3XS4)L 1 L2(23)

The trivial concept.The trio.The sextet.The sextet-line.The 212-type involution of M24.

The trio and a sextet refining it.The projective line over GF(23).

0 21 + 12.3 2456

(D 21+9, 1 . 24L3(2)

26.3 24L3(2)

A 21+6. 1 .2555

25.1 24(2 X S4)

E 24.1 24(2 X S4)

r 1 1 24(2 x S4)X 2 1 24S4

A 1 . 1 . 24(2 X S3)

The hexad.The octad.The octad.The duad.The syntheme.The syntheme.The duad in a hexad.The 27-type involution and one of its pairs.The hexad with a hexagon graph on it.


EXPLANATION OF TABLE 2

For each M- or G-orbit, we give its name, its stabiliser (in which, for M, thefirst composition factor gives the 211-stabiliser, and for G, the first and the firsttwo composition factors give the 21+ 12- and 21+12. 3-stabilisers), and the conceptcorresponding to it. The definitions of the various concepts are as follows:

In M24, a trio is a set of three disjoint octads of the Steiner system S(5, 8, 24).A sextet is a set of six tetrads, the union of any two of which is an octad.A sextet-line is a set of three sextets refining a trio, forming a line in the

7-point projective plane corresponding to the seven sextets refining the trio.In M22. 2, a hexad is a member of the Steiner system S(3, 6, 22).An octad is the set of eight points fixed by a 27-type involution.A syntheme is a splitting of a hexad into three duads.

To simplify the calculations, we find a group 23. 212.S5 X L3(2) that has asubgroup of index 5 in M, and one of index 7 in G. We may then obtaincommuting elements of orders 5 and 7, respectively in G but not M, and in Mbut not G. If we use these to transfer from one D-orbit to another, and use D totransfer within our D-orbits, some of our 21 equalities will be automatic, asthose on 3 sides of a rectangle will imply that on the fourth by commutativity.But this means the number of equalities that will need to be tested will besubstantially below 21.

Finally, there will be some tidying up to do. If we can show that, in theL112(2)- (or possibly Sp112(2)-) stabiliser of our set of standard vectors, the

subgroup that fixes one of them is no larger than M, it will follow that thisstabiliser has the same order as J4. One may then deduce that the involutioncentraliser is G, and from Janko's arguments it will follow that our group reallyis of J4 type. In other words, we will have proved the existence of J4, defined (torecapitulate) as the group generated by M and G, or alternatively as thestabiliser of the set of D-images of our 36 representative vectors. We may thenwish to complete the circle by showing that this group contains U3(11). Eventhough our original generators may not be recoverable (owing to several changesof basis) it will still not be too difficult to build the group.

There will still remain the question of uniqueness. It is only by the assumptionof the existence of a subgroup U3(11) and Thompson's conjecture that the grouphas been characterised. To prove uniqueness, one needs to work with represen-tations that are known to exist unconditionally. For example, the 1333-dimen-sional representation, when restricted to M, G or D, splits respectively 45 +1288, 693 + 640, and 45 + 288 + 360 + 640, with the obvious fusions from D toM or G. If one defines 1333 X 1333 matrices for D, and adjoins additionalgenerators for M and G, one will be able to fix proportionality factors betweenthe 45- and 640-spaces, and the others, but given the factor between the 288-and the 360-space in M, one will not be free to choose it in G. By morecomplicated arguments, the proportionality factor can be determined to withinmultiplication by a divisor of a power of 2. (Of course 2 is not a prime in thep.i.d. of algebraic integers of Q(V7 ).) It seems likely that further reasoningon these lines, possibly combined with experimental work, can prove uniqueness.

276 SIMON NORTON

The principal people to whom acknowledgment should be made for assistancein carrying out the work here described are R. A. Parker, J. H. Conway, D. J.Benson and J. G. Thackray.

Addendum.' The existence proof for J4 has now been completed by a methodslightly different from that outlined. The main steps are as follows (we writeJ = <M, G>):

(1) Work by Parker indicated that if v, is the standard vector fixed by M, andw a vector of the 12-space E invariant under M, then in the exterior square ofthe 112-space the J-images of v, A w generate a subspace of codimension 1221.Thackray's proof of this on February 20, 1980 was the last step to be carried out.The complete splitting of this exterior square is in fact I + 1220 + 3774 + 1220+ 1, and it is believed that one of the ordinary 1333-dimensional characterssplits 1 + 112 + 1220 modulo 2.

(2) If, in the above, wo and wd are elements of E corresponding respectively toan octad and a dodecad of M24, then the spaces of all x such that v, A x,wo A x, wd A x, lie in the above 4995-space were verified to be of dimension 12,6 and 2 respectively. This dimension is clearly a J-invariant, and if we define J'to be its L12(2)-stabiliser then J < J'.

(3) Elements of M and G were found that generate what is in fact thecomplete J4-stabiliser of w0, L = 210 L,(2). The splitting of the 112-space underL was found to be 1 + 5 + 10 + 40 + 40 + 10 + 5 + 1. It was checked that theunique invariant 16-space lies in the 56-space corresponding to v1, and that itspointwise M-stabiliser is trivial. It was also verified that if w0 and w1 areelements of E corresponding to disjoint octads, then their corresponding 6-spaces have just one nonstandard nonzero vector in common, and its M-imagesgenerate the 56-space of v1.

(4) We can now show that the J'-stabiliser, and hence the J-stabiliser, of v, isexactly M. From the results above it can be proved that an element of J' fixes Epointwise if and only if it fixes the 16-space of w0 pointwise. It then follows thatthe space of vectors fixed by all such elements is invariant under <M, L> = J,and since the action group of J' on E is at most M the result follows. As vi is theonly nonzero element of the 112-space fixed by 02(M), it has an odd number ofJ'-images, showing that the Sylow 2-subgroups of J', J, M and G are isomor-phic.

(5) Finally we show that the J'-centraliser of Z(G) = (1, t) (say) is exactly G.By virtue of Janko's characterisation of J4 in [1] this shows the isomorphism ofJ', J and J4. The quotient of the kernel of I + t by its image is of dimension 12,and it may be seen that the action group of C,.(t) can only be 3 M22. 2. Anargument using a minimal normal subgroup of C,.(t)/t shows that C,,(t) fixesthe 10-space generated by the orbit O. The result follows quickly.

The uniqueness problem has also been solved: the argument shown aboveapplies with M, G and D replaced by L and the J4 stabilisers of (respectively)the unordered and ordered pairs (w0, v1 + w0). The last of these groups is

'This new material was completed after the conference and was not reported during theconference.


L n M. The character restrictions of the 1333-character are respectively 465 +868, 45 + 840 + 448 and 45 + 420 + 420' + 448.

Thackray's machine computation has been repeated to eliminate a possibleerror, while Benson has found a presentation for J4 by following the originalplan.

It is intended to publish the proof in more detail in the near future. Theacknowledgements made prior to this addendum should be repeated here.

REFERENCES

1 . Z. Janko, A new f i n i t e s i m p l e g r o u p o f o r d e r 221 . 33.5 7. 113.23 29 31 .37-43 whichpossesses M24 and the full covering group of Mu as subgroups, J. Algebra 42 (2) 1976, 564-596.

2. J. G. Thackray, Reducing modules in nonzero characteristic, Lecture AMS Summer ResearchInstitute on Finite Group Theory (University of California, Santa Cruz, June 25-July 20, 1979).

3. J. A. Todd, Quart. J. Math. (Oxford Series) (2) 21 (1970), 421-424.



SCHUR MULTIPLIERS OF THE KNOWNFINITE SIMPLE GROUPS. II

ROBERT L. GRIESS, JR.1

The purpose of this note is to make a current report on the Schur multipliersituation for finite simple groups. A similar report was made several years ago[4]. At this writing, the known finite simple groups consist of the alternatinggroups, the groups of Lie type and 26 sporadic simple groups. The Schurmultipliers of all these groups have been determined, the last case, F2, havingbeen completed only recently. See Tables 1, 2 and 3 for a summary of thisinformation. Finally, we give references for the computations of the Schurmultiplier which have appeared in the literature.

Attributions. For the alternating groups, see Schur [17] (and see [10] for amodern account).

The finite Chevalley groups, Steinberg variations (except type 2A", n even),Suzuki groups and Ree groups of type F4 groups have the p'-part of theirmultiplier described by the theorems of Steinberg [18]-[21], where p is thecharacteristic of the relevant finite field. Furthermore, Steinberg had determineda finite list of possible exceptional groups (one with multiplier ap-group) amongthe (untwisted) Chevalley groups. His work, and the work of others, settled theexact list of exceptional groups. Steinberg informs us that he will publish hiswork on the exceptional Chevalley groups and on the family 2A, n even. Theresults in this area not due to Steinberg are due to Thompson and Burgoyne,independently, for A2(4), Fischer, Rudvalis and Steinberg for B3(3), and Griessfor G2(3), G2(4), F4(2) and the twisted groups 2A" (except for n even, n > 4,which is due to Steinberg), 2D", 3D4, 2E6, 2F4 and 2F4(2)' [5].

Using other methods, multipliers of some twisted groups of Lie type havebeen handled by Alperin and Gorenstein [1] for the Suzuki groups 2B2(22n+1)and the Ree groups 2G2(32"+') and by Ward [22] for the Ree groups 2F4(22nt1),n > 2. We also mention some general results of Curtis [3] and Grover [8] whichapply to groups with a BN-pair.

1980 Mathematics Subject Classification. Primary 20C15.'Supported in part by NSF grant MCS-77-18723 (02).


279

280 R. L. GRIESS, JR.

TABLE 1

Alternating groups

Alt(n) Multiplier

n<3 1

n=4,5andn>8 Z2

n=6,7 Z6

TABLE 2

Finite Groups of Lie Type

The multiplier ofG(q), q= p", is R X P where R is a p'-group and P is a p-group.

Group of Lie type R R x P (when P is zO 1)

A,(q) Z(r+1,q- 11) Z2 (1, q) = (1, 4)Z2 X Z3 (1,9)

Z2 (2, 2)Z3XZ4xZ4 (2,4)

Z2 (3, 2)B,(q), I > 2 Z(2,q - 1) Z2 (2, 2)

Z2 (3, 2)Z3xZ2 (3,3)

C1(q), I > 2 Z(2,q-1) Z2 (2, 2)Z 2)32 ( ,

Dr(q),1 > 4 Z(4,q'-1), 1 odd Z2 X Z2Z(2gl_1) x Z(2q'_1), 1 even

(4,2)

E6(q) Z(3,q- 1)

E-7(q) Z(2,q-1)E8(q) 1

F4(q) Z2 (4,2)

G2(q) Z3 (2, 3)Z2 (2, 4)

2A,(q), I > 2 Z(1+ 1,q+ 1) Z2 (3, 2)Z4xZ3xZ3 (3,3)Z3 X Z2 X Z2 (5,2)

2B2(q) Sz(q) I Z2 x Z2 (2,8)2D,(q), 1 > 4 Z(4,q'+ 1)

2E6(q) Z(3,q+ 1) Z3 X Z2 X Z2 (6,2)

2F4(q) 1

2F4(2)' 1

2G2(q) 1

3D4(q) 1

SCHUR MULTIPLIERS OF FINITE SIMPLE GROUPS. 11

TABLE 3

Sporadic Groups

281

Symbol forSporadic Group (Discoverer) Order Multiplier

M (Mathieu's groups) 24325.11

M12 26335.11 Z2

M22 2'325.7.11 Z12

M23 2'325.7.11.23 1

M24 210335.7.11.23 1

J1 (Janko's groups) 233.5.7.11.19 1

J2 (= HJ, Hall-Janko) 2'33527 Z2

J3 2'.335.17.19 Z3

J4 (Janko) 221335.7.113.23.29.31.37.43

Held (Held) 2103352.73.17 1

HiS (Higman-Sims) 2932537.11 Z2

McL (McLaughlin) 2'36537.11 Z3

Suz (Suzuki) 2133'527.11.13 Z6

I (Conway's groups) 22139547211.13.23 Z2

.2 21836537.11.23 1

.3 2103'537.11.23 1

F22 (Fischer's 3-transposition groups) 21739527.11.13 Z6

F 11 13 23218313527 17 123

FF4

. .. .

221316527311.13.17.23.29 Z3

LyS (Lyons) 283'567.11.31.37.67

Ru (Rudvalis) 21433537.13.29 Z2

O'S (O'Nan) 2934 5,7'. 11. 19.31 Z3

F2 (Fischer's (3, 4) -transposition group) 241313567211.13.17.19.23.31.47 22F1 (Fischer-Griess) 2463 2059761 12 13 317.19.23.29.31.41.47.59.71 1

F3 (Thompson) 2153'0537213.19.31 1

F5 (Harada) 21436567.11.19 1

For the sporadic groups,the published results are due to Burgoyne and Fong[2] and Mazet [13] for the Mathieu groups, Janko [11] for J1, McKay and Wales[15] for J2 and J3, McKay and Wales [14] for HiS, O'Nan [16] for O'S, Harada[9] for F5, Lempken [12] for J4, Griess [6], [7], for Held, Suzuki, ' 1, ' 2, 3, F22,F23, F24 (Norton was the first to construct a 3-fold cover of F24), F2, Fl.

In addition, Thompson has settled the multipliers of McL, LyS and F3 andFeit, Lyons and Rudvalis settled the multiplier of Ru.

We shall collect proofs of the unpublished results and publish them in thenear future. The individuals associated to these results have given permission todo so. We understand that Mazet is preparing a manuscript on the multiplier ofA2(4) = L3(4) - M21.

282 R. L. GRIESS, JR.

REFERENCES

1. J. L. Alperin and D. Gorenstein, The multiplicators of certain simple groups, Proc. Amer. Math.Soc. 17 (1966), 515-519.

2. N. Burgoyne and P. Fong, The Schur multipliers of the Mathieu groups, Nagoya Math. J. 27(1966), 733-745; Correction, ibid. 31 (1968), 297-304.

3. C. Curtis, Central extensions of groups of Lie type, J. Reine Angew. Math. 220 (1965), 174-185.4. R. Griess, Schur multipliers of the known finite simple groups, Bull. Amer. Math. Soc. 78 (1972),

68-71.5. , Schur multipliers of finite simple groups of Lie type, Trans. Amer. Math. Soc. 183

(1973), 355-421.6. , Schur multipliers of some sporadic simple groups, J. Algebra 32 (1974), 445-466.7. , A construction of FI (to appear).8. J. Grover, Covering groups of groups of Lie type, Pacific J. Math. 30 (1969), 645-655.9. K. Harada, The automorphism groups and the Schur multiplier of the simple group of order

21436567.11.9, Osaka J. Math. 15 (1978), 633-635.10. B. Huppert, Endliche Gruppen. I, Die Grundlehren der Math. Wissenschaften, Band 134,

Springer-Verlag, Berlin, 1967.11. Z. Janko, A new finite simple group with abelian Sylow 2-subgroups and its characterization, J.

Algebra 3 (1966), 147-186.12. W. Lempkin, The Schur multiplier of J4 is trivial, Arch. Math. 30 (1978), 267-270.13. P. Mazet, Sur le multiplicateur de Schur du groupe de Mathieu Mme, C. It Acad. Sci. Paris 289

(1979), 659-661.14. J. McKay and D. Wales, The multiplier of the Higman-Sims group, Proc. London Math. Soc. 3

(1971), 283-285.15. , The multipliers of the Janko groups, J. Algebra 17 (1971), 262-272.16. M. O'Nan, Some evidence for the existence of a new simple group, Proc. London Math. Soc. 32

(1976),421-479...17. 1. Schur, Uber die Darstellung der symmetrischen and der alternieden Grippe durch gebrochere

lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250.18. R. Steinberg, Generateurs, relations, et revetements de groupes algebriques, Colloque sur la

Theorie des Groupes Algebriques, Bruxelles, 1961, pp. 113-127.19. , Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33-56.20. , Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875-891.21. , Lectures on Chevalley groups, Yale University Notes, 1967.22. H. N. Ward, On the triviality of primary parts of the Schur multiplier, J. Algebra 10 (1968),

377-382.

INSTITUTE FOR ADVANCED STUDY

UNIVERSITY OF MICHIGAN


2-LOCAL GEOMETRIES FOR SOMESPORADIC GROUPS

M. A. RONAN AND S. D. SMITHI

The theory of buildings, developed by J. Tits [5], provides a satisfyinggeometric interpretation and classification for the groups of Lie type and it,together with the recent work of F. Buekenhout [1] on geometries for thesporadic groups, is what has stimulated the research presented here.

If A is the building associated with a group G of Lie type and characteristic p,the stabilizer P of a vertex of A is a maximal parabolic subgroup; these are themaximal p-local subgroups, and are always p-constrained, and contain a Sylowp-subgroup of G. In our work we consider specifically the case of diagramgeometries (see [1]) for the sporadic groups, in which the stabilizer of a vertex isa maximal 2-constrained, 2-local subgroup, although not always containing aSylow 2-subgroup. Some very interesting geometries associated with sporadicgroups are obtained in this way.

Remarks and notation. A hollow node o indicates that the 2-local subgroup Pof G does not contain a Sylow 2-subgroup. In every such case one obtains ann-split cover of each classical subgeometry containing that node, by which wemean that every vertex v corresponding to that node in a classical subgeometryis replaced by a set of n vertices each of which is incident to everything to whichv is incident.

A square node indicates that although there are no vertices in the geometrybelonging to it, there are, nonetheless, in certain subgeometries, objects to whichit can be associated. To these objects there is associated a 2-local subgroupwhich is contained in the 2-local stabilizing the subgeometry. For example, inM24, with may be associated all 2-central involutions, a centralizer of which(21+6L3(2)) is contained in an octad stabilizer, 24L4(2).

Adjacent to each node of a diagram we write A to mean that if v is a vertexbelonging to that node, then A )4 B = Stab v, and A is the kernel of the actionon link(v).

1980 Mathematics Subject Classification. Primary 20D05; Secondary 20B05, 20020.'Both authors were partially supported in this work by the National Science Foundation.


283

284 M. A. RONAN AND S. D. SMITH

octads trios sextets

"2 4L4(2) S3xL3(2) SP4(2)

24 26 263

Following the notation of Conway [2], an octad is a block of the Steinersystem S(5, 8, 24), a trio is a set of three mutually disjoint octads, and a sextet isa set of six mutually disjoint tetrads (4-sets). Each sextet is a parallel class oftetrads, and each tetrad is contained in a unique sextet. The geometry wedescribe also appears in the work of Shult and Yanushka [4] on near n-gons.Incidence is as follows (octad = o, trio = t, sextet = s):

o I t if o is one of the three octads of t,o I s if o is the union of two tetrads of s,t I s if t is three pairs of tetrads of s.If one fixes a sextet it is clear that the octads and trios incident with it are the

duads (ab), and synthemes (ab)(cd)(ef) of a set of six elements; as such theyform the points and lines of the unique generalized quadrangle of order (2, 2)having collineation group Sp4(2) = S6.

If one fixes an octad, the trios, sextets, and involutions of type 1828 which fixthe octad pointwise, form the points, lines and planes of a PG(3, 2) (see [2]). Inthis sense the involutions belong to the square node of the diagram, yet each oneis associated to a unique octad. It turns out to be impossible to assign some set Sof vertices to the square node, because the above geometry satisfies the intersec-tion property (see [1] and [6]), and hence for v E S, link(v) would be the Sp6(2)building; counting incident pairs (octad, v E S) one obtains 331759.15, a con-tradiction.

Let V denote the 24-dimensional vector space over GF(2), spanned by the 24symbols on which M24 acts, and let G c V be the Golay code. The quotient ofG by the all l's vector is an 11-dimensional space on which M24 acts irreducibly;octads, trios and sextets are 1, 2 and 4-spaces respectively. In the space V/ G,the sextets are 1-spaces spanning a hyperplane, and M24 acts irreducibly on this11-space (which is dual to the previous module); octads and trios are 6-spacesand 3-spaces.

3 6E8 9E8 D4 coord.

frames1 8 4 2 1t

n8(2) S3xL4(2) S3xSP4(2) M24

21+8 22+12 24+123 211

2-LOCAL GEOMETRIES FOR SOME SPORADIC GROUPS 285

Using the work of Conway [2] on the Leech lattice A, one sees that thesubgroup 2"M24 of 1 stabilizes a coordinate frame comprising 24 1-spaces onwhich M24 acts faithfully. Any 8 of these 1-spaces which form an octad ofS(5, 8, 24) span an E. sublattice of A; any 4 span a D4 sublattice. Thus trios andsextets correspond to direct sums of three E. sublattices and six D4 sublattices asindicated in the diagram above. _

Letting bars denote images mod 2A, A = A/2A is a 24-dimensional vectorspace over GF(2); the images of the 3 E. sublattices forming ®3 E8 intersect ina 4-space, and the 6 D4 sublattices in a 2-space. Moreover the coordinate framesbecome 1-spaces, of A4 (see [2]), and so the vertices of our geometry may beregarded as certain 8, 4, 2 and 1-spaces of A, as indicated by the diagram above.These subspaces all lie in A2 U A4 and are therefore totally singular under thequadratic form Q on A, defined by Q(v-) = v v/16 (mod 2), where v E A.Moreover the 4, 2 and 1-spaces lie in A4, and therefore those contained in anyE. are totally singular for the nondegenerate quadratic form Q' on E. definedby Q'(v-) = v v/32 (mod 2). This verifies the Q'(2) part of the diagram, andthe M24 part follows from the discussion above, this verifying the full diagram.We remark that the square node corresponds to the set of 4-spaces in A4 whichare not included in our geometry, and each of which lies in a unique E8.

Monster

O10(2) S3XL5(2) SP4(2)xL3(2) S3XMZ4 1

210+16 25+10+20 23+363 22+11+2221+24

The existence of the above geometry is dependent on the existence of theMonster, which has not yet been established.2 The diagram suggests, however,that there may be a representation over GF(2) in which the vertices of ourgeometry correspond to certain 10, 5, 3, 2 and 1-spaces, reading from left toright on the diagram.

X24M24 S3xL4(2) SP4(2)'xL3(2)211 28+6 23+12 21+123

The 2-local structure of F24 gives rise to a geometry belonging to the diagramabove. The m subgeometry for U4(3) 2 is very interesting, and its verticescan be regarded as certain 1, 2 and 3-spaces over GF(4) inside the U6(2)geometry as was shown us by W. M. Kantor.

2Recently R. Griess has given a construction of the Monster.

286

M22

M. A. RONAN AND S. D. SMITH

hexads quintetsSPk(2)1 0-W

4424

23

4

pts. t triples-- is aa complete triplesystem with 5 points

and ) = 10 triples.

D is a 2-splitcover of the Sp4(2quadrangle.

Fix two points of the Steiner system S - S(5, 8, 24) and let S" denote theremaining 22 points. An octad of S will be called an octad (resp. hexad) of S" ifit meets S" in 8 (resp. 6) points, and a sextet of S five of whose tetrads lie in S"will be called a quintet. Incidence is as follows:

olhifon h = 0,o I q if o is the union of two tetrads of q,h I q if h contains a tetrad of q.

For a fixed hexad h = (abcdef), the duads (ab) correspond to quintetsincident with h, and the synthemes (ab)(cd)(ef) correspond to three quintetswhich (considered as sextets of S) correspond to a trio of S, which in turncorresponds to a disjoint pair of octads of S" incident with h (6 + 8 + 8 = 22);thus thea subdiagram is evident. Now fix an octad o; the unique involution ofM24 fixing o pointwise and transposing the two points outside S" determinestrios and sextets which form a projective plane of order 2 (see M24 above); thesecorrespond to hexads and quintets incident with o, so the .- subgeometryfollows. For a quintet q, the hexads and octads incident with q obviouslycorrespond to tetrads and pairs of tetrads of q, thus verifying .-o.

If we take the 11-dimensional hyperplane of the Golay code G for M24 andintersect it with the hyperplane of V (see M24) perpendicular to the vector 12022fixed by M22, then we have a 10-dimensional vector space over GF(2) on whichM22 acts irreducibly. Octads and hexads correspond to two types of 1-spaces,and quintets to 4-spaces.

.2

2 is the subgroup of 1 fixing a vector A E A2, see [2], and since A is singularunder Q (see 1 above) we obtain the 22-dimensional irreducible moduleM = A /<A> for 2. The 8-spaces E8 for 1, which lie in X -L split into two orbitsunder -2, those which contain A and those which do not. In the former case weobtaina 7-space of M, and in the latter case A picks out a unique 4-space F ofE8 n A4 (corresponding to the square node of the - 1 diagram) having theproperty that A + F C A2. The 2 and I-spaces of A4 n A' give 2 and 1-spacesof M, on which 2 acts transitively, and the vertices of our geometry correspondto 7, 4, 2 and l-spaces of M, as indicated by the diagram. Incidence is con-tainment, except that between a 7 and 4-space incidence is intersection in a3-space. The L4(2) part of the diagram is now obvious, and the M22 part followsfrom the discussion above from which it is seen that the 7 and 4-spacescontaining a vector of A4 n A 1 correspond orbits of 8-spaces and henceoctads of S, under M22; the full diagram follows.

BM Sp8('2 )2 2(2) S xMxL3 3 22

9+165

2 \ 23+32 22+10+20 21+22

15(2)

25+5+10+10

The 2-local structure of the Baby Monster gives rise to a geometry belongingto the diagram above although we do not have a GF(2) module with which todescribe it.

J4 M24

211

15(2)

210

10

S3xL3(2) M222

23+12 21+123

The existence of the above geometry is dependent on the existence of J4,which has not yet been established. A possible GF(2)-module of dimension 112is currently under construction by S. Norton,3 and it appears that the vertices ofour geometry may correspond to subspaces of dimensions indicated by thediagram above.

3Recently, Norton and others in Cambridge constructed J4.

288 M. A. RONAN AND S. D. SMITH

HS L3(2)

43

Sp4(2)

24

tS 0

is a 2-splitcover of theprojective planeof order 2.

is as in M22.

The 2-local structure of HS leads to the above geometry. A 20-dimensionalGF(2)-module can be obtained from_A (see 1) by taking <A, p 1 /<A, u> whereAEA2,1LEA3fA1and A+1LEA3.

Ru L3(2) S5

23+8 211CEEP is the G2(2) hexagon

0 is a 4-split cover ofthe projective plane oforder 2

2tC 0 is a 2-split cover of

the complete triplesystem on 5 points.

He Sp4(2) SP4(2)

263 263

L3(2)

21+6

1 2 4Suz

Sp4(2)' S3xL2(4) U4(2)

24+63 22+8 21+6

It is well known [2] that 3Suz is a subgroup of 1, and its center < > acts fixedpoint freely on A. Thus one can regard A as a 12-dimensional vector space overGF(4), call it U, and the quadratic form Q on A gives rise to a unitary form onU (one can obtain this form directly from A via U(x, y7 = 0, w, 1, w2 E GF(4) ifx y/8 = 0, 1, 2. 3 (mod 4) respectively, where x,y E A). We take as vertices ofour geometry the 2, 4 and 8-spaces of the 1 geometry (above), which are fixedby w; these can be thought of as certain totally isotropic 1, 2 and 4-spaces of U,as indicated by the diagram. Considering one of our 4-spaces F of U as anE. C A (see 1 above) fixed by w, we see that the quadratic form Q' on E. givesrise to a nondegenerate unitary form on F, for which the 1 and 2-spaces of ourgeometry in F are totally isotropic; this verifies the U4(2) part of the diagram.


Now considering a 1-space of our geometry, the 2 and 4-spaces containing itcorrespond to 4 and 8-spaces of the 1 geometry containing the corresponding2-space, and fixed by w. Since w acts trivially on the Sp4(2) quadrangle of such 4and 8-spaces, the Sp4(2) part of the diagram follows immediately.

Final remarks. In [3] Quillen proves that the building for a group G of Lie typeof char p is homotopy equivalent to the partially ordered set of p-subgroups ofG; it would be interesting to know to what extent this is true, for p = 2, for thegeometries and groups given here. In [6] Tits shows that the universal 2-cover ofcertain diagram geometries (in particular that for 3Suz above) is a building. Itwould be interesting to know what properties the universal 2-covers of ourgeometries have, and in particular, what the structure of their automorphismgroups is.

REFERENCES

1. F. Buekenhout, Diagrams for geometries and groups, J. Combinatorial Theory Ser. A 27 (1979),121-151.

2. J. H. Conway, Three lectures on exceptional groups, Finite Simple Groups, Powell and Higman(eds.), Academic Press, New York, 1971.

3. D. Quillen, Homotopy properties of the poser of nontrivial p-subgroups of a group, Advances inMath. 28 (1978), 101-128.

4. E. Shult and A. Yanushka, Near n-gons (to appear).5. J. Tits, Buildings of spherical type and finite BN pairs, Lecture Notes in Math., vol. 386,

Springer-Verlag, Berlin and New York, 1974.6. , A local approach to buildings (to appear).


PART IV

Representation theory of groups of Lie-type


PROBLEMS CONCERNING CHARACTERS OFFINITE GROUPS OF LIE TYPE'

CHARLES W. CURTIS

Introduction. In this note, we give a brief survey of some of the known resultsabout representations and characters (over the field of complex numbers) offinite groups of Lie type, and call attention to some open problems. This articleis a supplement to the expository paper [1], to which the reader is referred forbackground, unexplained notation, references to the literature, and a moresystematic presentation. We wish to refer also to the expository article of Lusztig[12] (to which [1] was intended as an introduction) and the lecture notes bySrinivasan [14], as well as lectures by Alvis, Kilmoyer, Lusztig, Springer andSrinivasan at the Santa Cruz conference, where other related results and prob-lems were discussed.

1. Orientation. In order to develop the representation theory of finite groupsof Lie type (in C), we view them from the standpoint of reductive groups overfinite fields, following the point of view of Steinberg in his memoir [16] (withsome changes in notation).

I.I. Finite groups of Lie type. Throughout the paper, k denotes a finite field FQof characteristic p, and G a reductive group over k, which is always taken tomean a connected reductive group over the algebraic closure k of k, with ak-rational structure given by a Frobenius endomorphism F: G -- G such thatthe group of fixed points GF = {x E G: Fx = x) is finite. The finite groups GFobtained in this way are the finite groups of Lie type. They include the finiteChevalley groups, and the twisted versions of Chevalley groups. As examples tokeep in mind, we have

G = GL (k ), F: (xv) --+ (x;? ), GF = GLn(q)

or

G = GLn(k), F: (xi,) --3.'(x;?)-1, GF = Un(q)

1980 Mathematics Subject Classification. Primary 20G05.'This research was supported in part by NSF grant MCS 76-07015.

C American Mathematical Society 1980

293

294 C. W. CURTIS

Each reductive group over k contains a pair (Bo, To) consisting of an F-stableBorel subgroup B0 and an F-stable maximal torus To C B0. These define a split(B, N)-pair of characteristic p in the finite group G', with Borel subgroup Bo,

N = N(TO)F, and Weyl group W = N(T0)F/ To , together with a splitting of Boas a semidirect product Ba = TO Uo , U0 = R. (Ba) (unipotent radical of Bo).Then Uo = OO(BO ), and is a Sylowp-subgroup of GF.

The F-stable subgroups P > B0 are the standard parabolic subgroups, and arein bijective correspondence with the parabolic subgroups W. of the Weyl group,where J is a subset of the distinguished generators, and

Wj = <J>, P = Pj = BoWjBo.

Each F-stable parabolic subgroup P has a Levi decomposition P = L V, whereV = R (P), and L is an F-stable Levi factor. If P H Wj C W, then LF is afinite group of Lie type with Weyl group Wj,

VF = OQ(PF), and pF = NG,(VF).

Another important family of subgroups are the F-stable maximal tori (T).These are distributed into GF-conjugacy classes, which are in bijective corre-spondence with the conjugacy classes in W (in the untwisted case), and with theI-cohomology group H 1(F, W) in general.

1.2. Cuspidal characters. Harish-Chandra's philosophy of cusp forms applies tofinite groups of Lie type GF as follows. Starting from a character ip E Irr(LF),for an F-stable Levi factor of an F-stable parabolic subgroup P, we lift T to acharacter of pF, with VF < ker (so as to simplify the computation ofintertwining numbers e.g.

GF GF

WjwWj

if P = Pj).Everything would be relatively straightforward if all irreducible characters of

GF were components of induced characters 1 GF as above, for proper F-stableparabolic subgroups.

Some irreducible characters are missed by this construction, however; theseare the cuspidal characters of GF, and are characterized by the condition

I°F) = 0

for all unipotent radicals V of proper F-stable parabolic subgroups P. Byanalogy with the Lie group situation, the cuspidal characters are said to form thediscrete series of GF.

(1.2.1) THEOREM (HARISH-CHANDRA). Each E Irr(GF) is either cuspidal, or isa component of GF, for a cuspidal character q, of L F for an F-stable Levi factor Lof an F-stable parabolic subgroup P. Moreover the pair (q), L F) associated with isessentially unique, in the sense that if qo E Irr(LF), q E Irr(LF), with ip and T'cuspidal, then

(7"GF,cGF) 0

unless L F LF and q; , q)' in GF.

CHARACTERS OF FINITE GROUPS OF LIE TYPE 295

Thus we have the following main problems, neither of which is completelysolved, whose status is part of the subject of this paper.

(I) Construction of the discrete series.(II) Decomposition problem: for a cuspidal character T E Irr(LF), to decom-

pose c ' into irreducible components.1.3. The MacDonald conjectures (1968) and their solution. About ten years ago,

MacDonald conjectured the existence of families of virtual characters (RT(9))of GF, parametrized by pairs (T, 0) consisting of an F-stable maximal torus andlinear characters 0: TF -q, C, with the property that for T minisotropic, and 0 ingeneral position, ± RT (0) is irreducible and cuspidal.

In 1955, Green had constructed the families {RT(0)} for the groupsby a profound combinatorial analysis, which led to the determination ofIrr(GL (q)). It was not clear whether a similar analysis was possible for otherChevalley groups. Bhama Srinivasan succeeded, in 1968, in constructing thefamilies (R7 (0)) and decomposing them to find the character table for thesymplectic groups Sp4(q), followed in 1974 by a determination of the charactersof G2(q) by Chang and Ree. In all these cases, a crucial step in the constructionof the { RT (9)} was Brauer's criterion for virtual characters.

At the Vancouver International Congress of Mathematicians (ICM), Lusztigannounced a new approach, based on considerations from algebraic topology, tothe construction of the discrete series for In the same lecture, heoutlined a fundamentally new construction of the virtual characters RT(0) in thegeneral case, using the action of GF on algebraic varieties X over k, which led toordinary representations of G F on the 1-adic cohomology groups H'(X) withcompact supports. The definition of the virtual characters RT(0) involves theLefschetz character

l(x, X) 1)' Tr(x, H,'(X)), x E GF.

The computation of the values of RT (9) is based in part on the cohomologicalmethods developed by Weil, Grothendieck and Deligne for investigating thenumber of rational points on varieties defined over finite fields.

Lusztig's results, still partly in conjectural form in his ICM lecture, wereproved in his joint paper with Deligne [4], and solved the MacDonald conjec-tures, among other things, as the following summary indicates.

Properties of the RT (0) (Tan F-stable maximal torus and 0: T F ---> Q.(i) Orthogonality . (R,G.(9), RT, (0')) = 0 unless TF- T'F and 0 --- 0' in GF.(ii) Behavior on semisimple elements. For s semisimple in G F, RT (0)(s) _9G£(s)StG(s)-I, where StG is the Steinberg character of ((GF, and is defined by

StG(S) = (- 1)a(G)-a(Z.(. ))Sts)G(l)

s semisimple, where a(G) is the k-rank of G.(iii) Behavior on unipotent elements. For a fixed F-stable maximal torus T,

RG(0)1.g unipotent is independent of 0, and defines a function QT : Gunipotent -' C(Green's functions). (Another construction of the (R,.(0)) starting from the QTwas given by Kazhdan [7].)

(iv) Degree.

deg RT(0) _ ±I GFI p.1I TFI.

296 C. W. CURTIS

(v) Irreducibility. ± R5(0) is irreducible if 0 is in general position, i.e. W8 0

for all w 1 in the Weyl group W(T)F = N(T)F/TF of T.(vi) Discrete series. ± R' (O) is irreducible and cuspidal provided that 0 is in

general position, and T minisotropic (i.e. contained in no proper F-stableparabolic subgroup).

Returning to our main problems, the construction of the discrete series of GFwill involve the still unsolved problem of decomposing RT (9), for T a miniso-tropic torus (for 0 not in general position e.g. 0 = 1).

The components of R G( 1are called unipotent representations of G F; theyplay a role in character theory analogous to the unipotent classes among allconjugacy classes.

2. The decomposition problem.2.1. Hecke algebras. Let H be a subgroup of a finite group G, and let

4, E Irr(H). Let e be a primitive idempotent in CH such that CHe affords I;then CGe affords the induced character 4,G, and

EndCG(CGe)° = eCGe.

The semisimple algebra eCGe is the Hecke algebra H(G, H, O. There is abijection --+ w = P Hoc, H, >

from the irreducible components E ,G and theirreducible representations {w} of H(G, H, p), such that the character values(x), x E G, are given in terms of the representations of H(G, H, ii):

(2.1.1) THEOREM (REE). Let E 4, G correspond to w. Then for x E G, C theclass sum containing x,

ZG(x) w(eCe)

1gEGw(ege)w(eg_e)

These ideas are applied to the Chevalley groups GF using the theory ofgeneric algebra.

2.2. Generic algebras. Let W be a Coxeter group, with distinguished generatorsR = ( w 1 . . . . . w.). Let (XI, ... , X } be indeterminates over Q, with X; = Xj ifw; -- wj in W. Form an algebra Aw over o = Q[X1, ... , with basis{aW} WE w and multiplication given by

awaW = awW if l(w,w) > I(w),

aWaW = X,aWW + (Xi - 1)aW, l(w,w) < 1(w).

Then Aw is an associative algebra, first introduced by Tits to study theconnection between H(GF, Bo, 1Bo) and the group algebra of the Weyl groupC W, suggested by the Bruhat decomposition: B° \ GI BO *--3- W.

A homomorphism f: o - Q defines a specialized algebra Awl = Q ®, Aw,obtained by applying f to the structure constants of Aw. Here are someexamples.

(2.2. 1) (i) AwJ QW for f: X; --* 1.(ii) C ® A0 = H(GF, Bo B. 1B), where W is the Weyl group of GF, for a

suitable homomorphismf.


(iii) C ®Q Akaf = H(GF, Bo , 0), for an adjoint Chevalley group, for suitablychosen f, where Wa is the stabilizer of 0 in W (R. Kilmoyer [10]).

(iv) C ®Q AWf = H(GF, LF, A), for A any irreducible cuspidal unipotentcharacter of LF, for a suitable choice of W and f (see G. Lusztig [12]).

Note that the decomposition of 9GF (and 1Bo) are the minimal cases of theDecomposition Problem (see § 1.2). The structure of H(GF, LF, fl for an arbi-trary irreducible cuspidal character of LF has recently been determined byHowlett and Lehrer [6].

Using (2.2.1), (i) and (ii), it follows that the algebras C W and H(GF, Bo , 1B)are isomorphic (the Deformation Theorem of Tits). Therefore we have abijection from Irr(W) to the irreducible characters of H(GF, BF 1,9F andhence to the irreducible components of 1Bo.

(2.2.2) THEOREM. For each q7 E Irr(W) there exists a polynomial dq,(X) EQ[X] such that if GF is an untwisted Chevalley group with Weyl group W, andis the character in 1 Bo corresponding to qp, then

deg $, = dq,(q).

The generic degrees d., have been computed for all indecomposable Weylgroups. It turns out that the degrees of the irreducible components of theinduced characters in (2.2.1) (iii) and (iv) can also be expressed in terms of thegeneric degrees dq, (viewed as polynomials in Q[XI, .... (See also Howlettand Lehrer [6] for further comments on generic degrees.)

2.3. Problems.(2.3.1) Representations of generic rings. (Ak, (9, k, where k is an algebraic

closure of K = Q(X1, ... , X.).) For W of classical type A. - D., Hoefsmitfound the irreducible representations (and the generic degrees) by an ingeniousextension of Young's construction of the representations of the Weyl groups. Incase K = Q(X), it is known that the irreducible representations of A, ® K arerealizable in K with some exceptions (Benson and Curtis), but their constructionremains unsolved. Recently Kazhdan and Lusztig constructed a family ofrepresentations of generic algebras using graphs associated with Coxeter groups,which exhibit connections with singularities of Schubert varieties, and results ofJantzen and Joseph on primitive ideals in enveloping algebras (see [8], [9]).

(2.3.2) Character formulas. Apply information about representations of genericalgebras to evaluate characters of irreducible components of induced representa-tions associated with the generic algebra (see §§2.1 and 2.2).

(2.3.3) Interpretation of generic degrees. The generic degrees dq,(X) have beencomputed using character tables of individual Weyl groups. Do they have someintrinsic significance, arising from geometrical considerations involving theCoxeter groups?

(2.3.4) Special functions over finite fields. The decomposition of Hecke algebrasis equivalent to finding analogues of spherical functions on finite groups. Besselfunctions over finite fields were introduced by Gelfand and Graev to decomposeH(GF, Uo , qp) for qp a nontrivial character on Uo , in case G = SL2. Orthogonalpolynomials related to Hecke algebras of Chevalley groups have been studied byDunkl and Stanton (see [15]). Other formulas involving Gauss sums, and arisingfrom the study of Hecke algebras, were proved by Helversen-Pasotto [5]. Are

2 98 C. W. CURTIS

there connections between these problems on "special functions" and zetafunctions of algebraic varieties, as Gauss sums are related to zeta functions ofhypersurfaces?

3. Character formulas.3.1. Values on regular semisimple elements. We begin with some known results.(3.1.1) (Surowski [17]). Let GF be an untwisted Chevalley group, and let P1 be

an F-stable standard parabolic subgroup associated with a parabolic subgroupW1 of W. Let s be a regular element in TF, where T is a F-stable maximal toruscorresponding to the conjugacy class of w in W. Then

lPf(s) = 1W (w).

(3.1.2) (Lusztig [13]). Let be an irreducible component of 1Ba correspondingto an irreducible character y, of W (see §2.2). Let T be an F-stable maximaltorus. Then there exists a virtual character of W (depending on the GF-con-jugacy class of T) such that if s E TF is regular, then

k(s) _ ('P, Ow.(In particular, k(s) is independent of q. In [13] Lusztig proved that there exists aclass function on W behaving as in (3.1.2); it is easily checked, however, that(, qp) E Z for all rp E Irr(W), so that is a virtual character of W.)

(3.1.3) (Deligne and Lusztig [4]). Let s be regular semisimple in TF, for anF-stable torus T, and let be an irreducible unipotent character (i.e. E RT(1)for some F-stable torus T'). Then

(s) _ RT(1)).

Problem. Decomposition of RT (1) into irreducible components. For the Coxetertorus T (corresponding to the class of a Coxeter element in W) Lusztig [11]proved that for a suitably chosen variety XT, for which (H(XT)) afford theirreducible components of RT(1), we have

0) HH(XT) and H.(XT) are disjoint, if i =j.(ii) F: HH(XT) -* HH(XT) is semisimple (for GF untwisted), and the eigen-

spaces of F are irreducible G F-modules. Some of them are cuspidal, andprovided a start towards Lusztig's determination of unipotent characters (theirnumber and their degrees) for each type of group.

3.2. The Steinberg character. Solomon and Tits proved that the Steinbergcharacter StG of GF is afforded by the top homology group in the rationalhomology of the combinatorial building .(G) of G. This geometrical interpreta-tion can also be used to compute the character values of StG. This can mostefficiently be done by introducing the spherical building JB (G) of a reductivegroup G over k (see Curtis, Lehrer and Tits [3]). As a simplicial complex, JB (G)is the d-fold suspension of i.(G), where d is the k-rank of the connected centerof G. The finite group GF acts on JB (G), hence on the rational homology ofJZ (G). The representation of GF afforded by the top homology group of JB (G)is StG. The values of StG can then be found using the fact that for x E GF, thevalue of the Lefschetz character at x of the homology representation of GF onH,(JB (G)) is the Euler characteristic of the fixed point set JZ (G)X. If x E GF isnot semisimple, then JB (G)z is contractible, and it follows that StG(x) = 0. On


the other hand, if x E GF is semisimple, the fixed point set JB (G)X can beidentified with the spherical building JB (ZG(x)°), from which one obtainsStG(x).

Problem. Use the theory of homology representations to construct otherrepresentations of GF, and calculate the character values. (See [1, §4] forreferences to Lusztig's work on homology representations of GL (q), and [2] fora construction of the dual of a character of GF using the homology of 0(G) witha suitable coefficient system. The duality operation is discussed in Alvis's articlein these PROCEEDINGS.)

REFERENCES

1. C. W. Curtis, Representations of finite groups of Lie type, Bull. Amer. Math. Soc. (N.S.) 1(1979), 721-757.

2. , Homology representations of finite groups, Proc. Second Conf. on Representations ofAlgebras, Carleton Univ., Lecture Notes in Math., No. 25, Springer-Verlag, Berlin and New York,1980.

3. C. W. Curtis, G. I. Lehrer and J. Tits, Spherical buildings and the character of the Steinbergrepresentation, Invent. Math. 58 (1980), 201-210.

4. P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math.(2) 103 (1976), 103-161.

5. A. Helversen-Pasotto, L'identite de Barnes pour les corps finis, C. R. Acad. Sci. Paris Sir. A-B286 (1978), A297-A300.

6. R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalized Hecke rings,Invent. Math. 58 (1980), 37-64.

7. D. Kazhdan, Proof of Springer's hypothesis, Israel J. Math. 28 (1977), 272-286.8. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent.

Math. 53 (1979), 165-184.9. , Schubert varieties and Poincare duality (to appear).

10. R. Kilmoyer, Principal series representations of finite Chevalley groups, J. Algebra 51 (1978),300-319.

11. G. Lusztig, Coxeter orbits and eigenvalues of Frobenius, Invent. Math. 28 (1975), 101-159.12. , Representations of finite Chevalley groups, CBMS Regional Conf. Ser. in Math., vol.

39, Amer. Math. Soc. Providence, R. I., 1978.13. , On the reflection representation of a finite Chevalley group (to appear).14. B. Srinivasan, A survey of representations of finite Chevalley groups, Lecture Notes in Math.,

No. 764, Springer-Verlag, New York, 1979.15. D. Stanton, Some q-Krawtchouk polynomials on Chevalley groups (to appear).16. R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. No. 80, 1968.17. D. Surowski, Permutation characters of finite groups of Lie type, J. Austral. Math. Soc. (to

appear).



THE RELATION BETWEENCHARACTERISTIC 0 REPRESENTATIONS

AND CHARACTERISTIC pREPRESENTATIONS OF FINITE GROUPS

OF LIE TYPE

R. W. CARTER

Any discussion of the decomposition on reduction mod p of the irreduciblecomplex representations of the finite groups of Lie type must necessarily betentative in our present state of knowledge, since neither the irreducible complexrepresentations nor the irreducible modular representations are fully understood.

As far as the modular theory is concerned, we know how to parametrise theirreducible modular representations but do not know their characters or theirdimensions except in certain special cases. As for the complex theory, Deligneand Lusztig have constructed families of generalised characters which includemost of the irreducible characters and Lusztig has constructed further irreduci-ble characters in subsequent work, but these do not as yet constitute all theirreducible complex characters.1

We shall therefore concentrate on the question of how the Deligne-Lusztigcharacters in general position might be expected to decompose on reductionmodulo p. This is known only in very special cases. However by looking at theexperimental evidence and comparing it with the decomposition of the principalseries modules for the restricted enveloping algebra, one can get quite a goodidea of how the Deligne-Lusztig characters might be expected to decompose intomodular irreducibles.

1. The irreducible modular representations. Let G be a simple simply-con-nected algebraic group over the algebraic closure K of the finite field Fp and

1980 Mathematics Subject Classification. Primary 20C20, 20G40.

'Since the manuscript was prepared Lusztig has succeeded in determining the degrees of allcomplex irreducible representations of finite Chevalley groups of adjoint type, provided the numberof elements in the base field is sufficiently large.

0 American Mathematical Society 1980

301

302 R. W. CARTER

assume G is defined and split over F,,. Then the Frobenius pth-power map a isan endomorphism of G and G. g E G; g° = g) is a Chevalley group overF 2P.

For example one could take G = SL (K), and then one has G. = SL (p).Let T be a maximal torus of G. Then T is isomorphic to K* x X K* (1

factors) where K* is the multiplicative group of K. Let X be the set of1-dimensional rational representations of T. Then X = Z ® ®Z (1 factors)and X is called the lattice of weights. Let XR = X ® R and let W = Y,G(T)/T.W is called the Weyl group. It operates on T by conjugation, so also on X. Ittherefore acts on XR. XR admits a W-invariant positive definite form whichmakes it into a Euclidean space.

The torus T acts on the Lie algebra of G, which decomposes into a direct sumof 1-dimensional T-submodules. The nonzero elements of X arising in this wayare the roots. These form a finite subset 1 of X. (D contains the negative of eachof its elements.

Let L, be the hyperplane in XR orthogonal to r and wr be the reflection in Lr.Then W is the group of isometries of XR generated by wr for all r E (D.

The connected components of XR - Ur L, are called chambers. We pick oneof these chambers C and call it the fundamental chamber. The set X + = X nC is called the set of dominant weights. The nonzero dominant weights whichcannot be expressed as a sum of two nonzero dominant weights form a Z-basisyi, ... , y, of X. They are called the fundamental weights. A weight 2;_1n;y, liesin X + if and only if each n; > 0.

THEOREM (CHEVALLEY [2]). There is a 1-1 correspondence between irreduciblerational G-modules and dominant weights.

The irreducible G-module with dominant weight A will be denoted by MA.We now consider irreducible KG°-modules. Let Xp = (V_1n,y;; 0 < n; <p).

XP is called the set of restricted weights. I X0 = p'.

THEOREM (CURTIS, STEINBERG [3], [11]). For each A E XP, MA remains irreduci-ble as a G°-module. The M., A E XP, form a complete set of nonisomorphicirreducible KG°-modules.

EXAMPLE. Type A2. G = SL3(K). The restricted weights XP lie in the regionshown in the diagram.

2We consider Chevalley groups over the prime field for convenience. It would be possible togeneralise the ideas in this article to Chevalley groups over arbitrary finite fields.

CHARACTERISTIC 0 AND P REPRESENTATIONS 303

2. The restricted enveloping algebra. Let gc be the simple Lie algebra over C ofthe same type as G. gc contains root vectors e r E 0, and elements h r E 0, ina Cartan subalgebra. Let ',1 be the universal enveloping algebra of gc and %be the Kostant ring in L. This is the subring generated by the elements e;`/k!for all r E I, k > 0. Let 0,1K = % ® K. iK is called the hyperalgebra, and itssubring u generated by all elements e, ® 1, r E I, is called the restrictedenveloping algebra. u is finite dimensional, and its dimension is pd"" Sc.

THEOREM (CURTIS [3]). The M., ,\ E XP, may be regarded as u-modules andthey form a complete set of irreducible u-modules.

For each X E XP there is a universal u-module generated by a highest weightvector of weight X. We denote this by Z. and call it a principal series u-module.Z. has dimension pN where 2N = I. Z. contains M,, as an irreducible quotientand all its composition factors have form Mµ for certain it E XP. We wish toconsider the multiplicity of M, as a composition factor of Z., and to do this wemust introduce the affine Weyl group.

3. The affine Weyl group. The 1-dimensional representations of T give rise to1-dimensional representations of the Lie algebra of T, so elements of X may beregarded as acting on the Cartan subalgebra he of gc spanned by the elements

Then we have L,_('\ EXR;\(h,)=0).Letp=yl+ + y, bethe sum of the fundamental weights. We translate the hyperplanes L, to passthrough -p instead of 0. Let L,'0 = (X E XR; (X + p)(h,) = 0). We also considerparallel hyperplanes, translated by multiples of p. Let

L,,k= (XEXR;(X+p)(h,)=kp), kEZ.

Let w,,k be the reflection in L, k. Then the group of isometries of XR generatedby the w,,k for all r E I, k E Z is called the affine Weyl group W..

The connected components of XR - U,,kL,k are called alcoves. The set A, isan alcove where A 1 is the set of X E XR satisfying the conditions

(X + p)(h,) > 0 for all fundamental roots r,,

(X + p)(hR) < p where R is the highest short root.

W,, operates on the set of alcoves and given any two alcoves A, A' there is aunique a E Wa with a(A) = A'. Let XR and (XR)p be defined by

XR = n;y,;n, ER,n, 301

1

(XR)p = f n;y1;n; ER, 04 n; <p - 1}.r 1

An alcove A is called dominant if A n XR is nonempty and restricted ifA n (XR)p is nonempty.

EXAMPLE. Type A2. G = SL3(K). The decomposition of XR into alcoves isshown in the diagram.

304 R. W. CARTER

11A1

W,, is generated as a Coxeter group by the reflections in the 1 + 1 walls of A 1.It has a natural partial ordering defined as follows. Given a, a' E W,, we saya < a' if a, a' have reduced expressions in terms of the Coxeter generators suchthat the first is a subsequence of the second. This partial ordering on WQ inducesa partial ordering on alcoves in which A 1 is the smallest.

If W E W, A E XR we define w.A = w(A + p) - p. If A is an alcove then w.Ais an alcove. If the alcove A is restricted w.A need not be restricted. Howeverthere is a unique v E X such that w.A 1 + pp is restricted. This v is called pw. Wewrite A. = w.A1 + ppw. Then A. is a restricted alcove and every restrictedalcove has form A. for some w E W. There may, however, be repetitions. Thenumber of restricted alcoves is I WI/f where

f = I (=- W; Aw = Ai}

f is equal to the order of the centre of the group Gc of the same type as G overC.

Any alcove has form A,,, + pv for some v E X. We write Ax, = Aw + pv. v isuniquely determined by Ax,,, but there are f choices for w.

There is a unique maximal restricted alcove, and this is Awo where wo is theelement of W of maximal length.

Let c?, = (Aw; w E W) be the set of restricted alcoves and Cps be the set ofdominant alcoves A satisfying A < A. Then I (?, I < I d,1.

The cardinalities of l?r Cps in some low rank examples are as follows:

A 1 = SL2, I d, 1 = 1, ?s l = 1,A2 = SL3, I?,I = 2, I?sI = 2,B2 = Sp4, V e, I = 4, Qs = 4,A3=SL4, I?.I =6, jQj =8,G2, I ?, I = 12, 1

I = 16.


The alcoves in (fr, d. for types A1, A2, B2, G2 are shown in the followingdiagrams.

AI

A2

B2

306 R. W. CARTER

Let A = wo.A + ppwo and A = (X; A (=- A). Then if A is an alcove so is A. If Ais restricted so is A. In fact we have

Aw = Awow, AwY = Awow,wo(y).

The map A -* A gives a duality which reverses the order relation on the restrictedalcoves.

4. The composition factors of the principal series u-modules. We shall subse-quently assume that p is greater than the Coxeter number of W and consider thecomposition factors of Zµ where µ E X lies in some alcove. We begin with someexamples.

Type A,. G = SL2(K), f = 2. Zu has dimension p. Zµ has two compositionfactors Mµ, M(p_2)P_µ of dimensions µ + 1, p - I - µ respectively.

-P (p-2)P-A µ (P- OP

Type A2. G = SL3(K), f = 3. Zµ has 9 composition factors. 3 of its irreduciblefactors occur with multiplicity 1 and 3 occur with multiplicity 2. Their weightsare placed as shown in the diagram.

Type B2. G = Sp4(K), f = 2. Zµ has 20 composition factors. Two of itsirreducible factors occur with each of the multiplicities 1, 2, 3, 4.

We now consider the general case. Results concerning the multiplicity withwhich MA occurs as composition factor of Zµ were proved by Humphreys [5].More recent results of Jantzen [8] suggest it is desirable to split up thesemultiplicities in a manner which depends on the alcove structure of the group inquestion.

CHARACTERISTIC 0 AND p REPRESENTATIONS 307

THEOREM (HUMPHREYS, JANTZEN). There exist nonnegative integers c(A', A) forA' E CT, A E Q A' < A such that the composition factors of Zµ are given by

Zµ H 2 2 c(Aw.v, A)M,(n)(w,v)EWXX AEQ,

Aw,,EQ, Aw,GA

where q = wow.p + ppwow + pwo(v) and a is the element of W. such that a(Aw.r,)= A. Note that q c Awow,wo(v) _ Aw,v

In particular the number of composition factors of Zµ is

f 2 2 c(A', A).A'E9, AEQ,A'' A

The matrix of coefficients c(A', A) in low rank cases is as follows.

Al (1).

A 2

r(

l\1 11

3

B2

1

1

1

1

1 1

1 1

1

1 1

1 1

1

1

1

1

1

1

2

1

1

1

1

3

2

1

1

1

1 2 3 4 5 6 7 8 11 13 15 16

1 1 1 1 1 1 1 1 2 3 3 3 42 1 1 1 1 1 1 2 3 3 2 43 1 1 1 1 1 2 3 2 2 3

4 1 1 1 1 2 2 2 1 2

5 1 1 1 1 2 1 1 2

6 1 1 1 1 1 2

7 1 1 1 1 1 2G2 8 1 1 1 1 2

9 1 1 1 1

10

11 1 1 1 1

12

13 1 1 1

14

15 1 1

16

308 R. W. CARTER

5. The conjectural decomposition of the Deligne-Lusztig characters. Deligneand Lusztig [4] construct a generalised character RT,B of G. for each a-stablemaximal torus T of G and each 1-dimensional complex representation B of T. IfB is in general position then there exist numbers EG = ± 1, ET = ± I such thatEGETRTO is an irreducible character of G.

Now the multiplicative group K* is isomorphic to the subgroup of C* of allcomplex roots of unity of order prime to p. We choose such an isomorphismembedding K* as a subgroup of C*.

We assume B is in general position. Then we can find a weight µ E A, suchthat the character µ of T, when restricted to T and interpreted as a map from T.to C*, coincides with 9. The maximal torus T of G. determines an elementz E W by which a split torus in G. must be twisted to derive T. [1, Part E].

Let EGETRTB be a modular character of G. obtained from the irreduciblecomplex character EGETRTB by reduction modulo p. The composition factors ofany KGB-module with character EGETRTO are uniquely determined. We firstconsider the situation in a number of examples. We assume in these examplesthat B is a generic character of T. This is a somewhat stricter condition thanthat B should be in general position, viz. not fixed by any nonidentity element ofthe appropriate Weyl group. We require in addition that the weight µ E Al issufficiently far from any wall of the alcove Al to ensure that various weightsobtained from µ by reflections and small deformations remain within theinterior of their respective alcoves.

Type A1.G= SL2(K),f = 2. _(i) Suppose T = 1. Then dim(EGETRTA) = p + 1. There are two composition

factors Mµ, M(P_1)P_,1 of this module. We have dim Mµ = µ + 1 anddim M(P-1)P-µ = p - µ

-P (P-1)P-1+ 1+ (p-1)p

(ii) Suppose T 1. Then dim(EGETRTB) = p - 1. There are two compositionfactors Mµ, M(p_3)p_,1 of this module. We have dim Mµ = µ + 1 anddim M(P-3)p-µ = p - 2 - µ.

-P (p-3)p-1+ 1+ (P-1)p

We see that in both the above cases the weights of the composition factors ofEGETRTO are obtained from those of Zµ by making small deformations. Thedeformations which need to be made depend upon the twisting element z E W.

Type A2. G = SL3(K), f = 3. EGETRTO has 9 composition factors. 3 of thesehave weights in the top restricted alcove. The reflections of these three weightsin the bottom alcove also occur as highest weights of composition factors, aswell as 3 additional weights in the bottom alcove obtained from the former 3 bysmall deformations. The precise positions of the weights depend on the twistingelement T. For more precise information we refer to Humphreys [6].

CHARACTERISTIC 0 AND p REPRESENTATIONS 309

Type B2. G = Sp4(K), f = 2. Humphreys has conjectured, on the basis ofstrong numerical evidence, that EGETRTO has 20 composition factors, arranged inclusters as shown in the diagram. [6].

In general we may expect that EGETRTO has a similar decomposition pattern toZµ, with the difference that the weights of the composition factors must bedeformed slightly. The way in which these weights need to be deformed hasbeen suggested by Humphreys [6] when the alcove concerned is restricted.

CONJECTURE. Let µ E A i give rise to a character 0 of T. in generic position.Then the composition factors of eG ETRT.e are given by

2 Z c(A,,,,, A)MQ(n)w,jEWXX AE&,

A,., E 9,, A_ GA

where q = wow.µ + ppw0w + pwo(v) + wowwor(ewok,k,o + v) and c = w-1(pw)

We note that q E Aw,o = Awow,wo(v) The term wowwoT(ewo,,,,,,o + v) representsthe small deformation necessary in q. Its dependence on w and T is backed by afair amount of evidence, but its dependence on v must be regarded as moretentative since the number of cases in low rank involving_ contributions fromnonrestricted alcoves is small. As before a E W. satisfies a(Aw, v) = A.

Finally we c. nsider the question-what are the coefficients c(A', A) whenA' E e5, A E Q, and A' < A? We conjecture that these coefficients are relatedto the polynomials Py ,(q) recently introduced by Kazhdan and Lusztig [9] wherey, z are two elements of a Coxeter group satisfying y < z.

CONJECTURE. Suppose the alcoves are labelled by elements of W. in such a waythat the alcove wo.A 1 is labelled by 1 E W0. With this labelling let A' E Cps belabelled by y E W. and A E Q, be labelled by z E W.. Then c(A', A) = Py z(1).

310 R. W. CARTER

The polynomials P ,,(q) may be defined inductively on 1(z) as follows:

(i) P1.1(q) = I.(ii) Let z = sz' where 1(z') = 1(z) - 1. Then

P . = 41 `P + `P z - u(x z')q(l(2')-r(X)+1)/2P

X

where the sum extends over elements x satisfyingy < x < z', 1(x) in 1(z') mod 2,sx < x; where µ(x, z') is the coefficient of q(qz'>-qx)-1)/2 in P,, Z. and where

c0 ifsy>y,

=1 if Sy <.Y.

The polynomials P, z(q) in low rank cases for Wa for those pairs of elementsy, z needed to calculate c(A', A) are as follows.

Al

((

(1).

A2 l l 1 I'

B2

A3

1 1 1

I 1 l

1 1

1 1 1 1 l+q 1+2q1 1 1 1 1+ q

1 1

1 1

1 1

l 2 3 4 5 6 7 8 11 13 15 16

1 1 1 1 1 1 1 1+q I + q+q2 I+q 2+ q' I+ q3 +q4 1+q+q'+q42 1 1 1 1 1 1 +q I + q+q2 I +q 2+ q3 l + q3 l+q+q3+q43 I I I I I l+q I + q+q2 I +q2 1 + q3 1+q+q34 1 1 1 1 1+q I +q 1 +q2 1 I+q5 I I I I I +q 1 1 I + q

6 1 I 1 1 1 l+q7 1 1 I 1 1 t+qG28 1 1 I 1 I+q9 1 l 1 I10

Il 1 1 1

12

13 I 1 1

14

15 1 1

16

It appears therefore that the Deligne-Lusztig characters in generic positionwill have the same number of composition factors on reduction modp as theprincipal series modules for the restricted enveloping algebra. There is also anintriguing connection with the composition factors of the Weyl modules for thealgebraic group G. These are reductions mod p of the irreducible rational


modules for GC. Jantzen [8] has shown that Weyl modules for G with highestweight in general position in the lowest p2-alcove have the same number ofcomposition factors as the principal series u-modules. Another paper of Jantzen[7] discusses the decomposition behaviour of Weyl modules with highest weightsin the restricted region. Comparison of these two papers of Jantzen suggeststhere is a remarkable duality between the decomposition of Weyl modules in therestricted region and the lowest p2-alcove. Lusztig has conjectured [1Q] that thecoefficients of the inverse of the decomposition matrix for Weyl modules can beexpressed in terms of the Kazhdan-Lusztig polynomials.

Finally Verma (1974) gave a conjecture on the way in which the Deligne-Lusztig characters should be expressible in terms of characters of Weyl modules.The relation between Verma's conjecture and our conjecture about the decom-position of the Deligne-Lusztig characters into modular irreducibles is notobvious.3

REFERENCES

1. A. Borel et al., Seminar on algebraic groups and related finite groups, Lecture Notes in Math.,vol. 131, Springer-Verlag, Berlin and New York, 1970.

2. C. Chevalley, Seminaire Chevalley. Vols. I, II, Classification des grouper de Lie algebriques,Paris, 1956-1958.

3. C. W. Curtis, Representations of Lie algebras of classical type with applications to linear groups,J. Math. Mech. 9 (1960), 307-326.


5. J. E. Humphreys, Modular representations of classical Lie algebras and semisimple groups, J.Algebra 19 (1971), 51-79.

6. , Weyl groups, deformation of linkage classes and character degrees for Chevalley groups,Comm. Algebra 1(1974), 475-490.

7. J. C. Jantzen, Zur Charakier-formel gewisser Darstellungen halbeinfacher Gruppen and Lie-Alge-bren, Math. Z. 140 (1974), 127-149.

8. , Uber das Dekompositionsverhalten gewisser modularer Darstellungen halbeinfacherGruppen and ihrer Lie-Algebren, J. Algebra 49 (1977), 441-469.

9. D. Kazhdan and G. Lusztig, Representations of Coxeler groups and Hecke algebras, Invent.Math. 53 (1979), 165-184.

10. G. Lusztig, Some problenr. in the representation theory of finite Chevalley groups, thesePROCEEDINGS, pp. 313-317.

11. R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33-56.12. J. C. Jantzen, Zur Reduktion modulo p der Charakiere von Deligne and Luszlig (to appear).

UNIVERSITY OF WARWICK, ENGLAND

3Since the preparation of this paper Jantzen has announced a proof of Verma's conjecture forcharacters in generic position. His article [12] also succeeds in comparing the Deligne-Lusztigcharacters with those of the ZA.

SOME PROBLEMS IN THEREPRESENTATION THEORY OF

FINITE CHEVALLEY GROUPS

GEORGE LUSZTIGI

In the first section of this paper, I will present a classification of the unipotent(complex) representations of a finite Chevalley group and state a conjecture ontheir character values. The second section is a review of results of Kazhdan andmyself [3], [4]; these lead to some questions which are formulated in the thirdsection. In particular, I will state a conjecture on the character of modularrepresentations of a finite Chevalley group.

1. Classification of unipotent characters (see [1], [5], [6], [7]). Let G be analmost simple algebraic group defined and split over the finite field Fq (q =power of a prime p). Choose a maximal torus and a Borel subgroup B D T suchthat T and B are both defined over Fq. The G(Fq)-conjugacy classes of maximaltori in G which are defined over Fq are in 1-1 correspondence with theconjugacy classes in the Weyl group W(T)/T. Let T. be a maximal torusdefined over Fq, corresponding to w E W. The virtual character RT (l) of G(Fq)(see [1] and the lectures of Curtis and Srinivasan) will be denoted R. We haveRw = Rw, if and only if w, w' are conjugate. By definition, an irreduciblecharacter p of G(Fq) is unipotent if <p, Rw> 0, for some w E W. For exampleall components of R1 = IndB(Fgj(1) are unipotent characters; it is well knownthat they are in 1-1 correspondence with the irreducible characters of W. Foreach irreducible character X of W, we denote by Xq the corresponding irreduci-ble character of G(Fq) contained in R1, and we define two polynomials PX(Z),PX(Z) by

PX(q) = dim( W1 -12 X(w)R.),W

PX(q) = dim(X;).

1980 Mathematics Subject Classification. Primary 20D06, 20C15; Secondary 20C20.'Supported in part by the National Science Foundation.


313

314 GEORGE LUSZTIG

The Polynomials PX, PX are known in all cases. They coincide, when G is of typeA, but not in general. P. has integral positive coefficients, while the coefficientsof fix are, in general, only rational numbers. It is an experimental fact that thehighest power of Z dividing PX divides also P. We say that X is special if thehighest power of Z dividing PX coincides with the highest power of Z dividingPX. If X is special, we have (see [7])

PX(Z) = Z' + higher powers of Z,

PX(Z) = ,'-, Z6 + higher powers of Z, (1)

where n is of the form 2` (c > 0), 3!, 4!, or 5!. We attach to such X the finitegroup I'X of order n, isomorphic to (0252)`, `G' 3, C54 or C55. Let us now define forany finite group I', a finite set M(F) as follows. M(F) consists of all pairs (x, a)where x is an element of F defined up to conjugacy and a is an irreduciblecharacter of the centralizer Z(x) of x in F. The set M(F) has a canonical pairingM(I') x M(F) -* C, defined by

{(X, (1), (y, T)} = I Z(X)

2 a(gyg-l) T(g-'Xg)g(=-r

x'gYg - ' - gyg

The following result gives a classification of all unipotent characters of G(Fq)and a uniform formula for their degree. (In the case where G is of type F4 or E8,we must make the assumption, which is probably unnecessary, that q is suffi-ciently large.)

THEOREM ([5], [6]). There exists a bijection

{ unipotent characters of G(Fq)) H u M(FX)X E Wspecial

with the following property. If pxo is the unipotent character corresponding to(x, a) E M(FX), then

dim(pp o) _ 2 f (X, a), (y, T) } PX (q) (2)(y. T)EM(rx)

XEW

Moreover, pi l = Xq, for X special.

We now state a conjecture which strengthens (2).Problem I. For any px o as above, and anyjw E W, we have

<px,o, R.> = {(X, (1), (y, T)}X (w)(y T) E M(rx)

XE W0=Xy

This is essentially equivalent to a character formula for pX o on semisimpleelements. The collection of characters (px Q) in 1-1 correspondence with M(I'X),with X a fixed special character of W, is the Fq-analogue of what, over local

REPRESENTATION THEORY OF FINITE CHEVALLEY GROUPS 315

fields, Langlands calls an L-packet. This collection consists of 22c, 8, 21 or 39characters according to whether I'x is (c 2)c, c 3, S4 or C25S. The classification ofunipotent characters given above remains valid (with the appropriate modifica-tion) when G is no longer assumed to be split.

The starting point for the following problem is the observation that thedimension of any irreducible complex character of G(Fq) is given by a poly-nomial in q of degree equal to half the dimension of a unipotent class C in G,and such that the exponent of the largest power of q dividing this polynomial isthe dimension of the variety JB of all Borel subgroups containing a fixedunipotent element u E G. Let C' be the G-conjugacy class of u. (C # C' ingeneral.) It is likely that C' can be described as follows.

Problem II. Let p be an irreducible complex character of G(Fq). Show that thereis a unique unipotent class C' in G which has the property that >2gEC'(Fq) P(g) # 0and has maximal dimension among unipotent classes with this property. Twounipotent characters of G(Fq) give rise to the same unipotent class in G if and onlyif they correspond to the same special representation of W.

2. A review of results in [3], [4]. Let W be a Coxeter group and let S be its setof simple reflections. We consider the Hecke algebra 3C over the ring of Laurentpolynomials Z[q 1/2, q- 1/2]; 3C is defined as follows. 3C has basis elements Tw,one for each w E W; the multiplication is defined by the rules

Tw Tw, = Tww,, if l(ww') = l(w) + l(w'),

(Ts + 1)(T,, - q) = 0, if s E S.

Here l(w) is the length of w. Let < denote the standard partial order on W. Thefollowing result was proved by Kazhdan and myself in [3, 1.1].

THEOREM. For any w E W, there is a unique element C. E 5C such that

CC = Z (- 1)!(w)-!(y)g1(w)/2-1(y)Py.w(q-)Tyy'w

_ 2 (- 1)1(w)-!(y)q-1(w)12+1(y)py.w(q)Ty.(3)l

Yew

where Py w is a polynomial in q of degree < Z(l(w) - l(y) - 1), if y < w, andPw,w = 1.

While no explicit general formula for P3 , is known, there is an algorithm bywhich Py w can be computed in any given case. In the case where W is a Weylgroup, there exists [3] a cohomological interpretation for the coefficients of thepolynomials Py w. Consider the Schubert variety Jew corresponding to w E W.This has singularities, in general, hence it does not usually satisfy Poincareduality. However, by a general construction of Goreski, Macpherson andDeligne, one can attach to w some new cohomology which does satisfyPoincare duality. The corresponding local cohomology groups of w havedimensions given by the coefficients of Py w; in these terms, the equality (3)simply expresses the fact that local Poincare duality is satisfied by this newcohomology. A similar interpretation holds in the case where W is an affineWeyl group. In particular, in these cases, Py w has positive coefficients.

316 GEORGE LUSZTIG

Given y, w E W, we say that y -< w if the following conditions are satisfied:y < w, l(w) - l(y) is odd and deg Py w = -'(1(w) - l(y) - 1). Using -< , we shalldefine an equivalence relation '"LR on W. Given x, x' E W, we say x <LR x' ifthere exists a sequence of elements of W: x = x0, x1, ... , x,, = x' such that foreach i (1 x1 or xr _ is <x_1, x.s > x1. We saythatx -LRx'ifx <LRx' and x' <LRX.

The equivalence classes for '"LR are called the 2-sided cells of W. By [3, 2.3],for any x E W, the subspace of 3C with basis Ty (y <LR x) is a 2-sided ideal of3C. It follows that the subspace spanned by the 7T for y in a fixed 2-sided cellcan be regarded as a quotient III' where 13 I' are 2-sided ideals in 3C and,therefore, it is a 2-sided 3C-module.

3. Three more problems. Assume first that W is the Weyl group of G as in § 1.Each 2-sided cell of W gives rise to a 2-sided 3C-module, hence (by specializingq -* 1) to a 2-sided W-module. These give a decomposition (over Q) of the2-sided regular representation of W.

Problem III. Two irreducible characters X', X" of W appear in the same 2-sidedcell of W if and only if XQ, X?' correspond to the same special representation of W.

In the remainder of this section G (as in § 1) will be assumed to be simplyconnected. The Fq-rational structure on G will not play any role so G is nowregarded as an algebraic group over Fq. Let X(T) be the character group of Tand let Q be its subgroup generated by the roots. Let W. be the group of affinetransformations of X(T) generated by W and by translations by elements inp.Q. Then W. is an infinite Coxeter group: its standard set of generators consistsof those of W, together with the reflection in the hyperplane (q E X(T)Jao (rp)= p), where ao is the highest coroot.

In [3, 1.5], Kazhdan and I formulated a conjecture on the characters of theirreducible quotients of Verma modules of a complex simple Lie algebra. I wishto state a modular analogue of that conjecture. Let p E X(T) be defined by thecondition that p takes the value 1 on each simple coroot. An element w E W. issaid to be dominant if - wp - p takes > 0 values on each simple coroot. Forsuch w, let L. be the irreducible representation of G, of finite dimension over Fwith highest weight - wp - p. Let V. be the Weyl representation of G over FPobtained by reducing modulo p the irreducible representation with highestweight - wp - p of the corresponding complex group. (It is well defined in theGrothendieck group.) We assume that ao (p) < p.

Problem IV. Assume that w is dominant and it satisfies the Jantzen conditionao (- wp) < p(p - h + 2), where h is the Coxeter number. Then

ch L. = 2 (- l)t(w)-rcr)Py,w(1)ch Vy. (4)y E W dominant

Y <W

From this, one can deduce the character formula for any irreducible finitedimensional representation of G (over Fr), by making use of results of Jantzenand Steinberg. The evidence for this character formula is very strong. I haveverified it in the cases where G is of type A2, B2 or G2. (In these cases, ch Lw hasbeen computed by Jantzen.) One can show using results of Jantzen [2, Anhang]

REPRESENTATION THEORY OF FINITE CHEVALLEY GROUPS 317

that this modular conjecture contains as a special case the conjecture ofKazhdan and myself on Verma modules in characteristic zero.

Our final problem concerns unipotent classes in G. It is closely related to theconjecture in [7]. We assume thatp is sufficiently large.

Problem V. Show that each 2-sided cell in W. meets some proper standardparabolic subgroup of Wo. Hence there are only finitely many 2-sided cells in Wo.Show that they are in a natural 1-1 correspondence with the unipotent classes in G.This correspondence should have the following property: a 2-sided cell in W. isfinite if and only if the centralizer of the corresponding unipotent element containsno nontrivial torus.

REFERENCES


2. J. C. Jantzen, Moduln mit einem hochsten Gewichl, Lecture Notes in Math., vol. 750, Springer-Verlag, Berlin and New York.

3. D. Kazhdan and G. Lusztig, Representations of Coxeler groups and Hecke algebras, Invent.Math. 53 (1979), 165-184.

4. , Schubert varieties and Poincare duality, Proc. Sympos. Pure Math., vol. 36, Amer. Math.Soc., Providence, R. I., 1980, pp. 185-203.

5. G. Lusztig, Representations of finite Chevalley groups, CBMS Regional Conf. Ser. in Math., vol.39, Amer. Math. Soc., Providence, R. I., 1979.

6. , Unipotent representations of a finite Chevalley group of type E8, Quart. J. Math. OxfordSer. (2), 30 (1979), 315-338.

7. , A class of irreducible representations of a Weyl group, Proc. Kon. Nederl. Akad. SeriesA, 82 (3) (1979), 323-335.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY


REPRESENTATIONS IN CHARACTERISTIC p

LEONARD L. SCOTT I

We will be discussing here topics in three related areas: maximal subgroups,irreducible representations, and group cohomology.

1. Maximal subgroups. I will assume in this section that the finite simplegroups can be classified and even that their irreducible representations incharacteristic p can someday be determined. What I wish to demonstrate is thatthis will carry us a long way toward the determination of their maximalsubgroups.

It should not come as a total surprise that such a problem is tractible, sincethe corresponding question for complex Lie algebras and connected Lie groupswas solved some time ago by E. B. Dynkin [8]. Let us review Dynkin's solution.

First of all, Dynkin treated the exceptional types separately, and I surelyexpect that the same will be necessary in the finite group case, requiring internalclassification theory type arguments rather than representation theory. Thisreduced Dynkin essentially to studying the maximal subgroups of SL(n, C),O(n, C), and Sp(n, C). Next, he reduced to the case of an irreducible subgroupby simply noting that any irreducible subspace has an obvious stabilizer (whenthere is a form around, the irreducibility forces the subspace to be eithernonsingular or totally isotropic). This had the pleasant by-product for Dynkin ofalso reducing to the case of a semisimple subgroup, since any connected abeliannormal subgroup would have to act by scalar multiplications; in the finitegroups case the maximal local subgroups associated with primes distinct fromthe characteristic would still have to be determined.

Next Dynkin reduced the problem from the semisimple to the simple2 case(i.e., the problem of finding all maximal connected subgroups which wereirreducible and simple) by using the tensor product decomposition for irreduci-ble representations of a direct product. Any product of two or more terms would

1980 Mathematics Subject Classification. Primary 20G05; Secondary 20B15, 20B35, 20C20, 20C30,20G10, 20J06, 20G40, 18G99, 14F05, 14L17.

1 Supported by the National Science Foundation.'Read as "quasisimple". Dynkin's "simple" groups can have finite centers. In the finite case also

one needs irreducible representations for central extensions as well as for the simple groupsthemselves.


319

320 L. L. SCOTT

have to be contained in SL(s) x SL(t) with s + t = n and 1 < s < n; theanalogous subgroup for Sp(n) is Sp(s)_X 0(t) and for O(n) there are twopossibilities: O(s) X O(t) and Sp(s) X Sp(t). This reduction goes through inspirit for the finite case, though the list of possibilities is larger-e.g., SL(s) wr S3,3s = n. (Incidentally, Dynkin checks as he goes along that each group in his listof obvious possibilities is indeed maximal.) Also one reduces not to a quasisim-ple group but to an automorphism group containing it.

Finally in the simple case Dynkin has available a complete classification ofthe simple groups in his category together with a complete determination of theirirreducible representations. He also has available for each irreducible represen-tation the knowledge of whether such a representation has a symmetric oralternating bilinear invariant, or none at all (denoted + 1, -1, or 0 in his table).This settles the question as to whether our subgroup can be put in O(n) or Sp(n)or neither, which up until now had been hanging. The analogous informationwill certainly be required in the finite case (even more because of the two typesof quadratic forms) and will just have to be considered part of the representationtheory problem. The corresponding information for hermitian forms is alsorequired, though this would at least follow from complete knowledge of thecharacters.

Having come this far, Dynkin now decides, in a brilliant application of thelaw of excluded middle, to decide which of his (theoretically classified) irreduci-ble simple groups is not maximal (in O(n), Sp(n), or SL(n)). This makes itpossible to list at least some version of the results on a single page (see the tableat the end of this paper). There are just four infinite families and fourteenindividual exceptions. More detailed information on the exceptions is availablein longer tables [8] I have not given here. For another exposition of Dynkin'swork, see Tits [26].

Now let us assume that we have adapted Dynkin's program to successfullyfind all maximal subgroups of the finite simple groups of Lie type and theircontaining automorphism groups, and somehow manage to treat the sameproblem for the finitely many sporadic groups. The question arises now as towhat we do with the symmetric and alternating groups, perhaps the mostinteresting case of all. The answer, happily, is that we may almost be done atthis point! I have listed the general forms for possible maximal subgroups in anappendix. As one might expect, the only undetermined possibilities (assuming aclassification of the finite simple groups) involve a primitive embedding of anautomorphism group of a simple group. If said simple group is not an alternat-ing group of lower degree, then we would know all of its primitive permutationrepresentations at this point. Thus we would be in a good position to inductivelydetermine all possible exceptions to maximality, completing our classification.

2. Irreducible representations. We will be discussing here representations ofgroups of Lie type in their natural characteristic p. Little has been done withrepresentations in fields of characteristic I op, though Alperin has suggestedthat this theory should parallel the modular theory of the symmetric groups,which has a substantial literature [23].

In characteristic p the irreducible representations all are restrictions of ra-tional (polynomial) representations of the ambient simply connected algebraic

REPRESENTATIONS IN CHARACTERISTICp 321

group G over the algebraic closure k of GF(p) (Steinberg). To keep an examplein mind, the group G is SL(n, k) if the original group was SL(n, p'). This is quitea serious class of examples, with the irreducible representations (or even theirdegrees) of SL(5, p) still unknown. (To my knowledge no one in the field hasactually sat down and tried directly to construct representations. It would be agood problem (suggested originally by M. O'Nan) to try and find a generalrecipe for the irreducible representations of SL(n, 2) simply from a combina-torial point of view. Something like Young diagrams is what I have in mind.)

Very recently (just a few weeks before this conference) George Lusztig madethe first serious conjecture regarding the characters of the irreducible representa-tions of G. To give the "character" of a representation of an algebraic group isto describe it on a maximal torus (the diagonal matrices in SL(n, k), in thatexample). The representation on such a torus T is a direct sum of 1-dimensionalrepresentations or "weights"; from this point of view the "roots" are therepresentations associated with the root groups, positive roots corresponding toroot groups in a fixed Borel subgroup B, e.g.,

1

in

The Weyl group W = NA(T)/T obviously acts on the weights and permutes theroots. Each irreducible representation has a unique highest weight with respectto the ordering X > µ iff X - µ is a sum of positive roots. The abelian group ofweights carries a natural positive definite symmetric bilinear form ( , ) and thehigh weight X of each irreducible representation is "dominant" in the sense that(X, a) > 0 for each positive root a. In this way the dominant weights completelyparameterize or label the irreducible representations of G, even though littlemore is known about them. To describe Lusztig's conjecture and its context weshall go back first to the classical theory of Weyl and Kostant which successfullydescribes the character of the irreducible module Va with high weight X when kis replaced by the complex numbers.

First of all, let µ be any weight of T and regard p as a 1-dimensionalrepresentation of B. The Lie algebra b of B correspondingly acts on µ, and wemay consider the corresponding induced representation Zµ = QL(g) ® 6) µ forthe Lie algebra g of G. Here Qt (g) denotes the universal enveloping algebra. Themodule Zµ is also a rational T- (and even B-) module and is freely generated by1 ® p over 2L (u-), the universal enveloping algebra for the group U- generatedby the negative root groups. In the example this group is

s 1

* * 1

Because of this, the multiplicity of any T-weight P in Zµ is just p(v - µ) = thenumber of ways v - µ can be written as a sum of positive roots. The module Z,is called the Verma module associated with µ. The formulas of Weyl and Kostant

322 L. L. SCOTT

describe V. as a simple alternating sum

VA] = Z (- l)'(wl[ ZWA]WE W

in the Grothendieck group of T-modules and thus give its character. Herew X = w(X + p) - p where p is half the sum of the positive roots, and (- 1)1(w)is the determinant of w in its action on the weights ( expressed in terms of thelength function 1(w) of reflection group theory). The proof of this result inHumphreys' book [10] shows quite clearly the importance of showing that theonly Zµ in the same "block" as V. are of the form Zw.A (Harish-Chandra).3(Each Zµ is indecomposable, and even has a unique irreducible quotientmodule.) This gets one to the point where there must be at least some expression[ VA] = EwE W c(w)[Zw.A]. Because of this, Humphreys and Verma pushed acorresponding block-theoretic investigation in the characteristicp case using theaffine Weyl group WP (the semi-direct product of W with pZ0, the latter actingby translation on the weights, so that WP X = W- X + pZO; here Zc denotesthe root lattice).

Recently [1] H. H. Andersen (sharpening results of Humphreys, Kac-Weisfeiler, and Jantzen) has proved that the high weights of any two irreduciblerepresentations of G in the same block must also belong to the same orbit underthe affine Weyl group Wp.° This implies that there must be at least someexpression EwEW c(w)[ZW.A] for the irreducible representation in characteristicp(the simplest way to make sense of this in characteristic p is to think of thecharacter of Zw.A in terms of the partition function p we discussed earlier).Essentially Lusztig's conjecture gives the coefficients c(w) for the most im-portant weights when the prime p is large relative to the root system. I havegiven a precise statement in an appendix. Implicit in its philosophy are results ofJ. C. Jantzen [12] which allow one to obtain formulas for all weights from just afew well placed ones, also parameterized by elements of W.

Lusztig's conjecture is analogous to an earlier conjecture [15] of Kazhdan andLusztig in characteristic 0 regarding irreducible modules which are the quotientsof Verma modules associated with nondominant weights. The main ingredientsfor the characteristic 0 (resp., characteristicp) conjecture are certain polynomialsP,,., defined for each pair of elements w, w' of the Weyl group (resp., affineWeyl group), the values of these polynomials at 1 giving the requisite doublyparameterized system of coefficients (the various c(w)'s above). The polynomialsPw w were apparently first found in the representation theory of generic Heckealgebras, arising naturally in lifting Springer's Weyl group representations tothese algebras. Since this conference took place, they have been shown to bePoincare polynomials for a new geometric cohomology theory of Goreski,

3Two indecomposable modules are in the same block if they can be joined by a chain ofindecomposable modules with a nonzero homomorphism (either direction) between successive terms.

4 Recently S. Donkin has completed the determination of the blocks [28]. They are described asorbits of W, or its analogues for higher powers of p, depending on the power (plus one) of p dividingA+p.

REPRESENTATIONS IN CHARACTERISTIC P 323

MacPherson, and Deligne, applied to Schubert varieties. (See Lusztig's article inthese PROCEEDINGS, where he also notes that the characteristic p conjectureimplies the characteristic 0 conjecture, through a translation principle of Jant-zen.)

Previously a Poincare polynomial interpretation in terms of group cohomol-ogy had been given by David Vogan (cf. [2] and §3), but with coefficientsinvolving the unknown irreducible modules, and assuming a conjecture he saysis equivalent. In spite of these drawbacks Vogan's interpretation does formallyimply the characteristic 0 conjecture, and conceivably could be instrumental inits proof. I have described a characteristic p analogue in §3.

The theory is too young to say definitely where proofs might come from, so Iwill just survey some of the other approaches that have emerged so far. Onereally untried possibility I have already mentioned to you: directly constructrepresentations. Another major avenue of attack is to decompose known non-irreducible representations. The main possibilities here are the Weyl modules,which may be described by a suitable reduction mod p from a characteristic 0module (k ®z % v+ in terms of the Kostant Z-form) and a number of resultsin this direction have been obtained by Jantzen. Nowadays these modules maybe described as the duals of certain 0-dimensional cohomology groups(H°(B, -X ® R(G)) or H°(G/B, C(-X)), where R(G) is the affine coordinatering of G and C (- X) is the line bundle on GI B associated with X), and H.Andersen has already demonstrated the usefulness of considering the higherdimensional cohomology groups. (They are used heavily in the proof of hisresult on blocks cited above.) Another simple description of the Weyl moduleassociated with X is as the universal module with high weight X, the dual of the"induced" module -XIG = MorphB(G, -X) in the sense of algebraic groups.The equivalence of all these definitions depends on the vanishing theorem firstproved by George Kempf [16] and more recently by Andersen [2] and Haboush[27]. In any event the known results on the structure of these modules (asopposed to just knowing their composition factors) are very meager, the onlycomplete results being for type A (the group SL(2, k)), due to Carter and Cline,cf. [24]. (Carter and Cline actually give the lattice structure, though in generalone might be content with well-understood filtrations.) To give the reader someappreciation of where the structure theory of these modules is today, I mentionthe following open problem:

Let G be SL(3, k) and V its standard 3-dimensional module, with V*the dual. Describe the structure of the tensor product Sm(V) ®S"(V*) of symmetric powers for all integers m, n > 0.

The problem is open even with a reasonable bound on m and n (say m + n + 2< p2).

Still another approach is to look for modules in nature, meaning algebraicgeometry. Aside from the higher cohomology groups mentioned above, the mostinteresting phenomena of this kind to my mind are the B-filtrations of linebundles arising in George Kempf's study of Schubert varieties [16]. One mighthope to use these results in conjunction with the extension theory [6], whichmostly reduces the question of constructing G-modules to B. The theory of [18]

324 L. L. SCOTT

gives explicit constructions for bases of (duals of) Weyl modules, natural interms of algebraic geometry, for all classical groups. For type A Carter andLusztig [25] have used somewhat similar bases to obtain some partial results.Further material in this direction may be found in Towber [29] and James [23].

The Lie algebra of L(G) of G also plays some role in this theory; indeed, itwas recognized very early that it was enough to construct the irreduciblerepresentations of L(G). (Here "representations" are meant in the sense ofrestricted Lie algebras.) One surprising connection with the group case is thatthe projective indecomposable modules for L(G) lift to G modules for p largerelative to the root system (unknown for small p). This was proved by Ballard[3]; the context here is a theory of Humphreys [11], partly inspired by work ofJeyakumar. Recently Jantzen has shown for large primes that these modules arefiltered by Weyl modules. The ultimate role of these modules is difficult toassess, though it is at least clear we want to know more about them.

The restricted enveloping algebra of the Lie algebra has higher order ana-logues, the "hyperalgebras", introduced in the present context by Humphreys,developing suggestions of Verma. These have nice interpretations (cf. [5] for anexposition) in terms of the theory of infinitesimal groups. I will not say anythingabout this theory beyond the fact that if G, is the group scheme correspondingto a hyperalgebra (e.g., the restricted enveloping algebra of L(G)) then thecategory of modules for G, coincides with that of the hyperalgebra. Theadvantage of this point of view is that many group theoretic considerationssuggest themselves when we think of G which is a normal subgroup scheme ofG (for example, the Hochschild-Serre sequence, or the study of BG,-modules).Doubtless you thought G had no normal subgroups! Never fear, since theinfinitesimal scheme G, has only one element. But in spite of this it is useful forrepresentation theory.

I think on this note I will end this part of my exposition. There is one otherapproach, involving decomposing certain characteristic 0 representations offinite groups, which is discussed by Roger Carter in his lecture at this con-ference. The general role of characteristic 0 representations, aside from analogy,is not understood. For another instance where characteristic 0 representationsenter, see Green [9] where it is shown that the (infinite dimensional) injectiveindecomposable modules for the algebraic group G can be "lifted" to character-istic 0.

Addenda. Another approach was found by this author during the conference,partly inspired by some remarks of Humphreys. It offers a well-defined pro-gram, but seems a bit slow in a time of such dramatic events. An unpublished5result of mine asserts the injectives above have a filtration whose sections areinduced modules - XIG. One has [22] ExtG(L,, _XIG) = 0 for any irreduciblemodule Lµ whose high weight µ does not satisfy - wo µ > X. Consequently wecan form a submodule II of an indecomposable injective Q consisting of all"sections" - piG with µ < X. Next one shows that

dim ExtG(-XIG, IX,) = dim ExtG'(Soc(-XIG), IX,) =multiplicity of -XIc

5Added in proof. Recently S. Donkin has also found this filtration. His work will appear in Math.Z.

REPRESENTATIONS IN CHARACTERISTICp 325

as a section of Q. One shows also this is the characteristic 0 multiplicity, which isin turn the multiplicity of the irreducible socle of Q in - AIG as a compositionfactor (Green). Now it might be possible to inductively determineExt'(_X1G, la), using the cocycles from one answer to construct the "nextlargest" Ia as a maximal essential extension.

3. Cohomology. I will try to give a brief overview of some problems close tomy own interests and the theory of §2.

For algebraic group cohomology, it is important to study the structure of theinduced modules -AIG (or equivalently their duals, the Weyl modules) and tostudy how the injectives are built from these (cf. the addenda to §2). Thevanishing result on ExtG(LH, -AIa) mentioned in the addenda is actually validfor all Ext", n > 1, yielding a powerful dimension shift for computing algebraicgroup cohomology when the structure of the relevant Weyl modules is known.

Results of Cline, Parshall, Scott, van der Kallen [22] indicate how to computefinite Chevalley group cohomology in terms of algebraic group cohomology forlarge fields. Recently Bill Dwyer, using "split buildings" constructed by RuthCharney, has obtained some extremely promising stability results with respect tothe rank of the group [7]. At the moment, his results are stated for GL" andmodules related to the standard module, but they should easily generalize. Ileave the most definitive formulation as an open problem. Though there is stillwork to be done,6 it is now likely that entire families of finite group cohomologyproblems can be reduced to a finite number of cases by general methods.

Turning to David Vogan's work on the Kazhdan-Lusztig conjecture in char-acteristic 0, we can express his Poincare polynomial interpretation as follows, interms of algebraic group cohomology:

E q' dim A, L(-w. A))i>o

where y, w E W and A is any dominant weight. Here we agree that any Extgroup of negative degree is 0. The starting point of Vogan's investigation in [21]seems to be the observation that (a) implies the characteristic 0 conjecturethrough the application of an Euler characteristic formula. The analogue of thelatter in characteristic p is

L(- w. A)] = E E (-1)"dimExt4(-y A, V_y.a].-y.a n

Here A is again any dominant weight and L(- w A) is the irreducibile modulewith high weight - w A, but w, y come from Wa, and we assume - w A, -y Aare both dominant. The formula is easily proved by appealing to the fact [22]that - µ1c ® - p is B-acyclic for µ, p dominant and replacing L(- w A) by aninduced module.

6E.g., the stability results of [22] need to be treated for the twisted groups and one needs betterstability theorems for growth of the characteristic p. The latter problem at least reduces, usingmethods of [22], to algebraic group cohomology for a module twisted by the Frobenius endomor-phism.

Added in proof. The stability results of [22] have now been treated for twisted groups by G.Avrunin, Trans. Amer. Math. Soc. (to appear).

326 L. L. SCOTT

PROPOSITION. Assume A is in the "bottom alcove" C, and - w A is a dominantweight in the "bottom p2-alcove" C2 (see the Appendix; these are just thehypotheses of the Lusztig conjecture). If (.) holds for ally E W,, then so does theformula for [L_w.a] conjectured by Lusztig.

This follows easily by just comparing coefficients.It would be interesting to know if other of Vogan's results have analogues in

characteristicp. It would also be interesting to have some general calculations ofB-cohomology. As far as I know, complete results on H"(B, µ) with µ anarbitrary weight do not exist even for n = 2. Such calculations would also beextremely helpful for specific computations in the finite Chevalley group casementioned earlier.

To complete this exposition, I would like to come back once more to maximalsubgroups of finite groups. It is a theory of Bob Griess that interesting or"sporadic" 1-cocycles should lead to interesting subgroups by considering theelements of the group on which the cocycle is zero (the stabilizer of a vector inthe usual extension module corresponding to the cocycle). This is supported by anumber of theoretical results [19], [20], but no one has yet made an attempt tosystematically look at examples. A good starting point would beH'(SU(n, 22), A3V) where V is the standard module, which Wayne Jones hasshown to be 1-dimensional for n > 7.7 The same cohomology for larger fields is0, so that in some sense these cohomology groups are all sporadic. (The stablebehavior with respect to the rank is an instance of what one should be able toprove by generalizing Dwyer's results.) For some recent cohomology calcula-tions, see [14] and [4].

I would like to thank H. Andersen, J. Humphreys and G. Lusztig for severalconversations and my colleagues Ed Cline and Brian Parshall for numerouscontributions to this lecture.

Appendix: Statement of the Lusztig Conjecture (adapted from a lecture by H.Andersen). Let W be any Coxeter group and S its set of simple reflections. LetG denote the usual partial order on S in which y G w iff y has some reducedexpression which is a subsequence of a reduced expression for w (well-defined,cf., Bourbaki). Let l(w) denote the length of a reduced expression for w.

We are going to inductively define some polynomials PyW

in q for y, w E W.8We will have 0 unless y G w, and that the degree in q of P w is at mostZ(l(w) - 1(y) - 1). Set µ(y, w) equal to the coefficient in Py, of this largestpossible degree 1(l(w) - l(y) - 1) when the latter is a nonnegative integer.Define P., = I and Py w = 0 if y w. If y < w and Py w has been defined forally with smaller w, chooses with ws < w. Put

c = c(y, S)1 ifys < y,

=0 ifys > Y.

"The sporadic behavior actually occurs already for n = 6, where the same cohomology group is2-dimensional. One regards V as n-dimensional, so that dim A'V = 20 in this case.

81 am indebted to Roger Carter for catching an error in my original description of P,,,,, andapologize if any inaccuracies remain.

REPRESENTATIONS IN CHARACTERISTIC p 327

DEFINITION.

Py.W(q) = q -'P,,.. + q `Py.Ws

-q µ(z, ws)g(1(WS)-r(=) I)/2PyZ.Y «<Ws

is <z

In Kazhdan-Lusztig one finds the useful fact that y < w, sw < w, sy > ytogether imply Py, = P,, ,w (s E S; y, w E W). (We will just use this for givingan alternate form of the conjecture.)

Now suppose W is a Weyl group or affine Weyl group and Lµ denotes theirreducible module with high weight A. (I also write L(µ) sometimes.) Thecharacteristic may be 0 or p. Regard Win its usual reflection action, centered at- p, where p = half the sum of the positive roots, except regard the elements ofS as acting as reflections in the alcove containing - 2p (rather than the"opposite" alcove containing 0). We will assume throughout in the characteristicp case that p is at least the Coxeter number, which amounts to saying there is atleast some integral dominant weight interior to the lowest alcove C,. To beexplicit C, = f µl µ + 8 is dominant and (µ, a°) <p for all positive roots a).Our reflections S are acting through the walls of - C, in the characteristic pcase. We shall also need the "lowest p2-alcove" C2, which is defined by replacingp by p2 in the definition of C,.

We can now simultaneously describe the Kazhdan-Lusztig conjecture incharacteristic 0 and Lusztig's adaptation for characteristic p. Let A be a domi-nant weight and w E W (or WP in characteristic p). If the characteristic is p,assume A E C, and that - w A is a dominant weight in C2. The conjecture is

2 (- 1)r(wj-1(y)py'W(1)[Z-Y.;,

Y

In characteristicp this can also be written in the alternate form

L-WA] = 2 (-1)1(W)-1(r)Py.W(l)[ V_y.A].dominant

Formulas for all the other weights follow from work of Jantzen: a weight 1i is inthe upper closure of an alcove - w C, if no reflection in the walls of that alcovemoves µ closer to the origin -p. Every weight is in the upper closure of somealcove. Jantzen's theorem tells us the above formulas hold without change if wejust replace A by - w -' u. Thus for large p the conjecture gives formulas for allthe irreducible modules in the restricted range from which all others areconstructed via the tensor product theorem of Steinberg. The situation for smallp is unclear. According to an example of Jantzen, one cannot just use the sameformula for small p. (The example is G2 in characteristic 3.)

Appendix on maximal subgroups. Call a subgroup H of a direct productIl;E, G; diagonal if each projectionp3: f1;E1 G; --+ Gj is injective on H. If each p.is in fact an isomorphism (which means in particular that all GG are isomorphic)we shall call H a full diagonal subgroup.

328 L. L. SCOTT

LEMMA. Let H be a subgroup of a finite direct product II;EI Gi of nonabeliansimple groups, and assume each projection pj is surjective on H. Then H is thedirect product IIH. of full diagonal subgroups of subproducts II,,=-I G, where the 1.form a partition of 1.

PROOF. Certainly we may assume H 1, in which case there is a minimalsubset S C I with H n IIiES Gi 1. Set D = H n 11iES Gi and note that D isnormal in H, hence any projection of D is normal in any projection of H.Clearly, pj(D) 1 for j E S by minimality, so pj(D) = Gj. Minimality alsoimplies pp is injective on D. Thus D is a full diagonal subgroup of IDES Gj.

Now let E be the projection of H on IDES G.. Then D < E as noted above,since D is its own projection on this subproduct. But obviously a full diagonalsubgroup of a product of nonabelian simple groups is self-normalizing. ThusD = E and H is the direct product D X D' where D' = H fl IIiEI_S Gi. Theresult now follows by induction on the cardinality of the index set 1.

COROLLARY. Let H be as in the previous lemma. Then every subgroup K ofII,EI Gi containing H has the form

K = fl ps(H)SE°P

where 9 is a partition of 1 refining the partition we associated with H, and ps is theprojection of G on lliES Gi.

PROOF. By the lemma applied to K we have K = lsE6ps(K) for somepartition 9 of 1, with ps(K) full diagonal in 1IiES Gi. Since H C K we haveps(H) C ps(K). Also ps(H) = 11s n I ps n I,(H) where { I, } is the partition associ-ated with H. Since this product contains a full diagonal subgroup of IIiES Gi, itmust consist of only one term and coincide with ps(K). Q.E.D.

We can use the above results to get a very useful picture of the generalprimitive finite permutation group. Recall that such a group has a socle which iseither elementary abelian (an irreducible representation for the point stabilizer)or a direct product of isomorphic nonabelian simple groups .9 Any nontrivialnormal subgroup of a primitive group of course acts transitively.

THEOREM. Let H be a subgroup of a finite direct product G = IIiEI Gi ofisomorphic nonabelian simple groups. Then the transitive permutation representa-tion of G on G/H extends to a primitive permutation representation of some groupin which G is the socle if and only if either

(a) there is a partition 9 of I into subsets of equal prime cardinality with H thedirect product II s E As of full diagonal subgroups As of the subproducts II i E S Gi,or

(b) the subgroup H is a direct product II i E I Hi where Hi is a subgroup of Giwhich is an intersection G. n if, for some maximal subgroup Fli of a group Gi withG, C Gi C Aut Gi. Also, for each pair i, j of indices there must be an isomorphismof Gi with G. carrying Hi to H.

9The only case of a nonunique minimal normal subgroup N occurs when N is regular and thesocle is N X N (nonabelian).

The details of the proof are fairly straightforward from the previous results,and we leave them to the reader. We now aim at describing the possiblemaximal subgroups of the symmetric and alternating groups.

We recall that if A is an abstract group and B is a group acting on a set F,then the wreath product A wr B is the semidirect product (III, A) B obtainedfrom the action of B given by fb(Y) = f(Yb ') for f E 11r A (functions from F toA), b E B, and Y E F. If A acts on a set SZ, the usual set one has the wreathproduct act on is the disjoint union Ir 0 of 1fl copies of Q. However, there isanother natural action, namely, on the product III, St. This action is given by

gf)(Y) =g(YY(Y), gb(Y) = g(Yb ')

for g E III, 0, f E III,A, y r= IF, b E B. We shall call this the product action of thewreath product.

PROPOSITION. If A, B are transitive in their respective actions, and A actsprimitively but not just as a regular group of prime order, then A wr B is primitivein the product action.

PROOF. Suppose g, g' are distinct elements of a block A of imprimitivity. Let' E F be a coordinate where g and g' differ. Applying the stabilizer of g(y) in

the yth copy of A to g' yields at least one more element g" in A agreeing with g'everywhere but at y. Primitivity of A now implies A contains all such g".Transitivity of B now forces A to be all of III, SZ. Thus A wr B is primitive.Q.E.D.

THEOREM.10 Let M 6T. be a subgroup of the symmetric group S,,. Then someconjugate of M is contained in one of the subgroups listed below. Here I < m < nand p is prime.

(a) Sm X Sk, m + k = n (intransitive),(b) Sm wr Sk, mk = n (imprimitive),(c) Sm wr Sk, mk = n, m > 5 (product action),(d) V- GL(V),pk = n = IVI; V a vector space over GF(p),(e) (G wr SP) Out G, IGI°-1 = n, G a nonabelian simple group,(f) an automorphism group of a nonabelian simple group G < Q,,, containing G

and acting primitively (the full normalizer of G in

The group in (e) is the extension of G wr SP by the outer automorphism groupOut G obtained from the natural extension Aut G of a diagonal copy of G; theisotropy group is Aut G X S.

The proof is easy from the preceding results: if M is intransitive, imprimitive,or primitive but local we have cases (a), (b), or (d). Otherwise M is primitive andits socle is a direct product of isomorphic nonabelian simple groups. Case (b) ofthe previous theorem leads to cases (c) and (f) here, while case (a) there yieldseither (e) or (c) again.

10This theorem has been independently obtained by Mike O'Nan.

330 L. L. SCOTT

Table 111

Type of the group 13 Scheme of 13 N c

n>3 I I0-0-0-0. 0-0 3 (n+2) 0

2 1n>2) 0-o-0 0--0 3 (n+3) 0n 2 4

k2n (k+2s-1

k (n+l )k)k n > 1(B p-gyp... (-1)2n+1 k>l s=1 (k+s)

k>1 k 2k+5 (k+4)1

5 4

6(AI) 0 7 1

1 1

(A )--o- 0-- 189 1

S 3 --o- oo

1 1

(B4) 128 1

2(C3)1 .-O 90 1

2 1

(C ) -o 350 -13 2

1

(D5) 560 0

(D6)1 495 1

b1

(D6)2 0- 4928 0

(E6),1o--o-o-o-o 351 0

(E ) o- 17 550 06 2

b11

(E7), 1539 1

1

(E 7)2 0 27664 -1

b1

(E)

-o 365 750 17 3

b

(E ) -o 3 792 096 -17 4

b 1

"Reprinted from E. B. Dynkin, Maximal subgroups of the classical grows, Amer. Math. Soc.Transi. Ser. 2 6 (1957), p. 364.


BIBLIOGRAPHY

1. H. H. Andersen, The strong linkage principle, J. Reine Angew. Math. 315 (1980), 53-59.2. , The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B,

preprint (Aarhus).3. J. Ballard, Injective modules for restricted enveloping algebras, Math. Z. 163 (1978), 57-63.4. G. Bell, On the cohomology of the finite special linear groups. 1, II, J. Algebra 54 (1978),

216-238,239-259.5. E. Cline, B. Parshall and L. Scott, Cohomology, hyperalgebras and representations, J. Algebra 63

(1980), 93-123.6. , Induced modules and extensions of representations. I, 11, Invent. Math. 47 (1978),

41-51; J. London Math. Soc. (2) 20 (1979), 403-414.7. W. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. 111 (1980),

239-251.8. E. B. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Transl. Ser. 2 6

(1957), 245-378.9. J. A.-Green, Locally finite representations, J. Algebra 41 (1976), 137-171.

10. J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math.,vol. 9, Springer-Verlag, Berlin and New York, 1972.

11. , Ordinary and modular representations of Chevalley groups, Lecture Notes in Math., vol.528, Springer-Verlag, Berlin and New York, 1976.

12. J. C. Jantzen, Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen and Lie-Alge-bren, Math. Z. 140 (1974), 127-149.

13. , Darstellungen halbeinfacher Gruppen and ihren Frobenius Kerne, preprint (Bonn).14. W. Jones and B. Parshall, On the 1-cohomology of finite groups of Lie type, Proc. Con!. on

Finite Groups (Park City, 1975), Academic Press, New York, 1976.15. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent.

Math. 53 (1979), 165-184.16. G. Kempf, Linear systems on homogeneous spaces, Ann. of Math. (2) 103 (1976), 557-591.17. , The Grothendieck-Cousin complex of an induced representation, Advances in Math. 29

(1979), 310-396.18. V. Lakshmibai, C. Musili and C. S. Seshadri, Geometry of G/B, Bull. Amer. Math. Soc. (N.S.)

1 (1979), 432-433.19. L. Scott, Permutation modules and 1-cohomology, Arch. Math. 27 (1976), 362-368.20. , Matrices and cohomology, Ann. of Math. (2) 105 (1977), 473-492.21. D. Vogan, The Kazhdan-Lusztig conjectures, preprint (MIT).22. E. Cline, B. Parshall, L. Scott and W. van der Kallen, Rational and generic cohomology, Invent.

Math. 39 (1977), 143-163.23. G. James, The representation theory of the symmetric group, Lecture Notes in Math., vol. 682,

Springer-Verlag, Berlin and New York, 1978.24. E. Cline, A second look at the Weyl modules for SL2, preprint (Clark).25. R. Carter and G. Lusztig, Modular representations of finite groups of Lie type, Proc. London

Math. Soc. 32 (1976), 347-384.26. J. Tits, Sous-algebres des algebres de Lie semi-simple, Seminaire Bourbaki 1954/1955, expose

119, Paris, 1959.27. W. Haboush, A short proof of the Kempf vanishing theorem, Invent. Math. 56 (1980), 109-112.28. S. Donkin, The blocks of a semisimple algebraic group, preprint (Warwick).29. J. Towber, Young symmetry, the flag manifold, and representations of GL(n), J. Algebra 61

(1979), 414-462.

UNIVERSITY OF VIRGINIA


CHARACTERS OF FINITE GROUPSOF LIE TYPE. II

BHAMA SRINIVASAN

The aim of this article is to give a brief survey of the basic results in the theoryof Lusztig-Deligne representation and to indicate some open problems. Refer-ences for the Lusztig-Deligne theory are [4], [3], [21], [26], and [32].

1. Let G be a connected reductive algebraic group defined over Fq. Then wehave a Frobenius morphism F: G - G and the group G F of fixed points isfinite. There exists, up to GF-conjugacy, precisely one pair (To, Bo) such that Tois an F-stable maximal torus, B0 is an F-stable Borel subgroup, and B0 = T0U0where U0 is the unipotent radical of B0 and Uo is a Sylow p-subgroup of GF(p = char Fq). Let W(T0) = NG(To)/To. On W(T0) = W we have an equiva-lence relation: wl w2 if ww1(Fw)-l = w2 for some w W. (If GF is untwistedthen these equivalence classes are just ordinary conjugacy classes of W.) TheGF-conjugacy classes of F-stable maximal tori of G are in bijection with theequivalence classes of W. _

Fix a prime l p. Let Q, be an algebraic closure of Ql, the field of 1-adicnumbers. Suppose T is an F-stable maximal torus. Let TF = Hom(TF, Ql ). Welet B = TU be a Borel subgroup containing T, and note that B need not beF-stable. We define a variety X by: X= (g E G I g-1(Fg) E U). This is aclosed subvariety of G, and GF X TF acts on X by (g, t): h -+ ght-1 (g E GF, hE G, I E TF)._Thus, by functoriality, GF X TF acts on the 1-adic cohomologygroups HH(X, Ql). Let 0 E TF.

DEFINITION. RT(0) _ Q,)9Here, for any TF-module M, M9 is the part of M on which TF acts according

to 0. Each HH(X, Q,)9 is GF-stable and so RT(0) is a virtual representation of GF,i.e., an element of the Grothendieck group of GF. For some examples of RT(B),see [4], [21].

Let u be a unipotent element (p-element) of G F. Then the value of RT(0) at uis an integer independent of 0. So we may as well look at RT (1), and make thefollowing definition. (1 is the trivial character of TF.)

DEFINITION. QT (u) = Tr(u, RT(1)).

1980 Mathematics Subject Classification. Primary 20C15, 20G40.m American Mathematical Society 1980

333

334 BHAMA SRINIVASAN

The function QT on the set of unipotent elements of G F is called a Greenfunction.

The main properties of the RT(0) are the following:1.1 Character formula. Let g = su be the Jordan decomposition of g (i.e., s is

the p'-part and u is the p-part of g). Then

/Tr(g, RT(B)) C0(S)FxE F

Q oTxe(XSX ).

xsx-' E T1

Here CO(s) is the connected component of 1 of the centralizer C(s) of s in G..1.2 Orthogonality. If T, T' are two F-stable maximal tori of G, let N(T, T') _

(g E G I g-17g = T'), W(T, T') = T \ N(T, T'), W(T) = W(T, T). Let w de-note a representative for w E W(T, T') in N(T, T'). Then:

(RT (0), RT (0')) (w E W(T, T')FI''O' = B } 1,

IGuE £ QT(u)QT(u) = IN(T, T')F11TF1 ITFI.

F

u unipotent

In particular, if 0 E TF is regular, i.e., not fixed by any nontrivial element ofW(T), ± RT(0) is irreducible.

1.3 Dimension formula.

dim RT(0) = QT (l) = ±IGFI/I U0 I I TFI.

REMARK. This does not automatically follow from the Character Formula, andthe proof involves the properties of the Steinberg representation of G F

1.4. Every irreducible representation of GF occurs in some RT(B).REMARK. However, the traces of the RT(0) do not span the space of class

functions on GF, in general.1.5. Suppose T is contained in an F-stable Levi subgroup L of an F-stable

parabolic subgroup P of G. Then

RT(0) = IndpF(RT(B)),

where RT(B) denotes the lift of the virtual representation RT(B) of LF to PF. Inparticular, R G = Indgo(0), where 0 is regarded as a character of Bo .

2. There are two main open problems in the theory.(A) Decompose the RT(0) when 0 is not regular.(B) Find explicit values for the characters of the RT(0) (and of their con-

stituents).We now make some remarks on these problems.(A) In particular we can consider the decomposition of RT(1). The con-

stituents of the RT(l), for various T, are called unipotent representations. Lusztig(see [21]) has given a list of unipotent representations for each type of simplealgebraic group G. In the case of GL, they are parametrized by partitions of nand in the case of the other classical groups by certain combinatorial objectscalled "symbols". The unipotent representations are determined not by explicitlydecomposing the RT(1), but rather by inducing cuspidal unipotent representa-tions of parabolic subgroups and decomposing the induced representations. (For

CHARACTERS OF FINITE GROUPS OF LIE TYPE. II 335

the definition of a cuspidal representation see e.g., the article by C. W. Curtis inthese PROCEEDINGS.) An interesting feature of this is that the centralizer algebrasof these induced representations are specializations of the generic algebras whicharise when one decomposes R o(1), i.e., the permutation representation of GF onthe cosets of Bo (see [3]).

The decomposition of R o(1) has been described by Curtis, Iwahori andKilmoyer (see [3]) and that of R o(O), for any 0 E To, by Howlett and Kilmoyer[37]. Lusztig has decomposed RT(l) when T is a "Coxeter torus" (see [20]). Evenhere the situation is complicated and deep theorems of Deligne are used.

Lusztig has also given a classification of all the representations of GF, when Ghas a connected center. This can be described as follows. We have a reductivegroup G associated with G called the dual of G (see [19]). The group G' is alsodefined over Fq and we have a Frobenius morphism F: G -* G. For example,if GF = GL(n, q) then G*F = GL(n, q) and if GF = Sp(2n, q) then G*F =SO(2n + 1, q). Then there is a bijective map from the set of (isomorphismclasses of) irreducible representations of GF onto the set of pairs (s, qS) where sruns over a set of representatives for the semisimple classes of and ¢ is aunipotent representation of If X corresponds to (s, ¢) we have

dim X = (dim p)CG.(S)FL0

and dim qS, and thus dim X, is known in all cases.In the case of certain versions of the classical groups, this theorem is proved in

[19]. For the case of the exceptional groups (where q has to be assumed large, inthe case of E8), see [36].

(B) By the Character Formula we see that in order to compute the traces ofthe RT(0) it is sufficient to know the Green functions. (Of course, this is notenough to get the whole character table of GF.)

In principle one can calculate the Green functions of GL(n, q) and this will bedescribed below. Other cases where the Green functions are known explicitlyare: U(3, q) [33]; U(4, q], U(5, q) [23]; G2(q) (p 2, 3 [2]; p = 3 [6]); Sp(4, q)(q odd [30]; q even [6]); Sp(6, q) (q odd [5]). They are also known when G isclassical and T is a Coxeter torus (see [18]).

2.1 Description of Green polynomials for GL(n, q) ([10], [22]). Let G = GL,G F = GL(n, q). Both the conjugacy classes of maximal tori and the conjugacyclasses of unipotent elements in GL(n, q) are classified by partitions of n. Thuswe can denote the integer QT(u) by Qa,, where A and p are partitions of n; sinceit is also known that QT (u) is a polynomial in q, we write it as Qa,P(q).

If A is a partition of n, we write A F- n. Let x1, x2, . . . , x,N be indeterminateswhere N is sufficiently large, e.g., N > n. If A F- n we have a Schur function R.defined by

a(x;1+N-14ZZ+N-2. . . 4)A 1+ 2> ' N iT /

oESN 111<i<j N(xi - xj)

where A has parts (A1 > a2 >. . .> AN) (some of the parts may be zero) and


SN acts by permuting the variables. If p F- n and p has parts (pI > p2 > >PN)' let SP = sP sP2 s,N, where sm = x,' and so = 1. A famous theorem ofFrobenius says that

SP(XI, X2, .. . , XN) = XpRa(xl, X2, . . . , XN),XFn

where XP is the value of the character of S. corresponding to A, at the class of Sncorresponding to p. One can define functions Pa(xl, x2, ... , xN, t), called Hall-Littlewood functions, such that

SP(XI, x2, . . . , XN) _ XP {t)Px(XI, X2, . . . , XN, t)XFn

where XP (t) E Q [t], and XP (q-I) = q-"AQ,,P(q) where n,, = EN I(2) are theparts of the partition dual to A). A. O. Morris [22] gives a recursive rule forcomputing the XP (t). The reader can also find other references there for theGreen polynomials.

EXAMPLE. The tables of Green polynomials for G42, q) and GL(3, q) are asfollows.

p

12

2

lZ

l + q1-q

2

1

1

P 12 3

13 (l+q)(l+q+q2) 1+2q 1

12 1 - q3 1 1

3 (1 - q)(l - q2) 1 - q 1

If G is a classical group, it is known for large p that the QT G( are polynomialsin q (see [31]).

Problem. Given an explicit combinatorial description of the QT (u) if G is aclassical group.

2.2 Geometric description of Green functions. From the work of Springer andKazhdan it follows (for large p) that the QT (u) can be described in terms of"trigonometric sums" on the Lie algebra of G. There is a cohomologicalinterpretation for these trigonometric sums. The reader is referred to [29], [15]and [32] for details.

As was mentioned earlier, Problems (A) and (B) are two of the main problemsarising from the theory of Lusztig-Deligne. We will now mention various otherproblems which might be of interest.

(C) Groups with a disconnected center. The characters of GL(n, q) can becomputed in principle since we have a recursive rule for determining the Qa,P(q).But the characters of SL(n, q) and SU(n, q) are not known, except when n = 2

CHARACTERS OF FINITE GROUPS OF LIE TYPE. II 337

or 3 (see [7]). Lehrer [17] has described the splitting of the characters of GL(n, q)on restriction to SL(n, q), but this does not give information on the values of thecharacters of SL(n, q).

The group SL(2, q) has two irreducible representations of dimensioni(q + 1)

and two of dimension z (q - 1). The representations of dimension i (q - 1) werestudied by Hecke and have number-theoretic interpretations. Allan Adler (see[1]) has observed that when p = 3 (mod 4) the representations of SL(2, p) ofdimension '(p - 1) arise from the action of SL(2, p) on certain abelian varie-ties. One can ask whether other finite groups which arise from algebraic groupswith a disconnected center (e.g., SL, Sp, SO) have interesting representationswhich are analogues of these.

(D) p-adic groups. Let G be a Chevalley group over a p-adic field K, i.e., afinite extension of the field Qp of p-adic numbers. Lets be the ring of integersof K and 13 its maximal ideal. The group G(Z) is a profinite group and we haveG(Z) = lim G(Z/V").

Gerardin has constructed families of representations of G(C) corresponding toregular characters of unramified maximal tori (see [16] for details). For GL, thiswas done by Shintani [28]. Both these authors used Clifford theory, since thefinite group G(C/$") is an extension of a p-group by G(f. /'. Lusztig [35] hasalso constructed such families using l-adic cohomology. However, a complete setof representations has been constructed only in the case of SL2(Z), for p odd,and SL2(Z2), for p = 2 (see [25], [34]). In fact, even the conjugacy classes of thegroups G(Z/p"Z) are not known, except for SL2 (see [25]).

(E) Lifting. This problem arises from the "Base Change" philosophy ofLanglands. As before, let G be a connected reductive group defined over Fq andF: G -p G the Frobenius morphism. The idea is to look at the finite groups GF,GF" (e.g., GL(n, q), GL(n, q')) and compare the irreducible characters of GFwith the irreducible characters of GF' which are fixed by F. So there is ananalogy with the work of Glauberman, Dade and Isaacs ([9], [13]).

Now F acts on GFTas an automorphism of order m. Let H = GF". Let

A = <a> be a cyclic group of order m. Consider the semidirect product HAwhere a acts on h as F. Put H = G H. We have a norm map N: G -* Ggiven by

N(x) = x(Fx)(FZx) . . . (Fnr-1x).

If x E H, we can write (by Lang's Theorem) x = a(Fa)-1 for some a C G. Thena -1(Nx)a E H. The correspondence 0: xa -+ a -IN(x)a gives rise to a bijection(HA-conjugacy classes in Ha) H (conjugacy classes of H).

Notation. If M is a finite group, e(M) is the set of irreducible characters of M.DEFINITION. Let 4, E e(H). Then p e e(H) is a lift of if it extends to

Jj* E e(HA) such that 4,*(xa) = ± t/,(O(xa)) (x E H).

THEOREM (KAWANAKA [14]). Suppose p does not divide m. Let H = GL(n, q(resp. U(n, q'")), H = GL(n, q) (resp. U(n, q)). Then every i C e(H) lifts to some

E r(H). T h e m a p - * gives a bjection e(H) H E(H)'.

For GL, Shintani [27] has proved this theorem without the restriction that pdoes not divide m. Gyoja [11] has considered a general group G. He has shown


that if p does not divide m, p, q are sufficiently large, and 4' E e(H) is of theform ±RT(8) (8 a regular character of some TF C GF = H) then 4, lifts tosome E E(H).

Lusztig has pointed out (see [11]) that in the case where H = Sp(4, q),if = Sp(4, q2), the character 010 does not lift. One can therefore ask what ageneral theorem should be, as regards lifting characters of H to characters of H.

(F) Schur indices. Finally we mention the problem of finding the Schur indicesof the characters of groups of the form GF. The reader is referred to the papersof Gow [8], Helversen-Pasotto [12], and Ohmori [24]. Ohmori has shown that theSchur indices of the characters of GL(n, q) and SL(2n + 1, q) (q odd) are equalto 1. Gow has shown that SL(2n, q) (q - 1 mod 4) has characters of Schurindex 2. He has also computed Schur indices for certain characters of groups oflow rank. But the problem is open in general.

REFERENCES

1. A. Adler, On the automorphism group of a certain cubic threefold, Amer. J. Math. 100 (1978),1275-1280.

2. B. Chang and R. Ree, The characters of G2(q), Sympos. Math. vol. XIII (1974), AcademicPress, London and New York, 1974, pp. 395-413.

3. C. W. Curtis, Representations of finite groups of Lie type, Bull. Amer. Math. Soc. (N.S.) 1(1979), 721-757.


5. V. Downes, The characters of Sp(6, q), Thesis, Clark University, 1976.6. H. Enomoto, The characters of Sp(4, q), q = 28, Osaka J. Math. 9 (1972), 75-94; The characters

of G2(q), q = 38, Japan. J. Math. 2 (1976), 191-248.7. J. S. Frame and W. Simpson, The character tables of SL(3, q), SU(3, q2), PSL(3, q), PSU(3, q2),

Canad. J. Math. 25 (1973), 486-494.8. R. Gow, Schur indices of some groups of Lie type, J. Algebra 42 (1976), 102-120.9. G. Glauberman, Correspondence of characters for relatively prime operator groups, Canad. J.

Math. 20 (1968),1465-1488.10. J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80

(1955), 402-447.11. A. Gyoja, Liftings of irreducible characters of finite reductive groups, Osaka J. Math. 16 (1979),

1-30.12. A. Helversen-Pasotto, On the Schur index of representations of GL(n, Fq) C. R. Acad. Sci. Paris

282 (1976), 233-235.13. I. A. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95 (1973), 594-653.14. N. Kawanaka, On the irreducible characters of the finite unitary groups, J. Math. Soc. Japan 39

(1977), 425-450.15. D. Kazhdan, Proof of Springer's hypothesis, Israel J. Math. 28 (1977), 272-286.16. P. Gerardin, Construction de series discretes p-adiques, Lecture Notes in Math., vol. 462,

Springer-Verlag, Berlin and New York, 1975.17. G. I. Lehrer, The characters of the finite special linear groups, J. Algebra 26 (1973), 564-583.18. G. Lusztig, On the Green polynomials of classical groups, Proc. London Math. Soc. (3) 33

(1976), 443-475.19. , Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125-175.20. , Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976), 101-159.21. , Representations of finite Chevalley groups, CBMS Regional Conf. Ser. in Math., vol.

39, Amer. Math. Soc., Providence, R. I., 1978.22. A. O. Morris, A survey on Hall-Littlewood functions and their application to Representation

Theory, Combinatoire et Representations du Groupe Symetrique, Lecture Notes in Math., vol. 579,Springer-Verlag, Berlin and New York, 1977.

CHARACTERS OF FINITE GROUPS OF LIE TYPE. 11 339

23. S. Nozawa, Characters of U(4, q2), J. Fac. Sci. Univ. Tokyo 19 (1972), 257-293; Characters ofU(5, q2), ibid. 23 (1976), 23-74.

24. Z. Ohmori, On the Schur indices of GL(n, q) and SL(2n + 1, q), J. Math. Soc. Japan 29 (1977),693-707.

25. A. Nobs, Die irreduziblen Darstellungen der Gruppen SL2(Zp), insbesondere SL2(Z2). I, Com-ment. Math. HeIv. 39 (1977), 465-489.

26. J.-P. Serre, Representations lineaires des groupes fini algebriques, Seminaire Bourbaki 487,Lecture Notes in Math., vol. 567, Springer-Verlag, Berlin and New York, 1977.

27. T. Shintani, Two remarks on irreducible characters of finite general linear groups, J. Math. Soc.Japan 28 (1976), 396-414.

28. , On certain square-integrable unitary representations of some p-adic linear grows, J.Math. Soc. Japan 20 (1%8), 522-565.

29. T. A. Springer, Trigonometric sums, Green functions and representations of Weyl grows, Invent.Math. 36 (1976), 173-207.

30. B. Srinivasan, The characters of the finite symplectic group Sp(4, q), Trans. Amer. Math, Soc.131(1%8), 488-525.

31. , Green polynomials of classical groups, Comm. Algebra 5 (1977), 1241-1258.32. , Representations of finite Chevalley groups: A survey, Lecture Notes in Math., vol. 764,

Springer-Verlag, Berlin and New York, 1979.33. V. Ennola, On the characters of the finite unitary groups, Ann. Acad. Sci. Fenn. 323 (1963),

1-35.34. J. A. Shalika, Representations of the two by two unimodular grow, over local fields, Lecture

Notes, Institute for Advanced Study.35. G. Lusztig, Some remarks on the swercuspidal representations of p-adic semisimple groups, Proc.

Sympos. Pure Math., vol. 33, Amer. Math. Soc., Providence, R. I., 1979, pp. 171-175.36. , Classification of complex representations of finite Chevalley grows, Abstracts Amer.

Math. Soc., #773-20-16, 1 (1) (1980), p. 49.37. R. B. Howlett and R. W. Kilmoyer, Principal series representations of finite groups with split

BN-pairs, Comm. Algebra 8 (1980).


PRINCIPAL SERIES REPRESENTATIONS OFFINITE GROUPS

WITH SPLIT (BN)-PAIRS

R. W. KILMOYER

This paper is a summary of the main results of some work done jointly withR. B. Howlett. The detailed proofs of these results will appear in [4].

Let G be a finite group with a split (BN)-pair at characteristicp. In particularG = <B, N>, B n N = H < N. W = N/H = the Weyl group. B = XH, whereX is a Sylowp-subgroup of G and H is an abelianp'-group. Such groups includeall Chevalley groups both twisted and untwisted as well as all their reductiveversions and coverings. Thus, for example, these axioms include SL (q), GL (q)as well as PGL (q).

Let A be a character of H. Since B = XH is a semidirect product we canextend A to a linear character AB of B uniquely such that X is contained in thekernel of A. The problem we discuss in this paper is that of the decompositioninto irreducible components of the induced character ag G. It is worth noting thatin the Lusztig-Deligne notation ag = RH(A). Thus the series of representations(fig IA E H') which we are considering is just one of the series of Lusztig-De-ligne representations; namely, when the torus in question is the split torus. Thisseries is called the principal series of G.

As an example take G = SL2(q), q odd.

B= I\0

l/a)Ia,8 EFq,a O}, H= {(g 1Oaa EF9}.

X = {(0 8 1,8 E Fq ).

Then A may be regarded as a homomorphism of Fy' into C. Here Fq is the fieldof q elements and C is the complex numbers. One can easily see that lg = 1G +St where 1G is the trivial character and St is the Steinberg character; while if A isthe "± 1" character, then ag is the sum of two distinct irreducible componentsof degree (q + 1)/2. For all other A, ag is irreducible.

1980 Mathematics Subject Classification. Primary 20015.m American Mathematical Society 1980

341

342 R. W. KILMOYER

The Sylow p-subgroup X = T1aEA Xa where 0 is a set of positive roots for Grelative to H. Let II = (a,, . , a,) be the corresponding set of simple roots.W = <waIa E 0>, where wa is the reflection through the root a, a E A. Letqa = IXaI, qw = llga over a E 0+ such that w(a) E 0-. Here 0+, 0- denote thesets of positive and negative roots respectively. One knows that also qw =JB: B n Bwl. The qw (w E W) are sometimes referred to as index parameters.

Let Ea = BJ-'XbEB A(b)-'b, then the left ideal kGEX of KG affords theinduced character ag . We are taking k to be an algebraically closed field ofcharacteristic zero. On the other hand, EakGEE = H(G, B, A) is the centralizerring of ag . H(G, B, A) is called the Hecke algebra of G/B with respect to A. Theimportance of H(G, B, A) is that it contains the idempotents which act asprojections onto the primary components of ag. Thus by studying the represen-tations of H(G, B, A) we can get information about the irreducible componentsof ag .

Let (w) E N be a fixed representative for each w E W and put 8w(A) _gwEa(w)Ea. It is immediate from the Bruhat decomposition that 8w(A) = 0 ifAw A, while if S = (w E WIAw = A), then ( fw(A)Iw E S) is a basis forH(G, B, A). The action of Won IT is defined by the rule Aw(h) = A(whw-1).

The parameters qa(A). Although the subgroup S of W is always a reflectionsubgroup (generated by reflections) if G is a finite Chevalley group of adjointtype, S fails to be a reflection subgroup in general when G is not of adjoint type(for example Nevertheless, we can consider those reflections of Wwhich lie in S. For each such reflection we define a nonnegative power of p bythe following rule: Let a E 0 such that wa E S. If a is a simple root, let PQ bethe minimal parabolic subgroup Pa = B U BwaB. Then A 0 = X1 + X2 where X;are distinct and irreducible. It can be shown from a result of Curtis and Fossum[1] that the degrees of Xi and X2 must be h and hp" where h is prime top andy > 0. In this case we let qa(A) = p". If a is an arbitrary root such that wa E S,we can choose b E II and an element w E W such that w(b) = a. Then clearlywb E CK,(Aw), so we define qa(A) = gb(Aw) in this case. Of course one must showthat this depends only on a and A.

As a matter of fact, it turns out that there are only three possibilities for qa(A),namely qa(A) = 1, qa, or

3. The case qa(A) = I occurs already, for instance, in

our example above on SL2(q) with A = the "± I" character. In general whenqa(A) = 1 it should be regarded as a generalization of that example. If G is aChevalley adjoint group, then qa(A) is always equal to qa. The case qa(A) =can occur only in the unitary groups and the groups of Ree type.

Let t = (a E S l wa E S and qa(A) > 1). It can be seen that r is a rootsystem, although not necessarily a root subsystem of A. Let E be the unique setof simple roots for t such that E C 0+. Then (Ws, E) is a Coxeter system whereWs is the subgroup of S generated by reflections corresponding to the roots inF. Let D = (w E SIw(r+) = t+). One shows that S = DWs is a semidirectproduct, Ws < S.

THEOREM A. Let SH denote the inverse image of S with respect to the naturalmap N -+ W. Then A can be extended to a linear character of SH.

FINITE GROUPS WITH SPLIT (BN)-PAIRS 343

The extension obtained is clearly unique only up to the choice of some linearcharacter of S. The proof makes use of the parameters q,,(A) and their properties.We can define a generalized index parameter qw(A) for each w E S by the ruleqw(A) = llq,,(A) where the product is taken over all a E I'+ such that w(a) E F.

THEOREM B. Let Yw = A((w)-)(Vq_ )-' qw(A) - 3 . Then (Ywlw E S) is abasis of H(G, B, A), and one has ford E D, w e S and b c 2,

YwYd = Ywd' YdYw = Ydw'

Yww``, if w(b) > 0,

gb(n)Yww, + (qb(n) - l)Yw if w(b) < 0.

THEOREM C. The group D is an abelian p'-group.

(1)

(2)

The proof of Theorem C utilizes Theorem B to show that for any irreduciblecharacter a of D there exists an irreducible character K of H(G, B, A) such thatK(Ydw) = a(d)gw(A) where d E D, w E Ws. Then we show that the correspond-ing irreducible component of ag has multiplicity a(l) in ag and has degreeequal to a(1)IG: BI DI '(YwEw qw(A)) i. Since IG: BI is prime top and Xgw(A)is an integer it follows that I D I is prime to p. If a(1) > 1 it follows from atheorem of Howlett [3] that the corresponding component of ag has degreedivisible by p. But then p divides a(1) contradicting the fact that D is ap'-group.Hence D has no irreducible character of degree > 1 and is therefore abelian.

One can easily see that the structure of H(G, B, A) as an algebra is completelydetermined by (1) and (2) of Theorem B. Now let k[u] be the ring of polynomi-als over k. Choose some power q of p such that q,,(A) = q -O') for a E S. Wecould choose p = q, but if G is a Chevalley group over Fq, then the naturalchoice of q is best. Let ua(A) = u°(X) and uw(A) = llua(A) over a ` I'+ such thatw(a) E 1'-. We define 9, to be the k[u]-algebra with basis (awlw E S) such that(1) and (2) of Theorem B are satisfied with aw and ub(A) in place of Yw and qb(A).6T is called the generic algebra of S.

Under the specializations u -* q and u -* 1, Q specializes to H(G, B, A) andkS, the group algebra of S respectively, hence we obtain the known result thatthese two algebras are isomorphic. Moreover, if K denotes an algebraic closureof k(u), it can be shown that all irreducible characters of (?K = K ® Q taketheir values in the integral closure ( of k[u] in K. Extending the specializationsu -* q and u -* 1 to ( ® 6T we obtain the fact that the irreducible characters ofe, H(G, B, A), and kS are 1-l correspondence.

Note that the subalgebra of Q generated by (adl d E D) is just the groupalgebra k[u]D, while the subalgebra generated by (awIw E Ws) is the "stan-dard" generic ring of the Coxeter System (Ws, 2) as defined in [2]. Moreover,(T = k[u]D 063 where 63 is the subalgebra generated by (aw1w E Ws). How-ever, D acts on 63 as a group of automorphisms by conjugation. The representa-tion theory for QK = KD 063K is described by the following analogy ofClifford's theorem.

344 R. W. KILMOYER

THEOREM D. Let ' be an irreducible character of GJ " afforded by the irreducible63K-module V. Let C = (d E DI4Pd = 4,). Then V can be extended to a KC63K-module with character t' extending t . Form the induced e-module VQ =(TK ® KC ® ,KV and let Q be its character. Then ' is irreducible and all irreducible

&K-modules are found this way. One has QI4$K = ICI -'X dED4, d.

The essential point here is that the representation theory of )K is fairly wellunderstood. Corresponding to each irreducible character 4, of K there isdefined in [2] a "generic degree" d,(u) given by

(deg 1L')PX(u)

2 wEws uw(A) ' '(aw- 1)W(aw)

Here Pa (u) _ > w E ws uw (A) is a Poincare polynomial which is known in all cases.In particular P(u) = P1(u) is just the Poincare polynomial of G and P(q) =IG: BI. The generic degrees d,,(u) given above are known in all cases. See [4] fora complete list of references.

Now let X be an irreducible character of QK. We define a "generic degree"DX(u) by

DX(u) =(deg X)P(u)

JwES uw(n)-'X(aw-i)X(aw)

It can be shown that under the specialization u -+ q, DX(u) -+ DX(q) is thedegree of the irreducible component of ag corresponding to X, while underu -+ 1, DX(u) -+ DX(l) is the degree of the corresponding irreducible character ofS. Using Theorem D we can express DX(u) in terms of the known genericdegrees of BK as follows.

THEOREM E. Suppose X is an irreducible character of CTK, 4, = the restriction ofxto GJ3Kand C= {d E }.Then

DX(u) =P

(()ICI . dju).

We conclude with the following example. The generic ring JG3K always has twolinear characters B1 and B2 defined by 81(awb) = -1, 82(awb) = ub(A) for b c E.(This determines B1 and B2 since JG3K is generated as an algebra by (awbl b E E).)The generic degrees of BK corresponding to B1 and B2 are de (u) = uNa andde=(u) = 1. Here Na = llub(A) over all b c F'. B1 and B2 are clearly D-invariant,hence they can be extended (in IDI different ways) to irreducible characters X1and X2 of aK. The generic degrees of X1 and X2 are given by

dX (u) = qN,, P(u)PX(u)

dX=(u) = P(u)Pa(u)

Note that dd=(q) is prime top, while dX (q) is divisible by the highest power of pamong all components of ag . The extensions X1 of B1 are called the "generalizedSteinberg components" and the extensions X2 of B2 are called the "generalizedidentity components" of ag .

FINITE GROUPS WITH SPLIT (BN)-PAIRS 345

REFERENCES'

1. C. W. Curtis and T. Fossum, On centralizer rings and characters of representations of finitegroups, Math. Z. 107 (1968), 402-406.

2. C. W. Curtis, N. Iwahori and R. Kilmoyer, Hecke algebras and characters of parabolic type offinite groups with (BN)pairs, Inst. Hautes Etudes Sci. Publ. Math 40 (1972), 81-116.

3. R. B. Howlett, Some irreducible characters of groups with BN pairs, J. Algebra 39 (1974),571-592.

4. R. B. Howlett and R. W. Kilmoyer, Principal series representations of finite groups with split(BN) pairs, Comm. Algebra 8 (1980), 543-583.

CLARK UNIVERSITY, MASSACHUSETTS

'See the bibliography of [4] for a complete set of references.


CARTAN INVARIANTS ANDDECOMPOSITION NUMBERS

OF CHEVALLEY GROUPS

J. E. HUMPHREYSI

1. Introduction. If H is any finite group and p a prime, we can look for thecorresponding matrix C of Cartan invariants (multiplicities of modular irreduci-bles as composition factors of principal indecomposable modules) and for thematrix D of decomposition numbers (multiplicities of modular irreducibles inthe reduction mod p of ordinary irreducibles). These are related by the equationC ='DD. Both matrices break up into blocks according to the p-blocks (inde-composable two-sided ideals) of the modular group algebra of H. Of course, thishas little interest unless p divides I H 1.

When H is replaced by an entire family of groups of Lie type, with p taken tobe the characteristic of the underlying field of definition, it is less clear whatinformation we seek. We could ask for explicit algorithms to obtain C and D foreach individual group in the family. But we could also look for general patternsof behavior. Although C and D can be studied separately for each p-block, theretend to be very few of these [11].

So far only a few families of Lie type have been treated explicitly, mainly therank one types (SL2, Suzuki groups); see [1], [4], [4a], [5], [6], [8], [12], [20], [22],[23]. Whether C or D is emphasized, the methods have centered either on thecomparison of known tables of ordinary and modular characters or on theexplicit construction of projective modules. Direct comparison of charactertables will of course be less palatable as these tables grow more complicated. Soa purely combinatorial approach is unlikely to succeed. The example Sp(4, 5)exhibited in [16] may be instructive.

2. Chevalley groups and hyperalgebras. Here we describe briefly the frameworkin which we propose to work. For background information and detailed refer-ences see [14], [15], [16], [18]. Take G to be a simple, simply connected algebraicgroup, split over FP (for example, SL ), and write I',, for the group of rational

1980 Mathematics Subject Classification. Primary 20C20, 20G40.'Research supported in part by NSF grant MCS-79-02738.

® American Mathematical Society 1980

347

348 J. E. HUMPHREYS

points over Fp.. Besides these finite subgroups of G, it is advantageous to studythe hyperalgebras u" associated with certain infinitesimal subgroups of G(Frobenius kernels); here ul is just the restricted universal enveloping algebra ofthe Lie algebra of G. If we work over an algebraically closed field K ofcharacteristic p, we find a pervasive analogy between the representation theoryof u" and that of KI'", via the representation theory of G, as summarized in thetable below.

projective or injectiveindecomposables

intermediate modules

irreducible modules

G u" KI'"

Q(X, n) Q(X, n) R(X, n)

V(X) Z(X, n) reduction mod p

M(X) M(X) M(X)

The irreducible modules M(A) for G correspond to dominant weights A EX+; by restriction (resp. differentiation) those A whose coordinates with respectto fundamental dominant weights are less than p" yield precisely the irreduciblemodules for KI'" (resp. u"). The "Weyl modules" V(X) are obtained by reductionmodp from the usual characteristic 0 irreducible modules for the correspondingsemisimple Lie algebra, with character given by Weyl's formula. It turns out thatV(X) is a universal highest weight module for G, having M(X) as uniqueirreducible quotient. The modules Z(X, n) are universal highest weight modulesfor u", analogous to Verma modules.

The PIM's (principal indecomposable modules) Q(X, n) for u" are (at least forp > 2h - 2, h the Coxeter number) in a natural way G-modules whose restric-tions to KI'" are projective but not always indecomposable; then R(X, n) is adirect summand of Q(X, n). See [3], [14], [19]. For a given A E X +, Q(,\, n) isdefined for sufficiently large n (p as above), and there are natural embeddingsQ(X, n) -* Q(X, n + 1), allowing one to pass to the direct limit; this is just theindecomposable injective G-module having M(A) as socle (proved by S. Donkin[7], J. Ballard [3], J. C. Jantzen [19], using different arguments). Each of thesituations in the table has associated with it a Brauer duality C = ,DD, as shown(for G and u") by Jantzen [18], [19], J. A. Green [9], or the author [10].

When A E X + is written as a p-adic expansion A = 4 + pA1 +having coordinates less than p), one has Steinberg's tensor product theoremM(A) = M(A0) ® M(A1)(" 0 . There is an analogous expression for Q(A, n),but not in general for R(A, n).

If A has all coordinates equal top" - 1, M(A) is called a Steinberg module St".These modules are ubiquitous in the theory, and appear in all positions in theabove table (being projective/injective as well as irreducible).

3. Some general observations. (1) We are especially interested in findingpatterns applicable to all I'". Evidence suggests that p (except when very small)should play no special role; but specialization to small p may be difficult.Understanding the case n = 1 should help to predict what happens for arbitraryn, via the tensor product theorem; but this is tricky, even for SL2.

CARTAN INVARIANTS AND DECOMPOSITION NUMBERS 349

(2) We expect to find certain "generic" patterns, e.g., number and distributionof composition factors of PIM's. These patterns should degenerate in a sys-tematic (but complicated) way, as in the ordinary character theory [21].

(3) Results should be organized according to the geometry of the affine Weylgroup (relative to p): weights in a given alcove will behave similarly. This pointof view was first emphasized by D. N. Verma [24].

(4) All evidence suggests that strong analogies exist among the V(X), theZ(X, n), and the reductions mod p of ordinary representations (or Deligne-Lusz-tig characters) of F,,. In suitably general position, all of these should exhibit thesame composition factor behavior.

4. The matrix C. This can be viewed as purely a modular question. A basictechnique is to construct projective modules by tensoring with St', (see [1], [2],[5], [6], [12], [15], [20], [23]). For groups of rank 1 or 2, dimensions andcomposition factors of the Q(X, n) are computable. A key question is how towrite Q(X, n) as a sum of PIM's for r,,; in all cases R(X, n) occurs just once, and"usually" Q(X, n) = R(X, n). As an example, the dimensions of PIM's forSL(3, p) are mostly 6p3 or 12p3. Exact figures (for p > 3) are given below forlattice points X in various positions in the restricted alcoves. The numbers shownin Figure 1 (resp. Figure 2) stand for (dim Q(X, n))/p3 (resp. (dim R(X, n))/p3).For further cases, see L. Chastkofsky's contribution to these PROCEEDINGS.

1

Figure 1

1

Figure 2

The Cartan invariants can also be worked out for SL(3, p) and other smallcases; those for F1 are "deformed" versions of those for u1. This is illustrated inthe generic case of a PIM Q(X, 1) of dimension 6p3 for u, in Figure 3, with thecorresponding PIM R(X, 1) for KF1 = SL(3, p) in Figure 4. The locations ofweights are only shown schematically, to suggest "general position". Nongenericbehavior requires further study. Passage from n = 1 to arbitrary n also requires

350 J. E. HUMPHREYS

study. This has been worked out in detail only for SL2 and Suzuki groups. In thecase of SLz, M. Elmer [8] has shown how a systematic study of Cartan invariantsfor u makes possible a very simple transition to F,,, which was treated in a morecomplicated way by B. S. Upadhyaya [23].

Figure 3 Figure 4

5. The matrix `D. The Brauer characters of PIM's are much more uniform inappearance than those of the M(X): compare dimensions, for example. So it issensible to compare principal indecomposable characters with ordinary irreduci-ble characters, or Deligne-Lusztig (virtual) characters RT(9). Work of Ballard[2] shows that the character of R(X, n) is a Z-linear combination of certain s(µ)St,,, where St,, now denotes the Steinberg character of r,, and s(µ) comes fromthe Weyl group orbit Wu of a weight µ. If we parametrize finite tori in the usualway by elements of W (by twisting a split torus), a character of a maximal splittorus of G yields corresponding characters 0. of the finite tori T,(Fp,). In thissituation one sees [17] that 1W ,w ± RT (9W) = I W j s(µ) St,,, where W. is thestabilizer of µ and the sign is chosen to make the degree positive. To make thischaracter comparison effective one of course needs to know the characters ofPIM's in detail; Jantzen's work shows that this information will come from astudy of Weyl modules, for whose decomposition behavior Lusztig now has apromising conjecture (see his contribution to these PROCEEDINGS).

6. The matrix D. It is still of interest to see directly how an ordinary characterdecomposes mod p. As remarked already, we expect a strong analogy with thedecomposition behavior of Z(X, n), which should be "deformed" to yield theresults for F. Explicit deformations for SL(3, p) and Sp(4, p) are suggested in[13] (see also Carter's contribution to these PROCEEDINGS). We have alsosuggested a way to visualize what is going on by the use of a "Brauer complex"[14], [16], based on the geometry of the affine Weyl group and generalizing theBrauer tree of SL(2, p).

CARTAN INVARIANTS AND DECOMPOSITION NUMBERS 351

REFERENCES

1. J. L. Alperin, Projective modules for SL(2, 2"), J. Pure Appl. Algebra 15 (1979), 219-234.2. J. W. Ballard, Projective modules for finite Chevalley groups, Trans. Amer. Math. Soc. 245

(1978), 221-249.3. , Injective modules for restricted enveloping algebras, Math. Z. 163 (1978), 57-63.4. R. Burkhardt, Die Zerlegungsmairizen der Gruppen PSL(2, pf), J. Algebra 40 (1976), 75-96.

4a. , fiber die Zerlegungszahlen der Suzukigruppen Sz(q), J. Algebra 59 (1979), 421-433.5. L. Chastkofsky and W. Feit, On the projective characters in characteristic 2 of the groups

Suz(2'") and Sp4(2"), Inst. Hautes Etudes Sci. Publ. Math. (to appear).6. , On the projective characters in characteristic 2 of the groups SL3(2m) and SU3(2'"), J.

Algebra 63 (1980), 124-142.7. S. Donkin, Hopf complements and injective comodules for algebraic groups, Proc. London Math.

Soc. 40 (1980), 298-319.8. M. A. Elmer, On the modular representation theory of semisimple Lie algebras, Ph. D. thesis,

Univ. of Massachusetts, 1979.9. J. A. Green, Locally finite representations, J. Algebra 41 (1976), 137-171.

10. J. E. Humphreys, Modular representations of classical Lie algebras and semisimple groups, J.Algebra 19 (1971), 51-79.

11. , Defect groups for finite groups of Lie type, Math. Z. 119 (1971), 149-152.12. , Projective modules for SL(2, q), J. Algebra 25 (1973), 513-518.13. , Weyl groups, deformations of linkage classes, and character degrees for Chevalley

groups, Comm. Algebra 1(1974), 475-490.14. , Ordinary and modular representations of Chevalley groups, Lecture Notes in Math., vol.

528, Springer-Verlag, Berlin and New York, 1976.15. , On the hyperalgebra of a semisimple algebraic group, Contributions to Algebra: A

Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977.16. , Modular representations of finite groups of Lie type, Finite Simple Groups II, Chapter

12, Academic Press, New York, 1980.17. , Deligne-Lusztig characters and principal indecomposable modules, J. Algebra 62 (1980),

299-303.18. J. C. Jantzen, fiber Darstellungen hoherer Frobenius-Kerne halbeinfacher algebraischer Gruppen,

Math. Z. 164 (1979), 271-292.19. , Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine Angew.

Math. (to appear).20. A. V. Jeyakumar, Principal indecomposable representations for the group SL(2, q), J. Algebra 30

(1974), 444-458.21. G. Lusztig, Representations of finite Chevalley groups, CBMS Regional Conf. Ser. in Math., vol.

39, Amer. Math. Soc., Providence, R. I., 1978.22. B. Srinivasan, On the modular characters of the special linear group SL(2, p"), Proc. London

Math. Soc. 14 (1964), 101-114.23. B. S. Upadhyaya, Composition factors of the principal indecomposable modules for the special

linear group SL(2, q), J. London Math. Soc. 17 (1978), 437-445.24. D. N. Verma, Role of affine Weyl groups in the representation theory of algebraic Chevalley

grows and their Lie algebras, Lie Groups and Their Representations, Halsted, New York, 1975.

UNIVERSITY OF MASSACHUSETTS


DUALITY IN THE CHARACTER RING OF AFINITE CHEVALLEY GROUP

DEAN ALVIS

1. Introduction. Let G be a finite group with split (B, N)-pair of characteristicp. Let (W, R) be the Coxeter system of G. For X a virtual character of G, wemay define (following Curtis [5]) the truncation X(J) of X for J C R and the dualX* of X. X (j) is a virtual character of the parabolic subgroup P(J), andX* = E(-1)ISIX(GJ) summed over all J C R. We have (1G)(,) = lp(,), so thatX - X* generalizes the construction of the Steinberg character St = E(-1)1111 J)(as in [3]).

In this paper several general properties of the duality map X - X* areexamined. As the terminology suggests, duality has order two (X** = X) and isan isometry (if X is irreducible then ±X* is irreducible). The duality map alsointeracts well with the Harish-Chandra theory of cuspidal characters and theLusztig-Deligne virtual characters when G is the group of rational points of aconnected, reductive algebraic group defined over a finite field.

Since the conference, the author has learned that N. Kawanaka has indepen-dently proved Theorems 1 and 2.

The author wishes to thank C. W. Curtis for sharing ideas which led to thispaper.

2. Preliminaries. For G a finite group, char(G) will be the ring of virtualcomplex characters of G, and irr(G) will be the set of irreducible complexcharacters of G. Let ( , )G be the usual Hermitian form on char(G): (X1, X2)G =G I-1Y_gE

GX1(g)X2(g-1).

Now suppose (G, B, N, W, R) is a finite split (B, N)-pair of characteristic p.For J C R let P(J) = BW(J)B be the standard parabolic subgroup correspond-ing to J, where W(J) is generated by J. The structure of the subgroups P(J) isgiven in Curtis [4]. We have P(J) = L(J) V(J) (semidirect) where V(J) _OO(P(J)). The group L(J) has a split (B, N)-pair of characteristic p determinedby B(J) = B n L(J) and N(J) = N n L(J). The corresponding Coxeter sys-tem is (W(J), J).

1980 Mathematics Subject Classification. Primary 20C15.m American Mathematical Society 1980

353

354 DEAN ALVIS

For E char(L(J)) (J C R), let - be the element of char(P(J)) obtainedfrom via P(J) -* L(J) = P(J)/V(J). We define the truncation X(J) of X Echar(G) to P(J) by X(J) = E(X, y summed over E irr(L(J)). Note thatX(J) = xi where x, is the restriction of X(J) to L(J). Note also that X(J) is givenby

x(J)(g) = I V(J)I-' I x(gv), g E P(J).v E V(J)

If x is the character of G afforded by the CG-module M, then X(J) and X, areafforded by invV(j)(M), the fixed points of M under the action of V(J).

If K C J C R and E char(L(J)), then we may truncate to the standardparabolic subgroup P(J, K) = L(J) rl P(K) = L(K) V(J, K) of L(J), whereV(J, K) = L(J) n V(K) = OO(P(J, K)). The restriction of this truncation toL(K) will be denoted by J,K.

PROPOSITION. (1) If K C J C R, E char(L(K)) and x E char(G), then(X/)!,K = XK and

(y ,. L(J))-G = y -G./'(2) For J C R, X E char(G) and E char(L(J)), (x, J -G )G = (X(J) _)P(J) _

(x, f)L(J)'(3) Assume J, J' c R and E char(L(J)), ' E char(L(J')). Then

(J -G, J -G)G (JJF K' JJ,,K')L(K)'x E X(J,J')

where X(J, J') is the set of distinguished (W(J), W(J'))-double coset representa-tives in W and K = J n X(J') =X(K'),

(4) Let J, J' C R and suppose E char(L(J)). Then

(y _)!' 2 ( 'J,K)J

xEX(J,J')

where K is as in (3).

PROOF. (Sketch) The first claim of (1) is clear since invV(K)(M) _invv(JK)(invV(J)(M)) for any CG-module M. The second claim then follows fromthe first and part (2).

Part (2) follows directly from the definition of truncation.Part (3) may be proved using Mackey's Theorem [6, (44.5)] and the structure

of the intersections P(J) n XP(J') [4, Propositions 2.5 and 2.7].Finally, (4) is a consequence of parts (2) and (3).

3. Main results. Let G, W, R, etc. be as in the last section. We first state twogeneral results about the duality operation x --> X* _ Y,(-1)I'IX(j). Proofs of theseare outlined in [1].

THEOREM 1. Let J be a subset of R.(1) (CURTIS [5]) (X *)j = (Xj) * for X E char(G).(2) ( -G)* _ for E char(L(J)).

COROLLARY. If E irr(L(J)) is cuspidal, then (-1)IJIr.G.

THEOREM 2. The duality map char(G) -* char(G) is an isometry of order two.Thus x** = x, and ±X* E irr(G) if x E irr(G).

CHARACTER RING OF A FINITE CHEVALLEY GROUP 355

COROLLARY. The duality map permutes, up to sign, the irreducible components of-G for E irr(L(J)) cuspidal.

An explicit description of this permutation is known in the case of 1B ((q) for asystem of groups (G(q)) of type -(W, R). If q E irr(G) is the component of1B(q) corresponding to q) E irr(W), then (4,q)* = Xk,Q where 8 is the signcharacter of W (Curtis [5]).

Another character whose dual is explicitly known is the regular character p ofG. The author is indebted to T. A. Springer for communicating the results of [9]and G. Lusztig's suggestion of applying duality to them. Let e be the characteris-tic function of the set of p-elements of G, so e is a class function. Note thedefinitions of truncation and duality can be extended in the obvious way to classfunctions, so that Theorems 1 and 2 remain valid.

THEOREM 3 (SPRINGER). e =G x.

We apply Theorem 3 to obtain a relationship between the degrees of X and X*for X E irr(G). First note that EX(u) = IGI(X, e)G, where the sum is over allp-elements u of G. By Theorems 2 and 3 we have I G I(X, e)G = I G Ip(X*, P)G =

GIpX*(1). Thus X(l)-'Y-X(u) = IGIp(x*(1)/X(l)). Next, X(1)-1YX(u) can be writ-ten as YX(l)-'ICIX(x) summed over the conjugacy classes C of p-elements of Gwith x E C. It follows that X(1)-'EX(u) is an algebraic integer, henceG Ip(X*(1)/X(1)) is an integer. We may repeat the above with X E irr(G)

replaced by ±X* E irr(G). In summary, we have proved the following.

THEOREM 4. Let X be an irreducible character of G.(1) X(l)-'YX(u) = IGIp(X*(1)/X(1)), where the sum is over all p-elements u of

G.

(2) IX*(1)Ip- = x(1)p,.(3) X(l)-1Y_X(u) is, up to sign, a power of the characteristic p.

The next theorem sharpens the isometry result of Theorem 2. Let s be anelement of G whose order is prime to p, and let V(s) be the set of p-elementswhich commute with s.

THEOREM 5.

I X1(su)X2\SU) = 2 X1(SU)X2(su)U E V(s) U E V(s)

for any XI, X2 E char(G).

A special case of the theorem verifies a conjecture [8, p. 541] of N. Kawanaka.This conjecture has also been proved independently by Kawanaka.

COROLLARY. For any X E char(G), St(s)X*(s) _ uE V(s)X(su).

Now let us consider the case G = GF, where G is a connected, reductivealgebraic group over k = Fq, defined over Fq, with corresponding Frobeniusendomorphism F. Let T be an F-stable maximal torus of G, and let 9 be a linearcharacter of TF. Associated with the pair (T, 9) is a virtual character RT9 of G,

356 DEAN ALVIS

constructed by Lusztig and Deligne via the 1-adic cohomology of certainvarieties related to G [7]. The author is indebted to G. Lusztig for communicat-ing the following result.

THEOREM 6 (LuszTIG). (RT9)* = a(G)o(T)R-'29.

Note that G = GF has a split (B, N)-pair of characteristic p, the characteristicof Fq, in which B = _Bo for an F-stable Borel subgroup go of G and N = NT-011'for _T° c go an F-stable maximal torus. Also, a(G) = (-1)r where r is theFq-rank of _G, and o(T) is defined similarly.

Now, let X E char(G). For T, 9 as above and s a semisimple element of G,define

g(X, T, 9, s) = IZ(s)F: Z°(s)FI (RTBRR°8)c 9(t)

where Z °(s) is the connected centralizer of s in G and the sum is over allG-conjugates t of s in TF. Let AT, s)(x) be the rational function which whenevaluated at q gives

Al, s)(q) = a(T)a(Z°(s))I Z°(s)FI p,/I TFJ.The function f(T, s)(x) can be found using results of [11] and [12], and dependsonly on the action of F on the root system of Z°(s) and on the character groupsof the tori T and Z°(Z°(s)). By results of [7] we have

X(s) _'Ef(T, s)(q)g(X, T, 9, s)summed over the G-conjugacy classes of pairs (T, 9). This fact and Lusztig'sTheorem 6 imply the following.

THEOREM 7. For s e G= G F semisimple and X E char(G), X*(s) = St(s)X(s),where

XI(s) = Jf(_T, s)(q-t)g(X, j, 9, s)is the complex number obtained by replacing q by I /q in the f s in the expressionfor X(s).

The proof of Theorem 7 and the following corollary will appear elsewhere.

COROLLARY. Let X E irr(G), for G = G F as above. If the order q of the groundfield is sufficiently large, then the quantity X(1)-'EX(u) of Theorem 4 is of the form± q' for some nonnegative integer m.

The corollary confirms a conjecture of 1. Macdonald [10, 6.11] for q suffi-ciently large.

REFERENCES

1. D. Alvis, The duality operation in the character ring of a finite Chevalley group, Bull. Amer.Math. Soc. (N.S.) 1 (1979), 907-911.

2. A. Borel and J. Tits, Groupes reduciifs, Inst. Hautes Etudes Sci. Publ. Math. 27 (1965), 55-15 1.3. C. W. Curtis, The Steinberg character of a finite group with a (B, N)-pair, J. Algebra 4 (1966),

433-441.

CHARACTER RING OF A FINITE CHEVALLEY GROUP 357

4. , Reduction theorems for characters of finite groups of Lie type, J. Math. Soc. Japan 27(1975), 666-688.

5. , Truncation and duality in the character ring of a finite group of Lie type, J. Algebra (toappear).

6. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras,Interscience, New York, 1962.


8. N. Kawanaka, Unipotent elements and characters of finite Chevalley groups, Osaka J. Math. 12(1975), 523-554.

9. T. A. Springer, A formula for the characteristic function of the unipotent set of a finite Chevalleygroup (preprint).

10. , Cusp forms for finite groups, Seminar on Algebraic Groups and Related Finite Groups,Part C, A. Borel (ed.), Lecture Notes in Math., vol. 131, Springer-Verlag, Berlin and New York,1970.

11. T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and RelatedFinite Groups, Part E, A. Borel (ed.), Lecture Notes in Math., vol. 131, Springer-Verlag, Berlin andNew York, 1970.

12. R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. No. 80 (1968).



CHARACTERS OF PROJECTIVEINDECOMPOSABLE MODULES

FOR FINITE CHEVALLEY GROUPS

LEONARD CHASTKOFSKYI

Introduction. Let G be a semisimple, simply-connected, algebraic group de-fined over K, the algebraic closure of the finite field F . Denote the finite groupG(F) by F. The problem we are concerned with is that of finding the charactersof the projective indecomposable KI' modules, modulo a knowledge of theirreducibles.

Let RK(I') be the ring of Z-linear combinations of Brauer characters of IF andlet PK(F) be the subring spanned by the projective characters. Our technique isto construct elements in PK(F) which will be "almost" indecomposable, that is,they are indecomposable most of the time, and when they are not, they split intoa small number of indecomposables. We will state results giving a formula forthis decomposition and give a criterion for indecomposability to hold.

1. Preliminaries. As references we give [4], [5] and [9]. Let X be the weightlattice of G, Z[X] its group ring (written exponentially with basis elements e(X),X E X), and Z[X ]K" the ring of invariants under the natural action of W, theWeyl group of G. Elements of Z[X ]w can also be considered as elements ofRK(F). As an element of RK(r), 2aAe(pX) = 2aAe(X)F` = 2aAe(X), where Fr isthe Frobenius map.

We shall abuse notation and identify elements of these two rings, with thecontext making it clear which ring we are in.

Let X + be the subset of X whose coordinates (with respect to the fundamen-tal weights) are nonnegative and XP the subset of X + whose coordinates are lessthan p.

For X E X+ let ,(X) be the formal character (an element of Z[X]w) of theirreducible KG-module with highest weight X. The XP(X) for which X E X givethe irreducible Brauer characters of F. Denote the projective indecomposable

1980 Mathematics Subject Classification. Primary 20C15, 20G40.'Supported in part by a National Science Foundation grant.


359

360 LEONARD CHASTKOFSKY

character corresponding to XP(X) by (D.. Let p be the half-sum of the positiveroots of G and denote the Steinberg character Xp((p - 1)p) = I_ 1) by st.Every element of PK(F) is divisible by st. Write cI = st 0,,.

Let X(X) be the formal character of the irreducible 9 -module of highest weightX, where 9 is the complex Lie algebra of the same type as G. The sets (Xp(X):X E XP) and (X(X): X E X +) are Z-bases for RK(F) and Z[X]w, respectively.Denote the coefficient of Xp(.X) (resp. X(X)) of an element ri in RK(F) (resp.Z[X]W) by Multr (Xp(X), ri) (resp. Multg(X(X), ri)).

Z[X ]W is also a module over its subring Z[ pX ]W = (Z[X ]w)Fr = (,,Fr: q EZ[X ]W }. The following lemma is an easy consequence of Steinberg's TensorProduct Theorem.

LEMMA 1. A basis for Z[X]W over Z[pX]W is (Xp(X): X E Xp).

Notation. Denote the coefficient of XP(X) of an element ri E Z[X]W withrespect to this basis by MultG/p(Xp(X), '0-

Define a dot action ofW (resp. W') denote the group of transformations on X generated by W

acting with this dot action, together with all translations by pX (resp. pX'),where X' is the root lattice. Call a fundamental domain for W' an alcove.

It is known that for X E X+ we can write

X(X) _ bFrXp(µ)µ EX,

and

Xp(X) a,\µX(µ),µEX,

where a\µ, b\µ E Z[X]W. A necessary condition for a,\µ and b,,µ to be nonzero isthat X and µ are W-conjugate.

Set XP(X) = XP(-a0X), where ao is the longest element of W (so XP(X) is thecontragredient character to XM(X)). This definition extends by linearity to Z[X]w.When ri E Z[X ]W is considered as an element of RK(F), ri coincides with thecomplex conjugate of ri.

2. Construction of the characters "A. We will now construct the characters "Awhich are meant to be approximations to the (D,\. To motivate the constructionwe give the following lemma, which is a restatement of the orthogonalityrelations between irreducible and projective characters for F.

LEMMA _2. Let ' E PK(F) and write = st . Then Mult((Dµ, 'Y)Multr(st, Xp(µ)), for every µ E X.

PROOF. Mult((Dµ, (4', Xp( µ)) = (st q, Xp( µ)) = (st,Multr(st, qXp( z)). (The inner product in the middle quantities is the usual one.)

COROLLARY. For X E XP, q,\ is the unique element of RK(F) such thatMultr(st, t,\Xp( L)) = 8,\µ for all µ in Xp.

CHARACTERS OF PROJECTIVE INDECOMPOSABLE MODULES 361

We will construct "A = st q,\ so that q,\ has similar properties to q,\. In fact,we look for q,\ with the following property: (.) For every u E Xn,MultG/p(st, ,\X(L)) = 8,\,.

We require the following definitions. For µ E X, let s(µ) = 2,,_,e(,u'), thesum being over distinct conjugates of u under W.

For X andu are notW conjugate, 5C,,, is taken to be empty. Let hA,, = EhE X(h), where the sum is0 when %,, is empty.

We assume a fixed ordering of XP and let H be the matrix (hA,,), X, µ C v.

LEMMA 3. H is invertible over Z[X]w.

An analogue of the matrix H was considered by Hulsurkar [3]. His matrix wasessentially the same except the characters were replaced by dimensions, and thematrix was indexed by 0, T E W. Verma [9] and Jantzen [7] refined it to theversion where the characters appeared. In our version, the regular part of H(elements indexed by X, µ whose stabilizer in R' is trivial) corresponds to anumber of blocks of their version. There is, in addition, the irregular part. Themain point in Hulsurkar's paper was to show his matrix was invertible byshowing that with respect to some ordering the matrix is upper triangular withl's on the diagonal. This part of the proof carries over to our version, and theonly difficulty that remains is to show that this is also true for the irregular part.

Denote the X - u entry of H-1 by gA,,.

LEMMA 4. Define WA by

x bag s{(P - 1)p - v).µ, v

Then WA satisfies (.).

REMARK. Humphreys and Verma [6] have proposed a construction of G-mod-ules QA for X E XP which, when considered as modules for u, the restricteduniversal enveloping algebra of the Lie algebra of G, would be the P.I.M.'scorresponding to irreducibles with highest weight X. Ballard [2] showed thisconstruction worked for p > 3h - 3, h the Coxeter number of G, and Jantzen[8] has improved this result to p > 2h - 2. The formal characters of thesemodules QA are in fact precisely the characters "A = st "A which we haveconstructed.

3. Statement of results. In these statements we assume that p is "largeenough", that is, there is some fixed bound, depending only on the type of G,which p is larger than. The best bounds we are able to supply are relatively high,but in fact it should probably be no higher than h.

THEOREM 1. There exists a d, depending only on the type of G and not on p, suchthat if I<X + p, av> - kph > d for all roots a and all integers k, then "A = (DA.

Here the inner product is the symmetric nondegenerate one on X, and au isthe dual root to a. Again we do not specify d, but for the cases G is of type A2,B2 or A3, d = 1. The theorem says that if X is far enough away from the wall ofan alcove then 4)A We now see what we can say for X close to the wall ofan alcove.

362 LEONARD CHASTKOFSKY

THEOREM 2. The multiplicity of I in 4', is equal to Et bt.A4(,q)),71

the sum being over E X X.

The actual computation of the multiplicities from this formula is ratherinvolved. It has been carried out for types A2 and B2, and partially for A3.

Humphreys and Verma [6] conjectured that 'PA = (D,\ as long as X wasW-regular, in other words, that the condition in Theorem I could be replaced bythe condition "<X + p, a"> > 1 for all simple roots a." The only general resultsobtained previously were by Ballard [1] who showed that if µ = (p - 1)p + aoxis in the bottom alcove (i.e., µ E X+ and the sum of the coordinates of µ is lessthan p), that 4'A = (D,\. (Jantzen has apparently recently obtained formulaswhich are similar to ours.) Theorem 1 shows that the spirit of the conjecture iscorrect, that is, I does equal 'PA "generically." However, the author believesthat A3 will provide a counterexample to the actual conjecture by showing thatnonsimple roots may have to be included, and that G2 will show that d may haveto be taken greater than 1.

Similar results to Theorems I and 2 can be given for twisted Chevalley groups.We have also extended the results to the case where the group is defined overthe field of p" elements.

ADDENDUM 1. The restriction onp stated in the first paragraph of §3 can nowbe removed, so both theorems are true for all p.

ADDENDUM 2. Jantzen has pointed out an error in the author's calculations onA3, so that it does not provide a counterexample to Verma's conjecture.However, Jantzen has verified that G2 does provide a counterexample for bothreasons stated in the last sentence of the penultimate paragraph.

REFERENCES

1. J. W. Ballard, Projective modules for finite Chevalley groups, Trans. Amer. Math. Soc. 245 (1978),221-249.

2. , Injective modules for restricted enveloping algebras Math. Z. 163 (1978), 57-63.3. S. G. Hulsurkar, Proof of Verma's conjecture on Weyl's dimension polynomial, Invent. Math. 27

(1974), 45-52.4. J. E. Humphreys, Ordinary and modular representations of Chevalley groups, Lecture Notes in

Math., vol. 528, Springer-Verlag, Berlin and New York, 1976.5. , Modular representations of finite groups of Lie type, Proc. London Math. Soc. Sympos.

Finite Simple Groups, Durham, 1978.6. J. E. Humphreys and D. N. Verma, Projective modules for finite Chevalley groups, Bull. Amer.

Math. Soc. 79 (1973), 467-468.7. J. C. Jantzen, Uber das Decomposition-verhalten gewisser modularer Darstellungen halbeinfacher

Gruppen and ihrer Lie-Algebren, J. Algebra 49 (1977), 441-469.8. , Darstellungen halbeinfacher Gruppen and ihrer Frobenius-Kerne, J. Reine Angew. Math.

(to appear).9. D. N. Verma, Role of affine Weyl groups in the representation theory of algebraic Chevalley groups

and their Lie algebras, Lie Groups and Their Representations, Halsted, New York, 1975, pp.653-705.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

SOME INDECOMPOSABLE MODULES OFGROUPS WITH SPLIT (B, N)-PAIRS

N. B. TINBERG

1. Introduction and notation. In this report we examine certain indecomposablemodules of finite groups with split (B, N)-pairs. This paper is a continuation of[7] and was inspired by recent work by Green [2] and Sawada [4] on the modularrepresentations of such groups.

Let p be a fixed prime number. Assume G = (G, B, N, R, U) is a finite groupwith an unsaturated split (B, N)-pair of characteristicp and rank n; that is

(i) G has a (B, N)-pair [1, Definition 2.1, p. B-8] where H = B n N and theWeyl group W = N/H is generated by the set R = (wl, ... , of involu-tions,

(ii) B = UH is a semidirect product where U is a normal p-subgroup of B andH is abelian with order prime top.

The term "unsaturated" means "not necessarily saturated"; that is, we allowthe intersection of the N-conjugates of B to be larger than H.

Throughout this paper k will denote an algebraically closed field of character-istic p and all k-spaces are assumed finite dimensional. Generalizing Steinberg'swork in the algebraic group case, Curtis and Richen ([1], [3]) have shown thatthe set of "admissible pairs" or "weights" indexes the set of isomorphism classesof irreducible kG-modules when G has a saturated split (B, N)-pair. Let Y =Ind ,(ku) = kU where ku is k treated as a trivial U-module and let E _EndkG(Y). Sawada used the Curtis-Richen theory to show that these weightsalso index both the irreducible characters of E and the indecomposable compo-nents of Y. In [7] we were able to reformulate the results of Curtis, Richen andSawada and generalize the theory to include the unsaturated case. If v: N -4 Wis the natural homomorphism and J is any fixed subset of R, the parabolicsubgroup G. = (G1, B, N1, J, U) is an unsaturated split (B, N)-pair of char-acteristic p and rank JJJ. Here N. = v-t(Wj) and W. = <wJJww E J>. Themajor contribution of this paper is the application of results on the irreduciblemodular representations for G. to those of G itself.

1980 Mathematics Subject Classification. Primary 20C20.m American Mathematical Society 1980

363

364 N. B. TINBERG

We therefore set Y. to be kGj and let E. = EndkG(Y'). Our aim is to study theindecomposable components of Y = YR by examining those of Y. and inducingthem up to G. We summarize our results:

(1) We find necessary and sufficient conditions under which a multiplicativecharacter of E restricts to one of E. (3.1).

(2) If V is an indecomposable kG,-module which is a component of Y. thenVG must be a direct sum of indecomposable components of Y. This usefulformula is determined (3.3).

(3) The above results allow us to calculate the dimensions of the indecom-posable components of Y (4.2).

(4) We determine a character formula for the Brauer characters of thesecomponents (5.1) and find an irreducible character of G which corresponds tothe Steinberg character.

(5) The vertices of the indecomposable components are calculated using L. L.Scott's work [5] on permutation modules (6.1).

Let k* = k \ {1) and B = Hom(B, k*). If T is any subset of G then T9 =g-'Tg for any g E G. Let w E W and let (w) E N satisfy (w) H = w. If A is anysubgroup of G normalized by H then AM = A h(w) for any h E H so we writeA'V. If S and T are subgroups of G with S C T and M is any kT-module thenM IS denotes the restriction of M to S. We sometimes write pjS if p is thecharacter afforded by M (analogously for E and its subalgebras).

This short paper contains only the statements of results. For complete proofsthe reader is referred to [8].

2. Parabolic subgroups. For J C R, G1 = BWJB is an unsaturated split(B, N)-pair. Let wo denote the unique element of maximal length in W. For eachw,ERletU,= U n Uwo and H;=Hn<U;,Uw,>.For anyXEBletM(X)={w1l XjH; = 1).

DEFINITION. Let X E B, S C M(X) n J. We call (S, X) an admissible G1-pair.We denote by P. the set of all such pairs.

2.1 THEOREM. The set P. indexes:(a) the set {q1(S, X)} of irreducible characters of E1 (simple E1 modules are one

dimensional),(b) the set {M1(S, X)} of isomorphism classes of irreducible kGj-modules.Moreover YJ = Ia', p, Yj(S, X) is a decomposition of Yj into indecomposable

kG,1 submodules where each YJ(S, X) has head isomorphic to M1(S, X).

3. Induction and restriction formulae. The algebra E. can be considered asubalgebra of E and the characters of E restrict to those of E. in the following"natural" way:

3.1 LEMMA. Let 0: E1-4 k be any multiplicative character. There exists amultiplicative character q: E -4 k such that 0 = ,CIE,. In fact if 0 is determined bythe admissible G1-pair (S, X) and q is determined by the admissible G -pair (K, X')then

and S=KnJ.

MODULES OF GROUPS WITH SPLIT (B, N)-PAIRS 365

If {LxIX E B) is a full set of irreducible kB-modules, then each Lx is onedimensional and

kU B = EED Lx.

XEB

Hence

LGJ.

XEB

3.2 LEMMA. Let X E B. Then LX, _ E®,x)EPj YJ(S, X).

The following is obtained by 3.1 and 3.2:

3.3 THEOREM. Let (S, X) E PJ. Then if Y(K, X) = YR(K, X) we have

YJ(S, X)G = ® Y(K, X).K C M(x)S=KnJ

The above formula is extremely useful. Notice that taking J = M(X) it followsthat

3.4 COROLLARY. Each Y(K, X) is induced from YM(x)(K, X).

Moreover if j = R \ J and 0 is the empty set

3.5 COROLLARY. (1) YJ(J, X)G >2®CKCM(x) Y(K, X) and

(ii) YJ(O, X)G = E®g JCM(x) Y(K, X)and Y(J, X) can be identified as the unique component common to both expressions.

Let G1 be any subgroup of G containing U. Consider Yl = kG and itsrelationship to Y:

3.6 LEMMA. Let Gs (S C R) be the unique minimal parabolic subgroup contain-ing G1. Then

Y1 = 20 Y(K, X)SCK

xIBnG,-l

Take G1 = Gs:

3.7 COROLLARY. Let 1B be the identity character on B. Then(i) kGs > ScKcR Y(K, 1B) and

(11) kB = KEI)

CR Y(K, 1B)

4. Dimension of Y(K, X). The Weyl group of a (B, N)-pair is isomorphic to theWeyl group of a root system in Euclidean space (see [3, p. 439]) in such a waythat R corresponds to the set of fundamental reflections. We define 0 ={a;Iw, E R) to be the set of fundamental roots of this root system. If J C R letOJ = {ajw; E J). The following sets were first defined by Solomon [6] forarbitrary Coxeter groups:

366 N. B. TINBERG

DEFINITIONS. For each J C R set XJ = (w E W lw(0 J) > 0}, and VJ =(w E W l w(OJ) > 0, w(01) < 0} where J = R \ J. For each w E W, let q' =B: B n B w I. Clearly XJ = U J C K VK and this is a partition of XJ.

Before determining the dimensions we need the following proposition.Without loss of generality we can assume M(X) = R for X E B by 3.4.

4.1 PROPOSITION. For any J C R, YJ(J, X) has dimension one. MoreoverYJ(o, X) is irreducible for the empty set 0.

We use 3.5(i), 4.1 and the fact that JG: GKI = XwEXx qw to find the dimen-sions using an inductive argument on JKJ:

4.2 THEOREM. Let (K, X) be an admissible G -pair. Then Y(K, X) has dimension1WEV,r q

W

5. Brauer characters. Results of §4 can be applied to the case X = 1B. Let nKbe the Brauer character of Y(K, W. Then by 3.7(i) we have

1GGj = nKJcK

(J C R)

and

5.1 PROPOSITION. n J = YI J C K(- 1)I K ' I 1 GK. Notice that for J = 0, no is irre-ducible by 4.1 and is, of course, the Steinberg character.

6. Vertex of Y(K, X). Because Y is a permutation module and all charactersare E are known (see [4] or [7]) it is possible to calculate a vertex for theindecomposable components of Y using L. L. Scott's work [5]:

6.1 PROPOSITION. Let (K, X) be an admissible G -pair. Then U n UV- is avertex for Y(K, X) where wK, is the unique element of maximal length in K' _M(X) \ K.

REFERENCES

1. C. W. Curtis, Modular representations of finite groups with split (B, N)-pairs, Lecture Notes inMath., vol. 131, Springer-Verlag, Berlin and New York, 1970, pp. BI-B29.

2. J. A. Green, On a theorem of Sawada, Proc. London Math. Soc. 18 (1978), 247-252.3. F. Richen, Modular representations of split (B, N) pairs, Trans. Amer. Math. Soc. 140 (1969),

435-460.4. H. Sawada, A characterization of the modular representations of finite groups with split (B, N)-

pairs, Math. Z. 155 (1977), 29-41.5. L. L. Scott, Modular permutation representations, Trans. Amer. Math. Soc. 175 (1973), 101-121.6. L. Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376-393.7. N. B. Tinberg, Modular representations of finite groups with unsaturated split (B, N)-pairs,

Canad. J. Math. (to appear).8. , Some indecomposable modules of groups with split (B, N)-pairs, J. Algebra 61 (2) (1979),

508-526.

SOUTHERN ILLINOIS UNIVERSITY

Current address: OCCIDENTAL COLLEGE, CALIFORNIA

PART V

Character theory of finite groups


LOCAL REPRESENTATION THEORY

J. L. ALPERIN 1

1. Introduction. This title is our suggestion for the appropriate name for oursubject which is block theory and modular representations. This requires anexplanation and we shall offer one which is convincing to us. Our emphasis herewill be on conjectures, problems and questions; we shall not attempt a completesurvey. In fact, many important and active topics will not be mentioned; wehave made a very personal selection.

First, let's fix some notation. We let G be a finite group andp a prime. Let Rbe a complete discrete valuation ring with residue class field k of characteristic pand field of quotients K of characteristic 0. We assume that K and k are splittingfields for G. This is just the usual set-up for block theory.

There are three different approaches to representation theory: functional;ring-theoretic; module-theoretic. The first deals with matrix representations,characters, modular characters, determinants and functions on groups. It has avery analytic flavor. Idempotents, central idempotents, ideals, relative tracemaps and the like dominate the second approach while modules, projectivemodules, homological methods are used in the third approach. Since we areinterested in representations over K, R and k we now have nine differentsituations to consider: each of the three methods for each of three rings. This iseasily visualized as follows:

Functions Rings Modules

KR

k

The three methods compete and complement each other; no one of them seemsto be sufficient to deal with all the problems. Ideally one would like to start withkG modules and end up with results on values of characters; that is, work from

1980 Mathematics Subject Classification. Primary 20C20.'Supported in part by National Science Foundation Grant MCS-7904469.

o American Mathematical Society 1980

369

370 J. L. ALPERIN

the lower right corner to the upper left corner. However, things can go quitedifferently: there are results on modules which depend on Brauer's characteriza-tion of characters.

The p-local subgroups play a leading role in block theory: each of the maintheorems relates the representation theory of the p-local subgroups with therepresentation theory of G. This is exactly the same type of local to globalapproach that one finds in the study of the structure of groups. Interestingresults on the representations of local subgroups often are motivated by applica-tions of a global nature. Many of the vital questions we have are about theconnection of local and global information.

Thus, our subject is local in an arithmetic sense. We are studying representa-tions p-adically and modulo p when we bring in R and k. And it is also local inthe group-theoretic sense. Hence, it is local representation theory.

2. Characters. Brauer's work on block theory, stretching over decades, stronglysuggests the following problem as a reasonable choice for the main problem ofthe subject.

Problem A. Give rules which determine the values of the characters of G in termsof the p-local subgroups of G.

We shall always mean irreducible character when we speak of a character.The values of the characters at the elements of order divisible by p are the mostaccessible but other values, for example degrees, have many properties of a localnature. As a prime example of the sort of rule desired we have McKay'sconjecture:

CONJECTURE B. If P is a Sylow p-subgroup of G then the number of charactersof G of degree not divisible by p equals the number of characters of NC(P) ofdegree not divisible by p.

This is a remarkable idea; its truth has been verified in many special cases.We keep P as a Sylowp-subgroup henceforth. Recall that P is said to be TI if itintersects each of its conjugates in the identity. This is a situation that arisesoften in studying groups. Here one also can make a good guess.

CONJECTURE C. If P is TI then the number of characters of G not vanishing onthe nonidentity elements of P equals the number of characters of NC(P).

Of course no character of NC(P) vanishes on the nonidentity elements of P sothat statement is symmetric. We shall see in the next section that these last twoconjectures are special cases of a more general one. But now we want to bring inblocks to give a refinement of the McKay conjecture.

There is a canonical partition of the characters of G into subsets called thep-blocks of G. This partition is determined by congruences on the values of thecharacters. We fix a p-block b = (X ... . X,) of G. Moreover, let D be a fixeddefect group of b; it is unique up to conjugacy. The defect group plays a verysignificant role in the structure of the block.

For example, if I D I = p d and I P = p" then each character X; in b has degreeX(l) divisible by p"-d and this is best possible; there are characters in b of

LOCAL REPRESENTATION THEORY 371

degree not divisible by p"-d+i The height h, of the character x; of b is thenonnegative integer such that p"-d+" is the exact power of p dividing x;(1).There is a famous conjecture of Brauer's concerning heights:

CONJECTURE D. The defect group D is abelian if, and only if, every character inb has height zero.

Brauer's First Main Theorem on Blocks also shows the critical role of thedefect group: there is a canonical one-to-one correspondence between the blocksof G with defect group D and the blocks of N = NG(D) with defect group D.Hence, there is a block bN of N corresponding to b. We can now state therefinement of McKay's conjecture.

CONJECTURE E. The number of characters of height zero in b equals the numberof characters of height zero in bN.

McKay's conjecture is indeed a consequence of this one. This follows fromthe fact that the number of characters of degree not divisible by p in G equalsthe sum of the numbers of characters of height zero in all the blocks of G with Pas a defect group.

3. Local methods. Local subgroups are of central importance in representationtheory and in the study of the structure of groups. It has recently been shown [7]that the most basic results in both areas are special cases of general results inrepresentation theory. This opens the door to cross fertilization between bothareas. So far a number of ideas in local group theory have been shown to haverelevant generalizations in representation theory. Before making this precise let'sask a general question.

Question F. Which ideas from local group theory can be usefully generalized torepresentation theory?

The objects of study in the generalization of local group theory are thesubpairs, which are ordered pairs (Q, bQ) where Q is a p-subgroup of G and bQis a block of C(Q). The idea is to treat these as generalized subgroups! Theusual results on subgroups can be obtained by specializing all the blocks to beprincipal blocks (the block containing the principal character). The ideas ofcontainment, normality and conjugacy can be readily extended to subpairs.

One of the first results is that if R is another p-subgroup and R c Q thenthere is a unique block bR of C(R) such that (R, bR) c (Q, bQ) in the sense ofsubpair containment. In particular, if R = 1 then bR is a block of G. In this case,if bR = b, then we say that (Q, bQ) is a b-subpair. (This just means (bQ)G = b inthe usual sense.) The collection of b-subpairs of G is to be regarded as analogousto the collection of all p-subgroups of G.

With this in mind it is possible to generalize Sylow's theorem. First, let usrestate it in the following form: all the maximal p-subgroups of G are conjugate;a p-subgroup is maximal if, and only if, it is maximal in its normalizer. Thegeneralization is as follows: all the maximal b-subpairs of G are conjugate; ab-subpair is maximal if, and only if, it is maximal in its normalizer. Here the

372 J. L. ALPERIN

normalizer of a subpair turns out to be the inertial subgroup for the block of thesubpair and a subpair is also a subpair of that normalizer. The maximalb-subpairs are the subpairs conjugate with one of the form (D, bD) where D is asabove and bD satisfies (bD)N = bN.

The basic results and ideas on fusion generalize to this context. In particular,we can speak of the block b being controlled, that is, N controlling fusion in(D, bD). More precisely, this means that if (P, bP) and (Q, bQ) are subpairscontained in (D, bD) and (P, bp)8 = (Q, bQ) for g E G then there is c E C(P)and n E N with g = cn.

CONJECTURE G. If b is a controlled block then the number of characters in bequals the number of characters in bN.

This may be too bold and additional hypotheses may be required. Thisconjecture has a number of consequences. For example, Conjecture C wouldfollow quickly. Moreover, it is true, in analogy with a classical theorem ofBurnside, that if D is abelian then b is controlled. Hence, Conjectures G and Ewould imply half of Conjecture D by showing that if D is abelian then everycharacter in b has height zero.

As far as solving the other half of Conjecture D and also generalizingConjecture G to arbitrary blocks we just state another very general problem.

Problem H. Show that the number of characters in a block is determined by localinformation.

4. Reduction modulo p. By using the three rings K, R and k it is possible tospeak of reducing a KG-module modulop and obtain a kG-module which, whilenot unique, has well defined composition factors. This enables one to define thedecomposition numbers modulo p of any character of G. There is a very old andelusive problem which is easy to state.

CONJECTURE I. The decomposition numbers of a character x; in b are bounded bya function of ID 1.

This is just one example of a possible result tying together representationsover k and K. Here is another.

Question J. If G is a p-group does the cohomology algebra H'(G, k) determinewhich powers of p are degrees of characters of G ?

The multiplicities of the degrees are not so determined as is shown by cyclicgroups. It may be that further homological information must be assumed. It isalso possible to formulate a more general question for arbitrary G.

5. Modules. Let S . . . , St be simple kG-modules, one of each isomorphismtype. Let P1, ... , P, be indecomposable projective kG-modules so that P, has aunique simple quotient module and this is isomorphic with Si. Thus, the modulesP1, ... , P, are a complete set of representatives of the isomorphism classes ofindecomposable projective kG-modules. Let c. be the multiplicity of Sj as acomposition factor of F. so C = (c,,) is the Cartan matrix of G.


There is a unique direct decomposition kG = A + of the algebra kGinto a direct sum of indecomposable algebras, the block algebras. These corre-spond one-to-one with the blocks of G. Hence, we may assume A is the blockalgebra corresponding with b. The simple kG-modules which are simple asA-modules are said to be in b. The next conjecture is also an old one andequivalent with Conjecture I.

CONJECTURE K. The entries of the Cartan matrix corresponding with simplekG-modules in b are bounded by a function of ID1.

The determinant of the Cartan matrix has a nice module interpretation. LetK0(kG) be the Grothendieck group of projective kG-modules and let G0(kG) bethe Grothendieck group of kG-modules. The cokernel of the natural map ofK0(kG) to G0(kG) is a finite abelian group of order the determinant of theCartan matrix. It is a famous old theorem of Brauer that this determinant is apower of p; the best proof uses the characterization of characters. Thus, a resulton modules follows from a theorem on characters.

Question L. Is there a module-theoretic proof of this result?

Perhaps there is even a module-theoretic counterpart of the characterizationof characters though it hard to imagine one.

Finally, we shall give a conjecture suggested by Peter Donovan whichstrengthens Conjecture K considerably. Recall that two algebras are Moritaequivalent if their module categories are equivalent categories.

CONJECTURE M. Up to Morita equivalence there are only a finite number ofblock algebras with a defect group isomorphic with D.

Of course, if D is cyclic then all questions are answered. A demonstration ofthis conjecture would "prove" that there is a general theory of blocks extendingthe cyclic theory!

6. Experimental work. Our concern here is the study of projective kG-moduleswhere G is a specific group or one of a family of certain specific groups. Othertypes of questions about kG-modules for such groups would also fall under thesection heading but we shall limit ourselves here. The case of projective modulesin b when D is cyclic has been much studied, but the availability of the cyclictheory makes these results of a very special character and we shall not discussthem.

The case of p = 2 and G = A5, the alternating group of degree 5, wasanalyzed by Brauer decades ago and has since been rediscovered a number oftimes. However, recently there has been extensive research beginning with thecase of groups with dihedral Sylow 2-subgroups [1]. The motivation was togeneralize the cyclic theory in the case of principal dihedral blocks and this waslargely successful. This has led to other work using new ideas and now it seemslikely that the case of arbitrary dihedral blocks can be analyzed so that theexperimental work has been useful in getting theoretical results.

The case of abelian Sylow 2-subgroups was initiated with SL(2, 2") [5]. Thiswas continued by Landrock and Michler [11], [12] who came up with a couple of

374 J. L. ALPERIN

very surprising applications: a uniqueness theorem for the character table of thegroups of Ree type and the determination of the mod 2 decomposition numbers.Usually characters and decomposition numbers are determined long before onecan analyze projectives; Landrock and Michler showed one could go the otherway!

There are many applications of the experimental work. Examples have beenconstructed of interest in the theory of algebras (e.g. [6]). A number of interest-ing classes of modules have been uncovered, for example, periodic modules [4],algebraic modules [3] and simply generated modules [2].

There are many interesting and important directions for further research butwe shall just mention one: the case of G a group of Lie type and characteristicp.The evidence suggests results involving interesting combinatorics. The first caseto complete our knowledge is obviously the following one.

Problem N. Let G = SL(2, p"), describe the projective kG-modules and calcu-late all ExtkG(U, V) for all simple kG-modules U and V.

The last part is equivalent to describing the terms of certain minimal projec-tive resolutions which we shall discuss in the next section.

7. Resolutions. Let us recall the idea before discussing its relevance andimportance in local representation theory. An exact sequence

...where U is a kG-module and each P; is a projective kG-module, is a projectiveresolution of U. There are many such resolutions of U but there is one that iscanonical, the minimal projective resolution of U. It is characterized by theproperty that any other projective resolution of U is isomorphic with a directsum of the minimal one and a projective resolution of the zero kG-module.

The relevance of this idea in local representation theory first appeared in thecontext of blocks with a cyclic defect group [9]. It was shown how the notion ofthe Brauer tree was equivalent with certain minimal projective resolutions. Thistied together the representation theory with the results on periodicity ofcohomology. Green [10] and Peacock [13]-[15] have used resolutions as thestarting point in a striking module-theoretic reworking of the whole cyclictheory. In the case of principal dihedral blocks some striking double complexeshave been used to construct minimal resolutions [1]. These results and the keyrole resolutions play in the calculation of cohomology strongly suggest thatresolutions are very important for local representation theory.

A very useful invariant of a module, the complexity [4] is defined in terms ofthe rate of growth of the dimensions of the terms of resolutions. Indeed, let

.... p" _ ... -*P1-*PO -* U-*0be a minimal projective resolution of U. The complexity cG(U) is the leastnonnegative integer c such that there is a real positive number r with

dimk P" < rnc-1

for all sufficiently large n. It is quite easy to see that this is defined and is of alocal nature: cG(U) = cP(UP) where UP is the restriction of U to P. However,


much more is true [8]: cG(U) is the maximum of the complexities cE(UE) as Eruns over the elementary abelian p-subgroups of G. This result has a number ofcorollaries including Quillen's theorem on the Krull dimension of k)which was the original motivation. The result on complexity is the appropriategeneralization to arbitrary modules of Quillen's result on the trivial module k.There are a number of problems for further research in this area [8] but we wishto conclude with one of a very general sort.

We have seen in the third section how the basic results of local group theoryand of local representation theory can be synthesized into a single theory. Theprevious discussion shows some of the results dealing with the role of abeliansubgroups in representation theory. On the other hand, abelian subgroups arecentral in the local study of simple groups. Is this just an accident that theabelian groups are so important in two different areas? We think not.

Problem O. Give a synthesis for the theory of abelian subgroups in representationtheory and the theory of abelian subgroups in local group theory.

REFERENCES

1. J. L. Alperin, Minimal resolutions, Finite Groups '72, T. Gagen, M. Hale and E. Shult (eds.),North-Holland, Amsterdam, 1973, pp. 1-2.

2. , Projective modules and tensor products, J. Pure Appl. Algebra 8 (1976), 235-241.3. , On modules for the linear fractional groups, Finite Groups, Sapporo and Kyoto, 1974,

Nagayoshi Iwahori (ed.), Japan Soc. Promotion of Science, 1976, pp. 157-164.4. , Periodicity in groups, Illinois J. Math. 21 (1977), 776-783.5. , Projective modules for SL(2, 2"), J. Pure Appl. Algebra 15 (1979), 219-234.6. , A preprojective module which is not strongly preprojective, Comm. Algebra (to appear).7. J. L. Alperin and Michel Broue, Local methods in block theory, Ann. of Math. (2) 110 (1979),

143-157.8. J. L. Alperin and L. Evens, Representations, resolutions and Quillen's dimension theorem, J. Pure

Appl. Algebra (to appear).9. J. L. Alperin and G. Janusz, Resolutions and periodicity, Proc. Amer. Math. Soc. 37 (1973),

403-406.10. J. A. Green, Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1974), 197-213.11. Peter Landrock and Gerhard O. Michler, Principal 2-blocks of the simple groups of Ree type,

Trans. Amer. Math. Soc. 260 (1980), 83-111.12. ,

13. R. M. Peacock, Blocks with a cyclic defect group, J. Algebra 34 (1975), 232-259.14. , Indecomposables in a block with a cyclic defect group, J. Algebra 37 (1975), 74-103.15. , Ordinary character theory in a block with a cyclic defect group, J. Algebra 44 (1977),

203-220.



CHARACTERS OF SOLVABLE GROUPS

I. M. ISAACS

This is an expository lecture about some recent developments in the charactertheory of those groups which have an abundance of normal subgroups. Atten-tion will be directed primarily but not exclusively to solvable groups, althoughmuch of the theory goes through somewhat more generally.

I used the word "theory" in the preceding advisedly, since the last decade orso has seen the development of several general approaches which have been(and are being) applied to a variety of problems. I will discuss six distinct (butnot disjoint) areas of research. Although there are a number of important ideasnot contained among them, they will provide a convenient outline for thediscussion. (And besides, these are the areas I know best and in which I am mostinterested.)

The six topics are(A) Character correspondence.(B) Brauer characters.(C) or-special characters.(D) M-groups.(E) Character degrees.(F) Linear groups.

A. Character correspondence. This subject (and its name) originated inGlauberman [14]. Let H act on G with (IHI, JGJ) = 1. Write C = CG(H) andIrrH(G) = {X E Irr(G)IX' = X for all h E H). Glauberman proved that if H issolvable, then Irr(C)J = JIrrH(G)J. Actually, he showed much more: there is aninvariantly defined bijection ir(G, H): IrrH(G) -* Irr(C) which respects thenormal structure of H. When H is a p-group, ir(G, H) is easy to describe. Itcarries X E IrrH(G) to the unique E Irr(C) such that [Xc, ] 0 mod p. (Acomplete exposition of Glauberman's theorem can be found in my book, Isaacs[22].)

What has this to do with characters of solvable groups? Suppose we drop thecondition that H is solvable in Glauberman's theorem. Does anything still work?In this case, JH J must be even and so G1 is odd and G is solvable.

1980 Mathematics Subject Classification. Primary 20015.m American Mathematical Society 1980

377

378 I. M. ISAACS

Glauberman's method is hopeless here, but because G is solvable, a great deal ofinformation about the characters of G and its subgroups is available and this isenough to obtain results of the desired type.

The first progress in this direction was the unpublished work of Dade [5]where a general "Clifford theory" was introduced and a proof of the "otherhalf" of Glauberman's Theorem was sketched. Dade's work was quite abstractand general and it did not, it seemed to me, provide an explicit, charactertheoretic construction of a bijection IrrH(G) - Irr(C) for solvable G andarbitrary H. Using some of Dade's ideas, I was able to give such a constructionwhen I G I is odd. This appears in Isaacs [19].

A rough sketch of the method is as follows. Let K = [G, H] and let K/L be achief factor of G >1 H. Then U = LC < G and it suffices to construct abijection IrrH(G) - IrrH(U). Let q E IrrH(L). The object is to construct thebijection so that characters of G which lie over q map to characters of U whichlie over p. If q is not invariant in G, then by applying an appropriate inductivehypothesis to the inertia group IG((p) < G, the result follows. We may thusassume that q is invariant. It is then elementary to show that there are only twopossibilities:

(a) q extends to 0 E IrrH(K),(b) q" = e9 where 0 E Irr(K) and eL = e(p.

G

In situation (a) it is not hard to show that restriction defines the desiredbijection IrrH(GJ(p)-IrrH(UI(p). It is in situation (b) that most of the interestlies. In this "fully ramified" case we have e2 = 1K : Ll and the "vectorspace"K/L carries a symplectic form which is uniquely determined by p and thus isinvariant under G >c H. It is the analysis of this situation which is at the heartof Isaacs [19] (and ultimately of Dade's work also). A small part of this analysisappeared earlier, in Isaacs [17].

If JK : Ll is odd, then the theory works very well, and there is an invariantlydefined map Irr(G I (p) - Irr(U I (p) which is not only a bijection which respectsthe action of H, but which also has a number of other useful properties. If inaddition I G : KI is odd, then this map is exceptionally easy to describe: Xiff [Xu, ] is odd.

It was proved by Wolf [37] that if IGI is odd, then the net map IrrH(G) -Irr(C) obtained in this manner is independent of the choice of L and even morestrikingly: if in addition H is solvable, then this map is identical toGlauberman's.

What if IK : Ll is even in the above, fully ramified situation? Another methodof proof is available which is not sensitive to the prime involved in IK : L1. Thisalso gives a bijection with some desirable properties (for instance, if X - , thenX(1) = but not one which can, in general, be invariantly defined. This is

CHARACTERS OF SOLVABLE GROUPS 379

essentially in Dade [5] and explicitly in Dade [7]. An exposition and simplifica-tion can be found in Isaacs [24].

The technique of analysis of the characters of a solvable group which I havejust outlined, namely consideration of chief factors, with special attention paidto fully ramified ones, lends itself to a number of other problems, independentof Glauberman's theorem. I will mention a few of these.

McKay [27] conjectures that if G is simple and S E Sy12(G), then

{X E Irr(G)121 x(l)}I = I(J E Irr(Nc(S))121 f(1))1.

There seems to be no particular reason to restrict attention to simple groups andto the prime 2. In fact no counterexample is known for any prime or any group.In Isaacs [19], the odd order (solvable) case of the McKay conjecture wasproved. Using the even version of the theory in Dade [7], Wolf [38] proved theconjecture for all solvable groups. (This also appears in my expository paperIsaacs [24].)

Recently, Dade has proved the p-solvable case of the McKay conjecture, buthe informs me that this required the introduction of a great deal of new theorywith which I am not familiar. This result was independently obtained by Gres,and recently an amazingly short proof was found by Okuyama and Wajima [28].

Another variation on the McKay conjecture which was suggested by Alperin,is to replace the Sylow subgroup by the defect group D of a p-block and to limitattention to characters of G and of NG(D) which have height zero in corre-sponding blocks. In fact, the Okuyama and Wajima paper proves Alperin'sversion of the conjecture forp-solvable groups.

An application of the better behaved odd case of the theory of characters andfully ramified sections is that if X E Irr(G) is quasi-primitive where G is solvableand X(1) is odd, then G is primitive. This is a special case of a theorem of Berger[1] which does not require any condition on X(1) and does not use the generaltheory discussed here. In fact, the property of odd sections I used to prove thespecial case of Berger's theorem simply does not hold for even sections. I haverecently used this property of odd sections to obtain some new results aboutM-groups, which I will discuss later.

I will close this section with a pretty corollary of Glauberman's theorem andits "other half". If H acts on G and (IHI, JG`) = 1, then the permutation actionsof H on Irr(G) and on the set of classes of G are isomorphic.

B. Brauer characters. Let G be p-solvable. The Fong-Swan theorem (Swan[31]) asserts that if g E IBr(G), the set of irreducible Brauer characters of G forthe prime p, then there exists X E Irr(G) with X° = T. (We use the notation X°to denote the restriction of X to the p-regular elements of G.) One consequenceof this is that the set lBr(G) can be invariantly constructed from Irr(G), withouthaving to specify a maximal ideal and without reference to modular representa-tions. (I claim this as an excuse for including Brauer characters in this talkwhich is supposedly only concerned with ordinary characters.)

Among all X for which X° = q in the Fong-Swan theorem, is there one whichis canonical, one which can be chosen in some natural manner? If p = 2, Iproved (Isaacs [20]) that for ¢ E IBr(G), there exists a unique X E Irr(G) with

380 I. M. ISAACS

X° = q such that X is p-rational. (I.e., the values of X involve only p'-roots ofunity.) Furthermore, if N <G, then each irreducible constituent of XN inheritsthe property that it is p-rational and modularly irreducible. It seems to me that Xis a good candidate for the "canonical" lift of q.

When p = 2,p-rational lifts to Brauer characters exist but they are not uniqueand do not necessarily respect normal subgroups. In Isaacs [23] I found anothermethod for the construction of "canonical" lifts. This works for all primes and(fortunately) gives the same characters as the older method when p * 2. A seti (G) C Irr(G) is constructed for each p-solvable G. The map X - X° defines a

bijection from i (G) onto IBr(G) and i respects normal subgroups.I would like to sketch the construction of . Say that X E Irr(G) is p'-special

if p } X(1) and also p does not divide the determinantal order of any irreducibleconstituent of Xs for any S << G. It turns out that ° defines an injection fromthe set of p'-special characters into IBr(G). The image of this map is the set ofall q withp } (P(1).

For arbitrary E IBr(G), one can construct a subgroup H (unique up toconjugacy if certain properties are specified) and a Brauer character µ E IBr(H)such that µ c = p and p } µ(1). Let i E Irr(H) be p'-special with ii ° = µ andwrite X = pc. Then X° = p and we take 6 (G) to be the set of X E Irr(G) thusobtained.

The characters in (G) relate to other theorems about Brauer characters ofp-solvable groups. Fong [10] showed if p E IBr(G) and K is a Hall p'-subgroupof G, then there exists 0 E Irr(K) such that 9G = 4) ,, the projective indecom-posable character associated with q. Furthermore, 9(1) = q(1)p. No uniquenessresult is known about these "Fong characters" 0 and no useful characterizationof them among Irr(K) seems to be available.

The connection with i (G) is this. Given q E IBr(G), obtain the associatedsubgroup H as in the construction of i (G). We may assume that H n K is ap-complement in H. Let ifi E Irr(H) be p'-special with (11/c)° _ (p. Then

n K)K = 0 E Irr(K) and 0 is a "Fong character" for (p, i.e. B G = 4)T. Thissituation has not yet been fully explored.

Much of the manipulation involved in proving the results stated here does notdepend on modular representations. To a considerable extent it suffices to workwith ordinary characters. Also, much would work if the prime p were replacedby a set of primes i. For instance, one could define by looking at therestrictions of ordinary characters to it-regular elements. It seems likely thatmuch of the theory which I have described would continue to hold for arbitrarysets 7T in it-solvable (or even it-separable) groups.

C. ir-special characters. This topic links the first two and connects into otherareas also. It grew out of the Ph.D. thesis of Dilip Gajendragadkar, a recentstudent of Dade. Using techniques related to those discussed in part A of thislecture, Dade [8] constructed an injection Irr(N) -* Irr(G) where N is a systemnormalizer in the solvable group G. The problem posed by Dade to his studentwas to characterize the image of this map.

Gajendragadkar [11], [12] solved this problem very neatly. He generalized theclass of characters which I called p'-special in part B. For the it-separable group


G, let us say that X E Irr(G) is 'r-special if X(1) is a a-number and thedeterminantal orders of all irreducible constituents of Xs for all S 44 G arealso -7r-numbers. Gajendragadkar observed that essentially all of the properties Iproved forp'-special characters hold in the more general setting. (For instance, ifX is r-special, then its values are -rational, i.e. they involve only i-roots ofunity.) He also discovered what to me seems a remarkable result, though onceyou know it is true, it is not very hard to prove. He showed that if and -l are7r-special and 7r'-special respectively, then X = 1 is irreducible. Also, thisfactorization of X into ;r-special and -special characters is unique. Gajendra-gadkar showed that the image of Dade's map Irr(N) -* Irr(G) is the set of thoseX E Irr(G) which can be factored into p-special characters (where p runs overthe primes dividing I GI).

Suppose G is solvable. For every X E Irr(G), there is a unique minimal set ofprimes 7r such that X is vr-special. We write yr = 7r(X). Then if %, rl E Irr(G) withir(ia) n 7r(q) = 0, we have E Irr(G) and ir(k) = U vr(i). I believe thatconsideration of the set 7r(X) will prove to be a very useful tool for the analysisof the characters of solvable groups. What I have in mind is the possibility ofusing induction on i7r(X)i in order to study the character X.

In order to make such an induction work, one needs a sufficient condition forX E Irr(G) to factor into vr-special and ar'-special parts. I showed that if thereexists any normal series S : I = Go c Gl c c Gk - G with all factorsbeing 7r-groups or ir'-groups and such that the restriction of X to each G; ishomogeneous, then X can be factored, and each of the (unique) factors is alsohomogeneous on S. In particular, this works if X is primitive.

It follows from this factorization lemma that if X E lrr(G) is primitive for asolvable group G, then X factors into primitive p-special characters, one for eachprime. Conversely, every such product of primitive p-special characters willagain be primitive. I hope that this will contribute to the understanding ofprimitive characters and of those with the opposite property, monomial char-acters.

D. M-groups. An M-group is one in which every irreducible character ismonomial, i.e. is induced from a linear character of a subgroup. These groupsare necessarily solvable (Taketa [32]) and form a class of groups properlybetween supersolvable groups and all solvable groups. (This class is not, how-ever, a formation.) Numbers of sufficient conditions for a group to be anM-group are known (see my book and Seitz [30] for some of them) but no purelygroup theoretic characterization has been found. An indication of the difficultyof this problem is the observation of Dade [4] that every solvable group is asubgroup of an M-group.

In fact Dade's construction builds an overgroup involving no primes otherthan those already dividing the order of the original group. This suggests thequestion of whether a non-M-group can ever be embedded as a Hall subgroupof an M-group. The only evidence I know which bears on this is the result ofDornhoff [9] that a normal Hall subgroup of an M-group is again an M-group.

What about arbitrary normal subgroups of M-groups? This question was openfor quite a while and was finally answered in the negative by Dade [6] who

382 I. M. ISAACS

found an M-group of order 29. 7 with a subgroup of index 2 which is not anM-group. (Similar examples were found independently by van der Waall [33].)These examples depend on the prime 2 in a fundamental way. No odd analog ispossible and this leaves the question of whether or not normal subgroups of oddM-groups are necessarily M-groups.

(Chubarov [3] claims to have settled this affirmatively. The validity of hiswork is questionable, however, since he seems to obtain his theorem as aconsequence of a more general assertion which is, in fact, false.)

Some evidence in the positive direction was given by van der Waall [33] and Ihave just found some stronger evidence. My result is that if G is an M-groupand N aG and 0 is a primitive character of N, then 9(1) is a power of 2 and if9(1) > 1, then IG : NI is even. (Of course, in an M-group all primitive charactersmust have degree 1. My result shows that this is true about normal subgroups ofM-groups if either the order or the index of the subgroup is odd.)

It is interesting how closely tied this last result I mentioned is with the othertopics in this lecture. The main idea of the proof is to use the facts about oddchief sections which I mentioned in part A. A key reduction in the proof,however, is that 0 may be assumed to be q-special for some prime q. Thisdepends on character factorization as was discussed in part C. I believe that thissupports my contention that a true "theory" of characters of solvable groups isnow developing.

Price [29] raised the question of characterizing all "minimal non-M groups".These are groups such that every proper subgroup and homomorphic image isan M-group but which are not themselves M-groups. Price made a significantstart on this problem which has recently been completed by van der Waall [34].

E. Character degrees. In Isaacs and Passman [25] we began to study thequestion of what one can say about a group G if the set c.d.(G) of degrees of itsirreducible characters is known. I will mention just two aspects of this questionhere. In that early work with Passman, we found a function f(n) such that forany group G, if X(1) < n for all X E Irr(G), then there exists an abeliansubgroup A C G with I G : A I < f(n). (Actually, with appropriate interpretation,this result holds even for infinite groups.) For instance, the function f(n) = (n!)2will work. If G is known to be solvable, one can do much better and takef(n) = V'9")' for some fixed constant k. I would like to know if a polynomialbound exists for solvable groups. The only evidence I have that this might betrue is quite scant: for abelian by nilpotent groups, f(n) = n4 works and forsolvable groups with all Sylow subgroups abelian, f(n) = n2 works. (Theseresults can be found in my book.)

If G is an M-group, it was observed in Seitz [3q] that its derived length d.l.(G)is bounded above by Ic.d.(G)L. Seitz posed the question of whether or not this istrue for all solvable groups. I proved this if Ic.d.(G)l < 3. (In this case, one neednot assume G is solvable, it is a consequence of the hypothesis. See Isaacs [16].)My student Garrison [13] proved the result when Ic.d.(G)I = 4. In Isaacs [21] Ishowed that d.l.(G) < 31c.d.(G)l for all solvable groups and Berger [2] provedthat d.l.(G) < Ic.d.(G)l if I G I is odd. The full answer to Seitz's question is stillunknown. Also, I am not aware of any examples which show that d.l.(G) can be

anywhere near as large as Ic.d.(G)l if c.d.(G)J is large. A reasonable questionmight be for p-groups, say, can we ever have d.l.(G) > Ic.d.(G)I1/2 + 5, forinstance?

F. Linear groups. I have a special feeling for this subject since it was the topicof my thesis, though I have not worked in the area for some years now. I willdiscuss it only very briefly. Ito [26] showed that if G is solvable and has a faithfulcharacter of degree n < p for some prime p, then G must have a normal abelianSylowp-subgroup, except possibly if n = p - 1 is a power of 2 (in which case, ofcourse, p must be a Fermat prime).

Winter [35] imporved on Ito's result by showing that if G is solvable and doesnot have a normal abelian Sylow p-subgroup, then any faithful irreduciblecharacter of G must have degree divisible either by p or by some prime power

± 1 mod p. The principal question that I see in this area is to what extent isWinter's theorem true if G is onlyp-solvable rather than solvable?

Ito pointed out that the proof of his theorem goes through without essentialchange if G is only assumed to bep-solvable. In my thesis, Isaacs [15], I provedwhat amounts to the p-solvable case of Winter's theorem for faithful charactersof degree < 2p - 2. This result appears in improved form in Isaacs [18] where itis proved for faithful characters of degree < 2p. (Parts of this were obtainedindependently by Winter.) Surprisingly, it turns out that a p-solvable, non -p-closed group with a faithful character of degree n < 2p - 1 with n 2p - 2 isnecessarily solvable. Winter [36] took this one step further and did the casen = 2p + 1, where it also turns out that G must be solvable.

I will close by bringing us back full circle. A key step in the consideration ofthe foregoing problem, both in Winter's work and my own, is to reduce to thecase where G has a normal p-complement K. One is thus in the situation where ap-group P acts on ap'-group K and fixes a certain faithful irreducible character.This is the situation of Glauberman's theorem discussed in part A, and onethereby obtains useful information about CK(P). This technique was essential inmy thesis, where I found and used a primitive version of what later becameGlauberman's theorem. Unfortunately, I did not see the opportunities forfurther development of this idea which were visible to Glauberman when hecame upon it independently and at about the same time.

REFERENCES

1. T. R. Berger, Primitive solvable groups, J. Algebra 33 (1975), 9-21.2. , Characters and derived length in groups of odd order, J. Algebra 39 (1976), 199-207.3. 1. A. Chubarov, Monomial characters and subgroups, Russian Math. Surveys 33 (1978),

143-144.4. E. C. Dade, see page 585 of B. Huppert, Endliche Gruppen. I, Springer, Berlin, 1967.5. , Characters of solvable groups, mimeographed notes, University of Illinois, Urbana, Ill.,

1967.

6. , Normal subgroups of M-groups need not be M-groups, Math. Z. 133 (1973), 313-317.7. , Characters of groups with normal extra-special subgroups, Math. Z. 152 (1976), 1-31.8. , Caracteres venant des S-normalisateurs d'un group fini resoluble (preprint).9. L. Dornhoff, M-groups and 2-groups, Math. Z. 100 (1967), 226-256.

384 I. M. ISAACS

10. P. Fong, Solvable groups and modular representation theory, Trans. Amer. Math. Soc. 103(1962), 484-494.

11. D. Gajendragadkar, A characteristic class of characters of finite sr-separable groups, J. Algebra59 (1979), 237-259.

12. , A characterization of characters which arise from S-normalizers (to appear).13. S. C. Garrison, On groups with a small number of character degrees, Ph.D. Thesis, University of

Wisconsin, Madison, 1973.14. G. Glauberman, Correspondences of characters for relatively prime operator groups, Canad. J.

Math. 20 (1968), 1465-1488.15. I. M. Isaacs, Finite p-solvable linear groups, Ph.D. Thesis, Harvard University, 1964.16. , Groups having at most three irreducible character degrees, Proc. Amer. Math. Soc. 21

(1969), 185-188.17. , Extensions of group representations over nonalgebraically closed fields, Trans. Amer.

Math. Soc. 141 (1969), 211-228.18. , Complex p-solvable linear groups, J. Algebra 24 (1973), 513-530.19. , Characters of solvable and symplectic groups, Amer. J. Math. 95 (1973), 594-635.20. , Lifting Brauer characters ofp-solvable groups, Pacific J. Math. 53 (1974), 171-188.21. , Character degrees and derived length of a solvable group, Canad. J. Math. 27 (1975),

146-151.22. , Character theory of finite groups, Academic Press, New York, 1976.23. , Lifting Brauer characters of p-solvable groups. II, J. Algebra 51 (1978), 476-490.24. , Character correspondences in solvable groups, Advances in Math. (to appear).25. I. M. Isaacs and D. S. Passman, Groups with representations of bounded degree, Canad. J. Math.

16 (1964), 299-309.26. N. Ito, On a theorem of H. F. Blichfeldt, Nagoya Math. J. 5 (1953), 75-77.27. J. McKay, Irreducible representations of odd degree, J. Algebra 20 (1972), 416-418.28. T. Okuyama and M. Wajima, Character correspondence and p-blocks of p-solvable groups,

Osaka J. Math. (to appear).29. D. T. Price, Character ramification and M-groups, Math. Z. 130 (1973), 325-337.30. G. Seitz, M-groups and the supersolvable residual, Math. Z. 110 (1969), 101-122.31. R. G. Swan, The Grothendieck ring of a finite group, Topology 2 (1963), 85-110.32. K. Taketa, Uber die gruppen deren Darstellungen sick stimtlich auf monomiale Gestalt transfor-

mieren lassen, Proc. Japan Imp. Acad. 6 (1930), 31-33.33. R. W. van der Waall, On the embedding of minimal non-M-groups, Indag. Math. 36 (1974),

157-167.34. , On the structure of the minimal non-M-groups, Indag. Math. 81 (1978), 398-405.35. D. L. Winter, Finite solvable linear groups, Illinois J. Math. 15 (1971), 425-428.36. , On the structure of certain p-solvable linear groups. II, J. Algebra 33 (1975), 170-190.37. T. R. Wolf, Character correspondences in solvable groups, Illinois J. Math. 22 (1978), 327-340.38. , Characters of p'-degree in solvable groups, Pacific J. Math. 74 (1978), 267-271.

UNIVERSITY OF WISCONSIN-MADISON


LOCAL BLOCK THEORY IN p-SOLVABLEGROUPS

LLUIS PUIG

In [11 J. Alperin and M. Broue introduce a reasonable local structure in ap-block b of a finite group G. I am going to talk about how such a structurelooks when the group G isp-solvable.

To work with blocks of p-solvable groups, we have the old theory of Fong.Unfortunately this theory speaks about characters and we are interested inalgebras. But it turns out that it is possible to do the same work with algebras,even in a slightly more general way. So, I will start by talking about thisgeneralization, then employ it in my 'subject and, finally, I will describe theso-called nilpotent blocks in p-solvable groups.

1. On Fong's theory. Let p be a prime, G a finite group, F an algebraicallyclosed field of characteristic p and 0 a valuation ring of characteristic 0 suchthat F = l") /J((9). Let us call a block algebra of G any indecomposable directfactor A of t)G. To study a block algebra A of G, Fong's theory is particularlyefficient when G has a nontrivial normal p'-subgroup K: the image of 1 K in A isa direct product of matrix algebras over e so that G acts transitively on the setof factors and the problem decomposes into

(1) reduction to the stabilizer of one factor,(2) study of A when the image of 0 K is just a matrix algebra.

In both cases Fong constructs a bijection between the set of ordinary irreduciblecharacters in the block A of G and the set of ordinary irreducible characters insome block of a suitable group, in some sense smaller than G.

Let us restate the problem in a slightly more general way.Let A be an 19-order and f: G --* A' a group homomorphism. We have to

study the following two cases:Case 1. There exists a finite G-stable set E of pairwise orthogonal idernpotents in

A such that 1 = EeEE e and G acts transitively on E.Case 2. There exists a G-stable subalgebra D of A such that D ag M.(6) and

(p. n) = I.

1980 Mathematics Subject Classification. Primary 20D10.® Amencan Mathematical Society 1980

385

386 LLUIS PU1G

Clearly, Cases I and 2 may arise from some normal p'-subgroup K of G, butactually we do not need the existence of K. To study Case 1, I need thefollowing more or less well-known construction:

Construction 1. Let H be a subgroup of G, B an 6-order and g: H ---* B' agroup homomorphism. We denote by IndH(B) the double OG-module OG ®0H B®EH O G endowed with the product

(x®a(9 y)(x'®a'(9 y')= f 0 if yx' E H,it x ®ag(yx')a' ®y' ifyx' E H,

and with the group homomorphism

IndH(g): G -* IndH(B),

xH I xy®1®yyEG/H

That is, from the algebra structure point of view, Ind ,(B) is nothing but thematrix algebra MSG , HO(B). Moreover, notice that if B = Endo(N), where N is anO -torsion-free OH-module, then IndH(B) = Endo(IndH(N)).

PROPOSITION 1. Assume that Case I occurs. Let us choose e E E and setH = Ge (the stabilizer of e in G), B = eAe and g: H -* B' defined by g(x) = xe.Then A = IndH(B) and the diagram

A = IndH(B)

f 'S. 7' Indy(g)G

commutes. Moreover,(a) if f(G) generates A, then g(H) generates B;(b) if A is a block algebra of G, then B is a block algebra of H.

Notice that assertion (b) not only implies Fong's results about ordinaryirreducible characters but provides an equivalence between the O G-modules inthe block A and the OH-modules in the block B.

SKETCH OF A PROOF OF ASSERTION (b). As A = indH(B), B is a primaryalgebra and by (a) there is some block algebra B' of H which covers B. Then,setting A' = IndH(B'), A' covers A; but the homomorphism G -* A' forces theexistence of an algebra homomorphism A -* A'. A careful study of whathappens to e via these homomorphisms gives us the equality O -rank(B) _O -rank(B').

PROPOSITION 2. Assume that Case 2 occurs. Let us set B = CA(D). ThenA = D ®E B and there exist a finite central p'-extension G of G and two grouphomomorphisms r: G -* D' and g: G -* B' such that f(x) = r(.) ® g(. ), where. E G and x is the image of . in G. Moreover,

(a) if f(G) generates A, then g(G) generates B and r(G) generates D.(b) if A is a block algebra of G, then B is a block algebra of G.

As before in Case 1, assertion (b) provides an equivalence between theC G-modules in the block A and the OH-modules in the block B. Notice that,from the algebra structure point of view, A is still a matrix algebra over B.

LOCAL BLOCK THEORY IN p-SOLVABLE GROUPS 387

SKETCH OF PROOF OF ASSERTION (b). It is quite similar to the proof in Case 1,except that here there is no idempotent around. To produce a "good" idempo-tent, I employ the following "trick": let us denote by D° the opposite algebra;then, we have D° (&e D s Endo(D); but D - where (p, n) = 1, and soD = Z(D) ® Ker(tr); as G stabilizes this decomposition of D, we get in D°®E D a "good" idempotent fixed by G.

2. Local structure. Let us recall the definition of the Alperin-Broue subpairs. Iam taking here a slightly more general point of view. As before, A is an 0-orderand f: G -* A' is a group homomorphism. For any p-subgroup P of G, let us set

A(P) = AP/ Y, AQ + J(0)APQcP

and denote by BrP: A P A(P) the canonical map. In particular, f and Brpinduce a map fP: CG(P) -* A(P)'. Thus A(P) is an F-algebra, and whenA = 0 G, this is just a complicated way to define the group algebra of CG(P)over F: if A = 0 G, then A(P) a FCG(P). In some sense, A(P) can be viewed asa "residual algebra".

Now, when A is a block algebra of G, the (A, G)-subpairs are the pairs (P, e),where

(a) P is a p-subgroup of G such that A(P) m# (0),(b) e is a primitive idempotent of ZA(P).

There is a reasonable inclusion between (A, G)-subpairs and you can find thedefinition in Alperin's lecture, these PROCEEDINGS. Here I am interested in therelationship between (A, G)-subpairs and (B, H)-subpairs in Case 1, and be-tween (A, G)-subpairs and (B, G)-subpairs in Case 2. The answers are simpleand quite reasonable.

THEOREM 1. Assume that Case 1 occurs and that A is a block algebra of G.Then, there exists an injective map (P, e) H (P, s(e)) from the set of (B, H)-sub-pairs to the set of (A, G)-subpairs such that

(a) A(P)s(e) =(b) NG(P, s(e)) = CG(P) N,(P, e),(c) (Q, f) C (P, e) if and only if (Q, s(f)) C (P, s(e)),(d) (Q, f) and (P, e) are conjugate in H if and only if (Q, s(f)) and (P, s(e)) are

conjugate in G,(e) for any (A, G)-subpair (P', e') there exist a (B, H)-subpair (P, e) and

x E G such that (P, s(e)) = (P', e')x.

Actually, (b), (c), (d), (e) tell us in short that the categories of (A, G)-subpairsand (B, H)-subpairs are equivalent.

THEOREM 2. Assume that Case 2 occurs and that A is a block algebra of G. Thenthere exists a bijective map (P, e) H (P, t(e)) from the set of (B, G)-subpairs tothe set of (A, G)-subpairs such that

(a) is a simple F-algebra and A(P)t(e) = D(P) ®F B(P)e,(b) (Q, f) C (P, e)" if and only if (Q, i(f)) C (P, t(e))z, where z c G and x is

the image of z in G.

388 LLU1S PU1G

Now I am able to study the local structure in p-solvable groups. Let us assumethat G is p-solvable and that A is a block algebra of G. By Theorems I and 2, inorder to study the structure of the (A, G)-subpairs we may assume that Cases Iand 2 occur only in a trivial way; in particular, this means that the image of(SOP.(G) in A has to be of (9-rank one. As we may assume that G is faithfullyembedded in A, actually we may assume that Op,(G) c Z(G) and so we have

CG(OP(G)) = Z(G).Z(OP(G)).

In this case, (A, G)-subpairs mean nothing more than p-subgroups as thefollowing proposition shows:

PROPOSITION 3. Assume that A is a block algebra of G and that

CG(OP(G)) = Z(G).Z(OO(G))

Then, for any p-subgroup P of G, A (P) is a nontrivial indecomposable F-algebra,namely, I is a primitive idempotent of ZA(P).

3. Nilpotent blocks. Let us say that a block algebra A of G is nilpotent if, forany (A, G)-subpair (P, e), the quotient NG(P, e)/CA(P) is a p-group: from thelocal point of view, this seems to be the most simple structure. When the defectgroup is abelian, this is equivalent to saying that the inertial index is one, a casealready studied by Brauer. Actually, it is possible to study the irreduciblecharacters in such a block (and so Z(A)) without any hypothesis on the defectgroup: this is done in [3]. But whenever G is p-solvable, Theorems I and 2 andProposition 3 provide a direct and efficient way to study the structure of anilpotent block algebra A.

COROLLARY. Let us assume that G is p-solvable and that A is a block algebra ofG. Let P be a defect group of A in G. The following assertions are equivalent.

(a) A is a nilpotent block algebra of G.(b) There exist a P-stable p'-subgroup K of G and a P-stable block algebra D of

K such that A = indK P(D[P]) and the diagram

A = indK.P(D[P])IndK.P(g)

G

commutes, where D[P] is the twisted group algebra of P over D and g: K.P -*D[P] is the canonical map.

Notice that it is possible to construct an embedding of P in D[P] whichcentralizes D and so, actually we have

D[P] = D ®E OP.

Thus, from the algebra structure point of view, A is a matrix algebra over OP.

REFERENCES

1. J. Alperin and M. Broue, Local methods in block theory, Ann. of Math. 110 (1979), 143-157.2. M. Broue and L. Puig, Local structure in G-algebras, J. Algebra (to appear).3. M. Broue and L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117-128.

UNIVEASITE PARIS VII

CHARACTERS OF FINITE 7r-SEPARABLEGROUPS

DILIP GAJENDRAGADKAR

Our aim here is to present a special class of characters of finite 17-separablegroups which has some interesting properties. We also discuss the connectionbetween these characters and the Y-normalizers of finite solvable groups.

All the groups considered here are finite and all characters are complex.The proofs of the results quoted below can be found in [Gl] or [G2].

1. Definitions and basic properties. Let 17 be a set of primes. Let 17' denote theset of all primes that do not belong to 17. We call a positive integer a 17-number ifall of its prime divisors belong to 17.

A finite group G is said to be ir-separable if there is a chain G = Go D G1D D G = (1) such that G,/ Gi+1 is either a ¶-group or a i7'-group forevery i, i = 0, 1, . . . , n.

If X is a character of G, then the linear character det(X) denotes the determi-nant of any representation affording X and o(det(X)) is the order of det(X) in thegroup of linear characters of G.

DEFINITION 1.1. Write 3E (G) for the set of all X in Irr(G) such that(1) X(1) is a ir-number,(2) for every subnormal subgroup M of G and for every irreducible con-

stituent q) of Xt1,, o(q)) is a ir-number.The set is a generalisation of a similar set of characters considered by

Isaacs (see Definition 2.2 of [I]).Actually each X in X (G) is best thought of as the upper limit of a chain of

special characters each lying below the previous ones. More precisely we have

PROPOSITION 1.2. Let G be a 7r-separable group. Let G = Go D G1 DD G = (1) be a chain of normal subgroups of G such that each Gi/ Gi+

1is a

7r-group or a 7r'-group. Let X be a character in Irr(G) such that X(1) is a 17-number.Choose X< in Irr(Gi) such that Xo = X and X<+1 4 (i)GA., for each i. Then Xbelongs to X (G) if and only if o(X) is a ir-number for every i.

1980 Mathematics Subject Classification Primary 20C15; Secondary 20D10.m American Mathematical Society 1980

389

390 DILIP GAJENDRAGADKAR

If M is a subnormal subgroup of G and X is in Irr(G), it follows trivially fromDefinition 1.1 that every irreducible constituent of XM belongs to X (M).

For qq in X (M) write

X»(GIq) = {x EX»(G)Iq, 4 X,}.

Denote by G., the stabiliser of qq in G under conjugation.

PROPOSITION 1.3. Let q) be a character in X (M) where M is a normal subgroupof a Ir-separable group G.

(a) The set X, (G 1q)) is nonempty if and only if I G : G J is a ir-number.(b) Suppose I G : G91 is a i7-number. There is a linear character µ of G., such that

the map 0 -p (µB )G is a bijection from X (G., I q)) onto X (G I q)).

2. Values of characters in

PROPOSITION 2.1. (a) Let H be a subgroup of a Ir-separable G such that I G : His a ir'-number. Then the map X - XH is an injection from into X, (H).

(b) The group G has a normal i7-complement if and only if the map X - XH issurjective.

Write I G mn where m is a i7-number and n is a ir'-number. Let e be aprimitive mth root of unity. Then as an immediate corollary of Proposition 2.1we have

COROLLARY 2.2. (a) For every X in X (G) and every x in G, X(x) belongs toQ(e)

(b) If q is a prime in ?r', every X in is Brauer irreducible mod q.

In fact Q(e) is a splitting field for every X in

3. The multiplicative property. Since every 7r-separable group G is also 7r'-sep-arable, we have another special subset X,,,(G) of Irr(G).

THEOREM 3.1. Suppose G is 7r-separable. Let x, x' be characters in X (G) andX,,,(G) respectively. Then XX' is irreducible. Furthermore if Xx' = 4' for some

wehave

COROLLARY 3.2. Suppose G is solvable. For each prime p, let Xp be a characterin XX(G). Then llp"p is irreducible. Furthermore if flpxp - where 4'p lies inXP(G) for every p, then we have Xp = p for every p.

Note that the product IIpXp is not really infinite as XP(G) = (1) if p does notdivide I G I.

Denote by IIpXX(G) the set of all X in Irr(G) s.t. X = ;Xp where x. E XP(G)for every p.

4. Y-normalizers and IIpXX(G). Let Y be a saturated formation of solvablegroups containing the formation of nilpotent groups. Let D be an 9-normalizerof G. E. Dade has constructed a natural family of injections (Ph) from Irr(D)into Irr(G) (see [D3]). The construction of such injections for system normalizerswas originally announced in [D1]. Each injection in the family gives the sameimage Ph(Irr(D)) in Irr(G). Thus Ph(Irr(D)) is a characteristic subset of Irr(G).

CHARACTERS OF FINITE 7I-SEPARABLE GROUPS 391

The construction of Ph is quite complicated (see [D3]). Each injection Ph is acomposition of several injections called liftings. Each lifting itself is a `union' ofseveral other injections Bi which are character correspondences of the kinddescribed in [D2].

We describe below a characterisation of Ph(Irr(D)) which is independent ofthe construction of Ph.

Let J = ( T°) be an F-system of G of which D is the normalizer.DEFINITION 4.1. Let M be a normal subgroup of G. A character t in Irr(M) is

said to be 9-stable if t can be written as = II, where £ is a T°-invariantcharacter in XP(M) for each prime p.

We denote the set of all 9-stable irreducible characters of M by Irr(M)-.Write

Irr(G I Irr(M),j) = U Irr(G ).JEIrr(M)g

Let C(G) denote the centralizer of all the F-central chief factors of G. ThenC(G) is a characteristic subgroup of G containing the Y-residual G 'J of G.

THEOREM 4.2. Let N be a normal subgroup of G satisfying G'J G N G C(G).Then we have

Ph(Irr(D)) = Irr(G I Irr(N)g).COROLLARY 4.3. The group G belongs to 9 if and only if Ph(Irr(D)) = Irr(G).

Suppose F is itself the formation of nilpotent groups. In this case we haveC(G) = G. It is easy to see that the set of all 9-stable irreducible characters ofG is precisely II,XX(G). Therefore we have the following.

THEOREM 4.4. If D is a system normalizer of G, then we have Ph(Irr(D)) _11 XX(G).

Let D be a system normalizer of G. Let Do be the unique Sylow p-subgroup ofD. We consider Irr(DD) as a subset of Irr(D). By restricting Ph to Irr(DD) weobtain

Ph(Irr(DD)) = XX(G).

Thus Ph induces a bijection from Irr(DD) onto XD(G). The construction of Phcan be generalised to give a similar correspondence in p-solvable groups.

5. A splitting theorem. The special characters (viz. those in X (G) andPh(Irr(D))) seem to provide simple but elegant conditions for splitting exten-sions. Proposition 2.1(b) is an example of this. We given another example below.

Let D be an f-normalizer of G. The subgroup D is said to be a complementto G 6 in G if G = DG 6 and D n G 6 = (1). Various sufficient conditions forthis can be found in the literature. The following result gives a necessary andsufficient condition.

THEOREM 5.1. For a solvable group G, the following statements are equivalent:(a)G=DGgD n G6=(1).(b) If XI, X2 lie in Ph(Irr(D)), then every irreducible constituent of XIX2 also lies

in Ph(Irr(D)).(c) 2XEPh(Irr(D))X(1)2 = IDI.

392 DILIP GAJENDRAGADKAR

A similar theorem can be obtained for a p-solvable group G by replacing G byD°(G) and D by Do, where Do is the intersection of the normalizer of a Hallp'-subgroup of G with a Sylow p-subgroup of G.

REFERENCES

[D1] E. C. Dade, Characters and solvable groups, University of Illinois 1967 (preprint).[D2] , Characters of groups with normal extra special subgroups, Math. Z. 152 (1976), 1-31.[D3] , Caracteres tenant des T-normalisateurs d'un group fini resoluble, J. Refine Angew.

Math., Band 307/308, 53-112.[G1] D. S. Gajendragadkar, A characteristic class of characters of finite-separable groups, J. Algebra

59 (1979), 237-259.[G2] , A characterisation of characters coming from formalisers, J. Refine Angew. Math.

(to appear).[I] 1. M. Isaacs, Lifting Brauer characters of p-solvable groups. II, J. Algebra 51 (1978), 476-490.

TATA-BURROUGHS LIMITED, INDIA

ON CHARACTERS OF HEIGHT ZERO

MICHEL BROUE

Let G be a finite group, let p be a prime.One of the main ways to link characters of G with characters of p-local

subgroups of G is the following: whenever u is a p-element of G and X is acharacter of G, consider the function d"(X), which is the central function onCG(u) defined by

d"(X)(x) = 0 if x a CG(u)v',

d"(X)(s) = x(us) ifs E CG(u)o.

(Here I denote by CG(u),, the set of p'-elements of CG(u); notice that the letter dcomes from "decomposition", since for u = I the map d" is nothing but theordinary decomposition map.)

Let R be a complete set of representatives for the conjugacy classes ofp-elements. Then it is fairly trivial that

(1) the family of functions (d"(X))uER determines X,(2) whenever X and X' are characters of G, then

<X, X'>G = Y, <d"(X), d"(X )>co("), anduER

<d"(X), d"(X')>cc(") = <P), X '>G.

where I denote by X(") the product of X by the characteristic function of thep-section of u.

This construction is interesting because of the following consequence of thethree Brauer's Main Theorems:

(3) whenever X is in the principal block of G, then d"(X) is in the principalblock of CG(u).

This last assertion provides a very useful means to relate some characters ofG/OP(G) with some characters of CG(u)/OP.(CG(u)), since the kernel of theprincipal block of a group is the Op, of the group. Suppose for example that you

1980 Mathematics Subject Classification. Primary 20C20.O Amencan Mathematical Society t980

393

394 MICHEL BROUE

are working in the following situation: H is a subgroup of G which controls thefusion of p-elements in G, and which is such that

CG(u) = CH(u)OO'(CG(u))

for each nontrivial p-element u of H. Then the restriction from CG(u) to CH(u)induces a natural one-to-one correspondence between the characters of theprincipal block of CG(u) and the characters of the principal block of C,,(u), andthese correspondences induce a nice isometry between a subgroup of thecharacter group of G and the corresponding subgroup of the character group ofH-this precise situation has been studied in particular by Reynolds. Then, insome cases, you may apply coherence or other tricks to get results about thestructure of G/Oo-(G): see for example the the nonsimplicitycriteria of G. Higman, Tyrer, S. Smith, Puig, etc....

As in Alperin's article of these PROCEEDINGS and Puig's article of thesePROCEEDINGS our aim is to generalize to any block the methods and some of theresults of the p-local theory-at least, what in this theory can be expressed interms of principal blocks, as in the preceding paragraph. So, we have been ableto prove Sylow theorems, Alperin's fusion theorem, the Frobenius theoremabout nilpotent blocks; other generalizations are coming, as those concerningthe "minimal conjugation families" of Goldschmidt and Puig (see Note 1), orresults analogous to Glauberman's ZJ-theorem.

Today I am less ambitious. I just want to prove a generalization of a trivial,tiny property of characters in principal blocks, which would be ridiculous tostate even as a lemma, but actually is not ridiculous when stated for generalblocks. But I now need some notation.

Let K be the extension of the p-adic field Q. by the I G I th roots of unity, andlet 0, p, F be respectively the valuation ring of K, the maximal ideal of 0, theresidue field of 0. A block b of G is a primitive idempotent of the center ZFG ofthe group algebra FG; I denote by b the unique primitive idempotent of Z 0 Glifted up from b. Let me denote by cd(b) and call the codefect of b the valuationin p of the index in G of a defect group of b; the defect of b is denoted by d(b).An 0 -generalized character of G is a linear combination with coefficients in 0of characters of G. Such a function X is said to be in b iff X(bx) - X(x) for eachx in G; in this case, pd(b) divides X(1), and X is said to have height zero iffp-rd(b)X(l) is a unit in 0.

It is known now that we have to replace the p-elements of G by the b-Brauerelements (u, e), where u is a p-element of G which lies in a defect group of b,and e is a block of CG(u) which is a summand of the image of b by the Brauermorphism Br": ZFG -* ZFCG(u). Now we have to do the same construction as Idid at the beginning of my talk: whenever (u, e) is a b-Brauer element and X isan 0 -generalized character of G in b, we consider the function d(",`)(X), which isthe part of the function d"(X) associated with e-in other words, we have

X(eus) for s e CG(u)o.. Let R be a complete set of representativesfor the conjugacy classes of b-Brauer elements. Then

(b. 1) the family of functions (d(",`)(X))(".,) =R determines X,(b.2) whenever X and X' are 6 -generalized characters of G in b, then

CHARACTERS OF HEIGHT ZERO 395

<x, X'>c = <d(".e)(X), d("`)(X)>CG(u), and(u, e)ER

<d("(X), d(u.e)(X)>CC(u) = <X(".e)

X > ,

where I denote by X(""e) the central function on G which vanishes outside of thep-section of u and is such that X(""e)(us) = d for s in CG(u)v.

In [1], Brauer got some results about the numbers <X(",e), X)G that I got with aquite different method in [2]. My purpose here is, in part, to generalize one ofthese results.

It is well known that whenever X is an 0 -generalized character of G, thenX(u) = X(1) (mod p) for each p-element u of G. We can regard this congruenceas a property of characters in the principal block; let me complicate thisstatement. First of all, keeping the previous notation, we have X(u) = d"(X)(1).Then, if b is the principal block of G, the p-elements of G are in a naturalone-to-one correspondence with the b-Brauer elements; so we can write ourcongruence as X(1) (mod p) for any b-Brauer element (u, e). Fi-nally, let us denote by b(1) (resp. i(1)) the coefficient of b (resp. of i) on 1 whenexpressed on the natural basis of 0 G (resp. of 0 CG(u)); it can be checked thatb(1) = i(1) - I /I Op (CG(P))I (mod p), where P denotes a Sylow p-subgroup ofG. Hence our congruence can be written

d("'e)(X)(l) = X(1)i(1) b(])

(mod p) (C)

What I want to prove is that (C) holds for any block b. Notice that in generalthe numbers i(1) are no longer units of 0 . But, by a result of Brauer-which isactually a consequence of Lemma 3 below-we know that p-ed(b)b(]) is a unit in0, and since p`d(b) divides X(1), this gives sense to (C). A consequence of (C) is

PROPOSITION 1. The character X of G in b has height zero iff the functiond in e has height zero.

Now, by the characterization of elements with height zero which can be foundin [2], we get the following consequence

COROLLARY 2. Let X be an 0 -generalized character of G in b with height zero,and let (u, e) be a b-Brauer element. Then the number

pd(e)<X(u.e), X>u = pd(e)<d(",e)(X), d("'e)(X)>Cc(")is a unit in 0.

Proposition I and Corollary 2 are generalizations of some results of Brauer in[1]; indeed, in that paper he proved these results in the case where (u, e) was a"major subsection" of b, i.e. a b-Brauer element such that the defect groups of eare defect groups of b. The fact that those d(".e)(X) are nonzero was of essentialuse in Brauer's analysis of blocks with abelian defect groups and inertial index 1.Similarly, the generalization for any b-Brauer element is of essential use in [3],where Puig and I study nilpotent blocks without assuming that the defect groupsare abelian.

The proof of (C) I shall give here is surprisingly simple, compared to the longcalculations of [1]. It depends on the following lemma, which can be found in[2]:

396 MICHEL BROUE

LEMMA 3. Let a = EQEGa(s)s be an element of (Z(SG)b such that a(s) = 0whenever s (I G. Then p`d(b) divides a(1) and p_cd(b)a(1) is a unit in ® iff a doesnot belong to the Jacobson radical of Z (9 G.

Let us denote by wb: Z (9 G -* F the algebra morphism associated with b. Sincethe algebra (Z(SG)b is local, the following lemma results immediately fromLemma 3:

LEMMA 4. Let a be as in Lemma 3. Then we have

wb(a) = ( a(1)/b(1) ).

(I denote by - the reduction modulo p.)Now we can prove (C). Let x be an (9 -generalized character of G in b; let us

denote by d(x) the function d'(x), and by d(x)° the element of (Z (9 G )b definedby

d(x)° = Y, x(s-')s.sEG

Similarly, let us set

d(u'e)(X)o = Y, d(u'e)(X)(s-')s.

s E CG(u)

Thus, by Lemma 4, we have

wb(d(X)°) = (X(l)/b(1)).

But since x(us) = x(s) (mod p) for all s in CG(u)o., we see that

Bru( d(X)°) = dm(X)O,

and so

eBr.( d(X)°) = dlm'el(X)° .

Hence by Lemma 4 we get

we(Bru(d(X)°)) =d(ue)(X)(l)

e(1)

and since we o Br = wb, (C) is proved.

Note 1. A set of b-subpairs of G contained in a given Sylow b-subpair S of Gis a conjugation family iff it contains S and a conjugate of each essentialb-subpair. A b-subpair (P, e) is essential if f

(E1) P is a defect group of e as a block of PCG(P),(E2) NG(P, e)/PCG(P) contains a strongly p-embedded subgroup.

REFERENCES

1. R. Brauer, On blocks and sections in finite groups. II, Amer. J. Math. 90 (1968), 895-925.2. M. Broue, Radical, hauteurs,p-sections et blocs, Ann. of Math. (2) 107 (1978), 89-107.3. M. Broue and L. Puig, A Frobenius Theorem for blocks Inv. Math. 56 (1980), 117-128.

9 RUE BREZIN, F75014 PARIS, FRANCE

BRAUER TREES AND CHARACTER DEGREES

HARVEY I. BLAU

This note presents some very fragmentary results which I have obtained overthe past couple of years on the following two questions:

(1) Does a given tree occur as the Brauer tree of a block with cyclic defectgroup of some group algebra kG, and if so, to what extent does the treedetermine G?

(2) Does a given degree (as a function of some parameter, such as the order ofa Sylow subgroup) occur as the degree of an irreducible character of some finitegroup G, and if so, to what extent does it determine G?

These are big questions (almost nothing is known about (1)), but the classifi-cation of finite simple groups will make many aspects of them more accessible.The results below involve quite restrictive assumptions. Major classificationtheorems are not used in their proofs, except as noted.

Notation. G is a finite group, p a fixed odd prime dividing I G 1, P a Sylowp-subgroup of G with I P = p °. B. is the principal p-block of G.-

THEOREM 1. Assume that P is cyclic. There is no irreducible character X in B0with (p° + 1)/2 < X(1) <p° - 2.

This generalizes a special case of results of Feit [4], where JPJ = p. Thetechniques of [4], as refined in [2], carry over to the general cyclic case, and formthe basis of a proof.

P. Ferguson [6], [7] (see also these PROCEEDINGS) has extended results ofBrauer, Leonard, and Sibley to show that under essentially weaker assumptionson P (namely, CG(x) = CG(P) for all x E PS, and NA(P)/CA(P) is cyclic)there is no faithful X in any p-block with

p°2 1 < X(l) < 2p°3+ 1

and G/Z(G) is known or P < G if there is a faithful X with X(l) 4 (p° + 1)/2.It seems possible that in this situation the bound (2p° + 1)/3 may be extendedtop' - 2.

1980 Mathematics Subject Classification. Primary 20C15, 20C20.® American Mathematical Society 1980

397

398 H. I. BLAU

NEAR-THEOREM. Assume that P is cyclic. There is no irreducible character X inB0with (3p°+1)/2<X(1)<2p°-4.

When a = 1, this follows from [3]. For arbitrary a, not enough details of aproof are written down to justify calling the statement a theorem. I believeX(1) = 2p° - 4 does not occur, for p° > 7, but I do not have a proof yet, evenwhen a = 1.

DEFINITION. A tree is a star if there are no paths with more than two edges,i.e.

If G isp-solvable, then the Brauer tree of any block with cyclic defect group isa star with the exceptional node at the center (by results of Fong, Swan, andIsaacs). The converse is not always true. Here are some examples of simplegroups where the tree for B0 is a star. The exceptional node, if any, is denoted"exp", and the corresponding character degrees are listed.

Mi 1 (p = 5):

L2(2" = p + 1):

11

1 p+2 p+1exp

Sz(q) (p°Ilq + r + 1, where r =V2q , q = 2°dd)

r(q - 1)/2

q2 (q-r+1)(q-1)exp

r(q - 1)/2

BRAUER TREES AND CHARACTER DEGREES 399

In particular, the principal 13-block for Sz(8) is

1 35

exp

THEOREM 2. Assume P is cyclic. If the tree for B0 is a star, then every involutionin G/OO.(G) inverts some Sylow p-subgroup of G/OO,(G).

This is easily proved by applying to involutions and p-elements the formularelating class multiplication constants and character values. SinceNG(P): CG(P)I is the number of edges in the tree for B0, the Feit-Thompson

theorem instantly yields

COROLLARY. Assume P is cyclic. If the tree for B0 is a star with an odd numberof edges, then G is p-solvable.

THEOREM 3. Assume that P is cyclic, p° > 3, and the tree for B0 is a star. Thenthere is no character X in B. with X(1) = 2p° - 4.

OUTLINE OF PROOF. We may assume that p° > 13, that the tree for B0 has fouredges (so p = 1 (mod 4)), that the exceptional node is not at the center, and thatX is one of the (p° - 1)/4 exceptional characters:

x

exp

Also, G = G'.From Theorem 2, and modular theory, we compute X(t) = 2 for all involu-

tions t in G. We then use the following elementary

LEMMA. Let S be a 2-group, and X a complex representation of S withrational-valued character X. If the number of involution t such that X(t) = X(1)(mod 8) is even then I S 14 16 (and I S 14 8 if det X = 1).

Since X(1) - X(t) = 2p° - 6 - 4 (mod 8), the lemma applied to X yields16 } I G I. All members of the family of exceptional characters including X are inthe same 2-block. Now, methods of Brauer and Feit (see [5]), which bound thenumber of characters in a block in terms of the order of its defect group, implyp° 4 13, a contradiction.

400 H. I. BLAU

THEOREM 4. Assume that P is cyclic, that the tree for B0 is a star with four edgessuch that the exceptional node is not at the center, and such that the degree of the(conjugate) characters not on the real stem is I + p°:

t 1 +pa

Then p° = 13 and G/Oo(G) x Sz(8).

OUTLINE OF PROOF. We may assume G is simple. Let L, M be the irreduciblekG-modules (k a suitable field of characteristic p) which are the unique modularconstituents of , X respectively. The Green correspondence shows that thesymmetric summand of L ® L must contain M as a composition factor withmultiplicity at least (p° - 1)/4. Theorems I and 3 imply dimk M > 3p° - 4.This forces p° = 13, 3'(1) = 14, X(1) = 35. But this is now a situation treated in[1], where a straightforward argument using block separation shows that IGI _lSz(8)1 and that G is a CN-group. Thus Suzuki's theorem [8] ends the proof.

REFERENCES

1. L. Alex and D. Morrow, Index four simple groups, Canad. J. Math. 30 (1978), 1-21.2. H. Blau, Finite groups where two small degrees are not too small, J. Algebra 28 (1974), 541-555.3. , Degrees of exceptional characters of certain finite groups, Trans. Amer. Math. Soc. 249

(1979), 85-96.4. W. Feit, On finite linear groups, J. Algebra 5 (1967), 378-400.5. , Representations of finite groups, Lecture Notes, Yale University, New Haven, Conn.,

1969.6. P. Ferguson, On finite complex linear groups of degree (q - 1)/2, J. Algebra 63 (1980), 287-300.7. , Finite complex linear groups of degree less than (2q + 1) /3 (preprint).8. M. Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105-145.

NORTHERN ILLINOIS UNIVERSITY

A CORRESPONDENCE OF CHARACTERS

EVERETT C. DADE

1

We fix a valuation ring R having an algebraically closed field of fractions K ofcharacteristic 0, and a residue class field F of prime characteristic p. Then ap-block B of defect 0 for a finite group N contains exactly one irreducibleK-character (D of N, and the corresponding ring direct summand of RN ise((D)RN, where e((D) is the primitive central idempotent of KN corresponding to(D. We have the isomorphism of R-algebras:

(1) e(')RN = [R]1),where [R ]y denotes the algebra of ally x y matrices with entries in R.

Suppose the above group N is a normal subgroup of a finite group M suchthat:

(2) M/N is a p-group fixing B under conjugation.One easily verifies that:

(3) e((D) is a primitive central idempotent of RM corresponding to a p-block Beof M having a defect group D complementary to N in M.

Evidently D normalizes at least one p-Sylow subgroup S of N. The trivialcharacter is of S extends to the trivial character IDS of DS, and hence has asplit Clifford extension DS<ls>. Since is appears with p'-multiplicity in therestriction of (D to S, Theorem 4.4 of [2] implies that M<d> is isomorphic toDS<ls> and hence also splits. It follows easily that

(4) There is a unique extension of to an irreducible K-character V of M suchthat D acts on e(ls)V with determinant 1, for any p-Sylow subgroup S of Nnormalized by D and any KM-module V affording V.

The Brauer Correspondence gives us a unique p-block be of NM(D) having Das defect group and corresponding to Be. Since NM(D) is the direct productD X CN(D), there is a unique block b of defect 0 for CN(D) from which becomes as Be came from B. Let 0 be the unique irreducible K-character of CN(D)lying in b. Notice that, in the special case in which N is a p'-group, ¢ is thecharacter corresponding to (D in the Glauberman Correspondence of [4].

Now assume that(5) Both N and M are normal subgroups of a finite group G fixing B.

1980 Mathematics Subject Classification Primary 20C15.O American Mathematical Society 1980

401

402 E. C. DADE

By unicity G then fixes c, V, and Be, while NG(D), which covers G/M, fixesbe, b, and ¢. Our object is to show that the algebras e((D)RG and e(¢)RNG(D)are Mori ta-equivalent. More specifically, we shall construct a natural algebraisomorphism:

(6) [e((D)RG] I) [e($)RNG(D)] (1).It follows from (1) above and Lemma 4.8 of [2] that e((D)RG is algebra-iso-

morphic to [R[tor(G<(D>)]]W1), where R[tor(G<(D>)] is the twisted group algebraover R of the torsion subgroup tor(G<(D>) of the Clifford extension G<c>.Similarly e(¢)RNG(D) is algebra-isomorphic to [R[tor(NG(D)<0>)]].(1). So toconstruct the isomorphism (6) we need only provide an algebra-isomorphism

(7) R[tor(G<(D>)] ^' R[tor(NG(D)<O>)].For this, in turn, it suffices to find an isomorphism

(8) tor(G<(D>) _ tor(NG(D)<(D>)of extensions of the group tor(U(R)) of roots of unity in R by GINNG(D)/CN(D)

The existence of the unique extension V of (4) gives us a complement C totor(U(R)) in tor(M<(D>) acting trivially on V. Because G fixes V, the subgroupC is normal in tor(G<(X>) and tor(G<(D>)/C is isomorphic to tor(G<(De>) asextensions of tor(U(R)) by (G/N)/(M/N) _- G/M. Thus tor(G<(D>) is theresidual product of tor(G<(I >) and tor(G<(D>)/tor(U(R)) G/N. Similarlytor(NG(D)<O>) is the residual product of tor(NG(D)<4 >) and NG(D)/ CN(D ),where ¢e = to x ¢ is the canonical extension of ¢ to NM(D) = D X CN(D) asin (4). Since G/N is naturally isomorphic to NG(D)/CN(D), we can constructthe isomorphism (8) by giving an isomorphism:

(9) tor(G<(V>) =' tor(NG(D)<4 >)of extensions of tor(U(R)) by G/M ^f NG(D)/NM(D).

It follows from (1) that there is, to within isomorphism, exactly one RM-lattice W affording the character V. The natural epimorphism of R onto Fsends tor(U(R)) onto tor(U(F)), with the p-Sylow subgroup tor(U(R ))P askernel. As in §4 of [2], this epimorphism extends to a natural epimorphism oftor(G<(I >) onto tor(G<W>) with the same kernel tor(U(R))P, where G< W> isthe Clifford extension of the G-invariant indecomposable RM-lattice W asdefined in [1]. Applying (4) with M replaced by the inverse image of a p-Sylowsubgroup of GIN, we see that the p-part of the extension tor(G<(I >) splitsnaturally. So tor(G<(V>) splits naturally over tor(U(R)),, and thus is de-termined by its image tor(G<W>). Similarly, tor(NG(D)<4 >) is determined bytor(NG(D)<w>), where w is an RNM(D)-lattice affording 0e. So the isomor-phism (9) can be determined by an isomorphism:

(10) G<W> ^ NG(D)<w>of extensions of U(F) by G/M NG(D)/NM(D).

As in § 13 of [3], the lattice W has D as a vertex, and its Green Correspondenthas the form U ® w, where U is a unique indecomposable RD-lattice withvertex D considered as an RNM(D)-lattice. A theorem of Cline [1] gives us anatural isomorphism of extensions

(11) G<W> ^ NG(D).Evidently U is NG(D)-invariant, and NG(D) is the product of the twoextensions NG(D) and NG(D)<w> of U(F) by NG(D)/NM(D). Thus theisomorphism (10) will exist once we show

A CORRESPONDENCE OF CHARACTERS 403

(12) the extension NG(D) splits naturally.The proof of (12) is a long but routine calculation based on the classification

of indecomposable endo-permutation RP-lattices with vertex P, given for anyabelian p-group P by Theorem 12.5 of [3]. This completes the proof of (6).

The Morita-equivalence (6), of course, gives us a natural one-to-one corre-spondence between the set Irr(GI(D) of all irreducible K-characters of G lyingover c, and the corresponding set Irr(NG(D)I4)). It should be remarked that thiscorrespondence is determined uniquely by the choice of R, but that differentchoices of R in K can lead to different correspondences. So the correspondenceis only determined absolutely to within Galois automorphisms.

When N is a p'-group, we may choose for D any fixed p-Sylow subgroup ofM. In that case, as remarked above, the correspondence between (D and 'o is thatof Glauberman [4]. So we have obtained a natural one-to-one correspondencebetween Irr(GI(D) and Irr(NG(D)I4)) lying over that of Glauberman. As inTheorem 10.9 of Isaacs' paper [6], one can string together chains of thesecorrespondences to obtain one-to-one correspondences between the irreducibleK-characters lying in a p-block B0 of a p-solvable finite group H and those in thecorresponding block bo of the normalizer in H of a defect group of B0. Thus onecould prove the strong McKay Conjecture for p-solvable groups in this way.However, a more direct proof of that conjecture has been found by P. Gres [5],and, more recently, a very short proof has been found by Okuyama and Wajima[7]. Both these proofs are based on counting arguments, and neither gives acorrespondence of characters.

REFERENCES

1. E. Cline, Some connections between Clifford theory and the theory of vertices and sources, Proc.Sympos. Pure Math., vol. 21, Amer. Math. Soc., Providence, R 1., 1970, pp. 19-23.

2. E. C. Dade, Isomorphism of Clifford extensions, Ann. of Math. (2) 92 (1970), 375-433.3. , Endo permutation modules overp-groups. II, Ann. of Math. (2) 108 (1978), 317-346.4. G. Glauberman, Correspondences of characters for relatively prime operator groups, Canad. J.

Math. 20 (1968),1465-1488.5. P. Gres, Proc. 6th All-Union Conference on Group Theory, Chercassy, 1978.6. 1. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95 (1973), 594-653.7. T. Okuyama and M. Wajima, Character correspondence and p-blocks of p-solvable groups, Osaka

J. Math. (to appear).

UNIVERSITY OF ILLINOIS, URBANA-CHAMPAIGN


IRREDUCIBLE MODULES OF p-SOLVABLEGROUPS

WALTER FEITI

1. Introduction. Let F be an algebraically closed field of characteristic p andlet G be a finite group. If V is a finitely generated F[G] module let (V) denotethe isomorphism class containing V. Define (V) + (W) = (V + W) and(V)(W) = (V ® W). The Green ring A(F[G]) is the ring of all finite complexlinear combinations of elements (V), as V ranges over finitely generated inde-composable F[G] modules.

An element x E A(F[G]) is algebraic if it is the root of a nonzero polynomialwith integer coefficients. In other words if there exist integers a0, . . . , ak, not all0 with ao + + akx k = 0. A finitely generated F[G] module V is algebraic if(V) is algebraic.

This concept was first introduced by Alperin in [1], where he also showedamongst other things that if char F = 2 then the irreducible F[SL2(2")] modulesare algebraic. In [2] Berger announced that the irreducible modules of a solvablegroup are algebraic. The object of this paper is to give a relatively simple proofof Berger's result and to generalize it to p-solvable groups. This generalizationassumes that all finite simple groups are known at present and so depends on theclassifications of the finite simple groups. The precise statements follow.

Let us call a finite simple group G well behaved if for every prime p } G 1, theSp-group of the automorphism group of G is cyclic.

It is an exercise to verify that all known finite simple groups are well behaved.We will prove

THEOREM 1. Let F be an algebraically closed field with char F = p. Let G be ap-solvable group such that every simple group which is a subgroup of a factor groupof G is well behaved. Then every irreducible F[G] module is algebraic.

Throughout the remainder of the paper, F is an algebraically closed field ofcharacteristic p. An F[G] module will always mean a finitely generated F[G]module. The rest of the notation is standard.

1980 Mathematics Subject Classification. Primary 20C15.'This work was partially supported by NSF Grant #MCS 76-10237.


405

406 WALTER FELT

2. Algebraic modules. In this section we state without proof some elementaryproperties of algebraic modules. See [1].

LEMMA 2.1. Let V be an F[ G ] module. The following are equivalent.(i) V is algebraic.(ii) There exist a finite number o f F[G] modules W1, ... , Wm such that if W is

an indecomposable direct summand of V" = V ® 0 V for any n then W=W. for some i.

LEMMA 2.2. Let V, W be F[ G ] modules.(i) If V is algebraic and W V (i.e. W is a summand of V) then W is algebraic.(ii) If V, W are algebraic then V ® W and V ® W are algebraic.

LEMMA 2.3. Let H be a subgroup of G.(i) Let V be an algebraic F[G] module. Then VH is algebraic.(ii) If U is an algebraic F[H] module and VI UG then V is algebraic.

3. Some preliminary results. This section contains some necessary pre-liminaries. Lemma 3.6 is perhaps of independent interest and is related to someresults of Dade [3]. It is curious that although Theorem 1 refers only top-solvable groups, the proof uses properties of groups which are not p-solvable.

LEMMA 3.1. Let q be the power of a prime with q - 3 (mod 4) and let K = Fq.Let V be a vector space of dimension 2n over K and let f be a nondegeneratealternating bilinear form on V. Suppose that J E Sp(f) = Sp2,,(q) with J2 = - I.Then there exist a, b E K and an integer m > 1 such that if y = aI + bJ then

y2" _ - I and f'' _ - f, where fy(v, w) = f(vy, wy) for v, w E V.

PROOF. Since q 3 (mod 4) it follows that the K-algebra generated by J isisomorphic to Fq" Hence there exist c, d E K such that if x = cI + dJ thenx ' ' 1 = I. Since Jq = j-1 = - J this implies that

-I=(cI+dJ)(cI+dJq)=(cI+dJ)(cI-dJ)=c2I-d2J2=(c2+d2)I.Hence c2 + d 2 = - 1. Since J E Sp(f)

f(v, wJ) = f(vJ, wJ2) = -f(vJ, w)

and so

fx(v, w) = f(cv, cw) + f(cv, dwJ) + f(dvJ, cw) + f(dvJ, dwJ)

= (c2 + d2)f(v, w) + cd {f(v, wJ) + f(vJ, w)} = (c2 + d2)f(v, w)

= -f(v, w).Thus fx = - f. Let y be the 2-part of x. Then yen' I for some m > 1 asq + 1 - 0 (mod 4) and fy = - f as required.

LEMMA 3.2. Let p be a prime, let q be the power of an odd prime distinct from pand let K = Fq. Let V be a vector space of dimension 2n over K and let f be anondegenerate alternating bilinear form on V. Let P be a Sp-group of Sp(f).Assume that P acts irreducibly on V. Let c E K, c 0. Then one of the followingholds.

IRREDUCIBLE MODULES OFp-SOLVABLE GROUPS 407

(i) There exists x E GL(V) such that fx = cf, and x commutes with everyelement of P. Either some power of x is equal to cd2I for some d E K" withcd2 1orc= 1andx= - I.

(ii) p = 2, c a2 for a E K and P acts absolutely irreducibly on V.

PROOF. If c = a2 with a E K then x = aI satisfies (i). Thus it may be assumedthat c a2 for any a E K.

If x satisfies (i) then for a E K", ax satisfies (i) if c is replaced by cat. Thus itsuffices to prove the result for any fixed nonsquare c in K". Hence it may beassumed that

c2' 1 for some integer t ) 0. In particular c = - 1 if q = 3(mod 4).

Let G = GL(V). Suppose that (ii) does not hold. By Schur's Lemma CG(P)FQk as finite division rings are fields. If p 2 then k is even as dim V = 2n iseven. If p = 2 then P does not act absolutely irreducibly by assumption and so kis even since 0(2') is a power of 2. Hence in any case CG(P) contains a uniquecyclic subgroup A of order q2 - 1 which contains all nonzero scalars.

Choose J E A with J2 = CI. If f' = cf then x = J satisfies (i). Suppose thatf' cf. Then g = cf - f' 0. Clearly g is a P-invariant alternating bilinearform. Since P acts irreducibly on V it follows that g is nondegenerate. Bydefinition g' = - cg. There exists z E GL(V) with fZ = g. As Sylow groups areconjugate, z may be chosen so that PZ = P. Let J0 = JZ. Then J0 E CG(P) andso J0 E A. Thus by changing notation it may be assumed that J E A, J2 = CIand f' = - cf.

If - 1 = a2 for some a E K then x = aJ satisfies (i). Suppose that - 1 a2for all a E K. Hence q - 3 (mod 4) and c = - 1. Thus J2 = - I and f' = f.Hence J E Sp(f). Let x = y be defined as in Lemma 3.1. Then x satisfies (i).

LEMMA 3.3. Let q be the power of an odd prime and let K = Fq. Let V be avector space over K and let f be a nondegenerate alternating bilinear form on V.Let V = V, ® V2 with V2 = V,1 and dim V, = dim V2 = 2n. Let f. be therestriction off to V;. Let P; be a S2 -group of Sp(f,,) - Assume that P; actsabsolutely irreducibly on V. Then P, may be identified with P2. Let P = {(y, A yE P, = P2). Then there exists x E GL(V) such that fx = - f, x commutes withevery element of P and some power of x is equal to -d2I for some d E K with-d2 1.

PROOF. If - 1 = a2 for some a E K, the result follows from Lemma 3.2.Suppose that - I a2 for any a E K. Hence q - 3 (mod 4) and - d 2 1 forall d E K. We may identify V, with V2 and f, with f2. Define the lineartransformation J on V = V, ® V2 by J: (v v2) - (- v2, v,). It is easily seenthat J 2 = - I and f'' = f. Furthermore J commutes with every element of P. Ifx = y is defined as in Lemma 3.1 then x has the desired properties.

LEMMA 3.4. Let p be an odd prime. Let K = F21 let V be a vector space over Kand let f be a nondegenerate quadratic form on V. Let P be a p-group withP C 0(f). Assume that P acts irreducibly on V. If x e Z(P) - { 1) then vx vfor all v E V, v 0.

PROOF. Clear.

408 WALTER FEIT

LEMMA 3.5. Let q be a prime and let Q be an extra-special q-group withQI = q2"+'. Let Z = Z(Q) and let V = Q / Z. Let p be a prime distinct from q

and let F be an algebraically closed field of characteristic p.(i) Let f(x, y) = [x, y]. Then f defines a nondegenerate alternating bilinear form

from V to Z ^ FQ . If q = 2 then f(x) = x2 defines a nondegenerate quadraticform on V. Let A(Q) denote the group of all outer automorphisms of Q and letA0(Q) denote the subgroup consisting of all automorphisms which fix all theelements of Z. Then A0(Q) 4 A(Q) and is cyclic of orderq - 1. If q 2 then A0(Q) SP(ff) - SP2"(q) If q = 2 then A0(Q) = O(f) =02n(2) is an orthogonal group. In any case a subgroup H of Q is abelian if and onlyif the image of H in V is isotropic.

(ii) Let P be a p-group with P C A0(Q) and let G = QP be the semidirectproduct. If A is a linear character of Z with A 1, then (up to isomorphism) thereexists a unique irreducible F[G] module X,\ such that A is a constituent of thecharacter afforded by (XX)Z. Furthermore (X )Q is irreducible and every irreducibleF[Q] module which does not have Z in its kernel is isomorphic to some (Xx)Q_If His a maximal abelian subgroup of Q then H = Ho X Z, I H0I = q" and A Q =(X )Q where A(hz) = A(z) for h E Ho.

(iii) For i = 1, 2 let Q; be extra-special with Z; = Z(Q;). Then Z, - Z2 - Z.Let c c K" and let A1, A2 be linear characters of Z with Al = X2 `. Let Q = QI XQ2 and let Z(c) = {(z, z `)I z E Z) C Z(Q). Then X,\, 0 XX2 is an irreducibleF[Q] module with kernel Z(c). Furthermore Q/Z(c) is extra-special.

PROOF. (i) This is well known.(ii) The existence of a unique irreducible Brauer character q),\ afforded by an

F[Q] module such that A is a constituent of (c,\)Z is well known, as are all theother properties of Q in the statement. Then G = T(q),\). The existence anduniqueness of X,\ now follow from known results.

(iii) Straightforward verification.

LEMMA 3.6. Let P, Q, G = PQ, V, Z, Z(c) be defined as in Lemma 3.5. Let A, µbe linear characters of Z with A 1, µ 1. Then one of the following occurs.

(i) Let Aµ be the character of PZ defined by \tL(xz) = Aµ(z) for x e P, z e Z.Then

X'\

(ii) p = 2. Let Q, = Q for i = 1, 2. Let Q0 = Q1 X Q2/Z(-1), let PO ={(x, x)Ix E P), let Go = P0 Q0. Let X,\20 be the irreducible F[GO] module such that

(Xx)Q ® (Xx)Q2. Let X1 be the character of P0Z(Qo)/Z(-1) such thatA4((xz1, xz2)) = \2(zI z2) for x E P, z, E Z. Then

X'\2o ® X'\2o =

PROOF. The proof is by induction on I Q 1. Without loss of generality it may beassumed that P is a So-group of A0(Q).

Suppose that V contains a nonisotropic proper P-invariant subspace. Let Wbe minimal among such spaces. Then W = W n W 1 ® Wo for some P-in-variant space Wo (0) with Wo fl W 1 = (0). The minimality of W implies that


W = W°. Hence if W = W, then V = W, ® W2, where each W, is P-invariant,W, * (0) and W1 = W2. Since P is a S-group of AO(Q) it follows thatP = P, X P21 where P; acts trivially on Wj for i j. There exist extra-specialgroups with H/Z, ^- W, for i = 1, 2 where Z; = Z(H,), such that

(P1HI X P2H2)/Z0 = PQ

for some subgroup Z0 of Z, X Z2. Thus X,\ = XA ® X,2 and X. = Xµ 19 X,.,,where X., X,, are irreducible F[P;H1] modules which do not have Z; in theirkernels. Observe that if p = 2 and (i) is satisfied for a group then also (ii) issatisfied. Thus by induction it may be assumed that either I) or (ii) is satisfiedfor the groups P,H. for i = 1, 2. Hence either XN 0 X,ti = N'", for i = I or 2or (9 XA:O = (X)4 for i = 1 or 2. Since PH,Z2 fl PH2Z, = PZ,Z2 andP0H10Z(H20) n POH20Z(H10) = P0Z(H,0 X H 00) it follows from the tensorproduct theorem that 0 (Aµ)P2H = (7 µ)G and (A°)G-0 0 OX)G2 = G

and the result is proved in either case. Hence it may be assumed that V has noproper nonisotropic P-invariant subspace.

Suppose that (0) S V, g V with V, a P-invariant subspace. Then V, C; 1/11.Thus V,1 = V, ® V0 with V0 P-invariant. As V0 n Vo = (0) and V0 is iso-tropic this yields that V0 = (0). Hence V, is a maximal isotropic subspace of V.Furthermore V = V, ® V2 with V2 a maximal isotropic subspace which isP-invariant. Let M; be the inverse image in Q of V. Then M, is abelian.Furthermore M, = Mi0 X Z, where P normalizes MO. Let A, µ respectively bethe character of PM,, PM2 respectively with PM,O, PM20 in its kernel such thatA(xz) = A(z) for x e PM10, z e Z and µ(xz) = µ(z) for x E PM20, z E Z.Then A G - X,\ and µ G = X. As PM, n PM2 = PZ, the tensor product theo-rem implies that

XA ® X1,_ G ®jG = (Aµ)G.

Thus (i) holds. Therefore it suffices to prove the result in case P acts irreduciblyon V.

Let A = µ-`. Two cases will be considered.Case (I). There exists an automorphism a of G such that x° = x for all x E P,

Q° = Q, v° v for v E V, v 0 and z° = z` for z E Z.Case (II). The conditions of Case (I) are not satisfied.Suppose that Case (II) holds. By Lemmas 3.2 and 3.4, p = 2, c a2 for a E K

and P acts absolutely irreducibly on V. Let c = - 1. Let P0, Q0, Go be definedas in statement (ii). By Lemma 3.3 there exists an automorphism a of Go suchthat x° = x for all x E P0, Qo = Q0, v° v for v E V0, v 0, where VO =QO/Z(Qo) and z° = z 1 for z E Z.

In Case (I) change the notation and let P = P0, Q = Q0, G = GO, V = V0,X'\ = X'\0, Xµ = Xµ0 so that both cases can be handled simultaneously.

Let H = P(O)(Q0 X Q0), where P(0) _ ((x, x)lx E PO). Define Q(O) _((y,y)Iy E Q0) and Q(0) = ((y,y°)Iy E Q0). Then P(O) normalizes both Q()and Q('). Furthermore H/ Z(c) is extra-special, Z(c) C Q' and Q°/Z(c) isabelian. Since H c POQO x POQO, there exists an irreducible F[H] module Y =(Xxo 0 Xµ°)H with kernel Z(c).

410 WALTER FEIT

,Let Xµ be the linear character of P (0)Q (°)Z(H) with P (0)Q (°) in its kernel andA ((zl, z2)) = A(zl)µ(z2) = µ(zi-`z2). Then (Aµ") XQ°- YQ°x

°and so XtL

Y by Lemma 3.5(ii). By definition Q(O) n Q O C Z(H). Thus the Mackeydecomposition implies that

(X1(0 ® X0)Q(o)p(o)Z(H) = YQ(°)P(°)Z(H)

((^H _Q(°)P(°)Z(H)

(Aia' )Q(°)P(oZ(H) = NLP(°)Z(H))

Since Q(0)P(0)Z(H)/Z(c) - G° the result is proved.

LEMMA 3.7. Let p q be primes. Let Q be an extra-special q-group and let P bea p-group contained in A0(Q). Let F be an algebraically closed field of characteris-tic p and let V be an irreducible F[PQ] module which does not have Z(Q) in itskernel. Then V is an algebraic module.

PROOF. By Lemmas 2.3 and 3.6 either V ® V or V ® V ® V ® V is alge-braic. Hence by definition V is algebraic.

4. Proof of Theorem 1. Let W be an irreducible F[G] module and let q be theBrauer character afforded by W. It may be assumed that q is faithful and soOp(G) = <1>. The proof is by induction on p(1) = dimF W.

If q is induced by a character q)0 of a proper subgroup then q)0 is algebraic byinduction and so q is algebraic by Lemma 2.3. Thus it may be assumed that q isprimitive, i.e. q is not induced by a character of any subgroup. Let H be aminimal normal noncentral subgroup of G and let be an irreducible con-stituent of PH. As G is p-solvable, H is a p'-group. Therefore G = is theinertia group of . Since H is noncentral (l) > 1. Thus W - V ® W by Fong'stheorem where (l) = dimF V. If dimF V < q(1) then dimF W < p(1) also andso V and W are algebraic by induction. Hence W is algebraic by Lemma 2.2.Thus p(1) = (l) and so VH is irreducible. Hence if P is a Sr-group of G, thenVHP is irreducible. Furthermore VI(VHP)c and so by Lemma 2.3 it suffices toshow that VHp is algebraic. Thus by changing notation it may be assumed thatG = HP. Suppose that P C G° <G. It suffices to prove the result for G° byLemma 3.3 since V I(Vcfc

We will now consider two cases depending on whether H is solvable or not.Suppose first that H is solvable. The minimality of H implies that H is an

extra-special q-group for some prime q p. Thus W is algebraic by Lemma 3.7.Suppose finally that H is not solvable. Let Z = Z(G). The minimality of H

implies that Z C H' and H/Z - M1 X ... X Mk, where M, ^ M for all i andsome simple group M. Then (M,) is the set of all conjugate subgroups of M,and P acts as a transitive permutation group on the set (M,). There exists agroup H° with Z(H°) C Ho and H0/Z(H°) - M such that H is a homomorphicimage of H1 X ... X Hk with H; - H° for all i. Thus G is a homomorphicimage of d = P(H1 X ... X Hk), and P permutes the set (H,) transitively.Therefore PH = = IP;_, ,, where , is an irreducible character of H whosekernel contains all Hj for j i. Hence I G : k. There exists an irreduci-ble Brauer character q), of with (9)1)H = ,. Hence i = q. Since q isprimitive this implies that G. Hence k = 1 and H/Z is simple. As H/Z


is well behaved, a Sp-group of G is cyclic and so there are only a finite numberof indecomposable F[H] modules (up to isomorphism). Hence by Lemma 2.1.W is algebraic.

REFERENCES

1. J. L. Alperin, On modules for the linear fractional groups, Internat. Sympos. on Theory of FiniteGroups, Univ. of Tokyo Press, Tokyo, 1974, pp. 157-163.

2. T. R. Berger, Irreducible modules of solvable groups are algebraic, Proc. Conf. on Finite Groups,Academic Press, New York, 1976, pp. 541-553.

3. E. C. Dade, Une extension de la theorie de Hall el Higman, J. Algebra 20 (1972), 570-609.

YALE UNIVERSITY

FINITE COMPLEX LINEAR GROUPS OFDEGREE LESS THAN (2q + 1)/3

PAMELA A. FERGUSON

Throughout this talk, G denotes a finite group. If T and R are characters of G,then TR is a sum of irreducible characters of G. However, it is generally difficultto determine the multiplicities of irreducible characters in TR. In this talk wewish to discuss a case in which G has some characters (A1) such that themultiplicities of irreducible characters in certain products A;Aj may be de-termined. We will show how this information may be used to obtain stronginformation about G.

G satisfies Hypothesis 1 if G is a finite group with a nontrivial Sylow psubgroup P such that CG(x) = CG(P) for all x E P *. If G satisfies Hypothesis1, then P is an abelian trivial intersection group. Let N and C denote thenormalizer and centralizer of P respectively. C = P X V where V is the groupof p'-elements of C, and N/V is a Frobenius group. Let IN/CI = n andP I = q. We wish to discuss the following theorem.

THEOREM. Assume G satisfies Hypothesis 1, N/C is cyclic, and G has a faithfulirreducible complex character A such that A(l) < (2q + 1)/3; then one of thefollowing conditions is satisfied:

(i) PA G,(ii) G/ Z(G) _- PSL(2, q),(iii) P I = 3 and G1 Z(G) S4 or A 5,(iv) API = 5 and G/Z(G) A6,(v) P1 = 7 and G/Z(G) - AT

REMARKS ON THEOREM. The condition A(1) < (2q + 1)/3 is really a numeri-cal way of guaranteeing that AN has exactly one irreducible constituent whosekernel does not contain P.

If q = p, the theorem was proved by Brauer [1] and Tuan [8]. Clearly, theassumption that N/ C is cyclic is superfluous in this case.

If A(1) < (q - 1)/2, then Brauer [1], Leonard [6], and Sibley [7] showed thatPA G. They did not use the assumption that N/ C is cyclic. It would be nice to

1980 Mathematics Subject Classification. Primary 20C15.® American Mathematical Society 1980

413

414 P. A. FERGUSON

eliminate the assumption that N/C is cyclic for A(1) > (q - 1)/2. Work isbeing done on this problem.

The groups PSL(2, q) do have irreducible faithful complex characters ofdegree (q - 1)/2 for certain values of q.

Reductions. We assume q > p. First, we consider the case that A(1) =(q - 1)/2 and then (q - 1)/2 < A(1) < (2q + 1)/3. The proof for A(l) _(q - 1)/2 is much easier than that for larger values of A(1), but the plan ofattack is generally the same. In both cases we reduce to the following situation:V = Z(G), Z(G) is cyclic, n = (q - 1)/2, and G/Z(G) is simple. Even afterthese reductions, the argument is lengthy. (We must consider A,A,,,,, products ofexceptional characters in various blocks.) Therefore, we will assume for the restof this talk that V = Z(G) = 1, n = (q - 1)/2 = A(l), q > p, and G is simple.Again, the proof for this case serves as a model for the more general situation.The proof may be divided into three steps.

A. Decompositions. All the information in this section is found in Leonard [5]with slightly different notation.

We first describe the irreducible characters of N. Since N is a Frobenius groupwith N/PJ = n = (q - 1)/2; there are two irreducible characters A1, A2 of Nsuch that A,(1) = A2(1) = n. Moreover, A; vanishes off P for i = 1, 2. N/P cyclicimplies that there are n linear irreducible characters (X,11 = 1, . . . , n) whichcontain P in their kernels. (Al, A2, X111 = 1, . . . , n) is the full set of irreduciblecharacters of N.

Let Xm be a nonprincipal nonexceptional character of G which does notvanish on P'. We define the following nonnegative integers: em = (Xm, A; ),fl., = (Xm, e) for 1 = 1, ... , n, fm = 1%1 fmr, and dm = fm - em. Then

(Xm)N = em(A1 + A2) + fmlX!' (1)1=1

G simple implies that em > 1. If x E P*, then Xm(x) = dm. Now Xm(1) >em2n > q - I implies that A is an exceptional character.

Since n = (q - 1)/2, there are two exceptional characters Al, A2 of G. Wechoose notation so that A = A,. We define the following integers: a = (A,, A°),and b,=(A1,XI2) for 1=1,...,n. Then (A;)N=(a+e)A,.+aA)+1lb,X,where j i and e = ± 1. Now A,(1) = A(l) = n implies that we may choosenotation so that e = 1, a = 0, and b, = 0 for 1 = 1, ... , n. Therefore, we obtain

(A)N = A. (2)

Let A = (Xm Id m > 0, X, 1G) and C = (X,,, I dm < 0). Leonard [5] impliesin our situation that

dm=n-1 andAUC

(3)

2 dmfm=n-1. (4)AUC

Let R be a complement to P in N, then R = <r>. AN = Al implies thatdet A(r) = a"(n+1)/2 where a is a primitive nth root of unity. Now G simple

FINITE COMPLEX GROUPS 415

implies that det A = 'G. Therefore, n is odd. It follows that A2 = A, so thatA2 = Al*

We next obtain

((A,A2)N, X,) = 1 for! = 1, ... , n and (5)

((A,)N, X,) = 0 for t = 1, ... , n. (6)

We shall show how (5) is proved. (The proof of (6) is similar.) Equation (2)implies that (AIA2)N - AJA2 = A1A1 Since X, has P in its kernel and is linear,and A; vanishes off P, (A,A,, X,) _ (A A,X,) = (A,, A,) = 1.

The proofs of the following equalities are found in Leonard [5].

(AIA2 - A2, Xm) = dm and (7)

(A,A2 - A;, A, + A2) = -1. (8)

We may now decompose A2 and A1A2. If Xm E C, then (7) implies that(Xm, Ai) > -dm > 0. Now (6) implies that for I = 1, . . . , n;-dm((Xm)N, k) < ((Ai)N, X,) = 0. Therefore, fm = 0 if Xm E C. Hence, (1) im-plies that Xm(1) = -2dmn if Xm E C.

Let Xm E A, then (7) implies that dm < (A,A2, Xm), (1) and dm > 0 imply thatfm, > 0 for some 1. Now (5) implies that dmfm, = dm((Xm)N, X) < ((A1A2)N, X,)= 1. Therefore, dm = 1 if Xm E A. Now fm. = 0 for Xm. E C may be combinedwith dm = 1 for Xm E A and (4) to obtain I A fm = n - 1.

Since A2 = Al, (lG, A,A2) = I. Therefore, lG + 1A X,,, C A,A2. Equation (1)implies that I + 1A 2nem + 71A fm < A,(1)A2(1) = n2. Now lAfm = n - I im-plies that ,AI < 2A em < (n - 1)/2. However, em = fm - 1 for Xm E A and2A em < (n - 1)/2 implies that n - l - CAI _ EA(fm - dm) _ YA em < (n -1)/2. It follows that IA = (n - 1)/2 and em = I for Xm E A. Hence Xm(1)2n + 2 if Xm E A. Degree considerations imply that

A,A2 = lG + 2 Xm. (*)A

Now (6), (7), and (3) imply that

A,+l2A2-2 dmXm where 1,+C

B. IA(y)j is small if y is not conjugate to an element of P. Since n = (q - 1)/2is odd, we may choose x, E P * so that x, and x' 1 are representatives of thetwo distinct conjugacy classes of elements of P # in G. Let r = G - Ug Pg.

If y E t, then A(y) = Al(y) = A2(y) is an integer. Since G is simple, JA(y)I< n - 1. Let

dm 2Xm(y)ay=1+ 2AUC Xm(1)

Step A implies that

t + Xm(y) dm Xm(y)Ry_ 2n+2-C 2nd,.A

416 P. A. FERGUSON

Equation (.) implies that

`Ym(y) A 2(y) - 1=2n+2 2n+2

Equation (..) implies that

dm2nd y) = 2n (- dm Xm(y)) = AZ(y)2n A(y)

Therefore,

py = 1 + A2(y) - 1 + A2(y) - A(y)2n + 2 2n

Now routine calculations imply that

GI

(anA(y)C- 2 y n9

and

cX X y= I9 II (Qy + (q - n)A(y)

l n

Further, cX .X y and cX,X

y are both nonnegative. Substituting the expression forQy in cX .X y and cX and performing some arithmetic yields

A2(y) - (n + 1)A(y) + n ) 0,A2(y) + (n + 1)A(y) + n ) 0.

Elementary calculus applied to these inequalities and IA(y)I < n - 1 now implythat A(y)I < 1.

C. I G I is "small" so that G is known. Standard arguments applied to (A, A) _1 imply

A 2(l) +f A2(y) = 19I n (9)r

Now (*) and IA I = (n - 1)/2 imply that (A1A2, A1A2) = (n + 1)/2. (A;),,, _A. implies that if x E Pthen A,(x)A2(x) = (n + 1)/2. Therefore we obtain

A4(l) + A4(y) n2

1 ) I G I - ( n 2 1 )22 nlqGJ

= (n2g1)n

IGI (10)

Part B implies that E. A4(y) = Er A2(y). Now (9) may be used in (10) to obtain

n4+IGIn-n 2_ (n+l)nIGIq 2q

It follows that I G = nq(q + 1). We may now show that G is a Zassenhausgroup and n = (q - 1)/2 implies that G - PSL(2, q).

Concluding remarks. The obvious question is "how large can A(l) be so thatthe conclusion of the theorem holds?" The conjecture is q - 2. Feit [2] hasshown this if p = q and IZ(G)I is odd. If IZ I is even, the problem is open.Moreover, the procedure described above will not work.

FINITE COMPLEX GROUPS 417

REFERENCES

1. R. Brauer, On groups whose order contains a prime number to the first power. 1, 11, Amer. J. Math.64 (1942), 401-420, 421-440.

2. W. Feit, On finite linear groups, J. Algebra 5 (1967), 378-400.3. P. Ferguson, On finite complex linear groups of degree (q - 1)/2, J. Algebra 63 (1980), 287-300.4. , Finite complex linear groups of degree less than (2q + 1)/3, J. Algebra (submitted).5. H. Leonard, Jr., Finite linear groups having an Abelian Sylow subgroup, J. Algebra 20 (1972),

57-69.6. , Idem II, J. Algebra 26 (1973), 368-382.7. D. A. Sibley, Finite linear groups with a strongly self-centralizing Sylow subgroup, J. Algebra 36

(1975), 319-332.8. H. Tuan, On groups whose order contains a prime number to the first power, Ann. of Math. (2) 45

(1944), 110-140.

UNIVERSITY OF MIAMI


A CRITERION FOR CYCLICITY

PETER LANDROCKI AND GERHARD O. MICHLER

Introduction. It is the purpose of this article to give a criterion for the cyclicityof a Sylow p-subgroup of a finite group G using the structure of the principalindecomposable projective FG-module P which is the projective cover of thetrivial FG-module I of the group algebra FG of G over a splitting field F ofcharacteristic p > 0.

Let J = J(FG) be the Jacobson radical of FG, and let S(M) be the socle of a(right) FG-module M. The heart of an indecomposable FG-module P is theFG-module H(P) = PJ/ S(P). So far, the heart of the principal indecomposableprojective FG-module P, has played an important role in the theory of permuta-tion groups of prime degree, see P. M. Neumann [9]. The strongest version ofBurnside's theorem on nonsolvable permutation groups G of prime degree passerts that the heart H(P,) of P, is simple. Of course, in this case a Sylowp-subgroup is cyclic and a T.I. set.

Here we are interested in a converse direction. Suppose that a Sylow p-sub-group D of the finite group G is an abelian T.I. subgroup of G such thatNG(D)/CG(D) is abelian. If the heart of the principal indecomposable FG-mod-ule P, is simple, then D is cyclic (Theorem 6).

Concerning our terminology and notation we refer to Dornhoff [1], Felt [2],Gorenstein [3] and Green [4].

An application of this to the groups of Ree-type is in progress.

The criterion. In this section the proof of our criterion is given. It depends onseveral properties of the Green correspondence, which we state first.

Let G be an arbitrary finite group, and let F be a field of characteristic p > 0,which is a splitting field for G and all its subgroups. For any pair of FG-modulesX, Y and a subgroup U of G denote (X, Y)U := HomFU(X, Y).

(X, Y)1 G consists of all FG-module homomorphisms from X into Y factoriz-ing through a projective FG-module P, and

(X, Y)G (X, Y)G/ (` , Y)1.G.

1980 Mathematics Subject Classification. Primary 20C20, 20D20.'Supported in part by the Danish Natural Science Research Council.


419

420 PETER LANDROCK AND G. O. MICHLER

X ° Y denotes any extension of X by Y so that there exists an exact sequence0 - Y - X ° Y - X - 0. The Heller module QM of a finitely generated FG-module M is the kernel of a minimal projective resolution 0 F- M F- P, where Pis a projective cover of M, see J. A. Green [4]. In particular, also Q-1M exists.

The following subsidiary result is Felt's lemma [2, Theorem III. (5.9)].

LEMMA 1. (X, Y)l = (OX, 0

From J. A. Green [5, Lemma 3.2], we quoteI

LEMMA 2. If X or Y is a simple nonprojective FG-module, then (X, Y)'(X, Y)G

Notation. Let D be any p-subgroup of G and H any subgroup of G containingNG(D). Suppose that D is a T.I. subgroup of G. Then for every nonprojectiveindecomposable FG-module U there is a nonprojective indecomposable FH-module f U and a projective FH-module U' such that UH = f U ® U'. ByGreen's correspondence theorem (see Feit [2, Theorem III.(5.6)]) fU is uniquelydetermined by U up to FH-module isomorphism, the Green correspondence f isa bijection between the the isomorphism classes of the nonprojective indecom-posable FG-modules and the isomorphism classes of the nonprojective indecom-posable FH-modules. The converse off is denoted by g. In particular, gfU = U.

Since D is assumed to be a T.I. subgroup, Theorem III. (4.12) of W. Feit [2]and Theorem 4.5 of J. A. Green [5] assert

LEMMA 3. (a) If U, V are nonprojective indecomposable FG-modules, then(U, V)G (fU,fV)H

(b)QfU~fQ U.(c) If X, Y are nonprojective indecomposable FH-modules, then (X, Y)H =

(gX, 8Y)c(d) S2gX gQX.

The following subsidiary result is well known and due to S. A. Jennings [7].

LEMMA 4. Let D be a p-subgroup and let v be the rank of its Frattini factorgroup D/(D(D). Then

v = dimF(S(FD/I)) = dimF(ExtFO(I, I)),

where I denotes the trivial FD-module.

LEMMA 5. Let D be a normal abelian Sylow p-subgroup of the finite group Gsuch that G/CG(D) is abelian. Then the following assertions are equivalent:

(a) D is cyclic.(b) The socle S(P1/I) of PI/I is one dimensional.

PROOF. If D is cyclic, then PI is uniserial and G/CG(D) is abelian. Thereforeall simple composition factors of PI are one dimensional, and dimF S(P1/I) _1. Therefore it suffices to show that (b) implies (a).

Let v be the number of generators of the p-groups D. As D is normal,Villamayor's theorem (see [8, Theorem 11.8]) asserts that the projective FG-mod-ule PI and the projective FD-module FD have the same socle and Loewy series.

A CRITERION FOR CYCLICITY 421

Therefore by condition (b) and Lemma 4 we obtain

v = dimF(S(FD/I)) = dimF(S(P1/I)) = 1.REMARK. If G/CG(D) is not abelian, then, in general, the assertions (a) and

(b) of Lemma 5 are not equivalent.

THEOREM 6. Suppose that the Sylow p-subgroup D of the finite group G is anabelian T.I. subgroup of G such that NG(D )/ CG(D) is abelian. If the heart of theprincipal indecomposable projective FG-module PI is simple, then D is cyclic.

PROOF. Let E be the simple heart of PI, and let PE be the projective cover ofE. Then E is the unique simple FG-module extending I nontrivially. Further-more, I occurs only once as a composition factor of PE, and I does not occur asa composition factor of any indecomposable projective FG-module P, whereP PI or P PE. Therefore S0 E/ E = I ® M, where I is not a compositionfactor of the FG-module M. It follows that for every indecomposable FG-mod-ule Y containing I as a composition factor I either belongs to the head or to thesocle of Y.

Let b be the principal block of N = NG(D) and let B be the principal block ofG. By Green's correspondence theorem there is a bijection g between theisomorphism classes of the nonprojective indecomposable b-modules and theisomorphism classes of the nonprojective indecomposable B-modules, because Dis a T.I. subgroup of G.

Let IN be the trivial FN-module. Suppose that v = dimF(ExtFN(A, IN)) mt- 0for a simple FN-module A. By Hilton and Stammbach [6, p. 142] ExtFN(A, IN)

(OA, IN)'' . Hence by Lemmas 1 and 3 we have

v = dimF[(S2gA, I)' ] = dimF[(gA, SZ 'I)G].

As ExtFN(IN, IN) = ExtFG(I, I), we may assume that A z IN, because other-wise, dimF P1 = 3, and D is a cyclic group of order 3. Thus

0 = (A,I)N=(A,I)N(gA,I)G=(gA,I)Gby Lemmas 2 and 3. Similarly, (I, gA)G = 0. Hence I is not a composition factorof the indecomposable FG-module gA. Since SZ-'I has socle series s, it followsthat

v = dimF[(gA, SZ-'I)c] = dimF[(gA, E)G] (*)

As SZE/E = I ® M, where I is not contained in M, we obtain from (*) that

v = dimF[ (I , SZgAJ = dimF[(E, S2gA)G], (**)L E GI

because

dimF[(gA, E)G] = dimF[(E, 0gA)G].

For P a projective FG-module, let P denote the corresponding projectiveRG-lattice. By our assumption, the character of PI is of the form 1 + , where 1is the trivial character and is some irreducible character of G. Likewise, thecharacter of PE is of the form + q, where q) = 0, q some character.

422 PETER LANDROCK AND G. O. MICHLER

Furthermore, A lifts to a (unique) RN-lattice, A. Hence 1gA and 02gA lift(uniquely) to RG-lattices SZgi and SZ2 i as D is T.I. By (**) the character ofSZg is of the form v + X where X) _ (1, X) = 0. Moreover, dim,.,[(I, 02gA)]= v. Assume in the following that 02gA , I. _

By our observations above, the character of 020' is of the form v 1 + X',where (1, X) = 0. Hence (S22gA, I) , 0 as well and consequently, X' = w + X"where w 0 and X") = 0. Thus 02gA has exactly v + w composition factorsisomorphic to I, contains a submodule V isomorphic to (E)' and a factormodule isomorphic to (E)°. We now claim that 0 2ii contains an R-puresubmodule V such that (V + S 2taw)/S22i Zr = V and the character of V isTo see this, let Q be the injective hull of 0 2gA and let Xi C V, Xi -- E,i = 1, ... , w and V = Y-Xi. Let q)i e (PI, SZZgA) with gi(PI) = X.. Then q)i liftsto Vii: DI - 02 A C Q and q(PI) is contained in a direct summand of Qisomorphic to PE. Now the structure of PI and PE implies that Xi = A(PI) isR-pure in Q with character 1, and V = Y- ® Xi will do.

Thus (02gA / V, I) = 0 a contradiction as 0 2gA / V lifts to S22I/ V withcharacter vl + X".

Hence 02gA = I and v = 1. Moreover this determines A uniquely, andconsequently N satisfies condition (b) of Lemma 5, and the theorem follows.

Discussions with Jorgen Brandt have been useful.

REFERENCES

1. L. Dornhof1, Group representation theory. Part B, Marcel Dekker, New York, 1972. MR 50#458b.

2. W. Feit, Representations of finite groups, Lecture Notes, Yale University, New Haven, 1969.3. D. Gorenstein, Finite groups, Harper and Row, New York, 1968. MR 38 #229.4. J. A. Green, Vorlesungen uber modulare Darstellungen endlicher Gruppen, Vorlesungen Math.

Inst. Univ. Giessen, 1974. MR 50 # 13235.5. , Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1970), 197-213. MR 50

#2323.6. P. J. Hilton and U. Stammbach, A course in homological algebra, Springer-Verlag, Berlin and

New York, 1970. MR 49 # 10751.7. S. A. Jennings, The structure of the group ring of a p-group over a modular field, Trans. Amer.

Math. Soc. 50 (1941), 175-185. MR 3, 34.8. G. Michler, Blocks and centers of group algebras, Lecture Notes in Math., vol. 246, Springer-

Verlag, Berlin and New York, 1972, pp. 429-563. MR 48 # 11274.9. P. M. Neumann, Permutationsgruppen von Primzahlgrad and verwandte Themen, Vorlesungen

Math. Inst. Univ. Giessen, 1977.

AARHUS UNIVERSITET, DENMARK

UNIVERSITAT ESSEN, FEDERAL REPUBLIC OF GERMANY

A CHARACTERIZATION OF GENERALIZEDPERMUTATION CHARACTERS'

DAVID GLUCK

Permutation characters and permutation modules play an important role infinite group representation theory. It is therefore desirable to know whichgeneralized characters of a group are generalized permutation characters-that is,integral linear combinations of permutation characters. All characters of sym-metric groups are generalized permutation characters, as are all rationallyrepresented characters of p-groups. Several papers on generalized permutationcharacters have appeared but as far as we know ours is the first characterizationof generalized permutation characters for arbitrary finite groups. See the intro-duction of [6] for a discussion of some recent work on generalized permutationcharacters.

Our approach to the problem involves systematic use of p-integral linearcombinations of irreducible characters of a group G, rather than merely integrallinear combinations. Clearly a generalized character of G is a generalizedpermutation character if and only if it is a p-integral combination of permu-tation characters for all p dividing I G 1.

Before we state our main theorem we must give two definitions. For the first,let G be a group, p a prime number, and <x> a cyclic p'-subgroup of G. Let X bea rational-valued class function on G, and let Px be a p-Sylow of CG(x). Wedefine a class function Xx on Px by yz(p) = X(px) for p E P. For the seconddefinition, let H be a not necessarily normal subgroup of a group G and let 0 bea class function on H. We say that 0 is G-invariant if 0 has the same value onany two G-conjugate elements of H. In the statement of the main theorem whichfollows, Px will denote ap-Sylow of NG<x> containing P.

MAIN THEOREM. Let X be a rational-valued p-integral combination of irreduciblecharacters of G. Then X is a p-integral combination of permutation characters of Gif and only if Xx extends to an NG<x>-invariant p-integral combination ofrationally represented characters of P, for all p'-elements x of G.

1980 Mathematics Subject Classification. Primary 20015.'This report appears here with the permission of Academic Press. It will appear as part of a paper

with the same title in the Journal of Algebra, vol. 63, copyright 1980.® American Mathematical Society 1980

423

424 DAVID GLUCK

To give some indication of our proof of the main theorem we must introducethe notion of a rational p'-section. If (x> is a cyclic p'-subgroup of G, therational p'-section S. is defined to be the set (g E G I(gp.> -<x>), wheredenotes G-conjugacy. We have shown in [4] using the results of Dress [1] thatthe characteristic function IS, of S. is a p-integral combination of permutationcharacters of G. Therefore a class function X on G is ap-integral combination ofpermutation characters if and only if Xlsx is for each rationalp'-section S. Thismakes possible the reduction to p-groups in the main theorem. The maintheorem also leans heavily on the fact that all rationally represented charactersof p-groups are generalized permutation characters. It should be mentioned thatsome of our ideas resemble those of Roquette [7], although he did not studygeneralized permutation characters and we do not use his results.

Our main theorem certainly does not make it feasible to compute thegeneralized permutation characters of any finite group. However, the compu-tations are feasible in many nontrivial cases. For example, there should be noproblem in applying our theorem to GL(2, q) or SL(2, q). Furthermore there arenumerous individual groups, the Mathieu group M11 for example, whose gener-alized permutation characters can be computed very quickly from the maintheorem, but which are not covered by any previously known general results ongeneralized permutation characters.

REFERENCES

1. A. Dress, A characterization of solvable groups, Math. Z. 110 (1969), 213-217.2. A. Dress and M. Kuchler, Zur Darstellungstheorie Endlicher Gruppen I, Bielefeld, 1971.3. W. Feit, Characters of finite groups, Benjamin, New York, 1967.4. D. Gluck, A character table bound for the Schur index, Illinois J. Math. (to appear).5. J. Rasmussen, Rationally represented and permutation characters of nilpotent groups, J. Algebra

29(1974), 504-509.6. , Artin index of faithful metacyclic groups, J. Algebra 46 (1977), 511-522.7. P. Roquette, Arithmetische Untersuchung des Charakterringes einer endlichen Gruppe, J. Refine

Angew. Math. 190 (1952), 148-168.8. L. Solomon, The representations of finite groups in algebraic number-fields, J. Math. Soc. Japan

13 (1961), 144-164.9. , The Burnside algebra of a finite group, J. Combinatorial Theory 2 (1967), 603-615.

10. T. Yamada, Induced characters of some 2-groups, J. Math. Soc. Japan 30 (1978), 29-38.

UNIv1,RsrrY OF ILLINOIS


CHARACTER TABLES,TRIVIAL INTERSECTIONS

AND NUMBER OF INVOLUTIONS

MARCEL HERZOG

This note consists of three numerical problems and their solution. Theseproblems were chosen in order to focus attention to some more general theoremsproved recently by the author and by other group-theorists. Three conjecturesare stated, following discussions of Problems 1 and 2.

We shall use the following notation. An element of G is called 2-central if itscentralizer contains a Sylow 2-subgroup of G. The set of irreducible charactersof G is denoted by Irr(G) and Irr#(G) = Irr(G) - 1G. A Sylowp-subgroup of Gis denoted by Sp. A subgroup H of G possesses the TI-property (in short H is TI)if H n H9 = 1 for every g E G - NG(H). The group G is a TI-group if S2 isTI.

Here are the problems.

Problem 1. Let G be a finite group satisfying the following conditions.(1) G has one class iG of 2-central 2-elements and two classes fIG and f2 of

non-2-central 2-elements;(2) 1 S21 = 64 where S2 E WAG);(3) Y(fl) = Y(f2) for every Y E Irr(G);(4) ICG(i)I = 4ICG(fl)I; and(5) G has a character X of degree at most I G I satisfying

I i f1, f2X 91d -5d -d

where d is an odd positive integer.What can we say about G?

Problem 2. Find all simple groups with N involutions for the following valuesof N: (a) 105; (b) 165; (c) 5649553.


425

426 MARCEL HERZOG

Problem 3. Find in each Case A, B or C all simple groups with the followingtwo columns in their character table:

Case A Case B Case C1 1 1 1 1 1

31 -1 5 1 56 0

32 0 10 2 76 4

33 1 10 -2 76 -411 -1 77(t) 5

12 0 77 -3120 0

133 5

1

where each row may be repeated an arbitrary number of times, with theexception of row 77(t) which should appear an odd number of times and of row

which should appear an even number of times.We proceed with solutions and discussions of the problems.Problem 1: Solution and discussion. We claim that G is a TI-group; in fact, G is

an odd extension of Sz(8).How can we see that? Clearly i is an involution. Our first question is what is

A = I i G n S I, where S E Sy12(G )? Surprisingly enough, we can give a definiteanswer to this question. By (5) we get

(Xs, ls)s = d(91 - 5A - (64 - A - 1))/64 = d(28 - 4A)/64 > 0.Thus A < 7. But the coefficient of 1 s in Xs is an integer, hence it is equal to 0(remember d is odd), and we conclude that

A = 7. (6)

Next we compute S n f G By (3) if = (fl i)G so f, and f2 are elements oforder 4 and

IS nf1I=ISnfzI=(64-8)/2=28. (7)

Now denote by n(S, ctg x) the number of conjugates of S containing x E S andby n(x, ctd S) the number of conjugates of x contained in S. Then for eachx E S we get

I G: CG (x) I n(S, ctg x) = I G: NG (S) I n(x, ctd S).

Applying this formula to x = i and x = f (= fl or f2) and dividing we get, using(4), (6) and (7)

n(S, ctg i) = ICG (i) I n(i, ctd S) = 4 7= 1.

n(S, ctgf) ICG(f)I n(f, ctd S) 28

So

n(S, ctg i) = n(S, ctgf). (8)

CHARACTER TABLES 427

In particular, n(S, ctg f2) = n(S, ctgf). Clearly, f E S = f2 E S. By (8) we alsoget f 2 E S =f E S. Thus

f E Sr=fs E S. (9)

Next consider the square roots of i

B=If eEfGufzle2=i}I.Thus number is independent of the choice of i E i G and it follows by (4) that

B = IfG U fz I/IIGI = 21CG(I)I/ICG(fl)I = 8.Consider, finally, the set

(10)

= n {S E Syl2(G)Ii E S}and let C = 12 n iG1. Clearly C > 1. Moreover, by (9) and (10) 2 contains 8Celements of order 4. Thus

12:1=1+C+8C>8.But is a 2-group, hence 8112:1 implying C > 7 and 12:1 > 82 = IS I. Thus121 = S1 and consequently G is a TI-group.

This computation is a special case of the proof of

THEOREM 1 (D. CHILLAG AND M. HERZOG [2]). The TI-property for S2 can beread from the character table of a finite group.

In our proof we rely heavily on the classification of TI-groups by Suzuki [6].Problem 1 is an example of our treatment of the nonsolvable TI-groups. Welearned at this conference from Martin Isaacs, that the TI-property for anarbitrary SP can be read from the character table of a p-solvable group. M.Isaacs proved

THEOREM 2 (M. ISAACS). Let G be a finite p-solvable group and let P ESylp(G). Then P is TI in G iff

(1) G has p-length 1.(2) All p-blocks have defect 0 or full defect.

In view of Theorems I and 2 we suggestConjecture 1. The TI-property for Sp can be read from the character table of a

finite group.During this conference, Alan R. Camina and the author proved that the

property: "S2 is abelian" can also be read from the character table of a finitegroup [1]. Moreover, it was also proved that the property "Sp is abelian" can beread from the character table of a p-solvable group. These results suggest

Conjecture 2. The property "Sp is abelian" can be read from the charactertable of a finite group.

Problem 2: Solution and discussion. We shall determine all simple groups with,say, 165 involutions, or 1,000,165 involutions, or any "such" number. But not163 or 1,000,163 involutions, for these numbers I do not know the answer. Themain hint is: N = 1 (mod 4). It has been known for several years that if a2-group S possesses N ° 1 (mod 4) involutions, then S is one of the following:cyclic, generalized quaternion, dihedral or semidihedral. In my note [4], I

extended this result to arbitrary groups. That is, I proved

428 MARCEL HERZOG

THEOREM 3. Let G be a finite group with N involutions. If N - 1 (mod 4), thenS2 is cyclic, generalized quaternion, dihedral or semidihedral.

Thus, if N - 1 (mod 4) and G is simple, I G I > 2, then S2 is either dihedral orsemidihedral. These simple groups have been classified. Surprisingly enough,each such simple group is uniquely determined by N. Details will appear inanother note [5]. By checking the list, one discovers that A7 is the only simplegroup with N = 105, for N = 165 the the answer is MI, and PSU(3, 49) has5649553 involutions.

Let me conclude this discussion withConjecture 3. The order of a simple group is determined by its number of

involutions.Caution. A8 and PSL(3, 4) are both of order 20,160 and each contains 315

involutions.Problem 3: Solution and discussion. We shall consider here in some detail only

Case A. The first column is clearly the degrees column, and suppose that thesecond column belongs to the element u E G. The differences X(l) - X(u) areintegral and denote their greatest common divisor by k (k = 32 in our example).Finally, define, for each X E Irr G, d(X) by

X(1) - X(u) = d(X)k.We have in Case A

1 u X(1) - X(u) = d(X)k1 1 0=0 32

31 -1 32 = 1.3232 0 32 = 1.3233 1 32 = 1.32

What can we say about this simple group?(a) u is conjugate to u-1, hence G is of even order;(b) d(X) = 1 for each X E Irr"`(G);(c) X(1) - X(u) = 32 for each X E Irr#(G), hence 2 divides each such

difference. By Theorem 4.7.5 in [3], u is a 2-element.(d) Let v 1 be a 2-element of G and suppose that v is not conjugate to u. By

(c) and the orthogonality relations we get (X runs over Irr(G))

0 = E X(v)X(1) - E X(v)X(u) = 32 E X(v),x x x1

hence

E X(v) = 0. (+)x,I

On the other hand, X(v) is a sum of X(1) 2kth roots of 1, where o(v) = 2k, andso is X(l). We get

X(1) = X(v) (mod P)

where P is the prime ideal lying over 2 in the integers of Q(IGIVf ). Thus

X(v) X(l) X(1)2=I (mod P)x I x96] X '&I

CHARACTER TABLES 429

in contradiction to (+). Thus each nontrivial 2-element of G is conjugate to u. Itfollows that

(e) u is an involution and S2 is elementary abelian.Applying the classification of simple groups with an abelian S2 one gets

G = PSL(2, 32). In general, when d(X) = 1 VX 1, one gets G = PSL(2, 2")What happens in Case B? There k = 4 and d(X) is not equal to 1 in general.

However, the following property holds.

d(X) is odd `dX 1 of odd degree.

Under these conditions, one can show in a similar way that S2 is abelian. Inaddition to PSL(2, 2"), one gets now PSL(2, q), q - 5 or 11 (mod 16). Thecolumns of Case B correspond to PSL(2, 11).

What about Case C? There d(X) are not all odd for X 1 of odd degree.However, the following condition holds.

E {d(X)I X 1, X(1) is odd) is odd.

This condition suffices in order to show that S2 is abelian. Moreover, it turns outthat each nonabelian simple group satisfies this property. Thus we get

THEOREM 4. Let G be a nonabelian simple group. Then: S2 is abelian iff

E {d(X)I X 1,X(1)isodd) = 1 (mod 2).

The columns of Case C correspond to J,.These and related results are now in preparation for publication.

REFERENCES

1. A. R. Camina and M. Herzog, Character tables determine abelian Sylow 2-subgroups, Proc.Amer. Math. Soc. (to appear).

2. D. Chillag and M. Herzog, Defect groups, trivial intersections and character tables, J. Algebra, 61(1979), 152-160.

3. D. Gorenstein, Finite groups, Harper and Row, New York, 1968.4. M. Herzog, Counting group elements of order p modulo p2, Proc. Amer. Math. Soc. 66 (1977),

247-250.5. , On the classification of finite simple groups by the number of involutions, Proc. Amer.

Math. Soc. 77 (1979), 313-314.6. M. Suzuki, Finite groups of even order in which Sylow 2-subgroups are independent, Ann. of Math.

(2) 80 (1964), 58-77.

TEL-AVIV UNIVERSITY

REPRESENTATION THEORY ANDSOLVABLE GROUPS:

LENGTH TYPE PROBLEMS

T. R. BERGER

I will speak only of finite solvable groups. Many of the problems I discusshave easy extensions to p-solvable groups. Often p-solvable problems are notinherently harder than solvable ones, so I will not discuss the p-solvable case.

1. Length type problems. I want to discuss problems which may be looselycalled "length type problems." With some torturing such conjectures can usuallybe put into the following form: "Suppose that f and g are functions from groupsto nonnegative integers. It is conjectured that

f(G) < g(G)for all groups G". Sometimes the exact form of f or g is left open or is onlypartially specified.

In order to fix ideas, let me state such a problem.CONJECTURE. Let p be a prime. There is a linear function fp(k) such that if G is

a group and P is a subgroup of G of order p k contained in precisely one Sylowp-subgroup of G then the p-length lp(G) is bounded above by fp(k)

lp(G) < fp(k).By choosing P minimal among p-subgroups of G satisfying the hypotheses

and setting I P I = p k, it is clear that we may view fp as a function on groups sothat this is a length type problem.

Using techniques developed by Dade [21] it has been shown [57] that anexponential bound fp exists when p > 2. In principle, one should be able toextend these methods to include p = 2. Kegel informs me that if this case wereknown (with any bounding function fp) then locally solvable groups whichsatisfy Sylow's Theorem for the prime 2 would have bounded 2-length. It is notat all obvious that minor modifications of present methods can lead to a linearbound. In fact, it appears that a proof of the complete conjecture would reveal

1980 Mathematics Subject Classification. Primary 20D10, 20C15.m American Mathematical Society 1980

431

432 T. R. BERGER

quite deep information about the 2-series of a solvable group. Rae [94] has givensome results on the linear bound case.

REMARK. Before continuing my discussion of this conjecture, let me commenton general length type problems and why some of them are interesting. I thinkeveryone is aware of the Hall-Higman paper [52] and its treatment of p-lengthproblems. I would wager that most group theorists would view the methods ofthis paper as transcending the p-length inequalities given in it. I think all lengthtype problems can be measured by this type of scale. That is, a good conjecturefocuses attention on deep interconnections within a group; and a solution, to adegree, exhibits these interconnections. If we understood more, our conjectureswould focus on these interconnections rather than upon numerical comparisons.One advantage of the numerical approach is that it forces us to discover theinterconnections.

Just as with conjectures about primes, it is easy to make reasonable lengthtype conjectures. A solution to a crude problem will often, but not always, revealless than a solution to a more refined problem. It seems to me that existingcrude solutions should only be improved if either (1) the refined version isneeded or (2) the refined version will shed more light on general theoreticalquestions. In particular, to improve a certain bound from exponential to linearin one problem may be less important than changing a coefficient from 3 to 2 insome term of a bound in a different problem. Straightforward refinement ofexisting proofs is probably not worthwhile for its own sake.

In the case of the conjecture stated above, it is almost certain that proof of theexistence of a linear function f, would reveal far more than a solution giving anexponential function fP. I hope my remarks explain why one would want toattack or improve length type problems. Further, I hope I have indicated how toweigh the importance of the various length type problems.

II. Translation to representation theory. Once a conjecture has been statedgroup theoretically, it must be translated into a statement of representationtheory. This is often a difficult problem and takes us closer to the grouptheoretic interconnections we are supposed to reveal. One important point toremember is that our representation theoretic statement need only imply theoriginal conjecture, not be equivalent to it.

Back to our conjecture. Suppose that P is a p-subgroup of a group G, M is aSylow p-subgroup of OP,,(G) normalized by P and that P n M = 1. Assumethat we can choose g E OP.(G) such that g is centralized by P but not by M.Then Pg = P but M and Mg lie in distinct Sylow p-subgroups of G. Inparticular, if Q > PM is a Sylowp-subgroup then Qg Q but P < Q n Q. Ifthe conjecture is correct, then this cannot happen. Examples show that theelement g often exists so that we can make such an observation the basis for aproof of the conjecture. We now translate into representation theory.

Problem. Suppose that p is a prime, G is a group with OP.(G) = 1, and P is ap-subgroup of G such that P n OP(G) = 1. Assume that V is a faithful irreduciblek[G]-module over a field of characteristic unequal to p. Under what conditions isthere a vector v E V fixed by P?

REPRESENTATION THEORY AND SOLVABLE GROUPS 433

Observe that Op(G) (= M) is faithful and fixed point free on V. Thus v playsthe role of g in the translation. If we push far enough up the p-series, say

H = Op-p...p'p(G),"P's

and require that P n H = I then it is almost certain that the vector v existsfixed by P. A proof of this statement would yield a function in the conjecture ofthe form fp(k) = (m + 1)k + b. Thus the problem is one possible representationtheoretic interpretation of the conjecture.

III. A method of proof. Once we have restated a problem for representationtheory it usually has the following form. We are given a group G, a k[G]-moduleV over some field k, and a set of hypotheses 'C from which we wish to derive aconclusion (?. If we have been shrewd, the hypotheses 9C will be inductive andthere is a very general method which may be applied. This method does notrepresent an automatic solution but rather is a recipe which gives a naturaldivision into steps.

I will now outline the steps I would attempt to follow in completing a proof ofthe above problem.

A. Enlarge the field. Show that we may assume k is algebraically closed. Thismay not be possible and yet the rest of the method may apply but with greaterdifficulty. It is usually good to get as many roots of unity into k as possible.

B. Show that we may assume V is irreducible. This step usually is eitherobvious or of extreme difficulty. I mention it only because it parallels a laterstep.

C. Show that we may assume V is primitive. As a simple illustration supposethat V _ UIG where U is a k[H]-module for some subgroup H of G. ByMackey's Theorem

VIP '=' UI°I J -1®U ® XIHFnFI

where x runs over a set of H, P-double coset representatives in G. Suppose thatby mathematical induction (since IHI < IGI) we can find a fixed point u E Ufor H n P. If y,, y2, ... , y, is a transversal of H n Pin P then v= u 0y,+ + u 0 y, gives a fixed point of P in V. We can now conclude thatH = G and V is primitive.

Of course, in this problem and in most others, matters are not this simple.Sometimes we may not be able to choose H arbitrarily. We may be forced todiscover deeper group theoretic properties in order to continue. Let me illustratehow an attempt to complete this step leads to a representation theoretic theoremfor solvable groups.

There is an easy and nice way to obtain submodules of V which induce V viaClifford's Theorem. Assume that N 4 G and

VIN =VI ®VVED ... ®V, (..)where the V; are homogeneous components. Let T be the stabilizer in G of Vl sothat Vl is an irreducible k[T]-module which induces V (i.e. VIIG - V). It maybe that we can argue on groups like T but not on others. Thus we would be ableto show that t = I in (..) for all normal subgroups N of G, i.e. V is aquasiprimitive module. This leads to the following theorem.

434 T. R. BERGER

THEOREM 17. If G is a solvable group and V is a quasiprimitive k[G]-moduleover an algebraically closed field k then V is primitive.

This is one example of the kind of facts which can surface when analyzinglength type problems.

Let's return to the method once again. In most papers in the literature, this iswhere the method stops. Authors now resort to special hypotheses andtremendous ingenuity to complete proofs. In fact, W. Feit in a special lecturewill discuss a clever idea which extends and short circuits [14] the proof of atheorem of mine at this step. But the method does not stop here.

D. Show that V is tensor indecomposable. If dim V = 1 we surely can completeany proof of a true theorem. If dim V > 1 it may be that V decomposesnontrivially as a tensor product

V = VI ®k V2 ®k ... ®k V, (rrr)where the k[G]-modules V; are projective with possibly nontrivial factor sets andeach dim V > 1. The determination of these factor sets is a very special case ofa general theory of Dade [24], [25], [28]. Further, Isaacs [67], [69] has developeda character theoretic approach to the same problem. To simplify matters, wemay pass to a representation group G* of G: 1 -* Z - G* -* G -* 1 whereZ < and assume that all Vi and V are nonprojective Inany case, by assembling information obtained from the action of G* on V, wehope we can show t = 1 in (***).

If we view step B as passing from a sum to a summand and D as passing froma product to a factor, then categorically these two steps are "the same," but indifferent categories.

E. Show that V is tensor induction primitive (superprimitive). If we assume thatG is faithful on V then F(G), the Fitting subgroup, is of symplectic type. In fact,

F(G) = Z(G)Ewhere E is a normal extraspecial r-subgroup of G for a prime r, Z(E) < Z(G),and E/Z(E) is a chief factor of G.

Shift attention from V to E = E/Z(E). The group G acts naturally viaconjugation on the vector space E. The induction structure of G on E reflectsdown into the tensor induction structure of V.

For those who are unfamiliar, let me describe the tensor induced module.Suppose that H is a subgroup of G, x,, x2, ... , x, is a transversal of H in G, andU is a k[H]-module. We view UIG as the direct sum

UIG= U®x1®U®x2® ®U®x,with a certain action for G. With essentially the same action we may define ak[ G ]-module

UJ®G=(U®xl)®k(U®x2)®k ... ®k(U®x,)called the tensor induced module. If dim U = 1 then dim UJ®G = 1 and UI®G isnothing more than the transfer of G into H/Ho where Ho = ker U. If dim U >1 then dim U I ®G is usually very large.

Categorically speaking, steps C and E are "the same." Careful analysis of thisstep has lead to the notion of form primitivity and the structure of formprimitive modules [11].


Although it is not nice aesthetically, for the time being this is where themethod ends. In practice if I have been able to carry a proof through all thesesteps, I have been able to complete it. Further, one is lead directly to thebedrock cases if one assumes A-E can be performed and asks, "Now what?"

Philosophically speaking, the method should continue. There are at least twoideas in this direction. The most obvious one is to transfer attention to theGF(r)-space E and proceed through steps A-E again. One must reinterpretresults from E to V, and step A causes difficulty: what is happening in V whenwe pass from GF(r) to a larger field? The second idea is to link the geometry ofthe symplectic space E tightly to the structure of V so that representationproblems for V become geometric problems for E. Both of these ideas have beenused to some degree. Though I cannot state any firm conjectures, this topic is aworthwhile one for further investigation.

IV. General remarks on the method. This method is not new and is verygenerally applicable. You may have already performed steps A-C in someproof. My mission here is to delineate the steps and stress D and E. Occasionallystep D is used, but more frequently "cocycle fear" prevents this. There is now atheory and this need not be a problem. Serre used tensor induction some timeago to study cohomology but its use in finite group theory is quite recent.

Aside. Incidentally, is there a simple group with an irreducible character which istensor induced from a proper subgroup? It would have to be a character of largedegree.

This method also applies to composite nonsolvable groups. However, in stepE the extraspecial group E could instead be a quasisimple group. With theclassification at an advanced stage, and the representation theory of Chevalleygroups shaping up, the topics I am discussing could be viewed in the setting ofall finite groups.

My series of Hall-Higman Type Theorems is an attempt to develop thismethod by proving a sequence of theorems which should be generally applicableto a wide class of length type problems. Paper V [11] focuses most closely on thegeneral method. The others [7]-[13] attempt to build up a suitable catalog foruse after step E is completed. In her thesis, my student B. Hargraves hascompletely solved the regular orbit problem posed in these papers.

V. Length type conjectures. I have illustrated length type problems anddiscussed a general method for attacking them. Now I would like to state a fewmore of these problems.

(1) p-length problems. This is where it all begins. The Hall-Higman paper [52]relates the invariants of a Sylow p-subgroup of a p-solvable group G to thep-length of the group. For example, let dP be the derived length of a Sylowp-subgroup and I, be the p-length of a group G. If p > 2 then Hall and Higmanshow that dp > lP. Fletcher Gross and I [18] have shown that 2d2 > 12. Sharpen-ing this result would lead to an interesting study of {2, 3)-groups (a verycomplex class of solvable groups). Further, the improvement would strengthennilpotent length bounds for certain factorizable groups.

F. Gross suggested the following problem: Is the p-length of a group boundedabove by the number of conjugacy classes of elements of orderp?

436 T. R. BERGER

(2) Character degrees. Isaacs [68] has shown that if G has r distinct complexirreducible character degrees then d < 2r where d is the derived length of G. Itis conjectured that d < r and this is known when I G I is odd. Groups of evenorder are more subtle and consideration of this bound should lead to a betterunderstanding of the interaction of 2- and 3-layers of G.

The correct bounding function f such that d < f(r) almost certainly growsmore slowly than r. Are there much better bounds f than r when G is ap-group?

(3) Fixed point free automorphisms. For me, this is where it all started. Assumethat H = AG is a group with normal solvable subgroup G and complement Awhere (IA 1, 1 G 1) = 1. Suppose that A acts fixed point freely on G i.e. CG(A) = 1.It is conjectured that the Fitting length h of G is bounded above by the numberm of primes (counting multiplicities) dividing JAI. If A is solvable then Kurzweil[85], using a method of Dade, has shown that h < 4m. For nilpotent wreath-freegroups A I have shown [15] that h < m. The various papers on this problemhave all revealed nice interconnections within the groups in question.

Kurzweil [84] has recently investigated what happens when the group A isminimal simple. His bounds are almost certainly very weak. This line of researchshould shed some light on the action of simple groups on solvable groups.

In this same vein, one can assume that CG(A) > 1 and of a given isomorphismtype. Some interesting results have been obtained by Gagola [33] which go along way toward describing the structure of G. Under certain circumstances,such problems can be quite revealing.

(4) Bounds from classes of solvable groups. Dade [21] has shown that theFitting length of a solvable group is bounded as a function of the compositionlength of a Carter subgroup. Since the Carter subgroups are projectors of theclass 61 of nilpotent groups, we may say that there is a bound of the 6X-lengthof a group G as a function of the composition length of an 6X-projector. Suchbounds probably exist where 6X is replaced by other saturated formations.

Schunck classes and Fitting classes were devised in order to describe certainunique characteristic conjugacy classes of subgroups in solvable groups: namely,projectors and injectors. For such a class 3E, the connections between the3E-length of a group G and various invariants of i-projectors (injectors) have notbeen sufficiently explained. Translating length type conjectures into the lan-guage of classes of solvable groups suggests many new length type problems aswell as possible new group theoretic interconnections. As an illustration for theclass of p-groups such problems would be those arising from the properties of aSylow p-subgroup and the p-length of the group, i.e. p-length problems.

Is there a function f such that for any Schunck class 3E and any group G, the3E-length I of G is bounded above by f(m) where m is the composition length ofan i-projector of G? Are there examples of classes 3E where f exists but cannotbe linear? (polynomial?)

(5) M-groups. The analysis of M-groups is not a length type problem.However, I mention them because the methods I have discussed here apply tothem as well. In fact, the conjecture on character degrees arose from considera-tions of M-groups [103]. The main problem is to classify group theoretically theclass of M-groups. I would be surprised if any such simple characterization evenexists. However, there is a wide gap between classes of groups known to be

M-groups and the class of all M-groups. The problem then would seem to be todescribe group theoretically very large subclasses of the class of M-groups.

In another direction, little positive information is known about subgroups ofM-groups. Any solvable group may be embedded in an M-group. Normal Hallnr-subgroups of M-groups are M-groups [30]. But normal subgroups of M-groupsneed not be M-groups [29], [121]. However, Isaacs has evidence that in groups ofodd order, normal subgroups of M-groups may be M-groups.

Extending work of Price [93], van der Waall [1171, [1201, [1241 has recentlyshown that the known list of minimal non-M-groups is complete. The final partsof this work should soon be in preprint.

VI. Closing comments. Let 1 = Go < G, < 4 G, = G be a compositionseries of a general group G. The two major problems of finite group theory are:

(1) Determine the structure of possible composition factors of G. This is theclassification.

(2) Determine the ways in which composition factors may be joined to form agroup. This is the "cohomology problem."

The cohomology problem is a "many body problem" and thus is attackedindirectly. That is, the consequences and relations implied by a given cohomol-ogy are studied. Thus length type problems are part of (2). As such they havegeneralizations to all finite groups. A fruitful area for a local group theoristprobably would be generalized length type problems. Such problems are not old,mainly because they could not be formulated before now. Nonetheless, many ofthese "new problems" will be every bit as challenging as the "old problems." Forthose who are interested, I have attempted to give a reasonably completebibliography on length type problems and methods.

REFERENCES BY SUBJECT

(1) p-length problems. [16], [49], [50], [52], [54], [55], [56], [57], [94], [95].(2) Character degrees. [5], [35], [36], [68], [72], [76], [77].(3) Fixed point free automorphisms. [2], [3], [4], [6], [15], [16], [19], [33], [39],

[411-[481, [611, [631, [801, [811, [831, [841, [851, [861, [1081, [1091, [112], [113], [1261,[128], [130].

(4) Bounds from classes of solvable groups. [21], [58], [60], [107].(5) M-groups. [29], [30], [64], [65], [66], [781, [91], [92], [93], [97], [100], [1051,

[106], [110], [111], [115]-[125], [133].

(6) Methods. [1], [7]-[14], [17], [20], [22]-[28], [32], [34] [37], [38], [40], [51], [53],[59], [67], [69], [70], [71], [731, [74], [75], [79], [88], [89], [90], [96], [98], [99], [101],[102], [106], [114], [127], [129], [131], [132].

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(1972), 73-78.


PART VI

Combinatorics


GROUP PROBLEMS ARISING FROMCOMBINATORICSI

MARSHALL HALL, JR.

1. The collineations (i.e. automorphisms) of a combinatorial system will forma group. The interactions between the group and the system can yield informa-tion some times in one direction some times in the other.

This paper gives a sampling of problems from combinatorics which lead to (ormight lead to) problems on groups. The second section gives some basic facts oncombinatorial designs for the reader not already familiar with the subject. Thethird section quotes some general theorem on collineations.

The fourth section deals with problems that have arisen from Steiner triplesystems. It is conjectured that a Steiner triple system with collineations doublytransitive on its points is either a projective geometry over GF(2) or an affinegeometry over GF(3). Progress has been made on this but it is still unsettled. Acurious side issue involving the foundations of geometry arises. In an AG(n, 3)an affine geometry over GF(3) every triangle generates a Steiner triple systemS(9) with 9 points which is the affine plane over GF(3). But the converse is falseand there are systems in which every triangle is in an S(9) but the entire systemis not an affine geometry. There is no analogue for planes with 4 or more pointson a line. These curious pseudo-geometries can be associated with commutativeMoufang loops of exponent 3.

The fifth section deals with some problems arising from Hadamard matricesH,,. With every Hadamard matrix H. there is associated a 3-design H in whichevery triple of distinct points occurs equally often in the set of blocks. When canthis design have a triply transitive group G? C. W. Norman [17] has shown thatwith n = 4t and t odd, the only case arising is n = 12 and the group is the triplytransitive representation of M11 on 12 points. N. Ito [14] has come close toshowing that if n = 4t with t even then the design is an affine space over GF(2).A further curious construction of orthogonal sequences by R. Turyn [24] has acertain similarity to orthogonality relations in group representations. If it could

1980 Mathematics Subject Classification. Primary 05B05, 20B25.'This research was supported in part by NSF Grant No. MCS-7821599.


445

446 MARSHALL HALL, JR.

be shown that these sequences always exist it would prove that Hadamardmatrices H. always exist when n = 1, 2 or n - 0 (mod 4), the only possiblevalues.

Other problems which have not been discussed here are difference sets or thedetermination of all quadruply transitive permutation groups. These are majorproblems which space does not permit to be included.

2. Combinatorial designs.2 Although there are an enormous number of systemscurrently studied in combinatorics, most of what will be said here will deal withwhat are called partially balanced incomplete block designs, denoted as PBIBD.These will be referred to briefly as designs. These were introduced by statisti-cians in design of experiments. A design consists of points (a1) and blocks ( Bj)together with an incidence relation a;IBj (read a; is on Bj or Bj contains a;) whichholds between certain points and blocks. Here v = l { a; ) l is the number ofpoints, b = I(B j)l is the number of blocks. Each block contains k points andeach point is on r blocks, and if a;, aj are distinct points there are exactly Nblocks containing both of them. Examples of designs are the following

81:1,2,4, v=b=7,B2:2,3,5, r = k = 3,B3: 3,4,6, X= 1,B4: 4, 5, 7,

B5: 5, 6, 1,

B6: 6, 7, 2,

B7: 7, 1, 3.

B1: 1, 2, 3,

B2: 1, 4, 5,

B3: 1, 6, 7,

B4: 1, 8, 9,

B5: 2, 4, 6,

B6: 2, 5, 8,

B7: 2, 7, 9,

B8: 3, 4, 9,

B9: 3, 5, 7,

B10: 3, 6, 8,

B11:4,7,8,B12: 5, 6, 9.

v=9,b 12,r=4,k=3,X=1,

2 All the material and references in this section may be found in the author's book [9].

GROUP PROBLEMS ARISING FROM COMBINATORICS 447

B1: 1,3,4,5,9, v= b 11,

B2: 2, 4, 5, 6, 10, r = k = 5,B3: 3,5,6,7, 11, X = 2,

B4: 4, 6, 7, 8, 1,

B5: 5, 7, 8, 9, 2,

B6: 6, 8, 9, 10, 3, (2.3)

B7: 7, 9, 10, 11, 4,

B8: 8, 10, 11, 1, 5,

B9: 9, 11, 1, 2, 6,

B10: 10, 1, 2, 3, 7,

811:11,2,3,4,8.

The incidence matrix A of a design D with parameters v, b, r, k, X is

A = [ a;3 ], i = 1, ... , v, aid = I if aiIBj,(2.4)

j = 1, . . . , b, aid = 0 otherwise.

There are two elementary relations which the parameters must satisfy

bk = vr,(2.5)r(k - 1) = X(v - 1).

The first of these counts incidences of points on blocks in two ways. For thesecond if ai is a point, it appears on r blocks and in each of these is paired withk - I further points, whereas ai must be paired in a block with each of v - 1other points exactly X times.

The incidence matrix A satisfies the following relations

AAT=(r-X)I+V, (2.6)J ,,A = kJ b, AJbb = rJb

Here A T is the transpose of A, I is the v dimensional identity matrix andj = 4,W Jb.b, J,,,b are the matrices of all l's of the appropriate dimensions.

With B = (r - X)I + AJ we easily find

det B = (r - X)°-'(r + vX - X). (2.7)

We must have r > X unless every block contains every point. Also to avoidtrivialities we assume k > 2.

The statistician R. A. Fisher proved the following inequality

b ) v. (2.8)

This follows readily from the fact that B is nonsingular and so the rank of A is v.If b = v and so also from (2.5) r = k we say that the design D is a symmetric

design. Of the examples above (2.1) and (2.3) are symmetric designs but (2.2) isnot.

For symmetric designs D we may strengthen the relations of (2.6) and have

AAT=(k-X)I+AJ=ATA, (2.9)AJ=kJ, JA=kJ.


Among other things this means that for a symmetric design D, its dual D*obtained by interchanging the roles of points and blocks is also a design. Inparticular any two different blocks of D intersect in exactly A points.

For a symmetric design (or v, k, X design) it has been shown by Bruck, Ryser,and Chowla that necessary conditions are

(1) If v is even, k - X is a square.(2) If v is odd the equation

x2=(k -A)y2+(-1)v 2 1Xz2

has a solution in integers (x, y, z) not all zero. (2.10)

The first of these two conditions comes immediately from the fact that for asymmetric design det B = (k - A)°-Ik2 = (det A)2. The underlying theory ofthe second condition is the deep Hasse-Minkowski theory of the rationalequivalence of quadratic forms.

It has been shown by H. J. Ryser that if A is a nonsingular v by v matrixsatisfying either of the first two relations of (2.9) and either of the last two thenall four relations hold and also

k(k - 1) = A(v - 1) (2.11)

which is of course the second of equations (2.5) for a symmetric design.A collineation (or automorphism) of a design D is a mapping a which is

one-to-one on points and one-to-one on blocks, which preserves incidence. Moreprecisely

DEFINITION OF COLLINEATION. The mapping a of (a,) U (B.) onto itself is acollineation if and only if a, --+ (a,)a and B. -* (B.)a are one-to-one maps of pointsonto points and lines onto lines such that (a;)aI(BB)a if and only if a;IBj.

We may write

a = (P, Q) (2.12)

where P is the permutation matrix of a on points and Q is the permutationmatrix of a on blocks. We easily see that if A is the incidence matrix of D then

P-'AQ = A.It is also obvious that if al = (P11 Q0, a2 = (P2, Q2) then

(2.13)

aia2 = (PIP21 QIQ2) (2.14)

Clearly under this composition the collineations of a design D form a group.A combinatorial system which is essentially a design is an Hadamard matrix

H of dimension n. By definition H,, = [h,,], i, j = 1, , .. , n, every h, 1 and

nI,,.

Examples of Hadamard matrices are

HI =[1], H2

(2.15)

1 1 1 1

1 -1 1 -1 (2.16)1 1 -1 -1

1 -1 -1 1


and writing - for -1

1 1 1 1 1 1 1 1 1 1 1 1

1 - 1- 1 1 1 - - - 1-1 - - 1 - 1 1 1 - - - 1

1 1-- 1 - 1 1 1 ---1 - 1 - - 1 - 1 1 1 - -

1 - - 1 - - 1 - 1 1 1 -H12- (2.17)1 - - - 1 - - 1 - 1 1 1

1 1--- 1- - 1 - 1 1

1 1 1-- - 1 - - 1- 1

1 1 1 1--- 1 - - 1-1 - 1 1 1 - - - 1 - - 1

1 1- 1 1 1- - - 1--It is known that if n ) 3 then n is a multiple of 4, n = 4t.

Clearly if H. is an Hadamard matrix then permuting rows or columns andchanging the sign of rows or columns also yields an H. Such Hadamards arenaturally considered equivalent. Monomial permutations with ± l's P and Qdescribe permutation and sign change of rows and columns respectively and soH. and P -1 HH Q are equivalent. In particular if

HH (2.18)

we say that a = (P, Q) is an automorphism of H. Obviously the automor-phisms of an Hadamard matrix form a group.

If signs are changed in an H. = H4, so that the first row and column consistentirely of +1's, then deleting the first row and column, the remaining +1's(columns as points, rows as blocks) form a symmetric design with

v=4t-1, k=2t-1, X=t-1. (2.19)

These parameters always satisfy the Bruck-Ryser-Chowla condition. From adesign with parameters of (2.19), bordering with a row and column of l'smaking other entries -1 we obtain an Hot. The design in (2.3) and the H12 in(2.17) are related in this way.

3. General results on collineations of designs. The book Finite geometries [5] byP. Dembowski is an enormous encyclopedia on results not only on geometriesbut also on incidence systems in general and includes designs. A reference toalmost any result in finite geometries or designs prior to the 1968 publicationdate is likely to be included.

Among important general papers that have appeared since then are a paper2-transitive symmetric designs [15] by William M. Kantor and On collineationgroups of symmetric block designs [1] by Michael Aschbacher.

A symmetric design with X = 1 has parameters v = n2 + n + 1, k = n + 1,X = 1 and is a projective plane of order n.

THEOREM 3.1 (OSTROM AND WAGNER [18]). A projective plane with a collinea-tion group doubly transitive on its points is necessarily the Desarguesian plane overa finite field GF(q), q = p' and n = q.

THEOREM 3.2 (PARKER [19]). A collineation a of a symmetric design has thesame number of fixed blocks as fixed points.

There is an extension of the Bruck-Ryser-Chowla theorem to collineations ofsymmetric designs. We call a collineation group standard if every nonidentityelement has the same number of fixed points. Clearly a group of prime order isstandard.

THEOREM 3.3 (HUGHES [13]). Suppose that a symmetric v, k, A design has astandard collineation group G of order m, fixing N points. Putting t = (v - N)/mande=(t+N- 1)/2then

x2 = (k - X)y2 + (-1)1mN-'Xz2

has solutions in integers not all zero.

THEOREM 3.4 (AsCHBACHER [1]). Let p be a prime dividing the order of thecollineation group of a symmetric v, k, X design. Then either p divides v or p < k.

Kantor gives the name H to the design of (2.3).

THEOREM 3.5 (KANTOR [15]). H,, is the only 2-transitive symmetric design, not aprojective space, for which k is prime.

This is just a sampling of theorems on collineations of symmetric designs.There do not appear to be any general theorems on collineations of nonsymmet-ric designs.

4. Group problems on Steiner triple systems. A design with k = 3 and X = 1 iscalled a Steiner triple system. The equations (2.5) bk = rv and r(k - 1) _X(v - 1) in this case require

v = 1 or 3 (mod 6). (4.1)

In 1853 Steiner [23] posed as a problem whether the condition (4.1) wassufficient for the existence of a design. This was solved affirmatively by Reiss[20] in 1859. Both papers appeared in Crelle's Journal. These writers were notaware that the problem had been posed and solved by Kirkman [16] in 1847 inan article in the Cambridge and Dublin Mathematical Journal, nor untilrecently did anyone else seem to be aware of Kirkman's work.

Although badly misnamed, Steiner triple systems are quite interesting in termsof combinatorial and group properties. A Steiner triple system with v points willbe designated as S(v) where for completenesss we include S(1) an isolated pointand S(3) a single block with three points. The designs S(7) and S(9) areexamples (2.1) and (2.2) and are unique up to isomorphism. There are twononisomorphic S(13)'s, 80 S(15)'s and for larger values of v the number of S(v)'sincreases at an astronomical rate.

In investigating automorphisms of Steiner triple systems [7] I proved twotheorems.

THEOREM 4.1. Let S be a Steiner triple system in which, for every point x, thereis an involution a of S which has x as its only fixed point. Then every triangle of Sgenerates an S(9). Conversely suppose that S is a Steiner triple system in whichevery triangle generates an S(9). Then for every point x of S there is an involutiona of S which has x as its only fixed point.


THEOREM 4.2. Let S be a Steiner triple system and suppose that for each triple ofS there is an involution whose fixed points are precisely the three points of thetriple. Then every triangle of S generates an S(7) or an S(9).

It is reasonable to conjecture that a Steiner system S whose automorphismgroup G = G(S) is doubly transitive on the points of S is necessarily either aprojective space PG(n, 2) of dimension n over GF(2) or an af fine space EG(n, 3)of dimension n over GF(3). This conjecture is still unresolved and is aninteresting problem in group theory.

In my original paper I showed that if G(S) is doubly transitive then Snecessarily contains an S(7) or an S(9). If every triangle of S generates an S(7)then from the axioms of projective geometry S is necessarily a geometryPG(n, 2).

In the affine geometry AG(n, 3) every triangle generates an S(9). But con-versely if every triangle generates an S(9), S need not be an affine geometry. Inmy original paper I produced an S(81) of this kind, generated by 4 points butnot the AG(3, 3) which would have 27 points. Francis Buekenhout [3] showedthat if every triangle of a geometric system generated an affine plane, then theentire system was an affine geometry provided that every line contained at least4 points. This S(81) shows that the Buekenhout result cannot be strengthed toallow 3 points on a line.

Some important progress on the conjecture was made by J. I. Hall [11], [12].Two of his results are the following

THEOREM 4.3. If every triangle of S generates either an S(7) or an S(9), theneither all triangles generate S(7)'s or all triangles generate S(9)'s.

THEOREM 4.4. If G(S) is doubly transitive and contains a solvable normalsubgroup N then S is an affine geometry AG(n, 3) and the minimal normal N isthe group of translations.

If every triangle of S generates an S(9) then the points of S may be made intoan algebraic system in the following way: Choose an arbitrary point and call itthe identity 1. If 1, a1, a2 is a triple of S define

ai = a2, a2 = a1, a1a2 = a2a1 = 1. (4.2)

If aI and b, are not on a triple then consider the triples

1, a a2

1, b1, b2

a2, b2, c2.

Here we definea1b1 = c2. (4.4)

These rules clearly make the points of S into a commutative loop. These weresuggested to the writer [8] by R. H. Bruck and it was shown that under this rulethe Moufang identity holds.

(xy)(zx) = [ x(yz) ] x. (4.5)

Conversely from a commutative Moufang loop of exponent 3 a Steiner triplesystem with every triangle generating an S(9) can be constructed.


These commutative Moufang loops M have been extensively studied. R. H.Bruck [4] showed that if M is generated by n elements then it is nilpotent ofclass at most n - 1, and in particular is finite and contains a nontrivial nucleusN. If x E N then for any y, z

(xY)z = x(yz), (Yx)z = Y(xz), (Yz)x = y(zx). (4.6)

Using this fact the writer [8] was able to show that if every triangle of Sgenerates an S(9) and if G(S) is double transitive, then M is necessarily anelementary Abelian group, and S is an affine geometry AG(n, 3).

Quite recently it has been shown by J. D. H. Smith [21], [22] that thenilpotence class of M may be in fact the maximum. Independently this has alsobeen shown by L. Beneteau [2].

5. Group problems from Hadamard matrices. Consider an Hadamard matrixH. (n = 41) which has been normalized so that its first row (but not necessarilyits first column) consists entirely of +l's. Take the pair of matrices H,, and -Hand delete the first row of each of these. Then each of the further 8t - 2 rowshas 2t + l's and 2t - l's. Taking the 4t columns as points let us take as blocksB i , j = 1, ... , 8t - 2, those points which are +1 in the appropriate row. Theseblocks always form a 3-design H +. More precisely we have a design D = H +with

v=4t, b=8t-2, r=4t-1,k=2t, X2=2t- 1, X3=t- I.

Here v is the number of points, b the number of blocks, r the number of times apoint appears in a block, k the number of points in a block, A2 the number oftimes a pair of distinct points occur together in a block and X3 the number oftimes three distinct points appear together in a block. It is the constancy of A3which makes D a 3-design.

Numbering the columns of H12 00, 0, 1, ... , 10 the 22 blocks of D - Hit are

D= H12

B12:

B13:

B 14:

B15:

0, 2, 6, 7, 8, 10

1, 3, 7, 8, 9, 10

2, 4, 8, 9, 10, 1

3, 5, 9, 10, 0, 2

B1: oc, 1,3,4,5.9B2: oo, 2, 4, 5, 6, 10

B3: oo, 3, 5, 6, 7, 0

B4: oo, 4, 6, 7, 8, 1

B5: oo, 5, 7, 8, 9, 2

B6: oc, 6, 8, 9, 10, 3

B7: oo, 7, 9, 10, 0, 4

B8: oc, 8, 10, 0, 1, 5

B9: oc, 9, 0, 1, 2, 6

B10: oc, 10, 1, 2, 3, 7

B11: oo, 0, 2, 3, 4, 8

B16: 4, 6, 10, 0, 1, 3

B17: 5, 7, 0, 1, 2, 4 (5.2)

B18: 6, 8, 1, 2, 3, 5

B19: 7, 9, 2, 3, 4, 6

B20: 8, 10, 3, 4, 5, 7

B21: 9, 0, 4, 5, 6, 8

B22: 10, 1, 5, 6, 7, 9

This design has automorphisms which on the points are of the form

a = (oc0)(1, 2, 4, 10)(3, 7, 5, 6)(8, 9),/3 = (oc)(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10),y = (oc)(0)(l, 3, 9, $, 4)(2, 6, 7, 10, 8),6 = (oc)(0)(1)(2, 6, 9)(3, 7, 4)(5, 10, 8).

These permutations (in fact 6 is redundant) generate the Mathieu group M11 inits triply transitive representation on 12 points. Here <a2, p, y> or < /3, y, 6 > isthe automorphism group of the design 2.3 and is the group L2(11) of order 660in its unusual representation on 11 points. This is related to the fact which Iproved [10] that the automorphism group of H12 modulo a center (-I, -I) is thequintuply transitive Mathieu group M12

A natural question about designs Hn+ is when they can have automorphismstriply transitive on the points. C. W. Norman [17] showed that with n = 4t if t isodd, t > 1, then necessarily n = 12 and we have essentially the design (5.2) andthe group (5.3).

It is conjectured that if is triply transitive and n = 4t with t even, thenthe design Hn+ is an affine geometry over GF(2). Noboru Ito [14] has madeprogress towards proving this.

In this situation we will have blocks of the following shape which will beexplained below:

B1: a1

B2: a1

... al

... a,bs

cl

.

.

..

..b,,

c!,

B1 : cl ... c, d1 ... d,B2 : b1 ... b d1 ... d,

aB3: t

bt

ct

dt

_ aB3: t

bt

c dt t

2 2 2 2' 2 2 2 2'.................... ......................

B41-1-M, B4l-1-M,c a1w

bx

cy

dz'

C a1x

bw

cz

dy' (5.4)

CM

x=y=t-w,CM ,

z=w.

For each block B there is a complementary block B. Two blocks which are notcomplementary intersect in exactly t points. We consider two blocks B1 and B2which do intersect. Ito has shown that for G = the property of beingtriply transitive on points is equivalent to being rank three on blocks. In G thestabilizer GB,B2 of B1 and B2 which of course also stabilizes B1 and B2 movesfurther blocks B31 .... B41 _ 1 _ M and B3, ... , B4$ _ 1 _ M in orbits such that ablock B and its complement B are in the same orbit. It follows that each of theseblocks contains exactly t/2 of the (a), (b), (c), (d) points respectively. Thereremain precisely two orbits of length M, C1, ... , C. and complementsC1, . . . , C. which are distinct orbits. If C1 contains w, x, y, z points of theorbits (a), (b), (c), (d) then x = y = t - w and z = w and C1 contains


respectively x, w, z, y points of (a), (b), (c), (d). As Norman showed, theseparameters satisfy the relation

t2 = M(i - 2w)2 (5.5)

and so clearly M is a square. If M = 1 then w = 0 or t, so that modulo 2B, + B2 = b, - b,c, c, so is either C, or C,. Since the value of M will bethe same for any two intersecting blocks, in this case the design D = H,,' will bean affine geometry over GF(2). What Ito has shown is that for t even eitherM = 1 and we have the geometry or possibly M = 9.

THEOREM 5.1 (GOETHALS AND SEIDEL [61). If X, Y, Z, W are square circulantmatrices of order t and if

XXT+ YYr+ZZT+WWT+4t1, (5.6)

then

H=X YR ZR WR

-YR X -WTR ZTR(5.7)

satisfies

-ZR WTR X YTR

L -WR -ZTR YTR X

HHT = 4,141. 5.8)

Further if X is skew symmetric, i.e. X + XT = 21,, then also H is skew symmetricand H + H T = 214,. Here R is the matrix which reverses the order of coordinates.

R. Turyn [241 has combined this technique with 4 symbol orthogonal se-quences. For odd t we take a sequence a,a2 a, where each a, is one of ±1,± i, ±j, ±k, Defining the product of any two distinct symbols to be zero andall squares as I we require the sequence to have inner product zero with anycircular rearrangement, this

alai+m + a2a2+m . . . a,a,+m = 0 (5.9)

for m = 1, ... , t - 1, where subscripts are of course taken modulo t. Such asequence is called orthogonal. Examples of such sequences are

t =t =t =t =

1, 1,

3, 1, i, j,(5.10)

5, 1, i, i, j, -j,7, 1, i, i, -i,j, k, j.

We now suppose that A, B, C, D are symmetric circulant matrices of degree msuch that

A2+B2+C2+D2=4m1,,,. (5.11)


(For some purposes it is sufficient to take A, B, C, D all as the 1 dimensionalmatrix [1].) Given a 4 symbol orthogonal sequence of length t we constructX, Y, Z, W by means of the correspondence

1HA+Bi+Cj+Dk=A+Bi+Cj+DkHA,B,C,D,iH(A+Bi+C)+Dk)i=-B+Ai+Dj-CkH-B, A, D, - C,j H(A+Bi+Cj+Dk)j=-C-Di+Aj+BkH-C,-D, A, B,kH(A+Bi+Cj+Dk)K=-D+Ci+Bj+AkHD,C,-B,A.

(5.12)

That is standard quaternion multiplication sets up the correspondence. Giventhe orthogonal sequence we now use the correspondence (5.12) to set of tcolumns of depth 4 whose rows are respectively the first rows of X, Y, Z, W in(5.7) and the rest are derived by cyclic permutations of the first. For t = 5 thefirst rows of X, Y, Z, W are thus

1 i

X: A -B -B -C CY: B A A -D D (5.13)Z: C D D A -AW: D -C -C B -B

Between (5.8) and (5.11) we obtain H of dimension 20m and

HHT + 20mI20,,,. (5.14)

It is plausible that orthogonal sequences of all odd lengths t exist. If this wereso then Hadamard matrices H. would exist for all n = 0 (mod 4). The first n indoubt is 4.67 = 268.

There is a superficial similarity between the orthogonality of the sequencesand orthogonality relations in group representation. It may be that some clevertrick or deep perception can resolve this problem.

REFERENCES

1. Michael Aschbacher, On collineation groups of symmetric block designs, J. CombinatorialTheory 2 (1971), 272-281.

2. L. Beneteau, Free commutative Moufang loops and anticommutative graded rings, J. Algebra (toappear).

3. F. Buekenhout, Une caracterisation des espaces affins basis sur la notion de droite, Math. Z. 3(1969), 367-371.

4. R. H. Bruck, A survey of binary systems, Springer-Verlag, Berlin and New York, 1958.5. P. Dembowski, Finite geometries, Springer-Verlag, Berlin and New York, 1968.6. J. M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal, Canad. J. Math. 19

(1967), 1001-1010.7. Marshall Hall, Jr., Automorphisms of Steiner triple systems, Proc. Sympos. Pure Math., Vol. 6,

Amer. Math. Soc., Providence, R. I., 1962, pp. 47-66.8. , Group theory and block designs, Proc. Internat. Conf. Theory of Groups (Canberra

1965), Gordon and Breach, New York, 1967, pp. 115-144.9. , Combinatorial theory, Wiley, New York, 1967.

10. , Note on the group M12, Arch. Math. 13 (1962), 334-340.11. J. I. Hall, Steiner triple systems and 2-transitive groups, M. Sc. Dissertation, Oxford, 1972, pp.

1-41.


12. , Steiner triple systems with geometric minimally generated subsystems, Quart. J. Math. 25(1974), 41-50.

13. D. R. Hughes, Collineations and generalized incidence matrices, Trans. Amer. Math. Soc. 86(1957), 284-296.

14. N. Ito, On a conjecture of C. W. Norman, J. Combinatorial Theory Ser. A 27 (1979), 85-99.15. W. M. Kantor, 2 transitive symmetric designs, Trans. Amer. Math. Soc. 146 (1969), 1-28.16. Rev. Thomas Kirkman, On a problem in combinalorics, Cambridge and Dublin Math. J. 2

(1847), 191-204.17. C. W. Norman, A characterization of the Mathieu group M, Math. Z. 106 (1968), 162-166.18. T. G. Ostrom and A. Wagner, On projective and affine planes with transitive collineation groups,

Math. Z. 71 (1959), 186-199.19. E. T. Parker, On collineations of symmetric designs, Proc. Amer. Math. Soc. 8 (1957), 350-35 1.20. M. Reiss, Uber eine Steinersche combinatorische Aufgabe welchein 45st Bande diesel Journals,

Seite 181 gestellt wordenist, J. Reine Angew. Math. 56 (1859), 326-344.21. J. D. H. Smith, On the nilpotence class of commutative Moufang loops, Math. Proc. Cambridge

Philos. Soc. 84 (1978), 387-404.22. , A second grammar of associators, Math. Proc. Cambridge Philos. Soc. 84 (1978),

405-415.23. J. Steiner, Combinatorische Aufgabe, J. Reine Angew. Math. 45 (1853), 181-182.24. R. Turyn, Hadamard matrices, Baumert-Hall units, four symbol sequences, pulse compression and

surface wave encodings, J. Combinatorial Theory Ser. A 16 (1974), 313-333.

CALIFORNIA INSTITUTE OF TECHNOLOGY

Proceedings of Symposia in Pure MathematicsVolume 37 1980

GROUP-RELATED GEOMETRIES

ERNEST SHULT

One appealing feature of finite geometry to group theorists is the occasionalopportunity to substitute a geometric argument (or more precisely, a geometriccharacterization theorem) for a group-theoretic one. Geometric arguments oftenenjoy a direct simplicity which, at points where they could be introduced, mightwell contribute to the task of `revising' the ultimate classification of the simplegroup in the direction of shorter and more aesthetically inviting proofs.

A second appealing feature of finite geometry to group theorists is in thenature of the mathematics being devised. When one considers the "configura-tional arguments" with which group theorists have so often dealt for the pastcentury, it could well be said that group theorists have already been doinggeometry all along. Moreover, their work has done much to define what isimportant or relevant geometry (although one sometimes wonders how often inthe past that message has gotten through customs into the land of CombinatorialTheory).

Precisely how does a geometric argument replace a group-theoretic one?Unlike infinite group theory where so many of the theorems display the generalform: "The class A of groups contains the class B of groups," a great many ofthe theorems of finite group theory reduce to some sort of characterizationtheorem of the form: "A finite group with certain hypothesis is a (or is a centralproduct of) quasisimple group(s) on the following list." To provide a geometricproof of such a theorem one begins with a finite group with a set of hypothesesaccrued from its being a smallest counterexample to the theorem. The first stepis the building of some axiomatically describable geometry from the subgroupstructure. (Often there is more than one way to do this.) At the second step, oneinvokes a geometric characterization theorem, which now converts this subgroupgeometry into a known geometry. In addition, because of the way the subgroupgeometry is defined, the known geometry arrives with a known population ofmorphisms. The third step is then that of identifying subgroups of a (presumablyknown) automorphism group of a known geometry which are generated by thisknown population of morphisms. The geometric version of the group-theoreticcharacterization is then achieved by following successively these three steps, thatis, by following the arrows of the diagram:

1980 Mathematics Subject Classification. Primary 51E99; Secondary 20B20.OD American Mathematical Society 1980

457

458 ERNEST SHULT

Group (with hypotheses) Known groupJ.(Step 1) T(Step 3)

Underlying geometry --* Known geometry

of subgroups(Step 2)

(with certain knownmorphisms)

Step 1 is usually a matter of making a choice of geometries which is"judicious" in view of the availability of relevant theorems required for Steps 2and 3. It is in fact the prevalent unavailability of the latter theorems that makethis use of geometric arguments an only occasional event.

The type of theorem required for Step 3 is perhaps best envisaged by theextensive list of theorems which classify subgroups of Chevalley groups gener-ated by various classes of root subgroups and root elements, perhaps beginningwith work of McLaughlin [16], [17], Stark [23]-[25], Pollatsek [18], Wagner [30],and culminating in Cooperstein [9]-[11] and Kantor [15]. Sometimes, the geome-tries involved are not related to parabolic subgroups of Chevalley groups butrather to other less well characterized geometries. For example, recent work ofA. Wagner [31] on groups generated by homologies is related to the "deltageometries" described in Professor D. Higman's talk in this conference. Many ofthe above theorems utilize other "generational" theorems which bear an evenmore diffuse relation to their underlying geometries, and here one must mentionwork of Thompson [27] and Ho [13], [14] on quadratic pairs (see also Stark [26]),work of Aschbacher on odd transpositions [1], [2], Timmesfeld on root involu-tions [28], and Aschbacher's characterization of the symplectic and unitarygroups [3] in a way also related to delta geometries. It may also be that in Step 3,the action of the group on the geometry is known only in terms of its action as apermutation group, and here we may content ourselves with a single example,the recent and very beautiful characterization of the anti-flag-transitive projec-tive groups by Cameron and Kantor [8].

In Step 2, we are interested, of course, in theorems which can characterizegeometries; but to have any versatility for the purposes of Step 2 they mustcertainly start from very modest hypotheses. For example, if the geometry to becharacterized is ultimately to yield a building, the "axiomatically describablegeometry" that one should begin with ought to have simple hypotheses involvingonly shadows of one or two types of varieties in the diagram of the building. Thetheorem of F. Buekenhout and the author [6] involves only points and lines and(because we are in shadows of lines on points) the collinearity relation, andyields the polar spaces (essentially the buildings of type Group-theoreticarguments using this theorem at Step 2 appear in [22], in Aschbacher's char-acterization of odd-characteristic Chevalley groups [4], and could have providedas easily, the unitary-group characterization in Achhbacher's characterization ofthese and the symplectic groups cited above [3], [29]. Another characterization ofthe same set of buildings using only points, lines, quadrangles and the collinear-ity metric appears in Cameron's beautiful characterization of the near n-gonswhich are dual polar spaces [7]. Similar characterizations of geometries in termsof points, lines and local collinearity structures appear in Cooperstein's work

GROUP-RELATED GEOMETRIES 459

[12] yielding new characterizations of buildings of types A,,, D5 and E6. Finally,there are certain theorems characterizing generalized quadrangles but the stateof this art is far from perfect. In other rank 2 cases, use is made of Tit's theoremon Moufang quadrangles, hexagons and octagons. Mark Ronan's very nicetheorems give purely geometric hypotheses which ultimately realize the Moufangcondition [19]-[21] in generalized hexagons.

The problem of obtaining relatively simple axiomatic characterizations for theone or two-variety shadow geometries of other types of buildings today remainsan open and interesting problem.

The problem of obtaining relative simple (and hence versatile) characteriza-tions of still other types of geometries is like entering an even darker world.Geometries possessing Buekenhout diagrams which are not of spherical typepose many problems. For example, does the "freest possible" geometry consist-ing of points, lines and quadrangles with Buekenhout diagram

possess triangles (triplets of lines meeting pairwise at distinct points)?There are geometries (or geometric structures) well removed from the world of

Buekenhout which do appear in group-theoretic contexts, and do possess char-acterization theorems powerful enough to effect group-theoretic characterizationtheorems of the sort described at the beginning of this lecture. To save space Iwill not review the background presented in my lecture but only state that thefollowing theorem holds because of the characterization of the underlying2-graph.

THEOREM. If G is a triply transitive group whose 1-point stabilizer G. contains anormal subgroup Na of index 2 strongly closed in G. with respect to G, then Gcontains a normal subgroup N of index 2 with N n G. - Na.

A purely group-theoretic proof of this theorem due to Professor Wielandtexists and was mentioned in his lecture. To the author's knowledge, a nongeo-metric proof of the following characterization theorem is not known (seeProfessor Cannon's lecture).

THEOREM. Let (G, f) be a 2-transitive group in which(1) Ga is rank three on 12 - (a),(2) NG(Gap) = <t> x Gap where t is of order 2 and possesses fixed points.

Then either(1) G is Sym(5) on 6 letters,(2) The semidirect product of Sp(2n, 2) and the additive group of its vector space

V(2n, 2) acting on the 22n vectors, or(3) Sp(2n, 2) in one of its 2- transitive representations on 2"(2" ± 1) letters.

Perhaps this is enough to illustrate that geometric characterization theoremshave all shapes and still offer many open problems for group theorists to workon as well as offering possibilities for novel proofs in group theory itself.

460 ERNEST SHULT

REFERENCES

1. M. Aschbacher, On finite groups generated by odd transpositions. I, Math. Z. 127 (1972), 45-56.2. , On finite groups generated by odd transpositions. II-IV, J. Algebra 26 (1974), 451-459,

460-478, 479-491.3. , A characterization of the unitary and symplectic groups over finite fields of characteristic

at least five, Pacific J. Math. 47 (1973), 5-28..4. , A characterization of Chevalley groups over fields of odd order (to appear).5. F. Buekenhout, Diagrams for geometries and groups.6. F. Buekenhout and E. Shult, Foundations of polar geometry, Geometriae Dedicata 3 (1974),

155-170.7. P. Cameron, Dual polar spaces (to appear).8. P. Cameron and W. Kantor, 2-transitive and flag transitive collineation groups of finite projective

spaces (to appear).9. B. Cooperstein, Subgroups of E6(q) generated by root groups (to appear).

10. , Subgroups of exceptional groups of Lie type generated by long root elements, I. Oddcharacteristic (to appear).

11. , The geometry of root subgroups in exceptional groups. 1, 11 (to appear).12. , A characterization of some Lie incidence structures (to appear).13. C. Y. Ho, On the classical case of the quadratic pairs for 3 whose root group has order 3, Ph.D.

Thesis, Univ. of Chicago, 1972.14. , Chevalley groups of odd characteristic as quadratic pairs, J. Algebra 41 (1976), 202-211.15. W. Kantor, Groups generated by a class of elements central in long root groups (to appear).16. J. McLaughlin, some groups generated by transvections, Arch. Math. 18 (1967), 364-368.17. , Some subgroups of SL (F2), Illinois J. Math. 13 (1969), 108-115.18. H. Pollatsek, Irreducible groups generated by transvections over fields of characteristic 2, J.

Algebra 39 (1976), 328-333.19. M. Ronan, A geometric characterization of Moufang hexagons (to appear).20. , A combinatorial characterization of the dual Moufang hexagons (to appear).21. , A note on the 3D4(q) generalized hexagon (to appear).22. E. Shult, On subgroups of type Zp X Zp, J. Algebra 32 (1974), 119-131.23. B. Stark, Irreducible subgroups of orthogonal groups generated by groups of root type I, Pacific J.

Math. 53 (1974), 611-625.24. , Some subgroups of S2(V) generated by groups of root type I, Illinois J. Math. 17 (1973),

584-607.

25. , Some subgroups of S2(V) generated by groups of root type, J. Algebra 29 (1974), 33-41.26. , Another look at Thompson's quadratic pairs, J. Algebra 45 (1977), 334-342.27. J. Thompson, Quadratic pairs, unpublished.28. F. Timmesfeld, Groups generated by root involution I, J. Algebra 33 (1975), 75-134.29. , Personal communication, this conference.30. A Wagner, Groups generated by elation, Abh. Math. Sem. Univ. Hamburg 41(1974), 199-205.31. , Collineation groups generated by homologies (to appear).

KANSAS STATE UNIVERSITY


NEAR n-GONS

SAEED SHAD AND ERNEST SHULT

A linear incidence system is a set ' P of "points" and a special system L ofsubsets of 9,, called "lines" such that every line contains at least two points andevery pair of points lies on at most one line. In any linear incidence system thereis a natural metric in which collinear points are at "distance one" from oneanother, two noncollinear points are at "distance two" if they are both collinearto a common point, and so forth-that is, distance is graph-theoretic distancemeasured in the collinearity graph on 9..

A near 2n-gon is a linear incidence system (9, L) such that(N.1) Given any point p and any line L not containing p, there is a unique

point on L nearest p.(N.2) The collinearity graph on ' P has diameter n.The notion of a near 4-gon coincides with the notion of a generalized

quadrangle, but for n > 4, there are many sorts of near n-gons which are notgeneralized n-gons. Much of the starting point for the structure theory of nearn-gons begins with Yanushka's Lemma [4, Lemma 2.5].

Let a and c be two points at distance 2 in the near n-gon (9, L) and let b andd be two points collinear with both a and c. (It follows from (N.1) that b and dare not collinear and that a, b, c and d form a "diamond" as illustrated.)

b

a. . c

Yanushka's Lemma states the following: If any of the four lines defined bythe diamond abcd contains at least three points, then the set of a points of ' Pwhich are simultaneously within distance two of the four points abcd forms asubspace Q of (9, L) which is a generalized quadrangle and is the uniquemaximal subquadrangle of (9, L) containing both a and c.

1980 Mathematics Subject Classification. Primary 05B25.® American Mathematical Society 1980

461

462 SAEED SHAD AND ERNEST SHULT

These maximal subquadrangles afforded by Yanushka's Lemma are called"quad's" and if the hypotheses of the lemma hold in (9, , E) for every pair ofpoints at distance 2, we say "quads exist in (9, E)." If quads exist it is easy toshow that points, lines and quads satisfy the axioms of a Buekenhaut Geometrywith diagram

L

where "L" refers to linear space (every pair of "points" lies on a unique "line").If Q is a quad in (9,, E) and p is a point not in Q, one of two situations can

occur:(a) (The classical point-quad relationship) Q contains a unique point q nearest

p and the distance from any further point of Q top is its distance from q (in Q)added to the distance from p to q.

(b) (The ovoid point-quad relationship) Here p has distance d from a subset (9of points of Q and distance d + 1 from all remaining points of Q. The set (9forms an ovoid in Q, a set of points of Q which meets every line of Q at a singlepoint.

Among the known near n-gons is a "classical" type (in the sense that theyinclude the so-called classical geometries) and these are the dual polar spaces.Dual polar spaces are polar spaces in which the maximal subspaces are regardedas "points" and the sets of maximal subspaces containing a given second-maxi-mal subspace are the "lines". (Axioms N.1 and N.2 are easily verified.)

The classical near n-gons are characterized by the following very beautifultheorem of Peter Cameron [1].

THEOREM. Let (9, E) be a near n-gon in which quads exist and every point-quad relation is classical. Then (9, E) is a dual polar space.

One effect of this theorem is to shift the focus of the structure theory of nearn-gons to the ovoid point-quad relationship. Many generalized quadrangles donot possess ovoids (for example those of type Sp(4, q), q odd, or quadrangles oforder (s, t) with t > sZ - s when s > 1). There are six possible relationships of anonincident line-quad pair which space does not permit us to recite here, butsome of these require, for example, a partition of 9 into disjoint ovoids, andmany quadrangles will not admit such a partition. There is thus a dramaticeffect on the overall structure of a near n-gon induced by the ovoid-structure ofany of its quadrangles and this effect is under current study.

A near 2n-gon (9 , E) is said to be regular of order (s, t2, ... , t = t) if eachline contains 1 + s points, each point lies on 1 + t lines, and if, whenever twopoints p and q are at distance d (between 1 and n), then exactly 1 + td linesthrough q carry points at distance d - 1 from p. In this case (9, E) forms ametric association scheme with exactly

sd(1 + t)t(t - t2)(t - t3) . . . (t - td-1)(1+t2)... (1+td)

points at distance d from any given point. If t2 > 0 and s > 1, then quads exist,each with order (s, t2). The quads and lines through a fixed point p then form a

NEAR n-GONS 463

block design with parameters (v, b, r, k, A) in which r = t/t2 and b =1(1 + t)/ t2(1 + 12) must both be integers. In addition the following two inequal-ities can be proved, the first by eigenvalue techniques, the second by a simplegeometric argument (provided ff, L) is not a dual polar space):

t < s3 + t2(s2 - s + 1) (Haemers [2]), (1)

1 + t < (1 + 12)(1 + st2). (2)

For fixed values of n and s, only finitely many parameter sets yield integraleigenvalue multiplicities. The regular near hexagons with s = 2 were classifiedby Yanushka and Shult (4] and, in addition to the dual polar spaces andgeneralized hexagons with s = 2, include exactly two further nonclassical exam-ples: one of 729 points formed from the extended ternary Golay code, the otherfrom the 759 octads of S(24, 5, 8). All possible parameter sets for regular nearhexagons with s = 3 were determined by Shad [3] and five of the sevennonclassical parameter sets were eliminated by either the inequalities (1) and (2)or by arguments on the ovoid structure of the quads. One nonclassical parame-ter set for near octogons with s = 2 was shown to exist by Shad [3].

REFERENCES

1. P. Cameron, Dual polar spaces (to appear).2. W. Haemers, Eigenvalue techniques in desing and graph theory, Math. Centrum, Amsterdam,

1979.3. S. Shad, Characterizations of geometrics related to polar spaces, Ph.D. dissertation, Kansas State

University, 1979.4. E. Shult and A. Yanushka, Near n-gons and line systems, Geometriae Dedicata 9 (1980), 1-72.

KANSAS STATE Un7vERsrrY

ORTHOGONAL POLYNOMIALS,ALGEBRAIC COMBINATORICSAND SPHERICAL t-DESIGNS

EIICHI BANNAII

1. Spherical 1-designs. Let Std = ((x1, ... , xAXI + +xd = 1) be theunit sphere in Rd. A finite nonempty subset X in Std is said to be a sphericalt-design, if

d f pO dwO _ XX

p(x)

for all polynomials p of degree < t. This important concept of sphericalt-design, which may also be defined in several different but equivalent ways, wasfirst introduced by Delsarte, Goethals and Seidel [8] (1977). The reader isreferred to [8]-[10] for the equivalence of those definitions and the fundamentalproperties of spherical t-designs.

Many examples of spherical t-designs are, as naturaly expected, obtained fromvarious known nice structures such as regular polytopes, roots of reflectiongroups, Leech lattice, and so on. If d = 2, then the set of vertices of a regular(t + 1)-gon forms a spherical t-design in 5l2. The vertices of a regular 4-, 6-, 8-,12-, or 20-vertex polyhedron in R3 form a 2-, 3-, 3-, 5-, or 5-design in 523. Theroots of type E6, E7, or E8 respectively form a 4-, 5-, or 7-design in 526, 97, or 08.The 120 roots of the reflection group of type H4

3 3 5

form an 11-design in 04. The 196560 minimal vectors in the Leech lattice forman 11-design in 52241 and so on. Actually, many examples are obtained by usingfinite subgroups of the orthogonal groups 0(d).

1980 Mathematics Subject Classification. Primary 05B30, 05B40, 05A15; Secondary 33A65.'Supported in part by NSF Grant MCS-7903128 (OSURF 711977).


465

466 EIICHI BANNAI

THEOREM 1.1 [1]. Let G be a finite subgroup of 0(d) and let pi (i = 0, 1, ... )be the ith spherical representation of 0(d).

(1) If piI c are irreducible for i = 0, 1, ... , s, then for any x E SId, the setX = ( gx I g E G) is a spherical 2s-design in SId.

(2) In addition, if (p,.+1, P,)c = 0 for i = 0, 1, ... , s, then X is a spherical(2s + 1)-design in SId.

REMARK. The degree of

l r lPi=(ddi1 1 1

- ( d d i13

= dimR{homogeneous harmonic polynomials of degree i}.

In particular, po = 1, p1 = d and p2 = i d(d + 1) - 1.REMARK. Very recently, Goethals and Seidel [10] have shown that if x E SId is

chosen suitably, then the set X can have a larger t. If G is a finite reflectiongroup, then the sets X in Theorem 1.1 are m2-designs for any x E SId, but someX can actually become m3-designs for suitably chosen x E SId, where m1 = 1. <m2 < m3 < < and are the exponents of the reflection group. (For example, a19-design in 04 is constructed from H4; see [10] for the details.)

However the following basic questions still seem to be unanswered.Open problems. (1) Do spherical t-designs exist for arbitrary large t for some

d > 3? (2) Do spherical t-designs exist for arbitrary large t for a fixed d > 3?

2. Tight spherical t-designs.

THEOREM 2.1 (FISHER-TYPE INEQUALITY, DELSARTE, GOETHALS AND SEIDEL[8]). For a spherical t-design X in SId,

(d+ +s - 1)+(d++s 1 2) (_ degree ofpil ift=2s,IXI > J

J J o J

2(d+d s 1 1) ift =2s+ 1.

A spherical t-design X is said to be tight if the equality holds in either of theabove inequalities. If d = 2, then X is a tight spherical t-design if and only if Xis the set of vertices of a regular (t + 1)-gon. So, without loss of generality, wemay assume d > 3. The set of 240 roots of E8 and the 196560 minimal vectors ofLeech lattice are examples of tight 7- and 11-designs in SZ8 and 024 respectively.

REMARK. Let a tight 2s-design (or tight (2s + 1)-design) X exist in 'd withd > 3. Then

(1) The set ( 9, = (x, y)I x, y E X, x : ± y) consists of s elements, where (, )denotes the inner product in Rd.

(2) The polynomial P(x) := ll;.1(x - 9;) is a certain Jacobi (for t = 2s) orGegenbauer (for t = 2s + 1) polynomial.

(3) (Lloyd-type theorem) All the s zeros 9, of P(x) must be rational numbers.

THEOREM 2.2 (BANNAI AND DAMERELL [3], [4]). Let d > 3.(1) If t = 2s > 6, then there exist no tight spherical t-designs in 1d.(2) If t = 2s + 1 > 9, then there exist no tight spherical t-designs in 'd except

the case t = 11, d = 24 and I X I = 196560.

SPHERICAL 1-DESIGNS 467

PROOF. The proof is done by using the above-mentioned Lloyd-type theorem.Case (1) is an analytic proof: it uses the properties of orthogonal polynomials(see [3]). Case (2) is a number theoretical proof: one considers the Newtonpolygon of P(x) forp = 2 (see [4]).

The exceptional case t = 11 and d = 24 is related to many interestingmathematical objects. The kissing number Td is defined to be the largest numberof nonoverlapping unit spheres in Rd which touch another unit sphere [11].

THEOREM 2.3 (ODLYZKO AND SLOANE [11]). Tg = 240 and T24 = 196560. (AlsoT2 = 6, T3 = 12, but Td is not known for d ) 4, d : 8, 24.)

THEOREM 2.4 (BANNAI AND SLOANE [6]). (1) Tight 11-designs in 0124 and tight7-designs in Qg are unique up to orthogonal transformations.

(2) The arrangements of unit spheres for d = 24 and 8 that attain the kissingnumber are unique up to orthogonal transformations.

(Similar uniqueness results are obtained also for d = 23 and 7 (see [6]).)Observation. Compare the following two situations:(1) d = 8, 248 = 8 + 240, F3 (Thompson),(2) d = 24, 196883 = 24 + 299 + 196560, Fl (Monster).

Here 299 is the degree of p2 for d = 24 (see Theorem 1.1). This might suggestthat the monster can be constructed by constructing a nice lattice of degree196883 from the tight sphere packing in R24, as was done for F3.

3. Relations between orthogonal polynomials and algebraic combinatorics. Herewe collect several remarks which will indicate how current research in algebraiccombinatorics is going on, putting emphasis on the relations with orthogonalpolynomials.

(a) The concepts of (spherical) t-design and tight (spherical) t-design aredefined and studied not only on spheres but also on more general spaces, inparticular on compact homogeneous symmetric spaces of rank 1, i.e., compact2-point-homogeneous spaces.

(b) The discrete analogues, namely t-designs and tight t-designs on certainassociation schemes, have been defined and studied extensively (and previously),starting from Delsarte [7] (1973). In particular, the existence problem for perfecte-codes in P-polynomial schemes (= distance-regular graphs = finite 2-point-homogeneous spaces, roughly speaking) and tight t-designs in Q-polynomialschemes has been one of the central problems (see [7], etc. for the details). Heremany types of orthogonal polynomials appear.

(c) I believe that the classification problem of distance-regular graphs (ormore specifically, of (P and Q)-polynomial association schemes) with largediameters should be an interesting future problem, although it seems too ambi-tious and almost impossible at the present stage. Actually, all known families of(P and Q)-polynomial schemes with large diameters are related to the families offinite classical groups (including alternating and symmetric groups) in a verynice way, and it seems difficult to find other examples with large diameters.

(d) The concepts of spherical t-design and t-design are best understood fromthe viewpoint of (classical) analysis. The formula (1.1) is a special case of aquadrature formula (cubature formula) in approximation theory. The problem of

468 EIICHI BANNAI

classification of "tight t-designs on a finite 1-dimensional interval" is equivalentto the so-called Tchebycheff's problem (in classical analysis) which was solvedby S. Bernstein in 1937.

(e) The problems of studying the weight distributions of codes and designs,and also the spectra (i.e., eigenvalues and multiplicities) of (the adjacencymatrices of) distance-regular graphs are closely related to some concepts ofclassical analysis, such as Christoffel numbers of orthogonal polynomials, themoment problem, and so on (see [2], [5], etc. for the details).

(f) Finally, I would like to point out that the origin of these studies (i.e.,algebraic combinatorics) can be found in the works of I. Schur, and that someparts of them have been developed through the study of the centralizer rings(Hecke rings) of finite permutation groups (by H. Wielandt, D. G. Higman andmany other mathematicians).

REFERENCES

1. E. Bannai, On some spherical t-designs, J. Combinatorial Theory Ser. A 26 (1979),157-161.2. , On the weight distribution of spherical 1-designs, European J. Combinatorics 1 (1980).3. E. Bannai and R. M. Damerell, Tight spherical designs. I, J. Math. Soc. Japan 31 (1979),

199-207.4. , Tight spherical designs. II, J. London Math. Soc. 21(1980), 13-30.5. , Orthogonal polynomials and some problems in algebraic combinatorics, J. London Math.

Soc. (to appear).6. E. Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. (to

appear).7. P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep.

Suppl. 10 (1973).8. P. Delsarte, J. M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6

(1977), 363-388.9. J. M. Goethals and J. J. Seidel, Spherical designs, Proc. Sympos. Pure Math., vol. 34, D. K.

Ray-Chaudhuri (ed.), Amer. Math. Soc., Providence, R. I., pp. 255-272.10. , Cubature formulae, polytopes and spherical designs, Proc. Coxeter Symposium,

(Toronto, May 1979) (to appear).11. A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a

unit sphere in n-dimensions, J. Combinatorial Theory Ser. A 26 (1979), 210-214.



FINITE TRANSLATION PLANES AND GROUPREPRESENTATION

T. G. OSTROM

The object of this paper is to present a problem on the assumption that finitegroup theorists are looking for new fields to conquer.

Loosely stated, the problem is: Which finite groups can act on finite transla-tion planes? Let us make the question more definite. Let F be a field (finite orinfinite) and let V be a vector space of dimension 2d over F. A spread over V isa class of d-dimensional subspaces (the components of the spread) such that eachnonzero vector belongs to exactly one component. It is well known that everytranslation plane can be represented in such a way that the points are identifiedwith the elements of a vector space V and the lines are translates of a spreaddefined on V.

The subgroup of GL(2d, F) which preserves the spread (the image of acomponent must be a component) is the linear translation complement of theplane. Our question can now be restated as follows:

For which abstract groups G is there a representation of G as asubgroup of GL(2d, q) for some d and q such that G is a subgroup ofthe linear translation complement for some translation plane (spread)defined on a vector space of dimension 2d over GF(q)?

The question is still too broad to permit any hope of an answer that isreasonably near to being complete. Suppose that we restrict ourselves to non-solvable groups and look at the known cases.

The groups G then have subgroups of relatively small index of the followingtypes:

1. subgroups of GL(2, q d),2. direct products of groups of type 1,3. Suzuki groups,4. SL(2, 13) where (13, q) = 1,5. SL(2, 7) where (7, q) = 1.

So we might ask a more modest question: Are there any others?

1980 Mathematics Subject Classification. Primary 51A40, 20G40.m American Mathematical Society 1980

469

470 T. G. OSTROM

Actually there are some restrictions coming out of the fact that the group isacting as a collineation group on a projective (or af fine) plane that do give one atoe-hold:

1. The fixed point subspace of a nontrivial element of G cannot havedimension greater than d.

2. The fixed point subspace of an involution other than "-1" always hasdimension d.

3. The possible groups are known if some component, not invariant under G,is pointwise fixed by some p-element. Here p is the characteristic so that q is apower of the prime p.

Thus we might just suggest that one study subgroups of GL(2d, q) whichsatisfy conditions 1 and 2 above and do not contain p-elements acting in theway described in 3 above.

As a sort of a general strategy we propose starting with the following generalmethod of analysis. If G is nonsolvable, let Go be a minimal normal nonsolvablesubgroup and let H be maximal with respect to the conditions that H is normalin G, contained in Go but not equal to Go (so H is solvable). Then Go/H is adirect product of isomorphic simple groups.

We are so far short of a general solution that we might as well put in therestriction that G be irreducible and see what we can way under these circum-stances. There turn out to be two general possibilities:

I. H = Z(G0) and is, in fact, a subgroup of the Schur multiplier for G0/H.II. Go has a subgroup W which is normal in G and is a w-group for some

prime w. If Wo = W n Z(G0) then W/ Wo is elementary abelian.Case II breaks down into two subcases:IIa. W is elementary abelian and G is imprimitive as a linear group.IIb. W is extra special and G acts symplectically on W/ Wo.I do not know of any examples of Case II. If q and d are both odd and Case

II does not occur it turns out that Go must be SL(2, u) for some u (notnecessarily a power of the characteristic p) or a pre-image of AT Thus we knowquite a bit under these restrictions.

However this much information leaves very much in the open questions suchas the following: Hering has shown that SL(2, 13) acts on a certain translationplane with q = d = 3. Are there other planes where q is relatively prime to 13,where SL(2, 13) acts? Take some other value of u, say u = 11. Is there atranslation plane with (q, 11) = 1 on which SL(2, 11) acts?

What Hering did was to find a very explicit representation of SL(2, 13) inGL(6, 3). In this representation, each subgroup of order 13 has exactly twoinvariant 3-dimensional subspaces. The different 3-spaces invariant under thevarious subgroups of order 13 define a spread left invariant by SL(2, 13).

Let us leave the case where d is odd. It turns out that SL(2, q) has bothreducible and irreducible representations in GL(4, q). With some restriction onq, there are two types of (non-Desarguesian) translation planes for d = 2. In thereducible case, each p-group fixes a Baer subplane pointwise and the plane is aHall plane. In the irreducible case each p-group fixes a 1-space pointwise andhas a unique invariant 2-space which becomes a component of the spread. Theother components of the spread are 2-spaces invariant under subgroups of order3. This last is also due to Hering.

FINITE TRANSLATION PLANES 471

Thus the general program becomes: First develop general theorems whichscreen out as many as possible groups G. For some choices of q and d, try to getvery explicit representations of a group G which has survived the screening.Then try to show that G cannot act on a translation plane after all or that G canbe used to define a spread on which it acts.

I think it reasonable to hope that we can develop some more general"screening theorems"-perhaps with some restrictions on q and d. With ourpresent state of knowledge the last part appears to be very much ad hocdepending on the particular q, d and G being examined and depending on quiteexplicit representations.

What we need are representations from which we can determine the invariantsubspaces and fixed point subspaces of the various elements of the group.Possible constructions of planes seem more likely when there are invariantsubspaces of dimension d.

Rather than giving specific references, we are listing survey and expositoryworks by the author. Many more references are given in these works. Theresearch on which this was based was supported, in part, by the NationalScience Foundation.

REFERENCES

1. T. G. Ostrom, Finite translation planes, Lecture Notes in Math., vol. 158, Springer-Verlag, Berlinand New York, 1970.

2. , Classification of finite translation planes, Proc. Internat. Conf. on Projective Planes, T.G. Ostrom and M. J. Kallaher (eds.), Washington State Univ. Press, Pullman, Wash., 1973, pp.195-213.

3. , Recent advances in finite translation planes, Foundations of Geometry, selectedproceedings of a conference, P. Scherk (ed.), Univ. of Toronto Press, Toronto, 1976, pp. 183-205.

4. , Finite translation planes, an exposition, Aequationes Math. 15 (1977), 121-133.

WASHINGTON STATE UNIVERSITY


FINITE COLLINEATION GROUPS OFPROJECTIVE PLANES

CONTAINING NONTRIVIALPERSPECTIVITIES

CHRISTOPH HERING

1. Introduction. Although there has been considerable progress recently, untilnow only isolated results are known about collineation groups of finite projec-tive planes in general. However, there is a well developed theory of groupscontaining perspectivities. Here we have an almost complete system in whichonly a few gaps remain. Apparently it is possible to close the larger number ofthese gaps with the help of modern group theory.

Let G be a collineation group of a projective plane s,13 generated by perspectiv-ities. Let S3(G) be the lattice of substructures of s,13 left invariant by G and £1(G)the set of atoms in this lattice. One can easily analyze the possibilities for £1(G),and one finds that there are up to duality 16 types (see [2, Theorem 5.10]). Manyof these types can be explored with the help of standard methods, and questionsabout the structure of G, its action on s,13 and about the geometric structure ofthe set 3 of centers and t of axes of nontrivial perspectivities in G can beanswered to a satisfactory extent. There do remain, however, a few cases whichare not so easily accessible, in particular the cases that £1(G) consists of (a) justone subplane, or (b) a line (more precisely, that £1(G) consists of just onesubstructure, whose point set is empty and whose line set has cardinality 1), (c) aline a and a point on a, or (d) a line a and a point not on a. It is in these cases,which from the geometric point of view look most difficult, that group theory isespecially powerful. At least in the first two cases, the situation apparently canbe handled completely, under the additional assumption that G is finite and thenonsolvable composition factors of G are known. We describe this in detail forthe first case.

2. Finite groups containing perspectivities, for which £1(G) consists of asubplane. We define a projective plane to be a triple M, 2, 1) consisting of twosets 13 and 2 and an incidence relation I C 13 X 2 such that

1980 Mathematics Subject Classification. Primary 51A35, 20B25, 20D06, 20D08.o American Mathematical society 1980

473

474 CRISTOPH HERING

(I) if x, y E 2 and x : y, then there exists exactly one element P E 43 suchthat xIP and yIP,

(II) if X, Y E s,13 and X Y, then there exists exactly one element 1 E 2 suchthat X I land Y II, and

(III) there exists a subset X C 3 such that IXI = 4 and IX n (1)1 s 2 for all1 E 2 (here (1) =( P E $I PI1)).

Let ($, 2) be a projective plane. If P and Q are two different elements ofthen we denote by PQ the unique element of 2 incident with P and Q. Also, ifa, b E 2 and a b, then the unique element of $ incident with a and b isdenoted by a n b. Furthermore, we shall usually call the elements of $ pointsand the elements of 2 lines.

If s,13 C s,13 and £ C 2, then the pair (W3, S) is called a substructure of (s,13, 2)provided it has the properties

(a) PQ E £ whenever P and Q are two different points in ;3, and(b) a n b c s43 whenever a and b are two different lines in £.A substructure of (s,13, 2) is called a subplane of (S,13, 2) if in addition to

(a) and (b) it has the property _(c) there exists a subset 3r C 3 such that IX1 = 4 and IX n (1)I s 2 for all

1 E 2.Let G be a set of automorphisms of (s,13, 2). Then we denote by s,13(G) the set of

fixed points of G, by £(G) the set of fixed lines of G and by (G) the pair(s,13(G), £(G)). If A and B are two different fixed points of G, then AB is a fixedline of G. This together with the dual statement implies that 3(G) is asubstructure of (j3, 2), the fixed structure of G.

Let x E G. A line 1 is called an axis of x, if x fixes all points on 1. Likewise apoint P is called a center of x, if x fixes all lines passing through P. If x has anaxis or a center, then x is called a perspectivity. A nontrivial perspectivity hasexactly one center and one axis.

In the following, let ($, 2) be a projective plane, G a group of collineations of($, 2), ($3, £) the substructure generated by all centers and axes of nontrivialperspectivities in G, G the group induced by G on and P the subgroupgenerated by all perspectivities in G. We have

THEOREM 1. Assume that ($, 2) is finite, P : 1, and that G does not leaveinvariant any point, line or triangle in (j3, £). Then (j3, £) is a subplane. Also, jhas exactly one minimal normal subgroup M, G M s M, and one of the followingholds:

(a) M is nonabelian simple and the generalized Fitting subgroup is theproduct of a quasisimple group with $P, or

(b) IM I = 9 and 2 . 32 11 P I 1 23.34

Using the Feit-Thompson Theorem [1] we immediately obtain the

COROLLARY 1. If the hypotheses of Theorem 1 are satisfied, then G has evenorder.

The proof of Theorem 1 (see [2]) is quite elementary except for one point,where the Feit-Thompson Theorem is used. The main idea is to investigate verycarefully the properties of stabilizers G. in G of substructures X of (j3, 2) and

FINITE COLLINEATION GROUPS OF PROJECTIVE PLANES 475

fixed structures 3(X) in (13, 2) of subgroups X of G. Very useful for example arethe following simple facts:

LEMMA 1. Let Ll be a substructure of (43, 2) whose collineation group does not fixany point or line. Then Ll is a triangle, a subplane or the empty substructure.

LEMMA 2. Let a and /3 be perspectivities in G. If a/3 : 1, then (a/3) is not theempty substructure and not a subplane.

This implies

COROLLARY 2. Let X be an abelian subgroup of G normalized by a perspectivityT. If [T, X] :IL- 1 and (X) = (0, 0), then X contains an element x such that (x)is a triangle.

Because of Corollary 2, 3-elements play a special role in this investigation, andin some situations they are a little hard to deal with. However, in the finite casethese difficulties can be removed with the help of the following

LEMMA 3. Assume that (43, S3) is finite. Let x and y be two commuting elementsof order 3 in G. If 1,'3(x), 13(y) 0, then 43(xy) 0.

By the way, this is the only point, where the finiteness of the plane (S,13, 2) isneeded. All other arguments work for finite collineation groups acting oninfinite planes too.

Of course there remains the problem of determining which simple groups canbe isomorphic to the minimal normal subgroup M in Theorem 1. Also, if aspecific isomorphism type of finite simple groups occurs, we would like to knowto which extent it determines the geometric structure of (s,13, 2). For questions ofthis kind it would certainly be useful to have abstract characterizations of theperspectivities among the elements of G. Something in this direction we obtainwith the help of the following technical

DEFINITION. Let E be a simple group and a E Aut(E). We say that a hasproperty if for all x c E - @Ea there exists a subgroup H < Q2E<[a, x]> ofindex at most n such that the normal closure of H in <a, H> is a propersubgroup of E. Furthermore, we call a an abstract perspectivity if it has theproperty (;3).

With this definition we have

LEMMA 4. Assume in _Theorem 1 that I M I : 9, and let a be a perspectivity in G.Then the element a E G induced by a is an abstract perspectivity in G.

So our question leads to the problem of determining for each simple group Ewhich of the elements of Aut(E) are abstract perspectivities. This question hasbeen investigated by M. Walker and the author for the groups PSL(3, q),PSU(3, q), PSL(2, q), Sz(q) and groups of Ree type (see [3] and [4]). For thesegroups it was possible to locate all abstract perspectivities. It turns out that ineach case one can find a geometry (ASE, intimately related to E, such that theautomorphism group Aut(E) of E acts as a group of collineations on (ASE, andthe abstract perspectivities in Aut(E) are related to a very special type ofcollineations of (ASE. If E = PSL(3, q), then (ASE is just the projective geometry

476 CRISTOPH HERING

PG(2, q), and the special collineations are quasi-perspectivities. In the remainingcases we proceed in the following way:

Let E be a simple Chevalley group of rank 1 and characteristic p, and let Ca bethe set of Sylow p-subgroups of E. For two different elements P and Q in Cam, wedefine a subset b C Ca in the following way: b = (P, Q) if E is of type A 1 or2B2, b is the set of Sylow p-subgroups normalized by the subgroup of indexq - 1 in 92EP n 92EQ if E is of type 2A2(q), and b is the set of Sylowp-subgroups normalized by the subgroup of order 2 in REP n 92EQ if E is oftype 2G2.

Let 0 = b E and denote the incidence geometry determined by (a and 0 byCASE. Clearly Aut(E) induces a group of automorphisms of CASE. We call anautomorphism a of CBE a quasi-perspectivity if a fixes the block containing P andP° whenever P E Ca and P P°. Then we have

THEOREM 2. Let E be a Chevalley group of rank 1 and a E Aut(E). Then a hasproperty (;1) if and only if a induces a nontrivial quasi-perspectivity of CASE.

Unfortunately we cannot apply this to our geometric problem directly, be-cause abstract perspectivities possibly do not have the property (;1) but only(*3). This leads to a considerable amount of additional work and to someexceptional cases. However, it is always possible to locate the elements ofAut(M) which can act as perspectivities if M is one of the simple groupsmentioned above. Once this is done, one can in general establish an embeddingof the geometry (AS. into M, 2). This leads to the following result:

THEOREM 3. Assume that (s,13, 2) is finite, P : 1, and that G does not leaveinvariant any point, line or triangle. Assume furthermore that G contains a normalsubgroup M isomorphic to a finite simple Chevalley group of type A2 or of rank 1.Then one of the following statements holds:

(a) M, £) is desarguesian.(b) M = PSL(2, q), where q is an odd prime power, and each perspectivity : 1

in G is an involutory homology.(c) M = PSU(3, q) and either each perspectivity of G is a homology or each

perspectivity : 1 in G is an involutory elation.

Note that in Theorem 3 the subplane (q3, £) necessarily is desarguesian, if weassume that G contains both nontrivial elations and nontrivial homologies.

We also have

THEOREM 4. Assume that in Theorem 1 the unique minimal normal subgroup Mof G is isomorphic to an alternating group A. Then n < 7.

Furthermore, Reifart and Stroth [6] have considered the case that M isisomorphic to one of the 26 presently known sporadic simple groups. It turns outthat only the second Janko group J2 can possibly occur.

Of course it is very interesting which of the Chevalley groups of rank ) 2 canoccur. In particular, it seems important to know if the rank of the Chevalleygroups, which can be isomorphic to M, is bounded, because an answer to thiswould provide some evidence as to whether or not finite projective planesthemselves can be considered as low rank objects. This problem leads to

FINITE COLLINEATION GROUPS OF PROJECTIVE PLANES 477

questions on Chevalley groups that might be of general interest. For example, ina Chevalley group of higher rank is each abstract perspectivity necessarilycontained in a proper parabolic subgroup? Or, related to this: In a finite simpleChevalley group E does each element a not contained in any proper parabolicsubgroup necessarily have a conjugate a' such that E = <a, a'>?

The technique described here is not restricted to projective planes, but can beapplied to a great variety of geometries. For example it is equally powerful ingeneralized quadrangles (see Walker [7]) and inversive planes (see Niederdrenk[5]). Also, as mentioned before, many of the arguments apply in the same way tothe case of finite collineation groups acting on infinite planes. For example wehave

THEOREM 1'. Assume that G is finite, P 1, and G does not leave invariant anypoint, line or triangle in (13, 2). Then ("3, £) is a subplane and F'(G) is simple or a3-group.

PROOF. Use Lemma 3.3, Theorem 3.13 and the proof of Theorem 5.5 in [2].

REFERENCES

1. W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963),775-1029.

2. C. Hering, On the structure of finite collineation groups of projective planes, Abh. Math. Sem.Univ. Hamburg 49 (1979),155-182.

3. C. Hering and M. Walker, Perspectivities in irreducible collineation groups of projective planes. I,Math. Z. 155 (1977), 95-101.

4. , Perspectivities in irreducible collineation groups of projective planes. II, J. Statist.Planning Inference 3 (1979), 151-177.

5. C. Niederdrenk, Uber Automorphismengruppen van Mobiusebenen endlicher Ordnung, Geom.Dedicata (to appear).

6. A. Reifart and G. Stroth, On finite simple groups containing perspectivities (to appear).7. M. Walker, On the structure of finite collineation groups containing symmetries of generalized

quadrangles, Invent. Math. 40 (1977), 245-265.

UNIVERSITAT TUBINGEN


FURTHER PROBLEMS CONCERNING FINITEGEOMETRIES AND FINITE GROUPS

WILLIAM M. KANTOR

This note is intended as a very long footnote to the other finite geometrypapers in this volume. Areas of overlap between finite geometry (or combinator-ics) and finite groups will be listed which were either not mentioned, or wereonly briefly alluded to, at the conference. However, the descriptions andbibliography given here will also be brief : the goal is to indicate researchdirections, not to give a comprehensive survey.

1. Buekenhout and Tits discussed building-like geometries at this conference.Tits pointed out that they are readily constructed from arithmetic groups.However, flag-transitive examples may be harder to find, and may be of interestfor arithmetic groups when their diagrams are extended Dynkin diagrams.

The only known finite flag-transitive examples having such diagrams arerelated to sporadic groups: one example of8arising from the Lyons-Sims group, three of

arising from P J (6, 3) 2, PSU(6, 2) and 2+(8, 2) which are related to Fischergroups, and a fourth with the latter diagram (due to Ronan and Smith) arisingfrom Suz. The universal covers of the corresponding complexes are buildings (bya theorem of Tits), and deserve study.

Further interesting examples of building-like geometries must exist. Presuma-bly, finite examples even exist (undoubtedly with small groups) involving non-classical generalized polygons.

2. The study of generalized polygons has recently attracted a great deal ofattention. In addition to Tits' and Weiss' work on Moufang polygons and theirgeneralizations (discussed by Weiss at the conference), there have been major

1980 Mathematics Subject Classification. Primary 20B25.m American Mathematical Society 1980

479

480 W. M. KANTOR

and elegantly geometric characterizations of the generalized quadrangles [34]and hexagons [40], [26] which arise from Chevalley groups. As yet, 'F4(q)octagons have not been characterized geometrically.

However, no finite hexagons or octagons are presently known other thanthose arising from Chevalley groups. The existence of other examples is afundamental problem in this area.

Only a few types of nonclassical generalized quadrangles are known, as well.Most of the known ones can be obtained by modifying a parabolic subgroup ofa rank 2 Chevalley group. The elementary group-theoretic construction is foundin [22], where new quadrangles are then obtained using G2(q). Analogousconstructions exist for hexagons and octagons, but have yet to yield newexamples. Entirely new construction procedures are needed; however, it seemsvery likely that these would also be group-theoretic.

In general, finite quadrangles seem tighter (and hence scarcer) than projectiveplanes, hexagons much tighter still, and octagons so tight that very few typesshould exist. However, no Bruck-Ryser theorem is known. If a generalizedpolygon has s + I points per line and t + I lines per point (where s > I andt > 1), then some restrictions on s and t were obtained by Feit and Higman [14].For quadrangles and octagons, the inequality s < t2 has been known for a while[16], [17]. This year, s < t3 was finally proved for hexagons [15]. Much moreshould be possible (compare [33, §4]).

A curious variation of these inequalities has recently been suggested. Cameronfound a short combinatorial proof that a generalized quadrangle with 3 lines perpoint is necessarily finite, at the same time obtaining s < t2 (= 4) in this case. Iobtained the same results when there are just 4 lines per point, by a group-theo-retic argument.

Generalized polygons were a crucial ingredient in the determination of all2-transitive collineation groups of finite projective spaces [6]. Embeddings ofgeneralized quadrangles into projective spaces were classified in [3]. The corre-sponding problem for hexagons remains open. However, very restrictive embed-dings of generalized polygons into projective spaces were essential in [6] (andled, incidentally, to an elementary construction of the G2(q) hexagons).

3. There is undoubtedly a great deal left to be learned about the geometry offinite Chevalley groups, even in the case of rank ) 3. This was already madeclear by Tits 20 years ago [35], [36], and was discussed in detail by Shult at theconference.

Geometry permeates recent work on groups generated by long root elements[8], [9], [20], as well as applications of that work [7], [21]. An entirely geometricapproach was used in a different generation question [37]: the determination ofall primitive subgroups of GL(n, q) generated by reflections.

4. The characters of the centralizer algebras are known for the permutationrepresentations of PSL(n, q) on i-spaces for I < i < n - 1 [13], and of classicalgroups acting on maximal totally isotropic (or singular) subspaces [31]. Thesecomputations are highly geometric. The character values are polynomials in q,

FINITE GEOMETRIES AND FINITE GROUPS 481

which are related to classical hypergeometric functions. Combinatorial applica-tions of these results are given in [32]. Further results can be obtained usingHoefsmit's work [18] on the centralizer algebra of 1B.

The combinatorial analogues of permutation representations are coherentconfigurations, and in particular, association schemes. The latter have proven tobe fundamental in coding theory [12], [28], and have motivated the study of thecharacters of especially important examples (as in the preceding paragraph).

A survey of many group theoretic uses of coherent configurations is given in[4].

More recently, coherent configurations have led in other directions. Somewere discussed by Bannai at the conference. Another involves the Krein Condi-tion and its combinatorial consequences, as well as properties of irreducibleconstituents of permutation characters [6]. Yet another generalizes the graphtheoretic notion of coherent configuration to higher dimensional complexes [10].

5. Group theory has had important applications to coding theory (cf. [25]).The most familiar examples of this involve the Mathieu groups and the Golaycodes. The relevance of invariant theory to coding theory is discussed in [25] and[29]. (Further combinatorial applications of invariant theory are described in[30] and its references.)

Quadratic residue codes arise by taking the degrees(q + 1) representations of

SL(2, q) modulo a prime not dividing q. This led Ward [38] to study analogouscodes arising from Weil representations of Sp(2n, q).

Groups generated by transvections over GF(2) have very recently been crucialin the study (by Pless, Sloane and Ward) of codes over GF(3).

Liebler [24] has found that modular representations of symmetric groups yieldvery interesting codes. (Naturally, representations of symmetric groups arerelated to enormous numbers of combinatorial questions (see, e.g., [30])).

Recent results of E. Anders are especially promising: he obtained a codingtheoretic proof of the Bruck-Ryser theorem, as well as various generalizations ofit-including Hughes' results [19] concerning collineations.

6. Wielandt [39] showed that I G I < 24" for a uniprimitive permutation groupG of degree n. This has just been greatly improved by Babai [1], who proved that

G I < exp(4n112 (log n)2).

His proof is entirely combinatorial. Namely, he showed that any primitivecoherent configuration on n points has a "distinguishing subset" S of size< 4n 112 log n: for any distinct points x, y, there is an s in S such that (x, s) and(y, s) are in different relations. His original motivation for finding such an Scame from a problem concerning computational complexity. His methods prob-ably will have other group theoretic applications.

Cameron has pointed out that the classification of all finite simple groups willyield the inequality I G I < n clog log" except for some explicit, familiar types ofprimitive groups. A direct proof of this would undoubtedly have to be partlycombinatorial.

An entirely different type of inequality relates the rank of G to F*(G), whenthe latter group is simple [2], [27], [21]. Complete determinations have been

482 W. M. KANTOR

made for rank 2, and for rank 3 when F'(G) is an alternating or classical group[2], [11], [23]. The rank 3 representations of the remaining Chevalley groups haveyet to be determined; however, this is known to be a finite problem [27]. Thedetermination of all primitive rank 3 groups with F'(G) neither solvable norsimple remains open; combinatorial arguments will presumably be needed inorder to relate the underlying graph to the blocks of imprimitivity of F'(G).

REFERENCES

1. L. Babai, On the order of uniprimitive permutation groups (to appear).2. E. Bannai, Maximal subgroups of low rank of finite symmetric and alternating groups, J. Fac. Sci.

Univ. Tokyo 18 (1972), 475-486.3. F. Buekenhout and C. Lefevre, Generalized quadrangles in projective spaces, Arch. Math. 25

(1974), 540-552.4. P. J. Cameron, Suborbits in transitive permutation groups, Combinatorics, M. Hall, Jr. and J. H.

van Lint (eds.), Reidel, Dordrecht, 1975, pp. 419-450.5. P. J. Cameron, J. M. Goethals and J. J. Seidel, The Krein condition, spherical designs, Norton

algebras and permutation groups, Nederl. Akad. Wetensch. A81 (1978), 196-206.6. P. J. Cameron and W. M. Kantor, 2-transitive and antifag transitive collineation groups of finite

projective spaces, J. Algebra 60 (1979), 384-422.7. B. N. Cooperstein, Minimal degree for a permutation representation of a classical group, Israel J.

Math. 30 (1978), 213-235.8. , The geometry of root subgroups in exceptional groups. I, II (to appear).9. , Subgroups of exceptional groups of Lie type generated by long root elements. I, II (to

appear).10. C. W. Curtis, Homology representations of finite groups (to appear).11. C. W. Curtis, W. M. Kantor and G. M. Seitz, The 2-transitive permutation representations of the

finite Chevalley groups, Trans. Amer. Math. Soc. 218 (1976), 1-57.12. P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep.

Suppl. 10 (1973).13. , Association schemes and t-designs in regular semilattices, J. Combinatorial Theory 20

(1976), 230-243.14. W. Feit and G. Higman, The nonexistence of certain generalized polygons, J. Algebra 1 (1964),

114-131.15. W. Haemers, Eigenvalue techniques in design and graph theory, Math. Centrum Amsterdam,

1979.16. D. G. Higman, Partial geometries, generalized quadrangles and strongly regular graphs, Atti

Conv. Geom. Appl., Perugia 1971, pp. 263-293.17. , Invariant relations, coherent configurations and generalized polygons, Combinatorics, M.

Hall, Jr. and J. H. van Lint (eds.), Reidel, Dordrecht, 1975, pp. 27-43.18. P. N. Hoefsmit, Representations of Hecke algebras of finite groups with BN-pairs of classical

type, Ph.D. Thesis, Univ. of British Columbia, 1974.19. D. R. Hughes, Collineations and generalized incidence matrices, Trans. Amer. Math. Soc. 86

(1957), 284-296,20. W. M. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer.

Math. Soc. 248 (1979), 347-379.21. , Permutation representations of the finite classical groups of small degree or rank, J.

Algebra 60 (1979), 158-168.22. , Generalized quadrangles associated with G2(q) (to appear).23. W. M. Kantor and R. A. Liebler, The rank 3 permutation representations of the finite classical

groups (to appear).24. R. A. Liebler, On codes in the natural representations of the symmetric group (to appear).25. F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, North-Holland,

Amsterdam, 1977.

FINITE GEOMETRIES AND FINITE GROUPS 483

26. M. A. Ronan, A geometric characterization of Moufang hexagons, Ph.D. Thesis, Univ. ofOregon, 1978.

27. G. M. Seitz, Small rank permutation representations of finite Chevalley groups, J. Algebra 28(1974), 508-517.

28. N. J. A. Sloane, An introduction to association schemes and coding theory, Theory andApplication of Special Functions, Academic Press, New York, 1975, pp. 225-260.

29. , Error-correcting codes and invariant theory: New applications of a nineteenth-centurytechnique, Amer. Math. Monthly 84 (1977), 82-107.

30. R. Stanley, Invariants of finite groups, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 475-511.31. D. Stanton, Some q-Krawtchouk polynomials on Chevalley groups (to appear).32. , Some Erdos-Ko-Rado theorems for Chevalley groups (to appear).33. J. A. Thas, On generalized quadrangles with parameters s - q2 and t - q3, Geom. Dedicata 5

(1976), 485-496.34. , Combinatorial characterizations of the classical generalized quadrangles, Geom. Dedi-

cata 6 (1977), 339-351.35. J. Tits, Sur la geometrie des R-espaces, J. Math. Pures Appl. 36 (1957), 17-38.36. , Les ` formes reelles" des groupes de type E6, Seminaire Bourbaki No. 162, 1958.37. A. Wagner, Determination of the finite primitive reflection groups over an arbitrary field of

characteristic not two. 1, 11, III (to appear).38. H. N. Ward, Quadratic residue codes and symplectic groups, J. Algebra 29 (1974), 150-170.39. H. Wielandt, Permutation groups through invariant functions and invariant relations, Lecture

Notes, Ohio State Univ., 1969.40. A. Yanushka, Generalized hexagons of order (t, t), Israel J. Math. 23 (1976), 309-324.

UNIVERsrrY OF OREGON

PART VII

Computer applications


EFFECTIVE PROCEDURES FOR THERECOGNITION

OF PRIMITIVE GROUPS

JOHN J. CANNONI

1. Introduction. It appears that the classification of finite simple groups is nearat hand. Other papers in these PROCEEDINGS discuss the application of theclassification to the solution of a variety of long standing problems in the theoryof finite groups. In this paper we shall consider applications to a rather moreconcrete situation: Under what circumstances is it possible to design an effectivealgorithm for determining the isomorphism type of a finite group G? It isassumed that G is given explicitly, for example, by means of generating permuta-tions or matrices. As stated above, the problem is too general since it impliesthat the possible isomorphism types of finite groups are known in some fairlyprecise sense. However, such information is not even available for p-groups. It istherefore more profitable to consider the following three special cases of theproblem:

(i) Determine the isomorphism types of the composition factors of G. Theexistence of the classification means that this task is at least theoreticallypossible.

(ii) If G happens to be an "interesting" group (e.g. simple) or is closely relatedto an interesting group, determine the isomorphism type of G. We shall notattempt to specify just what is an interesting group beyond saying that it willusually be a member of a family which plays a noteworthy role in the theory offinite groups.

(iii) Verify the truth of a specific conjecture concerning the isomorphism typeof G. For example, an investigator may have good reasons for believing that Ghas a particular structure so that the possibilities for its isomorphism type areseverely restricted. A recognition algorithm can be specifically tailored for thissituation. The group theory programming language Cayley discussed elsewherein these PROCEEDINGS (Cannon [6]) is ideal for this kind of problem.

1980 Mathematics Subject Classification. Primary 20-04; Secondary 20B10, 20B20.'Thus research was supported by a grant from the Australian Research Grants Committee.


487

488 J. J. CANNON

In this paper we shall discuss approaches to the situation outlined in (ii), whenG is a permutation group. The reasons for considering this problem in thecontext of permutation groups are: the popularity of permutation representa-tions, the existence of a useful body of characterization theorems for permuta-tion groups, and finally the availability of some very powerful algorithms forcomputing structural information in permutation groups. More precisely, theproblem we address ourselves to may be stated as follows: Given a primitivepermutation group G of the type described in (ii) above, we seek an effectivealgorithm for recognizing its isomorphism type. Considering the power of thealgorithms currently at our disposal, it is convenient to consider groups whosedegrees do not exceed 10,000. This degree represents the current upper bound ofapplicability for the techniques in general. Certainly, some of the currenttechniques can be successfully applied to groups of larger degree and it isprobable that some of the remaining techniques can be extended to handlegroups of degree greater than 10,000.

The need to identify permutation groups arises in a number of differentsituations. A major source of problems of this kind is the study of specificfinitely presented groups by means of finite quotients. Using the Todd-Coxeteralgorithm one constructs a permutation representation on the cosets of somesubgroup. It is then often necessary to determine the structure of the permuta-tion homomorph. Using the techniques described in this paper we have de-termined all permutation groups of degree up to 50 that are homomorphicimages of the infinite group x2 = y3 = (xy)7 = e. Another source of permuta-tion group identification problems arises in the study of automorphism groupsof combinatorial structures. Fairly efficient algorithms have been developed forcomputing the automorphism groups of such structures as groups (Robertz [16]),graphs (McKay [14]), Hadamard matrices (Leon [12]), and self-dual codes (Leon[13]). If it has not already been done, one can expect the development of amethod for computing automorphism groups of block designs in the near future.Each of the above algorithms constructs a set of generating permutations for theautomorphism group and whenever the group is nontrivial one usually desires toknow as much as possible about its structure. Finally, if current attempts todevise methods for computing the Galois group of a polynomial are successful,this will be a fresh source of permutation group identification problems.

2. Computational techniques. For the remainder of this paper, G will denote apermutation group acting on the set St. Without much loss of generality, we mayassume that G is primitive, for if G is intransitive we analyze its transitiveconstituents, while if G is transitive but not primitive, we consider the groupobtained by letting G act on a system of imprimitivity in which the blocks are aslarge as possible. An algorithm described in Atkinson [2] may be used to testwhether G is primitive, and in the event that it is not, construct systems ofimprimitivity. The kernel of the action of G on a system of imprimitivity can berapidly computed using an algorithm of Butler [3].

Almost all algorithms for computing structural information for permutationgroups of large degree have as their basis the notion of base and stronggenerating set (Sims [18], [19]). Efficient algorithms for constructing stronggenerating sets for degrees up to 10,000 have been developed by Sims [17] and

RECOGNITION OF PRIMITIVE GROUPS 489

Leon [11]. Knowledge of a strong generating set for a group G immediately givesus the order and transitivity of G. Using strong generator techniques, Butler andCannon [4] have developed effective algorithms for computing normal closures,derived subgroups, derived series and lower central series. Assuming the availa-bility of a strong generating set for G, Sims [18], [19] describes extremelyefficient backtrack algorithms for computing centralizers of elements and inter-sections of subgroups. In a similar spirit, Butler and Cannon [5] have devised abacktrack algorithm for computing Sylow subgroups. Using the Sylow andsubgroup intersection algorithms it is a trivial matter to compute OO(G).

The techniques summarized above give us access to sufficient structure of Gfor the application of several very useful characterization theorems. At this stageit is perhaps useful to summarize the major algorithms we have at our disposaland give a rough indication of the relative cost of using them.

I (Cheap).Test whether regular or semiregular.Test whether transitive.Test whether primitive.Test whether alternating or symmetric.'Stabilizer of a sequence.Centralizer.

*Centre.*Intersection of subgroups.

II (Medium).Strong generating set.*Normal closure of a subgroup.*Derived subgroup.*Derived series.*Lower central series.*Sylow subgroup.*Core (and hence OO(G)).*Fitting subgroup.

III (Expensive).Conjugacy classes.Test whether simple.Normal subgroups.Computing in quotient groups (unless the quotient group can be constructed

naturally as a permutation group).

An asterisk on an item in this list indicates that a base and strong generatingset is assumed to be available. The costing of the items represents the author'scurrent state of knowledge and can be expected to change in the light of newdevelopments.

3. Outline of a recognition procedure.3.1. General strategy. In this section we shall give an indication as to how the

recognition procedure handles 2-transitive and rank 3 groups. Full details willappear in a later publication. Throughout this section G will denote a primitive

490 J. J. CANNON

permutation group of degree n given by a set of generators. G is assumed to bek-transitive but not (k + 1)-transitive and to have rank r. The theory of2-transitive and rank 3 groups has been extensively developed during the lasttwo decades so that there now exists a considerable body of characterizationtheorems for those simple groups which have such representations. By carefullymatching characterization theorems with the information that can be computedcheaply from the strong generators of G, we are able to devise a fast algorithmfor recognizing simple groups in these representations.

Flow Diagram of Recognition Procedure

is Clie scoop C alternating

or symanetrie'.

No

3-ti--nsitive

theorems

Ccapuoe a base and stronggenerating set for C anc hence

the values ci k and r.

k=2

2-tr2nsitivetheorems

rank 3theorems

YesFinish

k=1

general char.

then reins

3.2. Alternating and symmetric groups. It is highly desirable to recognize whenG is the alternating group A,,, or the symmetric group without having toconstruct a strong generating set. However, an algorithm which will alwaysrecognize A. or S,, without computing strong generators is often very expensivewhen G is neither A. nor 5,,. Thus the optimum algorithm will expend a limitedamount of effort on a sequence of tests (applied in order of increasing cost) thatdo not assume knowledge of strong generators, and then, if these tests areunsuccessful, settle the matter definitively by computing strong generators.

The first text examines a few permutations of the group in the hope of findingone, 1r say, which contains a p-cycle, p a prime and p < n - 3, such that thelengths of the remaining cycles of it are coprime top. If this test is unsuccessfulwe next compute an "approximation" to a strong generating set for G using theso-called "random Schreier method" (Leon [11]). This yields lower bounds onthe order and degree of transitivity. Bochert's order bound (Wielandt [20, p. 41])and the transitivity bounds of Jordan, Manning and Weiss (Wielandt [20, pp.39-40]) can now be applied. If these fail to show that G is A,, or S,,, theTodd-Coxeter Schreier method (Leon [11]) is used to extend the approximate

strong generating set to a full strong generating set. If G is not An or S, we nowhave strong generators available for the next stage of the analysis.

3.3. Triply transitive groups. Although 3-transitive groups are rare, it is possibleto recognize most of them using only the transitivity and order information thatis now available. Thus this step identifies PGL(2, q) on q + 1 letters; M11, M12,M22, M23 and M24 on 11, 12, 22, 23 and 24 letters respectively; M11 on 12 letters;Aut M22 on 22 letters; and the semidirect product of PSL(d, 2) and an elemen-tary abelian group of order 2d, where the product has degree 2d.

3.4. Doubly transitive groups. The 2-transitive representations of the finiteChevalley groups have been determined by Curtis, Kantor and Seitz [7]; thesolvable 2-transitive groups have been determined by Huppert [9]; and Hering isclose to finishing the determination of 2-transitive groups which contain aregular normal subgroup. (Hering's work assumes the classification.) The fami-lies of Chevalley groups having 2-transitive representations are as follows:

(A) PSL(d, q) acting on (qd - 1)/(q - 1) points, d > 3, q a prime power.

(B) PSL(2, q) acting on q + 1 points, q + 1 > 5, q a prime power.PSU(3, q) acting on q3 + 1 points, q3 + 1 > 10, q a prime power.R(q) acting on q3 + 1 points, q = 32m+1 > 3.Sz(q) acting on q2 + 1 points, q = 22,+1 > 8.

(C) Sp(2m, 2) acting on 2'-'(2'° ± 1) points, 2'-1(2' ± 1) > 6.

Apart from these a small number of other Chevalley groups have 2-transitiverepresentations.

The projective groups PSL(d, q), d > 3, are recognized using a very nicecharacterization theorem of O'Nan [15]. This theorem asserts that if G is a2-transitive group acting on the set St and if Ga, a E St, contains an abeliannormal subgroup A a which is not semiregular on St - a, then PSL(d, q) < G <PGL(d, q). The recognition process is now clear: we look for a prime power,q = p', which is consistent with the given degree and order. If there are nonethen G is not PSL(d, q); otherwise we extract strong generators for Ga andcompute Op(Ga) etc.

In order to apply characterization theorems for the groups listed in (B) and(C), the possibility that G has a regular normal subgroup must first be excluded.

LEMMA. Let G be a 2-transitive group of degree n containing a regular normalsubgroup N. Then n = p', p a prime, and N = OO(G).

This result provides an important reduction since it appears to be expensive todetermine whether or not an arbitrary primitive group contains a regular normalsubgroup. Having established that G does not contain a regular normal sub-group, we determine whether it is one of the groups of (B) (or a normalextension of one) by using a theorem of Aschbacher [1], or whether it is asymplectic group (C) by using a theorem of Shult.

Techniques have also been developed to determine the isomorphism type of Gwhen it is a known 2-transitive group containing a regular normal subgroup.Thus if the classification is complete, our procedure should recognize any2-transitive group of degree up to 10,000. Note that while our identification of G

492 J. J. CANNON

as a certain group usually involves the application of characterization theorems,it does not depend upon the completeness of the classification. If the procedureis given a 2-transitive representation of a hitherto unknown simple group, it willoutput a message saying that it cannot identify the group.

3.5. Rank 3 groups. The rank 3 representations of the classical groups havebeen determined by Kantor and Liebler [10], while Cooperstein has examinedthe exceptional groups. The solvable rank 3 groups have been determined byFoulser [8]. As in the case of 2-transitive groups, a regular normal subgroup of arank 3 group G must be an OO(G), for some prime p. The recognition procedurefor rank 3 groups makes heavy use of characterizations by subdegrees. Ourcurrent procedures are powerful enough to recognize most of the known simplegroups (and normal extensions of these) in their various rank 3 representations.However, further work remains to be done in order to satisfactorily handle rank3 groups containing a regular normal subgroup.

3.6. Groups of higher rank. Owing to the paucity of characterization theoremsfor higher rank representations of simple groups it is usually necessary to resortto abstract characterization theorems in such cases. The application of thesetheorems is considerably more expensive than the application of permutationgroup characterizations.

4. Implementation. The various parts of the primitive group identificationprocedure outlined above have been implemented using the permutation groupfacilities provided by the library of group theoretic routines discussed in Cannon[6]. The recognition program has been successfully applied to more than 2000groups (mostly having degrees in the low hundreds). Recognition times foralternating and symmetric groups were usually of the order of a second whilethe times for other groups ranged between 1 second and 100 seconds. (Times arein CYBER 76 seconds.)

Acknowledgements. I would like to express my appreciation to Peter Cameronand Bill Kantor for providing me with much useful information concerningpermutation group characterizations.

REFERENCES

1. M. Aschbacher, Doubly transitive groups in which the stabilizer of two points is abelian, J.Algebra 18 (1971), 114-136.

2. M. D. Atkinson, An algorithm for finding the blocks of a permutation group, Math. Comp. 29(1975), 911-913.

3. G. Butler, Computational approaches to certain problems in the theory of finite groups, Ph.D.thesis, University of Sydney, 1979.

4. G. Butler and J. J. Cannon, Computing in permutation and matrix groups. I: Normal closure,commutator subgroup and series (preprint).

5. , Computing in permutation and matrix groups. III: Sylow subgroups (preprint).6. J. J. Cannon, Software tools for group theory, these PROCEEDINGS, pp. 495-502.7. C. W. Curtis, W. M. Kantor and G. M. Seitz, The 2-transitive representations of the finite

Chevalley groups, Trans. Amer. Math. Soc. 218 (1976), 1-59.8. D. A. Foulser, Solvable primitive groups of low rank, Trans. Amer. Math. Soc. 143 (1969), 1-54.9. B. Huppert, Zweifach transitive, auflosbare Permutationsgruppen, Math. Z. 68 (1957), 126-150.

10. W. M. Kantor and R. A. Liebler, The rank 3 permutation representations of the finite classicalgroups (preprint).


11. J. S. Leon, On an algorithm for finding a base and strong generating set for a group given bygenerating permutations, Math. Comp. 33 (1980), 941-974.

12. , An algorithm for finding the automorphism group of a Hadanutrd matrix, J. Combina-torial Theory 27(1979), 289-306.

13. , Personal communication.14. B. D. McKay, Computing automorphisms and canonical labellings of graphs, Lecture Notes in

Math., vol. 686, Springer-Verlag, Berlin and New York, 1978, pp. 223-232.15. M. O'Nan, A characterization of L (q) as a permutation group, Math. Z. 127 (1972), 301-314.16. H. Robertz, Eine Methode zur Berechnung der Automorphismengruppe einer endlichen Gruppe,

Diplomarbeit, Rhein-Westfalische Technische Hochschule Aachen, 1976.17. C. C. Sims, Computational methods in the study of permutation groups, Computational Problems

in Abstract Algebra (Proc. Conf., Oxford, 1967), J. Leech (ed.), Pergamon, Oxford, 1970, pp.169-183.

18. , Determining the coryugacy classes in a permutation group, SIAM-AMS Proc., vol. 4,Amer. Math. Soc., Providence, R. I., 1971, pp. 191-195.

19. , Computation with permutation groups, Proc. Second Sympos. on Symbolic andAlgebraic Manipulation (Los Angeles, 1971), S. R. Petrick (ed.), Association for ComputingMachinery, New York, 1971, pp. 23-28.

20. H. Wielandt, Finite permutation groups, Academic Press, New York and London, 1964.

UNrvERsiTY of SYDNEY, AUSTRALIA


SOFTWARE TOOLS FOR GROUP THEORY

JOHN J. CANNONI

1. Introduction. Research in the theory of groups often gives rise to problemswhere the investigator either needs to know something about the structure of aparticular group, or has to perform calculations within a group or related object(e.g. a group ring or KG-module). While some problems are sufficiently uniqueto require "one-off" programs, it would seem that many computational prob-lems have a sufficiently similar structure to justify the development of programsof more general applicability. The standard example of a recurring calculation ingroup theory is the Todd-Coxeter process for enumerating cosets in a finitelypresented group. An example of a calculation that is common in the theory offinite simple groups is the determination of detailed structural information for aSylow 2-subgroup or centralizer of an involution. Provided that the order of thesubgroup is "reasonable", the machine can be used to compute conjugacyclasses, elementary abelian subgroups, normal subgroups, various types of series,etc.

The group theorist contemplating using the computer to solve a problemwhich involves actually computing within a group or related object is usuallyfaced with the problem of designing and implementing algorithms which can bevery complex. (Some group theoretic algorithms can take more than a year toimplement properly.) Clearly, unless the problem is of considerable importanceand impervious to attack by nonmachine methods, it may not be worth theperson's time producing the program. Equally distressing is the situation whereina number of different workers write their own programs to solve the sameproblem, either because they are unaware of the existence of similar programs orbecause the other programs are not transferable. This problem has been recog-nized by workers in other fields for some time and significant effort has beeninvested in the production of high quality portable implementations of the moreuseful algorithms in such areas as statistics (e.g. SPSS, IMSL and STATPAK)and numerical analysis (e.g. EISPACK and NAG).

1980 Mathematics Subject Classification. Primary 20-04.'This research was supported by a grant from the Australian Research Grants Committee.


495

496 J. J. CANNON

In recent years a subfield concerned with the design and analysis of grouptheoretic algorithms has emerged-a group theoretic analogue of numericalanalysis. A discussion of some of these algorithms may be found in Sims [15],[16]. This paper outlines the status of available implementations of some of thesealgorithms and gives an account of the group theory programming languageCayley.

2. Autonomous programs. We use the term autonomous to refer to a programwhich implements either a single algorithm, or a collection of algorithms,directed towards the solution of a single type of problem. Examples are Todd-Coxeter programs and programs for the interactive computation of charactertables. An autonomous program has its own input-output facilities and mostapplications require only that the user supply the appropriate data. Autonomousprograms usually cannot be called by other programs without substantialmodification.

Some autonomous programs known to me which are currently available fromtheir authors are: Todd-Coxeter (Alford and Havas, ANU, [6]), low indexsubgroups (Gallagher, RWTH Aachen or Cannon, Sydney), Reidemeister-Schreier (Havas, ANU, [9]), calculation of nilpotent quotients of finitely pre-sented groups (Havas and Newman, ANU, [13]), p-group generation (Ascione,Havas and Leedham-Green, ANU, [1]), Tietze transformations (Kenne, Havasand Richardson, ANU), calculation of algebraic invariants for knots (Havas,ANU, [8]), integer matrix diagonalization (Havas and Sterling, ANU, [10]) andinteractive character calculations (Esper et al., RWTH Aachen, [7]). Hunt [11],at the University of New South Wales, is currently developing a suite of routinesfor manipulating the character tables of very large simple groups. Hunt plans toinclude a file of character tables of finite simple groups with his package.

3. Group theory program libraries. A program library consists of a collection ofsubroutines implementing a range of algorithms. Typically the routines sharecommon data structures so that information is easily transmitted betweenlibrary routines. Normally a library can only be used by having the user writehis own driver program to handle input/output and to invoke the desired libraryroutines. The existence and availability of high quality program libraries greatlyfacilitates the exploitation of the computer by workers in a given area. Inparticular, the cost of implementing new algorithms can be greatly reduced if theimplementor is able to use "off-the-shelf" code implementing well-known algo-rithms.

The author and his group in Sydney have been engaged in the development ofa program library for group theory since 1972. A similar library is beingconstructed by Neubiiser's group in Aachen. Early accounts of the proposedscopes of these libraries may be found in Cannon [2] and Neubiiser [12]. TheAachen library includes programs for constructing normal subgroups, subgrouplattices, automorphism groups, character tables and representations of groups ofmoderate order. Highlights of the Sydney library are routines for computingstrong generators, conjugacy classes, normal subgroups, centralizers, normal-izers, Sylow subgroups and various kinds of series, in permutation and matrixgroups of very large order. Details may be found in Cannon [4].

SOFTWARE TOOLS FOR GROUP THEORY 497

Both the Aachen and Sydney libraries are written in FORTRAN and arerelatively portable, requiring less than one month's work to implement them ona new type of machine. The Sydney library now contains implementations ofsome 150 group theoretic algorithms. The benefit that can be derived from theavailability of such libraries is illustrated by noting that the complex primitivegroup identification algorithm outlined in Cannon [5] took less than a month toimplement, since all the group structure information required could be calcu-lated using existing library routines.

I understand that Leon and Pless at Chicago Circle are undertaking thedevelopment of a library of routines for various combinatorial and codingtheory calculations. This library includes some group theory algorithms and iswritten in PLC 1.

4. The group theory programming language Cayley. Although a range ofgeneral purpose programming languages is now available, the coding of evensimple algebraic algorithms in these languages is usually unnatural and cumber-some. While this may not deter the sophisticated computer user, it is clearly amajor impediment to the use of machines by the vast majority of grouptheorists. Secondly, the usefulness of a program library for group theory isenormously enhanced if the routines in it can be driven by a very high levelprogramming language. Motivated by these considerations, in 1975, I com-menced the development of a programming language (known as Cayley) whichis intended for group theory computations. The philosophy underlying thedesign of Cayley may be summarized as follows.

(i) Its syntax should provide a convenient system for expressing typicalcomputational schematas which arise in algebra. For example, in a conventionalprogramming language repeated execution is usually expressed by a constructsuch as

for i = 1to10do....However, in algebra, a more typical situation requires the performance of agroup of operations, once for each element of a finite set S. Thus Cayleyprovides the construct

for each x in S do ...where x is a variable which runs through the elements of S. This implies that thetypes of object on which the language operates include such things as element,set, group, field, module, mapping, etc.

(ii) The language should provide a group theorist with an easy means of usinga group theory program library. As indicated in the previous section, direct useof library routines normally requires considerable knowledge of the data struc-tures etc. used by the library. In my experience, a person with no previousknowledge of computing can use library routines through Cayley after quite ashort period of instruction. After a little practice, the novice user is able tosynthesize new algorithms using library routines as building blocks.

(iii) The language should provide facilities for interactive computing. Innumerous situations it is desirable to see the results obtained at each step beforespecifying what is to be done at next step. Cayley is therefore organized so that

498 J. J. CANNON

if it is run from a time-sharing terminal, each statement can be executed as soonas it is typed. This provides for a high degree of interaction between the user andthe machine.

Rather than catalogue the specific capabilities of Cayley, I will present anumber of short Cayley programs illustrating some typical computations.

EXAMPLE 1. Coset enumerations over a parametrized family of presentations.Problem [Due to B. H. Neumann]. Determine the orders of the groups having

presentations

<a, b, cl(ab)2 = C", aP = c,8, b4 = c''; p, q, a, /3 and yintegers; with two of a, /3, y equal to 1 and the

third equal to a nonnegative integer r>

for small values of p, q and r.CAYLEY PROGRAM.

g =free(a, b, c);forp=2to3do

for q=pto5dofor r = 0 to 3 do

rels = bTq'cT-r];print 'group defined by', rels, 'has order', index(rels, 0), '$N$N';rels = bTq'cT-1];print 'group defined by', rels, 'has order', index(rels, 0), '$N$N';rels = bTq'cT-1];print 'group defined by', rels, 'has order', index(rels, 0), '$N$N';

end;end;

end;

EXPLANATION. The first statement, g = free(a, b, c), defines g to be the freegroup on generators a, b and c. The statements beginning rels = define the valueof the variable rels to be the set of words on the right-hand side. The subsequentprint statements print the heading strings delimited by primes, the set rels, andthe index of the trivial subgroup in the group defined by taking the words in setrels as relators. The function index applies the Todd-Coxeter algorithm to thegroup defined by the first argument and the subgroup defined by the second(zero designates the trivial subgroup).

I am indebted to George Havas for this example.EXAMPLE 2. Detailed structure of small groups.Problem. Determine the elements, classes, subgroups, automorphism group

and character table of a small finitely presented group G, where

G=<a,b,cia3=b3=C3= 1, (ab)4 = (bc)4 =1,a(an)2=a,b(" 2=b,Ca=c)'.

CAYLEY PROGRAM.

g = free(a, b, c);g.relations: aT3 = bT3 = cT3 = 1, 1, cTa = c,

aT((a'b)T2) = a, bT((b'c)T2) = b;print relations(g);

print elements(g), classes(g), normal subgroups(g);print subgroups(g), automorphism group(g), character table(g);EXPLANATION. The g.relations statement imposes the given relations on g. The

objects named in the print statements are constructed if necessary and printed.EXAMPLE 3. Automorphism towers.Problem. Determine the automorphism tower for the direct product of the

quaternion group of order 8 and the cyclic group of order 2.CAYLEY PROGRAM.

g = trivial(O);

h = free(a, b, c); h.relations: aT2 = bT2 = cT2 = 1, (a, c) = (b, c) _1;

while not isomorphic (g, h) dog = h;h = automorphism group(g);print h;

end;EXPLANATION. The function trivial(O) creates the group of order 1, while the

next two lines set up defining relations for the direct product of the quaterniongroup and the cyclic group of order 2. The statements between while and endwill be executed until groups g and h become isomorphic, i.e. until the automor-phism tower becomes stationary.

EXAMPLE 4. Nonisomorphic groups having the same character tables.Problem [M. Isaacs]. Are the character tables of the split and nonsplit

extensions of the elementary abelian group of order 8 by L2(7) the same? Weobtain the split extension as a subgroup of index 15 in the alternating group ofdegree 8, and the nonsplit extension as the finitely presented group.

<a, b, c, x, yla2 = b2 = c2 = 1, (a, b) = (a, c) = (b, c) = 1,

x2 = y3 = (xy)7 = 1, (x, y)4 = abc,ax= b,b''=bc,c''=b,(c,x)=(a,y)=1).

CAYLEY PROGRAM.

"we first do the split extension";a8 = alternating (8);s = low index subgroups(a8, 0, seq(15, 15));print 'split extension', character table(s[l]);clear;"now do the nonsplit extension";g = free(a, b, c, x, y);g.relations: aT2 = bT2 = cT2 = 1, (a, b) = (a, c) = (b, c) = 1,xT2 = yT3 = (xy)T7 = 1, (x, y)T4 = aTx = b, bTy =cTy = b, (c, x) = (a, y) = 1;print `nonsplit extension', character table(g);EXPLANATION. The sentences enclosed by double quotes are comments and

have no effect. The function alternating(n) sets up generating permutations anddefining relations for the alternating group of degree n. The function low indexsubgroups (g, 0, seq(m, n)) computes the subgroups of group g whose indices lie

500 J, J. CANNON

between m and n. The subgroups are returned as a sequence (which we call s inthis instance). Subgroups in the sequence can now be accessed individually bymeans of a subscripting operation: s[l] will therefore be the first subgroup ofindex 15.

The character tables do turn out to be identical.EXAMPLE 5. Calculation of fusion patterns.Problem. Determine how the conjugacy classes of involutions in the Sylow

2-subgroup of Sp(4, 4) fuse in the whole group.CAYLEY PROGRAM.

library sp44;s2 = sylow subgroup(sp44, 2);c = classes(s2);fused = null;for i = 1 to length(c) do

x =representative(c[i]);if (order(x) 2 or x in fused) then loop else print 'class', i, 'with representa-

tive', x, 'fuses with';for j = i + 1 to length(c) do

y = representative(c[j]);if conjugate(sp44, x, y) then

print 'class', j, 'with representative', y;fused = fused join [y];

end;

end;end;

end;EXPLANATION. Cayley is allowed to access a permanent file (the library file)

which contains blocks of Cayley statements labelled in a unique manner.Usually such a block will contain the Cayley statements necessary to define agroup. Our program assumes that the library file contains a block definingSp(4, 4) and that this block is labelled sp44. The statement library sp44 causesthe library file to be searched for block sp44 and, once the block has beenfound, to execute the statements therein.

The function classes returns a sequence of classes, so that c[i] will have as itsvalue the ith class. We now require an element from the ith class and this isdone by the function representative which, given a set as its argument, returns anelement of the set. conjugate(g, x, y) is a boolean-valued function returning trueif elements x and y are conjugate in g. The set fused contains those representa-tives of Sylow 2-subgroup classes which fuse with earlier classes. The symbolnull denotes the null set.

EXAMPLE 6. Identification of a permutation group.Problem. Determine whether a 2-transitive group G of degree 156 is PSL(4, 5)

by applying O'Nan's characterization theorem [14]. This theorem states that if Gis a 2-transitive group acting on the set 2 and if Ga, a E 2, contains an abeliannormal subgroup A a which is not semiregular on 2 - a, then PSL(n, q) < G <PGL(n, q).


CAYLEY PROGRAM.

g = perm(156);g.generators: . . .

if (order(g) 2T7.3T2.5T6.13.31) then print 'g is not psl(4, 5)'; quit; elsegl = stabilizer(g, 1);al = core(gl, sylow subgroup(gl, 5));oml = orbit( g1, 2);if semiregular(a1, Oml) then print g is not psl(4, 5)'; else

print g is psl(4, 5)';end;

end;ExPLANATION. The program uses the fact that A" must be Op(G"). The

function orbit(g, i) computes the orbit of point i under group g and it is usedhere to define the set om 1 (52 - a) on which a 1 (A ") has to be tested forsemiregularity. The remainder should be clear.

An early definition of the syntax of Cayley may be found in Cannon [3].Subsequent work has resulted in a simpler, cleaner design for parts of thelanguage. The language is supported by the Sydney library described in §3,together with routines from the Aachen library, and modified versions of someof the ANU autonomous programs listed in §2. As in the case of the library, theCayley translator and interpreter are written in FORTRAN and have a similardegree of portability. The system comprising translator, interpreter and librarywill be available in 1981.

REFERENCES

1. J. A. Ascione, G. Havas and C. R. Leedham-Green, A computer-aided classification of certaingroups of prime power order, Bull. Austral. Math. Soc. 17 (1977), 257-274, 317-320.

2. J. J. Cannon, A general purpose group theory program, Proc. 2nd Internat. Conf. Theory ofGroups, Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin and New York, 1974, pp. 204-217.

3. , A draft description of the group theory language Cayley, SYMSAC'76, Proc. 1976 ACMSympos. Symbolic and Algebraic Computation, Assoc. Comput Mach., New York, 1976, pp. 66-84.

4. , The Cayley library of built-in functions, Technical Report No. 13, Computer-AidedMathematics Project, Dept. of Pure Math., University of Sydney, 1976, revised 1980.

5. , Effective procedures for the recognition of primitive groups, these PROCEEDINGS,pp. 487-493.

6. J. J. Cannon, L. A. Dimino, G. Havas and J. M. Watson, Implementation and analysis of theTodd-Coxeter algorithm, Math. Comput. 27 (1973), 463-490.

7. N. Esper, Ein interaktives Programmsystem zur Erzeugung der Rationalisierten charakterentafeleiner endlichen Gruppe, Staatsexamensarbeit, RWTH Aachen, 1974.

8. G. Havas, Computational approaches to combinatorial group theory, Ph.D. Thesis, Univ. ofSydney, 1974.

9. , A Reidemeister-Schreier program, Proc. 2nd Internat. Conf. Theory of Groups,Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin and New York, 1974, pp. 347-356.

10. G. Havas and L. Sterling, Integer matrices and abelian groups, Symbolic and AlgebraicComputation, Lecture Notes in Computer Sci., vol. 72, Springer-Verlag, Berlin and New York, 1979,pp. 431-451.

11. D. Hunt, A computer-based atlas of finite simple groups, these PROCEEDINGS, pp. 507-510.12. J. Neubuser, Some computational methods in group theory, Third Internat. Colloq. Advanced

Computing Methods in Theoretical Physics, Marseille, 1973, B-II-I-B-II-35.

502 J. J. CANNON

13. M. F. Newman, Calculating presentations for certain kinds of quotient groups, SYMSAC'76,Proc. 1976 ACM Sympos. Symbolic and Algebraic Computation, Assoc. Comput. Math., New York,1976, pp. 2-8.

14. M. O'Nan, A characterization of L (q) as a permutation group, Math. Z. 127(1972), 301-314.15. C. C. Sims, Some group-theoretic algorithms, Topics in Algebra, Lecture Notes in Math., vol.

697, Springer-Verlag, Berlin and New York, 1978, pp. 108-124.16. , Group-theoretic algorithms, A survey, Proc. Internat. Congr. Math., Helsinki, 1978, vol.

2, Academia Scientiarum Fennica, Helsinki, 1980, pp. 979-985.

UNIVERSITY OF SYDNEY, AUSTRALIA

THE COMPUTATION OF ACOUNTEREXAMPLE TO THE

CLASS-BREADTHCONJECTURE FOR p-GROUPS

VOLKMAR FELSCH

Introduction. This talk is intended to give only a brief report on the successfulsearch for a counterexample to the class-breadth conjecture (cf. formula (1)below) carried out by several colleagues at Aachen in the fall of 1978. Itdemonstrates that various computer programs played an essential role in thatproject. The results of the computations and the subsequent theoretical investi-gations will be described in detail in a forthcoming paper [Fel**].

The class-breadth conjecture. Let G be a p-group of order p" and nilpotencyclass c(G). The breadth b(G) of G is defined by pb(G) - max(I g' ; g (=- G ). Theso-called class-breadth conjecture stated that the relation

c(G) ' b(G) + 1 (I)is true for all finite p-groups. It has been discussed extensively in several papers[Lee69], [Ga172], [Vau74a], [Vau74b], [Mac77], [Mac78], and it has been provedfor certain classes of p-groups in these papers and in some earlier ones [Bur11,§99], [Kno5I], [Kno53], [B1a58], [Hup67, Kapitel III, Aufgabe 25]. Moreover, C.R. Leedham-Green, P. M. Neumann and J. Wiegold [Lee69] showed that

c(G) <p

p1 b(G) + I

holds for any finite p-group G.If we introduce the co-class cc(G) = n - c(G) and the co-breadth cb(G) _

n - b(G), then relation (1) is equivalent to

cb(G) < cc(G) + 1. (2)

Hence, as p cb(G) = min{ l C6(g)1; g E G), a promising strategy for a search forcounterexamples to the class-breadth conjecture was to investigate series ofp-groups with restricted co-class and increasing centralizers.

1980 Mathematics Subject Classification. Primary 20D15, 20-04.0 American Mathematical Society 1980

503

504 VOLKMAR FELSCH

Series of p-groups derived from space groups. C. R. Leedham-Green and M. F.Newman observed that certain space groups provide series of p-groupsof constant co-class.

A space group S of dimension d is an extension of a free abelian group T ofrank d by a finite group P which acts faithfully on T. P is called the point groupof S. In our context we are only interested in the special case that P is a p-groupfor some prime p and that P acts p-uniserially on T, i.e. that the P-submodulesof T with p-power index in T are ordered linearly. Then, if S = So > S, > S2> . denotes the lower central series of S, we get an infinite series ofp-groups G; = S/S; of increasing order and constant co-class cc(G;) = cc(Gc(p))for i > c(P).

The search for a counterexample. M. F. Newman [New78] pointed out thatthere is a group P4 < GL(4, Z) of order 26 which acts 2-uniserially on Z4 and, inaddition, has the property that for each space group S with point group P4 the2-factor groups G; = S/S; for i > c(P4) fulfill a certain condition which isnecessary for any counterexample to (1), namely they are covered by theirtwo-step centralizers CG,(S,/5,+2) with 0 < j < i - 2. An inspection of all4-dimensional space groups with point groups acting 2-uniserially on Z4 (whichhad been determined [Fin79], [Fin8O] from a complete list of 4-dimensionalcrystallographic space groups [Bro78]) showed, however, that there is no coun-terexample to (1) in the associated series of 2-groups of constant co-class.

The next dimension to look at was 8. Generalizing P41 J. Neubiiser and W.Plesken found a suitable 8-dimensional group P8 < GL(8, Z) of order 211 andnilpotency class 8, given by a presentation with 5 generators and 15 definingrelations and a matrix representation. It was then possible to settle the rest ofour investigation by a series of computer calculations using some of the grouptheoretical programs that have been developed during the last two decades.

To simplify the computations, we started with a run of G. Havas' Tietzetransformation program [Hav74], [Ken77] which reduced the presentation for P8to 5 short relations in 2 generators.

Then, in the first serious step, we used M. Pohst's implementation of W.Plesken's centering algorithm [P1e76], [Ple77] to calculate generating matrices(conjugate to those of P8) for a representative of each Z-class in the Q-class ofP8, where the Z-class or the Q-class of a subgroup of GL(d, Z) is the set of itsconjugates in GL(d, Z) under conjugation by GL(d, Z) or GL(d, Q), respec-tively. We got four Z-class representatives P8 t, say, with I < j < 4.

Subsequently, for each P$') we determined up to isomorphism all extensionsof Z8 by Ps(i) by means of an algorithm of H. Zassenhaus [Zas48], [Bro69] whichhas been implemented at Aachen by K.-J. Kohler. As we did not mind gettingsome isomorphism types more than once, we saved the work of computing thenormalizer of P8 in GL(8, Z). So we ended up with a set of altogether 40groups containing at least one representative for each isomorphism class ofspace groups with point group P8.

With the aid of the Canberra implementation [New76], [Hav76] of the nilpo-tent quotient algorithm [Mac74], [Wam74], [Bay74] it then was an easy task todetermine for each of these space groups the maximal co-class of the associated

COUNTEREXAMPLE TO THE CLASS-BREADTH CONJECTURE 505

series of 2-factor groups. It turned out that the values of these co-classes varybetween 5 and 11.

We now concentrated our efforts on the 2-factor groups G1 = S/Si of thegroup

S = (a, b, tl, t2, t3, t4, t5, t6, t7, t8 [ti, tj] = 1 for 1 < i <j < 8,

a4 = t4, b8 = 1, bab7aba3b7a3 = t2t5, (a3b4)2 = 1,

a 2(ab2)2(a3b6)2 - t-113131 3 4, at 1 Q t3t4t8' Qt2Q-1 t

2,

at3a-' = t1 113, at4a-1 = t4, at5a = t5, at6a-' = t6, at7a t7,

at8a - 1 = t3t4, bt1b t2t5 't6t7, bt2b -' = t4, bt3b -' = t5 116,

bt4b-' = t5, bt5b-' = t3t4t8 1, bt6b-' = t8 ', bt7b-' = t1, bt8b-' = t2t6)

which was the first one among the above space groups with cc(G) = 5 fori > c(P8). As we knew class and co-class for each G1, it remained to compute thebreadth or the co-breadth.

A second run of the nilpotent quotient algorithm for S provided us with apower-commutator presentation for G50 and hence for each G. with i < 50.Unfortunately, space requirements restricted the applicability of the conjugacyclass routines of the Aachen-Sydney GROUP system [Can74] to groups G1 withG, < 214, i.e. i < 9, and all these groups have centralizers small enough to

satisfy relation (2). Therefore we designed and implemented an algorithm for thecomputation of conjugacy classes and centralizers in large p-groups [FeI79].Applying it to G1 for increasing i, we found that cb(G29) = 7 = cc(G29) + 2. Inother words: The group G29 of order 234 has class 29 and breadth 27 and henceis a counterexample to the class-breadth conjecture.

Starting from these results it has been proved [Fel++] that G29 is the firstmember of a series of counterexamples which for each integer z contains a2-group G with c(G) > b(G) + z. On the other hand, to my knowledge it is stillan open problem if the class-breadth conjecture is true for odd primes or not.

ACKNOWLEDGEMENT. The extensive computer calculations referred to in thispaper have been done on the CYBER 175 of the "Rechenzentrum der RWTHAachen".

REFERENCES

[Bay741 A. J. Bayes, J. Kautsky and J. W. Wamsley, Computation in nilpotent groups (application),pp. 82-89 in [New741.

[B1a581 N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45-92.[Bro69] Harold Brown, An algorithm for the determination of space groups, Math. Comp. 23 (1969),

499-514.[Bro78] Harold Brown, Rolf Billow, Joachim Neubiiser, Hans Wondratschek and Hans Zas-

senhaus, Crystallographic groups in four-dimensional space, Wiley, New York, 1978.[Bur11] W. Burnside, Theory of groups of finite order, reprint of the 2nd ed. (Cambridge, 1911),

Dover, New York, 1955.[Can74] John Cannon, A general purpose group theory program, pp. 204-217 in [New74].

506 VOLKMAR FELSCH

[Fe179] Volkmar Felsch and Joachim Neubuser, An algorithm for the computation of conjugacyclasses and centralizers in p-groups, pp. 452-465 in [Ng79J.

[Fed..] Waltraud Felsch, Joachim Neubuser and Wilhelm Plesken, Space groups and groups ofprime-power order IV. Counterexamples to the class-breadth conjecture, J. London Math. Soc. (toappear).

[Fin79] Helga Finken, Raurrgmppen und Serien von p-Gruppen, Staatsexamensarbeit, RWTHAachen, 1979.

[FiiMO] H. Finken, J. Neubuser and W. Plesken, Space groups and groups of prime-power order II.Classification of space groups by finite factor groups, Arch. Math. (Basel) 33 (1980),203-209.

[Ga172] Joseph A. Gallian, On the breadth of a finite p-group, Math. Z. 126 (1972),224-226.[Hav74] George Havas, A Reidemeister-Schreier program, pp. 347-356 in [New74].[Hav76] George Havas and Tim Nicholson, Collection, pp. 9-14 in [Jen761.[Hup67] B. Huppert, Endliche Gruppen 1, Springer, Berlin, 1967.[Jen76J R. D. Jenks (ed.), SYMSAC '76, Proc. 1976 ACM Sympos. Symbolic and Algebraic

Computation (Yorktown Heights, N. Y., 1976), Assoc. Comput. Mach., New York, 1976.[Ken77] Peter Kenne and J. S. Richardson, Tietze. A programme to simplify group presentations,

Austral. Nat. Univ., Canberra, 1977.[Kna5lJ Hans-Georg Knoche, Uber den Frobenius'schen Klassenbegriff in nilpotenten Gruppen,

Math. Z. 55 (1951),71-83.[Kno531 , Uber den Frobeniusschen Klassenbegriff in nilpotenten Gruppen II, Math. Z. 59

(1953), 8-16.[Lee69] C. R. Leedham-Green, Peter M. Neumann and James Wiegold, The breadth and the class

of a finite p-group, J. London Math. Soc. 1 (1969), 409-420.[Lee..] C. R. Leedham-Green and M. F. Newman, Space groups and groups of prime power order

I, Arch. Math. (Basel) (to appear).[Mac74] I. D. Macdonald, A computer application to finite p-groups, J. Austral. Math. Soc. 17

(1974), 102-112.[Mac77] ... , The breadth of finite p-groups, I, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977/78),

31-39.[Mae7S] , Groups of breadth four have class five, Glasgow Math. J. 19 (1978),141-148.[New'74] M. F. Newman (ed.), Proceedings of the Second International Conference on the Theory

of Groups (Austral. Nat. Univ., Canberra, 1973), Lecture Notes in Math., vol. 372, Springer, Berlin,1974.

[New76] , Calculating presentations for certain kinds of quotient groups, pp. 2-8 in [Jen76].[New78] , Private communication, Aachen, 1978.[Ng79J Edward W. Ng (ed.), Symbolic and algebraic computation (EUROSAM '79, an international

symposium on symbolic and algebraic manipulation, Marseille, 1979), Lecture Notes in ComputerSci., vol. 72, Springer, Berlin, 1979.

[P1e76J Wilhelm Plesken and Michael Pohst, On maximal finite irreducible subgroups of GL(n, Z).1. The five and seven dimensional case; 11. The six dimensional case, Bull. Amer. Math. Soc. 82 (1976),757-758.

[P1e77] , On maximal finite irreducible subgroups of GL(n, Z). I. The five and sevendimensional cases, Math. Comp. 31 (1977), 536-551.

[Vau74a] M. R. Vaughan-Lee and James Wiegold, Breadth, class and commutator subgroups ofp-groups, J. Algebra 32 (1974), 268-277.

[Vau74b] M. R. Vaughan-Lee, Breadth and commutator subgroups of p-groups, J. Algebra 32 (1974),278-285.

[Wam741 J. W. Wamsley, Computation in nilpotent groups (theory), pp. 691-700 in [New74J.[Zas48] Hans Zassenhaus, Uber einen Algorithmus zur Bestimmung der Rawngruppen, Comment.

Math. Helv. 21 (1948), 117-141.

RHEINISCH-WESTFALISCHE TECHNISCHE HOCHSCHULE AACHEN, FEDERAL REPUBLIC OF GERMANY


A COMPUTER-BASED ATLAS OF FINITESIMPLE GROUPS

DAVID C. HUNT

Abstract. By choosing a reasonable data structure and working on a computer whichprovides a comfortable working environment it is possible to manipulate the charactertable of a large group much as one would the character table of a very small group byhand. The design described here has been developed through experience which includescalculating character tables of groups related to the Baby Monster simple group.

1. Now that the problem of classifying finite simple groups is probably closeto completion the question of future areas of research is one which in part led tothe 1979 Summer Institute of the American Mathematical Society. In this note Ishall discuss one approach to the question: that of creating an archive of factsabout finite groups. In principle one would like a record of all known informa-tion or an "atlas" such as is being developed in Cambridge, England. This willeventually contain information about ordinary and modular characters, aboutautomorphism groups and covering groups and about presentations in generatorrelation form and as matrices. A partial atlas which many group theorists wouldlike to have available in a convenient form is an accurate collection of, say 100,character tables of groups related to simple groups, both Chevalley andsporadic. Most of us have partial collections. Third copies of unchecked hand-written originals are common. Lack of uniformity is another problem. Theconcept "convenient form" has certainly changed with time. Clearly a hardcopy, i.e. a copy on paper, is important for some purposes but perhaps moreoften one wishes to perform calculations on a character table or tables and thenit is more convenient to have the tables stored in files on a disc attached to acomputer.

2. One impetus to start this project came from the clear need to be able to useseveral character tables at once in various ways while calculating the charactertable of the Baby Monster. The centralizer of one class of involutions in thisgroup has structure 2.E6(2).2 which means the character table comes in four

1980 Mathematics Subject Classification. Primary 20C15, 20D08.m American Mathematical Society 1980

507

508 D. C. HUNT

sections (ordinary and projective character values on and outside the commuta-tor subgroup). Each of these had to be calculated and other tables, principallythat of F4(2), had to be used concurrently. At this stage it was possible todetermine all the conjugacy classes of the Baby Monster and work is now inprogress to complete the character table essentially row by row (character bycharacter).

3. There are various criteria to be kept in mind in designing a system to beused both as an archival system and as a tool to generate new information. First,and perhaps most important, at a given point in time the system will containonly partial information about a particular group. In the early stages one mayonly have some of the conjugacy classes and a few characters. Even when thetable is complete one might not have recorded or even calculated factoriseddegrees, Schur-Frobenius indices, irrational character values or various powermaps between conjugacy classes. A second criterion is that the system should berelatively portable between computers. A third is that arbitrary precision calcu-lations must be able to be performed when necessary. Yet another criterion isthat others should be able to add programs of their own to the system withouttoo much trouble.

I do not claim to have found optimal solutions to all these problems but in thenext section I will describe the strategy currently being adopted. One item whichI have not found important is access to a very fast computer, as the computertime required to perform most of the calculations described in §5 is modest.

4. In order that the system be portable and easy to modify it has been entirelywritten in FORTRAN. A variable precision integer arithmetic package writtenin FORTRAN is available from Cambridge, England [2]. This has been in-cluded. To explain the way in which the first constraint has been satisfied I shalldescribe the central program which prints a copy of a character table. If thetable MON is to be printed only 4 files MON.HEAD, MON.TAB,MON.NAME and MON.CCL must exist. MON.HEAD is simply:

2 3 5 71113171923293141475971

4620 9 6 2 3 1 1 1 1 1 1 1 1 1

194 14 4

194194

111100

The first two rows are the group order. The third row is the number ofconjugacy classes and sizes of records in other files. The 4th row is the numberof characters and the number of irreducible characters. The last two rows giveinformation about the availability of additional information (such as powermaps). MON.TAB is a 194 X 194 random access file containing charactervalues. MON.NAME is simply names for each class IA, 2A, 2B, ... andMON.CCL is a vector containing the exponents of the primes dividing the orderof the centralizer of an element in a given class. If, as in this case, extrainformation is available, the program prints this as well. All other programs canalso access these same files.

A COMPUTER-BASED ATLAS OF FINITE SIMPLE GROUPS 509

The system is currently running under the UNIX operating system on aPDP11/70 computer owned by the Australian Graduate School of Managementat the University of New South Wales. UNIX is now available on very manycampuses throughout North America and Australia at least. There are specialfeatures of UNIX which make it particularly useful and which will be brieflydiscussed in §6. However, the system could be transferred to any computersupporting FORTRAN and provided with a disc system.

5. Programs which have been written so far include programs to print tables,to calculate inner products, to calculate multiplication constants, to form tensorproducts, to form induced characters, to restrict characters to subgroups, toreorganize tables in various ways, to correct individual entries, to form linearcombinations of characters and to divide characters by an integer. In additionthere are programs designed to solve sets of integer linear equations associatedwith splitting sums of two or more irreducible characters into components byrestricting to a subgroup and using the class fusion information. The onlyprograms which use any significant quantity of time is the inner productprogram and the multiplication constants program. One can run the first withcomplete accuracy, using the variable precision integer package, or else, to savetime with real arithmetic. The decision is made at execution time by oneparameter in a data file.

The method that has been developed for solving the systems of integer linearequations may be of some independent interest. A typical system consists of 40equations and about 40 variables, each variable lying between 0 and some smallinteger, often 1 or 2. The number of solutions, a priori, is thus about 1020. It isthus necessary to eliminate variables and a rather imprecise, but to datesuccessful, algorithm has been developed.

(i) Check if any equation has all coefficients divisible by an integer.(ii) Check if forming a simple linear combination of two equations leads to a

cancellation as in (i).(iii) Choose the variable all of whose coefficients are as small as possible.

Eliminate this variable by choosing the second largest coefficient and subtract-ing this equation from that with the largest coefficient. Choose a descendingsequence of coefficients and eliminate as usual.

Repeat (i), (ii), (iii) until the number of cases is less than about 10000, thenenumerate and obtain all solutions.

The advantages of this method is that it has been used to avoid integeroverflow (32767) for many systems of equations like the one described above. Itis clear that brute force leads to integer overflow very rapidly with such systems.

6. In this final section a little will be said about some of the special UNIXfeatures that have made the work involved in this project considerably easier toperform. A considerable amount of disc space is needed to store a charactertable in a random access file as described above. In the case of the Monstercharacter table the file consists of 194 X 194 x 30 = 1, 129, 080 characters or2207 blocks on the disc. This would be hard within an allocation of 3900 blocksif it were not for PACK and UNPACK, programs based on a Huffman code. AHuffman code is a two pass coding procedure. The first pass generates a

510 D. C. HUNT

frequency table of all symbols in the file (often only twelve for data files). It thenencodes by giving short codewords (such as 1 or 01) for common symbols andlonger codewords (such as 001101) for less common symbols. The coding is doneso that no codeword is the initial segment of any other codeword. This facilitatesthe very fast one pass decoding process. Using PACK 3 times the above file hasbeen reduced to 90,609 characters or 178 blocks on the disc. The UNIXcommands are all short and easy to remember, for example, LS: list all files inthis directory, CAT AAA: list all contents of the file AAA on the terminal, WCAAA: count the number of words and lines in the file AAA. A very interestingfeature is the pipe /\, which takes the output of one process and uses it as theinput to another. As an example LS A WC counts the number of files in thedirectory.

The manual [3] is easily readable, with most processes described in less than apage. A whole issue of the Bell System Technical Journal [1] was devoted topapers discussing the design and some of the applications of the UNIX system.

7. In conclusion it should be said that in no way is this project a competitor tothe Cambridge ATLAS project. It is hoped that all the ATLAS conventions willbe incorporated into this system. The length of time before the system can bemade available to others depends mainly on the effort that can be put into theproject.

REFERENCES

1. T. H. Crowley et al., UNIX time-sharing system, Bell System Tech. J., Vol. 57, No. 6, Part 2,July-August 1978.

2. J. Larmouth, Variable precision arithmetic in FORTRAN, University Mathematical Laboratory,Cambridge, England, July 1970.

3. K. Thompson and D. M. Ritchie, UNIX programmers manual, 6th ed., Bell TelephoneLaboratories, May 1975.

UNIVERSITY OF NEW SOUTH WALES


FINDING THE ORDER OF A PERMUTATIONGROUP

JEFFREY S. LEONI

In computational group theory, one frequently encounters the followingproblem: given a set S = {s1, s2, ... , Sm) of permutations on a finite setSt = {w1, w2, .... to.), determine the order of the group G generated by S. AsG I can be as large as n!, this problem can be quite difficult even when n is fairly

small. The approach which I will describe is called the Schreier-Todd-Coxeter-Sims (or STCS) method; it was originally proposed by Sims [6], [7], and I haveimplemented it (in Fortran) and tested it on a number of groups. It appears tobe quite effective for many groups with degrees as high as 10000. It combinesthe Todd-Coxeter coset enumeration process with an earlier approach to thegroup order problem, also due to Sims. This earlier algorithm, called theSchreier-Sims method, was exceedingly slow for degrees over a few hundred.Not only does the STCS method outperform the Schreier-Sims method forgroups of high degree, but it has the added advantage of yielding a set ofdefining relators for the group. Both methods are based on a theorem ofSchreier [2, Lemma 7.2.2] giving generators for a subgroup in terms of groupgenerators and subgroup coset representatives, and both give considerably moreinformation than merely the group order. As STCS is rather complicated, I shallpresent only an outline of the method (together with a brief discussion ofimplementation and performance) here; full details on my implementation maybe found in [3]. Some familiarity with coset enumeration is assumed; forbackground see [1].

Given an arbitrary generating set S, determination of I G I can be quitedifficult even when n is relatively small. To facilitate computation in permuta-tion groups of high degree, Sims [4], [5] introduced the concepts of base andstrong generating set. A base for G on St is a subset { 01, /3Z, ... , /3k} of St with

1980 Mathematics Subject Classification Primary 20-04, 20B99, 20F05.'Work partially supported by National Science Foundation grants MCS76-03143 and MCS77-

17372. Computing services used in this research were provided by the Computer Center of theUniversity of Illinois at Chicago Circle. Their assistance is gratefully acknowledged.


511

512 J. S. LEON

G$ $ ... $k _ <1> denotes the stabilizer in G of 01, /32, .... /ik). Astrong generating set for G relative to an ordered base (/31, /i2, ... , //'k) is asubset T of G such that G $ $ . . . $ = <T n GP P, ... , >, i = 1, 2, ... , k. Notethat a strong generating set provies generators for each group in the stabilizerchain

1=Gp...pkcGp...& c cGp C: G.

Naturally these concepts are most useful when the base size is much smallerthan the degree. Although there is no guarantee that a permutation group, evena primitive group, will have a small base, it turns out that many interestinggroups have bases with just a few points. For example the "baby monster",recently constructed by Sims and myself, has a four point base in its representa-tion of degree 13571955000. In permutation groups having a small base, theSTCS algorithm appears to be effective for degrees as high as 10000, andpossibly higher.

With a base and strong generating set for G, questions that might otherwise bequite difficult can be reduced to "orbit calculations". Orbit calculations areamong the easiest in computational group theory; those that are needed here aresimple to program and have time and space requirements that are linear in theproduct of the degree and the number of generators. The calculations in whichwe shall be interested are:

(i) Given a E St and H = <T > a subgroup of SS2, find the points of a H (theH-orbit of a).

(ii) Given /3, y E St, decide if y E QH and if so find a word u in T witha"=y

Problem (i) is solved in the most straightforward way possible: we apply thegenerators to a and record any new points, apply the generators to these newpoints and record any additional new points, and continue in this manner untilno further new points are obtained. Whenever a new point 8 is obtained as theimage of some previous pointy under generator tj, we set v(8) = j (v(a) is set tothe negative of the orbit number). The n-vector v is called a Schreier vector andprovides all the information required to solve problem (ii); given a point 8, set81 = 8 and 8j+1 = 8jXj' where xj = for j = 1, 2, ... until v(8p) < 0 for somep; then 8p is the first point in 8H and xp_1xp_2 x1 is a word in T mapping 8pto 8. (We will denote this word by u,,(8).)

Given a base and strong generating set for G on St, the group order problem issolved easily as follows.

k

G _ GO,...A_,: GO. RIi-1k

RiGs,

i-1

Since we have a generating set for G. , ... , p , determination of I G I is reduced toorbit calculations. The orbit of /3i under G$ ... A is called the ith basic orbitand denoted by 0('.

FINDING THE ORDER OF A PERMUTATION GROUP 513

We can determine if some permutation g of Sa lies in G as follows:g E G if and only if a8 E A(') and guy-' E G61 (u, vt't a Schreier

vector for G),gui-' E GG1 if and only if /32",' E 0(2t and guj'uz' e GO #, (u2 =

v(2) a Schreier vector for Ge ),

gu11 ... uk'_ i E Go,- pk 1 if and only if /3U1' ... E O(k) and gul' .. uk I

E Gp (uk = u000k l' vtk> a Schreier vector for G ).YI ... k l.1 .. Ok-1

Checking whether $f "' E AO and determining uj involve only orbitcalculations; assuming that containment holds for j = 1, 2, ... , k, eventuallywe obtain that g E G if and only if gui' uk' E G'8,_4 _ <1>. This in-volves nothing more than multiplication of permutations, although in practice itmay be the most timetaking step in the process as it requires application ofpermutations to all of SZ rather than to the base. Note that, if g (4 G, then one ofthe following occurs.

(1) $J' Dv) for some j (1 < j < k),(2) gut' . . uk' 1.

We have seen that the group order problem, among others, can be reduced tothe following fundamental problem.

Given 6J3 = 01102'' .. , /3k) and S= (s1, s2, ... , s,,,), extend J, and S to abase and strong generating set for G on Q.

I shall use the following notation (mostly due to Sims).

Gt') =GPIR:...a_1(i= 1,2,...,k+1),S ( ' ) 2, . . . ,

H(') <S(')> C- G(i = 1, 2, .. .,k+ 1),

AV) = pH(l) (the ith basic orbit),

n; = W41.

vt`) = Schreier vector for H(,),

u(')(S) = for 8 E Ot') (a word in S(') mapping /3, to S).

We may always assume

11 a S,Every element of S moves some point of .

(*)

If not, we merely delete the identity from S and/or add to appropriate pointsof U. Note that (*) implies H(k+1) = <1>.

LEMMA (Snots). The following are equivalent (assuming (*)).(i) JG3 and S are a base and strong generating set for G on St,(ii)G(j)=H(j),j=1,2,...,k+l,(iii) Hk) = HU+»,j = 1, 2, . . . , k,(iv) I

( J ' : Ht'+1>I = nj,j = 1, 2, ... , k.

514 J. S. LEON

Note that, if J3 and S('+') are already known to be a base and stronggenerating set for H('+'), then to show that J'v and S('' are a base and stronggenerating set for H(1 it suffices to show that H,') = H"') or, alternatively,that IH() : H(+pl = n,.

Both the Schreier-Sims and STCS algorithms are based on the followingresult.

THEOREM (SCHREIER). If G is a group generated by a set S, H is any subgroup ofG, U is a set of right cosec representatives for H in G, and if for g E G, k denotesthe representative in U for the right cosec Hg, then

H=<us(Us)-'Iu E U, s e S>.

The elements us(us)' are called the Schreier generators for H. Note that, if Gis a permutation group and H = Gs, then the Schreier generators are easilycomputable; in fact, if v is Schreier vector for G (with f first in its orbit), thenthe set of Schreier generators is {u,;(a)suo(a$)-'la E $ , s E S}.

Both the Schreier-Sims and STCS methods consist of checking that HO _II0+1) (or JH(° : H('+1)1 = n;) for i = k, k - 1, ... , 2, 1 (in that order). Inchecking H$) = H('+1), we already know that J and S('+') are a base andstrong generating set for H('+') (by the Lemma), and consequently we can verifycontainment in H('+') using the algorithm given earlier. In the Schreier-Simsmethod, the technique for verifying that Hs' = H('+') is to compute eachSchreier generator (8 E k", s E St') for H(') and check that itis contained in H('+'). Suppose that containment fails for some Schreier genera-tor g. Then one of the following holds.

(1)...,y!' AO) (i + 1 < j < k),

> >

In the first case, gu;+ E Hv); so we add gu,+l u,' 1 to S,reset i to j (noting that H(j+'), ... , H(k) are unchanged), and continue asbefore. In the second case, gui+1 uk' fixes fl? pointwise, so we add to IJ3 apoint moved by gu;+, uk', and we add gu,+, uk' to S. We then reset ito (the new) k and continue as before.

This method is predictable but slow; verifying that H(') = requiresmultiplying out approximately inn words in permutations of degree n (if G istransitive). We may view the Schreier-Sims method as verifying each relatoru(')(8)su(')(8'')-'u;+1 uk'. This large set of relators suffices to prove thatH() = Hbut often it is highly redundant. In the STCS method, we modifythe Schreier method in an attempt to reduce the number of redundant relators.We maintain a list 'R of verified relators in S ('JZ''> will denote the subset ofconsisting of relators involving only elements of S(''). Before adding a newrelator, we try to deduce that IH(') : H('+1)1 = n; by coset enumeration. Afterverifying only a small fraction of the relators, we may (and in practice, usuallydo) have enough relators to make this deduction; if not, we choose the nextrelator to be one not evidently a consequence of relators in 6A.


We shall need an integer-valued function GJ1Z, with IJL(j) >j for allj. WhenSTCS begins, 6, is initialized to the empty set. To verify that IH(') : H('+1)l =n;, we proceed as follows.

(1) Enumerate the group generated by S() subject to relators R(') on the rightcosets of the subgroup generated by S('+'), allowing `Jn,(n;) cosets to be defined.When the enumeration terminates, either 'JR,(n1) cosets will be defined or thetable will be closed; in the latter case, n; will divide the number of cosets sincethe relators RP hold in H(').

(2) If the coset table closes with n; cosets, we are done (IH(') : H('+')l = n;).

(3) Otherwise find words w1 and w2 (of minimal length) with 1"' 0 1'2 in thecoset table but with /3;" = /3;W'2 on St. (Such words exist as 'Jn,(n1) > n;; theprocedure for finding them is given in [3].)

(4) Necessarily w1w21 E G('+'). Check whether w1w2' E H('+') using thealgorithm described earlier.

(5) If w1w2-' E H('+'), add wlw2-'ui+1 uk1 to L, perform the collapse1'l = 1w'2 in the coset table, and go back to step (1). (In repeating step (1), theenumeration will be resumed from the point of interruption; note that thisrequires an interruptible Todd Coxeter coset enumeration program.)

(6) If w1wZ' E H('+'), we conclude IH(') : H('+ 1)1 n,.Note that we have found the element w1w2-'ui+1 uJ ' 1 which will be added toS.

To see that this process must terminate eventually, we note that for eachcollapse 1W1 = 1'2 in step (5), there exists a pair w1, w2 of words of length at most'Jn,(n1) with 1"'' 1'V2 before the collapse but 1', = 1w2 afterward.

The major problems in implementing the STCS algorithm involve the use ofinterruptible coset enumeration. In my experience, seventy to eighty percent oftotal execution time has gone into the enumerations. Several questions arise.

(1) What type of enumeration should be used? In [1] two basic strategies-theFelsch and HLT methods-are described.

(2) What function IJn,(j) should be chosen?(3) If the HLT method of enumeration is used, what lookahead strategy (if

any) should be employed?My interruptible coset enumeration program uses the HLT method (with

lookahead) because by [1] it appears to give the fastest execution times in amajority of enumerations; however, STCS employs coset enumeration in arather special way, and more study is needed on use of the Felsch method in it.

For the function IJL(j), my implementation uses a linear function a1 j + a2,where a1 and a2 are input parameters.

An HLT coset enumeration algorithm (with lookahead) operates in one of twomodes-define mode or lookahead mode. In the define mode, cosets areprocessed in the order in which they are defined. Processing a coset consists oftracing all the relators from that coset, defining new cosets as necessary tocomplete each cycle. When no more cosets can be defined, the algorithm entersthe lookahead mode. Relators are traced as before, with deductions or coinci-dences being processed, but no new cosets are defined. As described in [1], the

516 J. S. LEON

algorithm remains in lookahead mode until the first coincidence occurs (incre-mental lookahead) or until the end of the table is reached (complete lookahead);then it reenters the define mode if at least one coincidence occurred duringlookahead and terminates (unsuccessfully) otherwise. I have found that none ofthe strategies of no lookahead, complete lookahead, or incremental lookaheadworks particularly well in conjunction with STCS. I have developed a generaliza-tion which I call (1, 12, 13) lookahead; the enumeration program remains inlookahead mode until either 1, cosets have been processed during the lookaheador until 12 cosets have been eliminated through collapse (whichever occurs first);then it returns to define mode if the lookahead pass eliminated 13 or more cosetsand terminates otherwise. Note that no lookahead, complete lookahead, andincremental lookahead correspond, respectively, to (0, 1, 1), (00, 00, 1) and(oo, 1, 1) lookahead.

I have tested my implementation of the STCS algorithm extensively on fourprimitive groups with degrees between 315 and 6156 and with orders betweenthe square and cube of the degree and less thoroughly on a number of othergroups. The test runs used different values of a, (1.0 to 5.0), a2 (0 and 50), 11 (0to oo), and 13 (0 to ..211). Tests involved both (a) building up a base and stronggenerating set from a randomly chosen 2-element generating set and (b) verify-ing an existing base and strong generating set from which redundant stronggenerators have been removed. (I have developed an algorithm called therandom Schreier method, described in [3, §7], which very rapidly produces aprobable base and strong generating set, from which redundant strong genera-tors can be removed by the process described in [4]; hence verification of stronggeneration usually should suffice.) Complete results of the tests are given in theappendices of [3]; some selected results for Held's group of degree 2058 andorder 4030387200 are included in the appendix here. I have drawn the followingconclusions.

(1) The STCS algorithm appears to work most effectively in verifying anexisting strong generating set from which redundant strong generators have beenremoved.

(2) Optimal performance can be obtained with 'X (J) only slightly greaterthan j, say 1.lj or 1.2j, provided adequate lookahead is used. Small values of6)X(j) reduce memory requirements since )lL(n) by IS(')I + 2 arrays are usedfor the coset tables.

(3) Without lookahead (1, = 0), large values of OJIL(j) (3.0j or more in somecases) are required for good performance; this can result in huge memoryrequirements.

(4) Either complete or incremental lookahead (1, = oo) will produce poorexecution times; a finite nonzero value of 11 appears to be necessary for goodperformance.

A much more thorough discussion of the performance of the STCS algorithmappears in § 10 of [3].

Appendix. Performance of the STCS algorithm on Held's group of degree 2058.Table entries have the form t(L), where t = execution time in seconds on anIBM 370/158 and L = total length of defining relators. An asterisk indicatesthat the total length of the defining relators exceeded 4000.

/1

al a2

1.00 1

1.10 01.20 01.33 0

1.50 0

2.00 03.00 05.00 0


Verifying strong generation (12 - 13 - 1)

0 10 25 50 100 00

89 (2241) 51 (1013) 50 (1071) 23(294) 47(294)+ 35 (907) 31 (643) 19 (294) 21 (294) 59(294)+ 46 (1158) 20 (294) 20 (294) 21 (294) 75 (294)* 27(555) 21 (315) 21 (315) 25(315) 95(315)+ 20(315) 21(307) 21 (315) 26(315) 93(299)

114 (3240) 21 (315) 22(315) 23(315) 30(315) 118 (299)

29(398) 23 (315) 28 (315) 32 (315) 47 (299)

30(315) 35(315) 40(315) 42(299) 64(315)

Building up a base and strong generating set from randomly chosen2-element generating sets (71i,(j) = 1.2j + 50 and (11, 1, 1) lookahead)

1st random 2nd random 3rd random 4th random 5th randomgen set gen set gen set gen set gen set

50

200

s 104 (1505) 62 (703) 170 (1348)317 (1741) 115 (1254) 93(871) 56(353) 183 (1352)

Defining relators produced by STCS in verifying strong generation forHeld's groups (6J1i,(j) = 1.2j + 50 and (50, 1, 1) lookahead)

e2, (de)2, d5, c2, (ce)2, (cd)6, (cdcd-1)3, (cd-1)2(cd)2(cd-1)2(cd)2, b2,(bd2e)2, bcedcbd 2cdcd 2e, bdcd -1 bd -'cd -t(cd )2, (bd)4, (be)4, (bc)4,

(bcd 2cd -2c)2, bd -Iced 2 bd -lcdbcd -2cd -'cd -2ce,bdbcd -'bdcbd -'bd -2(cd )2cd -2, a 2, (ab)2,acedcacd -'cdbcbd -2cd -'cd -2cd 2e, adcdeadcd 2(bc)2d 2cd -2cd -'cde,

(ad -'cd)2, adbed -1bd -'adbd -1bd -2e, aedahd -leacbd 2(cd -2)2,acd -'abdcad -'cd 2cbd -'bd 2cd -2cdce.

REFERENCES

1. John J. Cannon, Lucien A. Dimino, George Havas and Jane M. Watson, Implementation andanalysis of the Todd-Coxeter algorithm, Math. Comp. 27 (1973), 463-490.

2. Marshall Hall, Jr., The theory of groups, Macmillan, New York, 1979.3. Jeffrey S. Leon, On an algorithm for finding a base and strong generating set for a group given by

generating permutations, Math. Comp. 35 (1980), 941-974.4. Charles C. Sims, Determining the conjugacy classes of a permutation group, Proc. Sympos. on

Computers in Algebra and Number Theory, Amer. Math. Soc., Providence, R. 1., 1970.5. _ , Computation with permutation groups, Proc. Second Sympos. on Symbolic and Alge-

braic Manipulation, Assoc. Comput. Mach., New York, 1971.6. , Some algorithms based on cosec enumeration, unpublished notes, 1974.7. , Some group theoretic algorithms, Topics in Algebra, Lecture Notes in Math., vol. 697,

Springer-Verlag, Berlin and New York, 1978.


PART VIII

Connections with number theory and other fields


MODULAR FUNCTIONS

A. P. OGGI

The following is an account of some parts of the theory of modular functionswhich may be of interest to group theorists.

1. The upper half-plane. Let denote the complex upper half-plane, i.e. the setof z = x + iy with y > 0. Any a = (° d) E SL(2, R) operates on . by a(z) =(az + b)/(cz + d). Then -a has the same action as a, and PSL(2, R) =SL(2, R)/ ± I = Aut(.) is the group of all (holomorphic) automorphisms of i.

The upper half-plane has a natural metric (the Poincare metric) with volumeelement

dV = dxAdY - idzAda (1)

27ry2 47ry2

This metric is invariant under PSL(2, R), since if z' = a(z) as above, we have

dz' = dz/ (cz + d)2 and y' = y/ (Icz + dl )2. (2)

The geodesics are the upper halves of lines or circles meeting the real axis (theboundary of .) orthogonally, i.e. vertical lines and semicircles with center on thereal axis. If D is a geodesic polygon, whose boundary is a simple closed curve8D consisting of r geodesic arcs and with interior angles al..... a,., then

rvolume of D = f dx/ (2lry) = (r - 2)/2 - a;/tot. (3)

ao i-tHere some of the corners may be cusps, either on the real axis or oo = + ioo,with interior angle 0.

Now let t be a discrete subgroup of PSL(2, R) such that the volume of I'N. isfinite. Then I' will have a fundamental domain Dr, bounded by geodesic arcs.The quotient Yl. = I'\ has a natural structure of Riemann surface, such that ameromorphic function on Yr is the same as a I'-invariant meromorphic functionon The local analytic structure is that of . t except at an elliptic fixed point z,where the stability group I'. = (a E t: a(z) = z) is (finite) cyclic of ordere > 1 and the local parameter is r,-invariant and vanishes to order e at z,

1980 Mathematics Subject Classification. Primary 10D05.'Partially supported by the National Science Foundation under grant number MCS 77-03719.


521

522 A. P. OGG

regarded as a point of !D. The inequivalent elliptic fixed points z1, ... , z5, oforder e 1 . . . . . es, correspond to corners of D. with interior angles a1 = 27T/e,.We can complete Yr to a compact Riemann surface Xr by adding cusps(parabolic fixed points) zs+1, ... , z1, corresponding to corners of D. on theboundary of here FZ, will be infinite cyclic and we put e, = oo. For example,if oo is a cusp and if c is the least positive number such that z H z + c is in F,then q = ez"'Zl` is the local parameter at oo. The genus g of X. is given by theformula [15, p. 42]

vol(F\ ) = 2g - 2 + (1 - 1/e;). (4)1=1

The modular group I'o = PSL(2, Z) has the well-known fundamental domainDo defined by - 1/2 < x < 1/2 and zl > 1, also Izj > 1 if x > 0; the volumeof Do is 1/6, the genus of Xro is 0, and we have e = 2, 3, oo, in agreement with(3) or (4). If F is a subgroup of F0 of finite index n, then Dr is the union of ncopies of Do, so of volume n/6, and only elliptic fixed points of order 2 or 3 canoccur, say n2 resp. n3 (inequivalent) fixed points of order 2 resp. 3 and no, cusps.Then (4) becomes

g = 1 + n/ 12 - n2/4 - n3/3 - n./2; (5)

one can also derive this from the Riemann-Hurwitz formula for the coveringXr _* X.

The modular curve Yro parametrizes isomorphism classes of elliptic curves, asfollows. Let E = CI L be an elliptic curve over C, where L = Zw1 ® Zwz is alattice; we assume that z = w1/wz E !D. Given another elliptic curve E' = C/L'and a similar basis wi, wZ for L', we see that E and E' are isomorphic if and onlyif L' = t L for some t E C (the map E -* E' being induced by u H tu), i.e.

wzl) = t(c d) `wz

for some (° d) E SL(2, Z), i.e. z' = a(z) for some a E F0. Thus we have abijection between Yro and the isomorphism classes of elliptic curves. The ellipticcurve E has 2e automorphisms, where e is the order of the stability group (F0)Z,i.e. E has six, resp. four, automorphisms if z -r ez"'/3, resp. z -r i, and otherwiseonly two.

The elliptic modular invariant j(z) is the isomorphism of Xro onto the Riemannsphere which is normalized by requiring it to have a simple pole of residue 1 atoo and to vanish at ez"'/3. In order to give j(z) explicitly, we consider for anyeven integer k > 4 the Eisenstein series

Ek(z) E (nz + m)-km,n

= 1 - (2k/Bk) a _1(n)gn, (6)n=1

MODULAR FUNCTIONS 523

where Bk is the kth Bernoulli number (B2 = 1/6, B4 = - 1/30, B6 =1/42.... ), a1(n) = E d' where d runs over the positive divisors of n, q = e2'",and the first summation in (6) is over all integer pairs (m, n) except (0, 0). Inparticular, E4 = 1 + 240E a3(n)q" and E6 = 1 - 504E' , a5(n)q". Then wehave

O(Z) = (E4(Z)3 - E6(Z)2)/1728 = q ll (1 - q")'`',00

M-1(7)

of weight 12. The product expansion can be proved in various ways; note thatO(z) is never 0. From (6), we see that Ek(z) is a modular form of weight k for ro,i.e. f = Ek is holomorphic in and at oo, and satisfies

f(a(z)) _ (cz + d)kf(z) (a e ro). (8)

Then 0 is a modular form of weight 12 (a cusp form since it vanishes at oo), and

00

j = E4/0 = 1/q + 744 + E c(n)q";n=1

(9)

this is so because j as defined by (9) is invariant under ro (being the quotient oftwo forms of weight 12), holomorphic in (since 0 is never 0) with a simple poleof residue 1 at oo; that E4 vanishes at e2'"'"3 follows from (8). Note that thecoefficients c(n) are positive integers. The first two are

c(1) = 196884, c(2) = 21493760. (10)

That the c(n) are integers (but not, so far as I know, their positivity) has beencrucial for arithmetic applications, such as the theory of complex multiplicationand class field theory; there has also been a great deal of work on congruencesand p-adic properties of the c(n). Serious consideration of the c(n) as positiveintegers (say by looking for a vector space of dimension c(n) naturally associa-ted to the situation) seems to have begun only recently, with very surprisingresults (cf. §6 below).

2. The modular curve X0(N). Let N be a positive integer and let ro(N) be thesubgroup of the modular group defined by the (, d) with c divisible by N. Itsindex in the full modular group is

(ro(1): ro(N)) = tp(N) = N11 (1 + 1/p), (11)PI N

where p runs through the prime divisors of N. Then Y0(N) = r0(N),\'!D parame-trizes isomorphism classes of pairs (E, C), where E is an elliptic curve and C is acyclic subgroup of order N of E (an isomorphism (E, C) A (E', C') being anisomorphism of E onto E' which carries C onto C'). This follows from an easygeneralization of the discussion in § 1 for N = 1; if E = C/ L and (wl, wZ) is abasis of L, we take C to be the group generated by w2/N (mod L).

524 A. P. OGG

Let X0(N) = Xro(N) be the compact curve obtained by adding cusps. Thegenus g = g(N) of X0(N) is given by (5), once we know the numbers of cuspsand elliptic fixed points. The pair (E, C) represents an elliptic fixed point oforder 2, resp. 3, if and only if its automorphism group is cyclic of order 4, resp.6, and we find

0 (if 41N),

n2 I +{ 4

(if 4 } N)(pN(

)1

P,

1 0 (91 N),

n3 11 1

39}N (12)

( +(pIN ))P

),(

where is the Legendre symbol. The number of cusps will be given below,together with a description of the fields of rationality of the cusps.

The Riemann surface X0(N) may be given the structure of a projective curvedefined over Q. (The function field is C(j, jN), where jN(z) = j(Nz). It is aclassical fact that F(j, jN) = 0 for a certain irreducible polynomial F(X, Y) EQ[X, Y]; the desingularization of this plane curve is XA(N).) Furthermore, thisrational structure is compatible with the modular interpretation of Yo(N): givena field K, Q C K C C, a point of Y0(N) is K-rational (i.e. has its projectivecoordinates in K in our model) if and only if it is represented by a K-rationalpair (E, C), i.e. the curve E is defined over K (by an equation y2 = 4x3 - g2x- g3 with g2, g3 E K) and the group C is K-rational, i.e. C° = C for anya E Aut(C/K). After the beautiful work of Barry Mazur [6], [7], it is nowknown that there are no such points over Q, i.e. Y0(N)(Q) is empty, except for afew (known) small values of N. Nothing is known about the determination ofYo(N)(K) for any number-field K Q.

The cusps (cf. [9]) are the r0(N)-orbits of P(Q) and in our model they are allrational in the cyclotomic field where N = e2n`/N. For each positivedivisor d of N, put t = (d, N/d); then we have y(i) cusps (d), where a E(Z/tZ)". An automorphism a of satisfies where s E(Z/NZ)", and in our model we have

a(d) - (d)Thus the cp(t) cusps (d) with the same "d" are conjugate, and their field ofrationality is Q(L). In particular, the number of cusps of X0(N) is

nx = (t) (t = (d, N/d)). (13)dJN

We will pay special attention to the cusps which are "multiquadratic", i.e.rational in a compositum of quadratic fields, i.e. for which the Galois group(Z/tZ)" is abelian of exponent 2, i.e. tj24. The number of these is

n (14)


where r is the number of prime factors of N, h is the largest divisor of 24 with h2dividing N, and s = S2 s3 is defined as follows. Write h = h2 h3 with h218 andh313. Then

=S2

3/4 (if 21 hzJIN),

1 (otherwise);

2/3 (if h3 = 911N),S3 =

1 (otherwise).

(We write MIN to mean MIN and M is relatively prime to N/ M.)

3. The normalizer of F0(N). Let ')t denote the normalizer of F = F0(N) inPSL(2, R) = Aut(o). Then B = ' Yt / F is the group of "modular" automorphismsof X = X0(N), i.e. those automorphisms induced by automorphisms of theupper half-plane. The set S of cusps of X is fixed by these automorphisms.

The group B was determined by Atkin, Lehner, and Newman [1], [8]; a neatdescription appears in [3]. In this section a reasonably conceptual treatment willbe given. A partial statement of the result is: B contains a subgroup W of"Atkin-Lehner involutions", of order 2', where r is the number of prime factorsof N, and (B : W) = h2 s, in the notation of (14).

One element of B is the "Fricke involution" wN, defined by the matrix (°N o)In terms of elliptic curves, wN sends the class of the pair (E, C) to that of thepair (E', C'), where E' = E/ C and C' = EN/ C is the image of the group EN ofall P E E with N P = 0 (which group is isomorphic to the product of twocyclic groups of order N) under the isogeny E --+ El C. The number of fixedpoints of wN is

h(-4N) (N 3 (mod 4)),(15)

h(-4N) + h(-N) (N - 3 (mod 4)),where h(- N) is the number of classes of primitive positive binary quadraticforms of discriminant - N. More generally, we have an "Atkin-Lehner involu-tion" wN, defined whenever N'MMN, i.e. N = N' N" with (N', N") = 1. It sendsthe pair (E, C' X C ") to (E/ C', EN. X C"/ C'), where C', resp. C ", is cyclic oforder N', resp. N". There is a formula generalizing (15) for the number of fixedpoints of wN.; cf. [10], for example. These wN, make up the group W. Each wN, isdefined over Q, since the construction procedure commutes with any fieldautomorphism.

Before attempting to define another kind of automorphism, let us recall thatthe group EN (isomorphic to the product of two cyclic groups of order N) has anatural skew-symmetric arv, satisfying eN(P Q) = = e2"'/^', if P, respairing - rv - PQ, is the point defined by w,/N, resp. w2/N (and w,, w2 is a basis for the latticedefining E). Suppose now that t2 divides N and let (E, C) be a pair as usual. LetQ generate C and choose P of order t with e,(P, Q) = ,. Let C be the groupgenerated by Q' = P + Q; it is cyclic of order N. For C' to be well defined, i.e.for it to depend only on C and not on the choice of generator, we need thefollowing: if we replace Q by a Q (for some a E (Z/NZ)") and hence P bya -1 P, then a -1 P + a Q generates the same group C'. Thus we need a2 = 1(mod t) for all a, i.e. t124.

526 A. P. OGG

Thus if t2 IN and t124, we have an automorphism u, corresponding to thematrix (N11 °), sending (E, C) to (E, C') as above. It is rational over sincethe construction is compatible with any field automorphism fixing ,.

Let B' be the group generated by W and by the u,; it is a subgroup of B andits elements are rational over K = QQh), where as before h is the largest integerwith h124 and h2IN. Then B' acts on the set S(K) of cusps (d) rational over K,i.e. for which t = (d, N/d) divides 24. This action is transitive, as follows. WriteN = Nl . . N where the Ni are powers of distinct primes; given a cusp (d),write similarly d = d, d, and t = t, t,. Now wN, replaces d,. by N;/ d; andleaves d fixed for j i. After operating by a suitable element of W, a given cuspis carried to one with d; < N;/d;, i.e. d; = t;, for all i, i.e. such that d = t. Thuswe assume that d = t, and that t124, since the cusp is to be K-rational. SincewN u,° wN ' is defined by (o °) and sends the cusp (°) to (°,), we see that anyK-rational cusp is in the B-orbit of the cusp (°).

Let us now show that B' is all of B. Note that a E 9L acts on P(Q), whoseelements are fixed by parabolic elements of IF, and so is a scalar times a rationalmatrix (° d) of positive determinant; it then acts on the set of cusps of IF, asclaimed above.

Let u E B,,, the stability group in B of the cusp oo. Then u is represented by

a =

Since

a

(x

0

yIX

r1 1 -i = 1

\0 1)a 0

E 'Yt,.

X2) E F,1

we have x2 E Z and similarly x - 2 CZ sox = ± 1, say x = 1. Thus anyu E B. is represented by a = (01 ), with y E Q. Similarly, if u E B fixes thecusp 0, then u is represented by

a' = I Ny, 1

'

if u fixes both 0 and oo, then a'aE

IF, and soy E Z and u = 1. Hence B actsfaithfully on the set of cusps. Since all cusps are rational in QQN), any u E B isrational over (If a E Aut(C), then u° = aua if a fixes thenu ° = u on cusps and so u ° = u.)

Actually, any u E B is rational over a compositum of quadratic fields, i.e.u° = u if a is a square in (Z/NZ)", the Galois group of To see this,suppose that a corresponds to a2 and choose y = (° d) E F. Then

2

ay a '0 1

(mod N),

so ay acts on cusps as does a. Hence u ° = aua -1 acts on cusps as does(ay)a(ay) ' = yay-' (where a E 't, represents u), so u° = u on cusps, whenceu°=u.

Thus any u E B is rational over K = Wh), where as always h is the largestdivisor of 24 with h2IN. It follows that the stability group B is cyclic of order h


(on (o that B = B', and that the order of B is h times the number IS(K)l ofmultiquadratic cusps (cf. (14)):

IBI = 2' h2 s =IS(K)l h. (16)

(Cf. [1] for the structure of B.)

4. Reduction modulo primes of X0(N). If 1 is a prime not dividing N, then bywork of Igusa the curve X = X0(N) has a good reduction modulo 1. The reducedcurve, denoted by X (mod 1) or just by X, is a nonsingular curve defined overthe field F, of 1 elements, with the same genus g = g(N) as in characteristic 0.Furthermore, we still have X = Y u S, where points of Y parametrize classes ofpairs (E, C) in characteristic 1, and points rational over a field K of characteris-tic 1 are exactly those containing a K-rational pair; the set S of "cusps" has thesame description as in characteristic 0; the cusps are rational over and theGalois action is as before.

The good reduction of X modulo 1 is a reflection of the fact that an ellipticcurve in characteristic 1 has as many points of order N as in characteristic 0,namely N2. This is no longer the case in characteristic p when pIN; then themapping p of E -p E is inseparable of degree p2. If p is purely inseparable, i.e. ifE has no points of order p, then E is called supersingular; otherwise (theordinary case) the group EP is of order p. Instead of pairs (E, C) we must nowspeak of isogenies E -p E' which are primitive (i.e. not divisible by any integer> 1) of degree N. For N = p, there are essentially only two p-isogenies (vs.ti(p) = p + 1 in characteristic 0), the Frobenius isogeny T: E which isalways inseparable, and its transpose satisfying o T = p, which is separableif E is not supersingular. (In coordinates, T(x, y) = (x", y"), and an equation forE (n) is obtained by applying the map a H a' to the coefficients of an equationF(x, y) = 0 for E.)

Assuming now that N = p M with p } M (the case p21 N is not so nice) wehave the reduction modulo p of X = X0(N), a la Igusa, Deligne and Rapoport[4]. The reduced curve X mod p has two components Z and Z', each isomorphicto X0(M) in characteristic p, with (say) a point of Z resp. Z' corresponding to apoint of X0(M) together with the Frobenius map T resp. its transpose Theintersection Z n Z' consists of the supersingular points of X0(M), where theunderlying elliptic curve is supersingular; at such points the p-isogeny can bethought of as either a T or a The cusps again cause no trouble, as they occuraway from Z n Z'. The intersections of Z with Z' are all transversal [4], so wehave:

Z Z' =I XO(M)SSI (17)

is the number of supersingular points on X0(M) in characteristic p. Thearithmetic genus pQ of a reducible curve satisfies the rule

p0(Z+Z')= -1in our case we appeal to the specialization principle, according to which thereductions have the same arithmetic genera as the original curves in characteris-tic 0. Thus p0(Z + Z') = g(pM) and p0(Z) = p0(Z') = g(M), and we get

I X0(M)SSI = 1 + g(p - M) - 2 g(M). (18)

528 A. P. OGG

Now we apply the genus formula (5), using the multiplicativity of n2, n3, n.. (cf.(11), (12), (13)). The main term is 1/ 12 times % (p M) - 2 tp(M) =(p - 1)ii(M) and the cusp term drops out, since n(p M) = 2 n(M); alsone(P - M) - 2 ne(M) = (ne(P) - 2) - ne(M) < 0. Thus

JX0(M)SSI > (p - 1). %'(M)/12. (19)

(The right side is (p - 1)/2 times the volume of X0(M) in characteristic 0.)The involution wp of X, reduced modulo p, interchanges the two components

Z and Z' (taking T to ); on the supersingular set Z n Z', it acts as theFrobenius. Since it is an involution, we recover the familiar fact that allsupersingular points of X0(M) are rational over Fp2. We then write

XO(M)ssl = rp(M) + 2 sp(M), (20)

where rp is the number of points rational over FP, and sp is the number ofconjugate pairs in Fp2 - F. Another specialization argument [11], [12] then gives

g(P)(p M) = g(M) + sp(M), (21)

where g(" (N) is the genus of the quotient of X0(N) by the involution wp.

5. Quotients of X0(N) of small genus. Let X X0(N) and suppose we have acurve X over Q, of genus g, and a map f: X - X, defined over Q and of degreed.

If 1 is a prime not dividing N, then we have a reduction modulo 1 of f, definedover F, and of degree d. Hence the number of points on X rational over a finitefield K is at most d times the number on X. Taking K = F,2, we have

IX(F,2)I < d IX(F,2)I < d(1 + 2gl + 12),

the latter inequality coming from the "Riemann hypothesis" for the curve Xover the finite field F,2. On the other hand, we have at least (1 - 1)ifi(N)/12supersingular points on X(F,2), by (19), and at least n' = 2' h s cusps, in the00

notation of (14). Thus we have

n. + (1 - 1)%(N)/12 < d(1 + 2gl + 12). (22)

If we fix d, g, and 1, then the inequality (22) gives an upper bound for N, inpractice fairly sharp. The method was devised in [10] to show that N < 71 if X ishyperelliptic or of genus < 1, when we can take d = 2 and g = 0. Assuming Nto be odd and taking 1 = 2, we have then 2' + ii(N)/12 < 10 and so N < 89.This is the main step in showing that X0(N) is hyperelliptic for exactly nineteenvalues of N, the largest being N = 71. Furthermore, the hyperelliptic involutionv (that involution such that X/(v) is of genus 0) is in the group B (in thenotation of §3) if N 37.

The exceptional case N = 37 is of some interest. Here we have g = 2 and so Xis automatically hyperelliptic. However, w37 has only two fixed points, by (15),so the genus of the quotient of X by (w37) is 1 and w37 is not the hyperellipticinvolution; it is not hard to see that the full automorphism group of X is thenoncyclic group of order 4. This is the only case known of an "exceptional"automorphism of X when the genus is > 2; it was proved in [13] that W = B isthe group of all automorphisms of X when N is square-free, N 37, and g > 2.

_The next simplest application of (22) is to determine for which N the genus ofX = X/ W is 0. If N < 300, we can read off those N from Table 5 in [2], and wefind

g(X/W)=0<=> N 1 -36,38-39,41 -42,44-47,49-51,54 - 56, 59 - 60, 62, 66, 69 - 71, 78, (23)

87, 92, 94 - 95, 105, 110, 119,

when N < 300. In particular, for primes p, let g+(p) be the genus of Xo (p) _Xo(p)/(wp); then we have

g+(p) = 0 r=p < 31 orp = 41, 47, 59, 71, (24)

a result mentioned already. By (21), these fifteen primes are exactly the primespfor which every supersingular elliptic curve in characteristic p is defined over F .

Given that g(X / W) = 0, we apply (22) with g = 0 and d = 2' and get(1 - 1)ii(N)/12 < 2'l 2, whenever 1 } N. Writing N = N1N2 N. as a productof powers of distinct primes, with (say) N, < N2 < < Nr, this is also

5 1212/ (1 - 1).r-i

(25)

In order to show that N < 300 (and so that the list of (23) is complete), weassume that N > 300 and show that the inequality (25) leads to a contradiction.If N is odd, we take 1 = 2; the right side of (25) is then 48. If r < 2, thenN < ii(N) < 4 48, so r 3; if r > 4, then N > 3 5 7 11 and the left sideof (25) is > 2 3 4 6 > 48. Hence r = 3 and Nl = 3 (by the same argument),so N < (3/4)% (N) < (3/4) 8 48 < 300.

Thus we can assume N to be even. If N = 2M with M odd, then the methodsof §4 (cf. [13, pp. 288-289]) show that M has the same property (the genus ofX0(M) divided by its Atkin-Lehner group is 0), so M < 119 and N < 300. Thuswe can assume that N is divisible by 4.

If 31 N, we use 1 = 3; the right side of (25) is now 54. As above, we get r = 3and hence N < (2/3)ifi(N) 5 (2/3) 8 54 < 300. Thus 31N. By the same argu-ment as before, if N = 3 M with 3 } M, then M < 92 (being on the list of (23)and divisible by 4) and so N < 300. Thus N is divisible by 32, and similarly by52, 72, , leading to a contradiction in various ways. (Results such as (23) canbe and were derived by other means, making use of upper bounds of classnumbers; cf. (15). But the method given here is much quicker and cleaner, andhas the advantage of being applicable to exceptional automorphisms, such as inthe case of N = 37.)

Finally, if N is square-free, we consider the problem of determining theautomorphism group of X + = X1 W, assuming that its genus is g + > 2. Anynontrivial automorphism of X + is necessarily exceptional, carrying the (unique)cusp onto a noncusp, since by a result of Helling [5], the normalizer Fo (N) ofF0(N) in G = PSL(2, R) is a maximal discrete subgroup of G (and, up toconjugation, these are the only maximal discrete subgroups of G commensurablewith the modular group). Taking now N = p to be prime, suppose first that X +is hyperelliptic. Then, by (22), with 1 = 2, d = 4, k = 0,

2+(p+1)/12<4.5

530 A. P. OGG

and sop < 216. After eliminating some values of p by rather ad hoc methods, wefind

X0 (p) is hyperelliptic <--> g+ (p) = 2

p = 67, 73, 103, 107, 167, 191. (26)

For these six values, Aut(X +) has only two elements. For any p (with g+(p)2), one can show that Aut(X +) is an elementary abelian 2-group, by a theoremof Ribet [14]; for g+(p) > 3 and p < 300, I have checked that Aut(X+) istrivial, but have not been able to prove this in general.

So far, the exceptional automorphisms have not been relevant to the study ofsporadic groups (cf. §6 below). However, group theorists are aware of thephenomenon of N = 37, and may yet find a use for it.

6. The monster. The monster M was discovered (although not invented) byFischer and Griess in 1973; if it exists, it is a sporadic simple group of order

IMI = 246. 320. 59.76. 112. 133. 17 19.23.29.31 41 .47.59.718 . 1053. (27)

One cannot help noticing (cf. (24)) that a prime p divides I M I if and only ifg+(p) = 0, and in 1975 a small prize was offered for an explanation [11]. It ishard to say now how seriously this was meant at the time; had I known thenthat the largest order of an element of M is 119 (cf. (23)), I should no doubthave been more certain that more than a mere coincidence was involved.

It is conjectured that M exists and has an irreducible rational character X2 ofdegree

X2(1) = 196883 = 47 59 71. (28)

Admitting this, then M is unique (Thompson) and the character table has beenconstructed (Fischer, Livingstone, Thorne). There are 172 irreducible rationalcharacters X1 = 1, X21 , X1721 ordered by increasing degrees, i.e. 172 rationalequivalence classes in M. (Recall that a, b E M are rationally equivalent ifa = x b' x -1 for some x E M and some integer s relatively prime to M 1.) In1978, J. McKay observed that

196884 = 1 + 196883,i.e. that the coefficient of q in j is

c(1) = X1(1) + X2(1); (29)

cf. (9), (10), (28). Continuing, Thompson [16] found for some small values ofn > 1, a rational character w,,, not too reducible, with

c(n) = (30)

the first few are

w1=X1+ X2,w2=X1+X2+ X31

w3 = 2X1 + 2X2 + X3 + X4,

w4=3X1+3X2+X3+2X4+ X51

w5 = 4X1 + 5X2 + 3X3 + 2X4 + X5 + X6 + X7


Assuming that we have w" for all n > 1, then for each x E M we have theThompson series

J(x) = l/q + w"(x)q". (31)n-1

Note that J(l) = j - 744 and that J(x) depends only on the rational equivalenceclass of x. Thus we have 172 holomorphic functions on , with a simple pole ofresidue 1 at oo and vanishing 0th coefficient there (call this "normalized at 00"),and with all Fourier coefficients in Z. A great deal of evidence concerning theother series J(x) was then collected, and they appeared to be modular functionson modular curves of genus 0 of the type discussed in §5. For all this, and muchmore, see Monstrous moonshine [3] by Conway and Norton; I understand thatAtkin and Serre, and perhaps others, were also involved in these developments.To be a little more precise, the conjecture is that for each x E M, there is adiscrete subgroup 0 of PSL(2, R) containing some F0(N) as a normal subgroup(the least such N is the level), such that X. is of genus 0 and J(x) is aHauptmodul for 0, i.e. is the isomorphism normalized at o0 of X. onto P(C).(Note that q = e2"1Z is the local parameter at oo, so z H z + 1 is the leasttranslation in 0.) Also, if n is the order of x and N is the level, then h = N/n isan integer dividing 24 with h2IN. (Such h, i.e. the difference between B and Winthe notation of §3, have been rather neglected in this account; they play a quitenatural role in [3].) In particular, taking x to be of prime order p, then theconjecture implies that Xo (p) is of genus 0 and so that p is among the fifteenprimes of (24), but until more is known about the monster, we are not likely tohave any idea why all fifteen appear. Similar remarks apply for the elements ofM of composite order.

To prove the conjecture, one first finds experimentally a correspondencebetween Hauptmoduln and elements x of M, or rather the rational equivalenceclass of x. Changing the notation, let

J(x) = l/q + c.(n)q"n-1

be the (known) modular function. Then c,(n) E Z. The claim is then thatx H c,(n) is a character of M for each fixed n > 1. (The w" were not unambigu-ously defined before, in (30), (31), so some such reversal of roles is necessary atthe present state of the art.) That x H c,(n) is a generalized character wasproved by Atkin, Fong, and S. Smith (it may be some time before we see theproof); that it is then a character for n sufficiently large was proved byThompson.

It should be emphasized that although the evidence that there is a deepconnection between modular functions and the monster is convincing beyondany reasonable doubt, nothing really has been proved, if you demand a theoreti-cal connection and not just a sorting out of data. Young mathematicians shouldrejoice at the emergence of a new subject, guaranteed rich and deep, with all thetheorems yet to be proved. It is particularly amusing that new light should beshed on the function j, one of the most intensively studied functions in all ofmathematics, by the most exotic group there is (or is not, as the case may be).

532 A. P. OGG

REFERENCES

1. A. O. L. Atkin and J. Lehner, Hecke operators on r 0(m), Math. Ann. 185 (1970), 134-160.2. B. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Lecture Notes in Math.,

vol. 476, Springer-Verlag, Berlin and New York, 1975.3. J. Conway and S. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3,

308-339.4. P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques, Modular Functions

of One Variable. II, Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin and New York, 1973,pp. 143-316.

5. H. Helling, On the commensurability class of the rational modular group, J. London Math. Soc.(2) 2 (1970), 67-72.

6. B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Etudes Sci. Publ. Math. 47(1978), 33-186.

7. , Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.8. J. Lehner and M. Newman, Weierstrass points of Io(n), Ann. of Math. (2) 79 (1964), 360-368.9. A. Ogg, Rational points on certain elliptic modular curves, Proc. Sympos. Pure Math., vol. 24,

Amer. Math. Soc., Providence, R. I., 1973, pp. 221-231.10. , Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449-462.11. , Automorphismes de courbes modulaires, Seminaire Delange-Pisot-Poitou, 16e annee

(1974-1975), no. 7.12. , On the reduction modulo p of X0(p M), U.S.-Japan Seminar on Modular Functions,

Ann Arbor, Mich., June 1975.13. , Uber die Automorphismengruppe von X0(N), Math. Ann. 228 (1977), 279-292.14. K. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math. (2)

101 (1975), 555-562.15. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc.

Japan, No. 11, Tokyo and Princeton, N.J., 1971.16. J. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular

function, unpublished, 1978.

UNIvERsrrY of CALIFORNIA, BERKELEY


A FINITENESS THEOREM FOR SUBGROUPSOF PSL(2, R) WHICH ARE

COMMENSURABLE WITH PSL(2, Z)

J. G. THOMPSON

1. If G is a subgroup of PSL(2, R) commensurable with F = PSL(2, Z), letg(G) be the genus of G, and let

T(G) = 2(g(G) - 1) + m + I J 1 - 1 ).

In this equation, m = mG is the number of cusps; ei = IE,I, xi is a generator forEi, and (x 1, ... , xk ) is a subset of G minimal with the property that everyelliptic element of G is conjugate to a power of some xi. Then

2vrr(G) = a(G)is the hyperbolic area of a fundamental domain for G acting on H, the complexupper half plane [2].

Let `3t, be the family of congruence subgroups of G, that is, the set ofsubgroups K such that for some N, K D F(N).1 The object of this paper is toprove that

18(K) 1

1 1

K'X IG : KI

2T(G),

in the sense that for any e > 0, there are only finitely many elements K of `3t,such that

g(K) 1

IG : KI2T(G) > E.

If we write

T(K) = 2(g(K) - 1) + R(K),

1980 Mathematics Subject Classification. Primary 20G20; Secondary 20G30.'Here

r(N)a{(a d)ESL(2,Z)Ja-l-b-c=d-1-0(modAmericanJMathematical

society 1980

533

534 J. G. THOMPSON

then since r(K) = I G : KIT(G), an equivalent formulation of (1.1) is that

limR(K)

= 0.(1.2)

KEK IG:KIThis assertion has a simple group theoretic interpretation. Let K E `3t, and letXK = X = I K be the corresponding permutation character of G on the cosetspace K \ G (G acts on the right). For each subgroup H of G, (XiH, 1H) is thenumber of orbits of H on K \ G, and so IG : KI/(X,H, 1H) is the average lengthof an H-orbit. I will show in §2 that (1.2) is a consequence of the followingassertions:

li(XKI u 1 u) = 0 f b li b U f G 1 3m

Kr=X IG:KI or every para c suo group o . .( )

1

Ei = E f lli ti b E f G 1 4K mr=X

I G KII

or every e p c su group o . .( )

Heuristically, (1.3) says that if K E `3t then with a probability approaching 1 asI G : K1 oo, a random coset of K in G lies in a U-orbit of cardinality largerthan N for any preassigned N, and (1.4) says that a random coset of K lies in aregular orbit of E.

2. The first reduction. In this section, I shall show that (1.3) and (1.4) imply(1.2). Let (ui, . . . , um) be a set of elements of G such that <u,> = U; is thestabilizer of c, and (c ... , cm) is a set of representatives for the orbits of G onQ U too).

Let P(K) be the set of (nonidentity) parabolic elements of K, so thatm

P(K) = U P;(K),

and every element of Pi(K) is conjugate in G to a power of u;. To say thatg E G, n E Z and U. satisfy (u;")g E K is to say that Kg-lu;" = Kg-'. Thus ifg,...... g;, is a set of representatives for the K \ G/ U; double cosets, andI Kg,1 U; : KI = n11, then every element of P,(K) is K-conjugate to a power ofprecisely one of the elements 1 < j < r;. But

r; = (XKI y,' lU)

is the number of U;-orbits on K \ G, and soMG

(XKIu,,1u) =mK

is the number of cusps of K. Now

R(K) = mK + R'(K),

where R'(K) is the contribution of the elliptic elements of K to r(K) and is yetto be dealt with. But if we assume that (1.3) holds, we get

R K _ R '(K)

Km IG:KI K=c IG:KI'

FINITENESS THEOREM FOR SUBGROUPS OF PSL(2, R) 535

in the sense that if one of the limits exists, so does the other, and the two limitscoincide.

Let Ei(K) be the set of (nonidentity) elliptic elements of K which areG-conjugate to an element of E;, 1 < i < k, where (x1, ... , xk) is as in §1.Then E(K), the set of (nonidentity) elliptic elements of K, is the disjoint unionof the E;(K).

Let y;i, ... ,y;,, be a set of representatives for the K \ G/E; double cosets.Then every element of E;(K) is K-conjugate to a power of precisely one of theelements y; where n,-,. = I Ky; jE; : KI. Of course, when n;j = e;, theny; jx;iyJ' = 1. In any case, however,

k

R'(K) = R; (K),i-i

Si

R;'(K) = 1

ei/nij is the order of The terms corresponding to n;j = e; are 0,but it is worthwhile to use the form for R;'(K) as given. The integer s; is easy tocontrol; it is

Si = (XKIEl 1E),

the number of E;-orbits on K \ G. AlsoSi

Zn;j=IG:KI,j-1

and so

R; (K) = (XKIE., 1E) - IG : KI/e;.

Now suppose (1.4) holds for E;. Then

lim R' (K) _ lim(XKIEI 1E) l _ 1

KE`7L I G : KI KE`7L I G : KI j e; e, e;

If (1.4) holds for all the E;, then (1.2) follows.

= 0.

3. The second reduction. The group GL(2, Q) acts on PSL(2, R) by conjuga-tion, and if G is commensurable with F, so is GX for all x E GL(2, Q).Furthermore, 3 (GX) = 'X(G)X, and the genus and the characters XK behaveproperly under conjugation by x, so we invoke a theorem of Helling [1] toreduce our problem to the case G C F0(f)+ for some square free f. Since3 (G) C .X (H) and T(G) = IH : GIT(H) for every discrete overgroup H of G,

we lose no generality by taking

G = F (f)+ . (3.1)

One advantage this reduction gives us is that in this case, m, = 1; F0(f)+ istransitive on cusps, so we take as our parabolic element

u=(11)

(mod ±I).0

536 J. G. THOMPSON

All parabolic elements of Fo(f)+ are conjugate to a power of u. Set = U.The notation x,, e;, E. is also fixed as in §1, where from now on, (3.1) holds. Forbrevity and by force of habit, I denote Fo(f)+ by G.

For each natural number n, let

G. = Fo(nf) n F(n). (3.2)

If K E 3C, then K 3 G for some n, and conversely if K 3 G. for some n, thenK E X. These assertions follow from the obvious containments

r(n) J G(n) _J F(nf).Note that G, = Fo(f) < G, and GIG, is an elementary abelian 2-group of rankv, where v is the number of primes dividing f.

4. The singular cosets. If S is a subset and X is a subgroup of a group Y, let

Ss(X, Y) = {Xyl y E Y and for some s E S, Xys = Xy}.

The elements of Ss(X, Y) are called the singular cosets of X with respect to S,or simply the singular cosets.

In case 1 Y : X I is finite, set

Es(X1 Y) = I SS(X, Y)1/1Y: X1.There is a useful monotonicity property of these functions.

LEMMA 4.1. If X1 C X2 C Y and I Y : Xil is finite, then

Es(X Y) < Es(X2, Y) for all S C Y.

PROOF. We argue that

I cSs(X1, Y)1 , IX2: X11 ' Ss(X2, Y)I. (4.1)

To see this, suppose X, y = Xi ys for some s E S. Then also X2y = X2 ys, so wehave a map

SS(X1, Y) SS(X2, Y),

Xly H X2y.

Since X2y contains 1X2 : X11 cosets of X1, (4.1) follows. From (4.1) we get

ES(XlY) = I SS(Xi, Y)I < IX2 : X111-5 s(X2, Y) I

IY:Xi1 IY:X'I1-Ss(X2,

Y) I = ES X2, Y)lY:X21

and we are done.In this paper, we are interested in the case

Y=G, X=KEAs for S, we pick a (large) positive number N. Let

GL, = GL = { u ' 1 < n < N).For each i = 1, ... , k, let &; be a set of elements of E; - { 1) which is minimalwith the property that every nontrivial subgroup of E; contains at least oneelement of &;. Thus, if e, has precisely v, distinct prime divisors, then 16;1 = vi,and each subgroup of E; of prime order is generated by a unique element of 6;.

Letk

=U &i.i-l

With all this notation, the sets S which concern us here are the subsets S of'?LN U 6. Since G is to be fixed at Fo(f)+ for quite a while, we set

Es(K, G) = es(K), Ss(K, G) = Ss(K).

And we set

E(K) = max( ,L (K), E5(K), s E 6).Notice that E(K) depends on N. By Lemma 4.1 we have

e(K)se(H) ifKCHCG. (4.2)

THEOREM 4.1. For every positive N and e, there are only finitely many K E `3tsuch that e(K) > e.

In the remainder of this section, I assume that Theorem 4.1 holds and derive(1.3) and (1.4).

For each S C '1L U 6, let %s(K) be the complement of Ss(K) in the cosetspace K \ G. Then

I'S(K)I = (1 - es(K))I G : KI.

By construction, L(K) is stable under U and every U-orbit on R(K) hascardinal > N. Thus, the total number of orbits of U on K \ G is

(XK,U' N

This gives us (1.3) by taking e -p 0, N -p oo.As for (1.4), for each i = 1, . . . , k, we have

156,(K)I -< v; max(ISy(K)I,y E 6i),

and so

0 ' e&.(K) < v;e(K). (4.3)

Let L (K) be the complement of S& (K) in K \ G, so that

I6As,(K)I = (1 - e&;(K))IG : K1.

By construction, R (K) is stable under E; and E; acts regularly on %;(K).Hence

(1 - e&,(K))IG : KI< (XKIE , 1E),

ei

(1 - es (K))I G : KIei

< vie(K)I G : KI +

whence (1.4) follows by letting e - 0.

538 J. G. THOMPSON

5. Properties of G. I turn now to the groups G of (3.2). The first assertion isthat

G,, < G. (5.1)

To see this, a typical element of G is represented by a matrix

b), 6 If, 6la, 6I d, f I c, ad - be = 6.p = 6-'/2( a (5.2)c

and a typical element of G is represented by integral matrices Q, X,

Q=I+nX, X= (1 + nx)(1 + nt) - n2fyz = 1.( ), (5.3)fz t

And P-'QP = (c D), where

A = 1 + 6-1(adnx - abnfz + cdny - bcnt),

B = 6-'(bdnx - b2nfz + d2ny - bdnt),

C = 6 -'(-acnx + a2nfz - c2ny + acnt),

D = 1 + 6 -' (-bcnx + abnfz - cdny + adnt),so (5.1) follows.

The groups G. play the role in G that the r(n) play in r, and they have similarproperties, for example

GnGm = G(n.m), G. n Gm = G[n,m] (5.4)

the proof of which is left to the reader. From (5.4) comes the decomposition ofG,/Gn "induced" by the Chinese remainder theorem:

G, / Gn = Gn / Gn X . . . X G,,, / Gn,

n = P,`ni, A i ni (5.5)

which gives

G,,/Gn = GGpi. (5.6)

Also, we have

GI/GP= PSL(2,p) if p { f. (5.7)

This isomorphism comes from restricting to G, the homomorphism r -PSL(2, p) which is induced by the ring homomorphism Z - Z/pZ.

As far as G is concerned, we obtain an exact sequence

1--+ Gn--+ G,--+ GIGn--+ T--+ I (5.8)

where T is an elementary abelian 2-group of rank v = number of primesdividing f.

For each K E K, let n = n(K) be the smallest natural number such thatK D G. Thus, we have

If K E T, n = n(K) andpIn, then K Z Gnlp. (5.9)

We use (5.9) to prove

If K E T, n = n(K), p In, p { f and p > 7, then the image ofK n G, = K, in PSL(2, p) is a proper subgroup. (5.10)


Let n = p m, p } m, and let

H=KnGm, L=KnGp..The direct decomposition G1/G = Gm/G X GP./G tells us that K1/HL isisomorphic to a section of both Gm/ G. and GP./ G,,. Since p > 7, Gp./ G. has nosection isomorphic to PSL(2, p), the only simple nonabelian sections beingPSL(2, q), for gl5m. Thus, neither Kl/HL nor L/G has PSL(2, p) as a section.Suppose (5.10) is false. Then HI G. has PSL(2, p) as a section. Since G.1 GGl/ Gp., we conclude that

Gm= H- Gm.

In particular, H acts irreducibly on each of the abelian groups G,,,p / G,,,p.. 1,i = 1, ... , e - 1. If h E H is chosen to map to an element of PSL(2, p) of orderp, then since p > 3, it follows that V E Gmp: - G,,,,..,, which then forces

Gw. _ (H n G,,,,) -

Since H -D G,,, we conclude that H D Gm. But K D H and n = n(K). This givesus (5.10).

Retaining the preceding notation, set

Since GP goes to 1 in the isomorphism (5.7), (5.10) implies that

n(M) = p (and not n(M) = 1). (5.11)

For further work, we need:

If L E 3C, n(L) = p, p } f and p > 7, then D= n 8EGL8 contains noelliptic elements 1. (5.12)

In any case, D n G1 = GP, since PSL(2, p) is simple. This implies that

[D, G1 ] C G.

Suppose x E D - (1) is of finite order e. We may assume that e is a prime.Since we are in F0(f)+, we have e = 2 or 3. Thus, x is represented by a matrix

X = 6-1/2(a d), trX = 0or-1.c

If tr X = -1, then as 6 1 f, while f is square free we have 6 = 1, x E G1, which isfalse. So x is an involution. Since [x, G1] C Gp, it follows that

a - d (mod p), b - c - 0 (mod p).

But a + d = 0 and p 2, so a - d - 0 (mod p). This is false, since ad - be =6 is a divisor of f, so is not divisible by p.

6. The characters of PSL(2, p). For each finite group F, define

e(F) = max (al , 11) - Vl , (6.1)C'0) I -

where a ranges over all the irreducible characters of F of degree > 1, and V

540 J. G. THOMPSON

ranges over all the cyclic subgroups. The characters of PSL(2, p) and PGL(2, p)are well known, and we find that

Pm e(PSL(2, p)) = lim E(PGL(2, p)) = 0. (6.2)

00 P-00

Suppose now that S C F and S LJ PSL(2, p), where F is either PSL(2, p) orPGL(2, p). Let X = ls, let V be a cyclic subgroup of F and let e > 0. We provethat

if p is large enough, then

(XI v, 1 y) 1 (6.3)X(1) lVI

<e.

Choose p so large that

(0I v' l v)1 < e/2 (6.4)0(l) I VI

for every nonlinear irreducible character $ of F. Since X is a constituent of theregular representation of F, we have X = A + it, where A(1) = 1 or 2, and everyirreducible constituent of µ has degree > 1. Let

µ = I a,4,;, 4); irreducible.

Since S 2 PSL(2, p), we see that µ O.Then by (6.4)

-$,21)e< (oilV, 1v) -

ill)< O;(1) 2

and so by linearity, together with a; > 0,

-tL2I)e< (µ,v lv)- IVI) < µ(2)e (6.5)

Now (Xlv,lv)=(µlv,ly)+a,aE (1,2),X(1)=µ(1)+b,bE{l,2)andso(6.5) may be rewritten as

- (x(l) - b) 2 < (XI v, l v) - a - (X(11 b) < (X(') - b)±

Hence

where

-A + < (XX(, IV)V l

< A 2 + r1,

X(])-b _ a _ bA -X(1) '

71

x(l) I VI X(l)

Since every nonprincipal irreducible character of PSL(2, p) is of degree >(p - I)/2, we are guaranteed that (6.3) holds provided p is large enough toforce (6.4), and if, in addition, p - I > 8/E.

7. A finite set of primes. Here we show that

Given N and e, there is M such that if p is a prime exceeding M andK E C with pl n(K), then e(K) < e. (7.1)

We proceed as follows. Let n(K) = p°m, p } m; let k = KGp. By (5.11), we have

n(K) = p if p 7 andp } f.

Let L be a subgroup of G which contains K and is maximal subject to n(L) = p.Let

D= n L9, F = G/D. (7.2)gEG

By (5.12), D contains no elliptic elements 1. Since GP C L, and GP < G, wehave D E `3C, n(D) = p. Thus, G1D/D = F1 PSL(2, p), and F, < F. LetS = L / D. By construction S Z F1, but every subgroup of F which contains Sproperly also contains F1, while F1 F, is an abelian 2-group.

Let C be the centralizer of F1 in F, so that C is also the centre of F. Bydefinition of D, this forces S n C = 1, SC = S X C. We argue that C = 1.Otherwise, SC D F1, whence (SC)' = S' D F' = F against S Z Fl. So F =PSL(2, p) or PGL(2, p).

Let X = is and let V1 be the image of U in F. Visibly, I V1I = p. Also,X(1) = IF: SI = I G : LI, so (6.3) gives

I(Xlu,lu)_ lI< (73IG : LI p E, )

if p is large enough.Let E be any elliptic subgroup of G, and let V be its image in F. Then

E I = I V I by (5.12), and so (6.3) gives

(XIE, lE)1 < e. (7.4)

IG:LI IEI

Taken together, (7.3), (7.4) show that E(L) < e if p is large enough. By (4.2), wehave E(K) < E(L), and (7.1) follows.

8. Further reduction. Let M be chosen to satisfy (7.1). Let `3td be the set ofK E 3C such that every prime dividing n(K) also divides d. By (7.1), the proof ofTheorem 4.1 will be complete if we show that 3CM! has only finitely many Kwith e(K) > e.

We proceed by way of contradiction. This produces for us a prime p < M anda sequence of groups K1, K2, ... in 3C such that n(K) = pan, with a1 < a2< . We can then use this sequence to show that already in XCp, Theorem 4.1is false. For each prime q p with q < M, let

p e(q) = exponent of a Sylow p-subgroup of Gl / Gq,

b = m ax {e(q)}.q #p

Now suppose K E 3CM,. Let n(K) = p" p; = p,e'n;, where p1 = p, e, _e. Then the exponent of

is a divisor of pb d for some d prime top.

542 J. G. THOMPSON

By definition of n(K), we have K 0 Let L = G,,,. We willshow that one of the following holds:

(a) e b + 3, L D Pick x E (Notethat C G,,.) Then x E L, and so

kEK,g,4EG,4.

Hencek-xg,,,

and so by definition of p d d = r, say,

k' = X' (mod G.).

However, the map x - x' induces an isomorphism between andGG/p / G,,, since e - b - 1 > 2, so the trouble spot at p = 2 is avoided. Thisisomorphism together with K D G,,, forces K D which is false. Sincee(K) < e(L), we have reduced the proof of Theorem 4.1 to the case where 3C isreplaced by `Xp.

9. Replacing G by G.. Let v be the p-adic valuation of Q; v(p'r/s) = i if r, sare prime top, v(O) = oo.

Since G = FO(f)+, we may embed G in PGL(2, Q) via

g = +b_1/2(a b) H (a b). Qxc d c d

Then we embed PGL(2, Q) in PGL(2, Qp) via the inclusion Q C Qp, and let G.be the closure of G in PGL(2, Qp). In general, if X C PGL(2, Q), X. will denotethe closure of X. The group G. is not difficult to identify.

(1) p } f, Gr, = IF,, U0 (mod Qp), where U0 is a subgroup of the unit group Zvof Zp.

(2)plf. Then

G = Fo(f) + (mod Q;).

Since Zp /squares is of order 2 for p 2 and is a four-group when p = 2, GG isfairly transparent. Notice that in case (2), 170(p) is a normal subgroup of G. andthe quotient group is an elementary abelian 2-group of order 2, 4 or 8.

It is well known that

G/Gp = G G, for all n,in an obvious notation (namely, G,,,,, _ (Gp.) ), and so if K E ICp and n(K) _p a, then

K/Gp. = whenever n > a.

So if S C c.L U , then

Es(K, G) = Es(K,,, Ge),

since Es(K, G) depends only on K/Gpa, G/Gpa.

Thus, to complete the proof of Theorem 4.1, we can work in G., and study theopen subgroups of it. Let U be the closure of U in G,,, so that U,, is the image inPGL(2, QQ) of ((o ')Ix E Zp). The image of (o i) in G is still denoted by u.

10. The structure of K/G,,,,,(K). I shall change notation slightly. If K is an opensubgroup of G,,, n(K) denotes the smallest natural number n such that K D G ,,.

Since the extension- 1I - - G,,.,, G,,,,,

splits if p 2 and n > I or if p = 2 and n 2, we identify G,,,,, with a subgroupof SL(2, Zr), and so reduce questions to matrix calculations.

Set K1 = K n G.,, / G, .(K).

LEMMA 10.1. K1 has a cyclic normal subgroup with cyclic quotient in thefollowing cases:

(a) p 2, 3,(b) p < 3 and n(K) > 10.

PROOF. As we are working in PGL(2, Qp), if p = 2 and x E G,;.1 - we

may choose a matrix representing x of the shape

(a d), a =- d = I (mod 4),

and so x2 (4 G,;,3. Since

[ Gt,.., G,.5] c GL.r+s,

we get (a) from Satz 11.11 of [3], and if p < 3 while n >, 10 then every subgroupof K1 of order at most p5 is abelian on at most two generators, so that Satze11.1 1, 11.12, 11.13 of [3] give us (b).

11. The critical subgroups. For each prime number 1, let 3C(l) be the set ofopen subgroups K of G such that K/K n G,,,1 has order 1 or 1. The object ofthis section is to show that Theorem 4.1 is a consequence of the followingassertion:

For each N and e, and each / E (2, 3, p), there are only finitely manyKE `3C(l) such that e(K, E. (11.1)

Suppose that (11.1) holds, and that K is an open subgroup of G. such thatE(K) > E. This means that one of the following holds:

(a) ER,n,(K) > E,(11.2)

(b) for some s E F9,e,(K) >E.

First, suppose that (a) holds. Let (K1, ... , Kh) be the set of all subgroups K,of K which contain K n G,,,, and satisfy IK; : K n G,,'11 = 1 or p.

Since K;/K n G,.1 = e and since the order ofis bounded by a function of M, M being the integer given in (7.1), it follows thath is bounded by a function of M, and so

h is bounded by a function of N and E. (11.3)

Consider next the singular coset Kg E StN(K, Then for some integert, 1 < t < N, Kgu ` = Kg. This gives us gu `g-1 - w E K. Since w° E G,,,1, it

544 J. G. THOMPSON

follows that w E K. for some i, which means that Kg E G,,). Since eachK; is a subgroup of K, it follows that

for some i E { 1, ... , h}, hI-5 qLN(K,., (11.4)

From (11.4) we get

(K) - I ;N(K)I (K),N IG,, : KI < IGG:KI

where

We argue thatH=

H < H*, where H* depends only on N and e. (11.5)

From (11.3), it suffices to show that I G : Kill I G : KI is bounded by a functionof N and e. But G. : K;I/IG : KI = IK : KI < IK: K n Gi1I. Since isbounded by a function of M, and since K/K n G,,,1 is isomorphic to a subgroupof (11.5) follows.

Let e, = e/H*. Since (11.2)(a) holds, it follows that e%(K) > el, and so by(11.1), K; lies in a finite set depending only on N and E. Thus, X. D for somen depending only on N and E. Since K D K, K itself lies in a finite setdepending only on N and E.

Similarly, if (I1.2Xb) holds, then K lies in a finite set depending only on Nand E.

From now on, we concentrate on the proof of (11.1).

12. The parabolic elements.

LEMMA 12.1. Suppose G. = PSL(2, Zr). For each integer n > 0, let K,,

1F0(p "),,. Then for each N and e, there is H(e, N) such that

if n > H(e, N).

PROOF. As a first attempt to find H, let H1 be the smallest natural number hsuch that p h > N. In proving Lemma 12.1, we restrict attention to those n > H1.

Now suppose K = K" and Kg is a singular coset with respect to -UN. Thenthere is t, 1 < t < N, such that Kgu' = Kg. The smallest such t is of the formt = p' for some integer i, since u°" fixes every coset of K. Let

g = (a b) (mod ± 1),c d

and let v(c) = j. Then gu°g-' _ (Y a) (mod ± 1) belongs to K. = 170(p '),,,whence v(y) > n. Explicit calculation gives y = -c2p', and so 2j + i > n. Inparticular, j -# 0, since p' < N H1. Thus, the orbits of U onK \ G of length < N are( all contained in cosets Kg, where g is represented by

1X

Ol ),v(x) > (n - i)/2,

and where i = H, - 1. The number of such cosets is at most p("+i)/2, while theindex of K in G is p"-'(p + 1).

Increasing N does not decrease G ), and so we may assume thatN > 1/e. Now choose H(e, N) such that

p(n+H,)/2 < epn-I(p + 1) if n > H(e, N).

This implies that

Is6LN(Kn, G )I < p(n+H,)/Z < IG : KnI if n > H(e, N)

and Lemma 12.1 is proved.

THEOREM 12.1. For each N, e there are only finitely many open subgroups K ofG,, such that

eq,N(K) > e.

PROOF. By §11, we may assume that K/K n G,,,I has order 1, 2, 3 or p. Wemay also assume that K is generated by K n G,,,,, together with an additionalelement of the shape gu'g-', where g c G,,, 1 < t < N. Set K, = g-'Kg. Then K,is generated by K, n Gv 1, together with u', 1 . t < N. We may also of course,restrict our attention to those K such that e%(K) > e. Then

eqLN(K) > e and KI C PSL(2, Zp).

We may therefore assume that G PSL(2, Zr). Note that this forces GPGL(2, Zr). We assume notation is chosen so that K = K,. Let

j(K) = j = min v(c)

where (, d) (mod ± I), ranges over K. By Lemma 12.1, we may restrict attentionto those K such that

j(K) < H(e, N). (12.1)

Since K = <u', K n G,I>, each element of K is of the shape

k= (1 0)(ax 1 0 a-')(0 1)'

a, x, y E Z,,, v(a) = 0,

and we can choose such an element with v(x) = j. Then we choose i such thatup"EK,

pZ<p'<N.(It is harmless to assume N >P2 .) Then

kupk-' = k' = 1 - aZxpi aZp'E K.

-xaap' 1 + aZxp'

Since (o ) E K whenever v(w) > i, we may choose w so that

(1 - a2xp')w + a2p' = 0.This gives

0k'-k(01)

aZxp'

-xZa2p' (1 - aZxp')-'

(a, 0 )I\

ix1 a

(12.2)

E K_

546 J. G. THOMPSON

Repeating this construction with k1 in the role of k, we get

k2 =1 - a,xip' 0

-x2a2p' (1 - a2,x1p')-'

a2

0I1x2 a2'

E K.

Thus,

[ k 1' k2 ]=(

X 11 E K,where

X = -x1al - a21a2x2 + a1a2x1 + a2x2.

We need v(X). Since a2 = 1 - p', x2 = we have

X = -xla1 - a21(I - a1x1p')(-1)xiaip'

+a,x1(l - a2,x,p')2 + (1 - aix1p')(-1)xiaip'x2 'a2(a2 2a - 1) + x3 2'a4(-a2 + a + I).1P ' - l P > > >

If p * 2, we get that

v(X) = i + 2v(x1).If p = 2, then u(a2 - 2a1 - 1) = 1, whence

v(X) = I + i + 2v(x1).

Thus(' 1) E K if v(z) ' 31 + 4j + 8, where 8 = O if p 2, 8 = 1 if p =2.Letting k2 play the role of k, we get that K contains an element k3 of the

shape

a3 0k3= , v(a3)=0, v(a3-1)=3i+4j+8.

0 a3

Hence

K D Go 3i+4j+8,

and so K lies in a finite set by (12.1) and (12.2). This completes the proof ofTheorem 12.1.

13. The elliptic elements. We need some properties of the groups

Bn ` r(P),,/r(P"),, C. = G,I/GV,n

Of course, when p < f, then B" = C", but we must contend with the case pl f, too.

LEMMA 13.1. Suppose x c r(p'), - r(p'+'),,, y e r(pj),, - r(pj+'),,, and thatif p = 2, then i, j > 2, while if p 2, i, j > 1. Suppose also that n > i + j + 1 ifp = 2, and n>i+jifp 2. Suppose also that [x,y]Er'(p"),,.Let .,ybetheimages of x, y in B,,. Then (x> and (y> have a nontrivial intersection in B".

PROOF. Let x1 = x° -', y, = y°"-'-', and let x y1 be the images of x1, y, inB". Our hypotheses guarantee that x1, y, both have order p. We must show thatxj = y; for some integer i, so suppose false.

FINITENESS THEOREM FOR SUBGROUPS OF PSL(2, R)

Choose X0, Yo E SL(2, Zp) which represents x, y respectively, and with

X0=I+p'X, Yo=I+p'Y.Let

X= a Q y= (a blly S (c dl

Since <x1 > and <y, > are distinct subgroups of B" of order p, it follows that

X and Y are linearly independent (mod p).

On the other hand, since [x, y] C I'(p ")L, it follows that

X Y - YX -T= 0 (mod 4) if p = 2,

X Y - YX 0 (mod p) if p 2.

We also have the relations

(1 + p'a)(1 + p'8) - p2i/3y = 1,

(I + p a)(1 + p'd) - p2jbc = 1,which yield

Now

Y= a(ya

a+8+p'a8-p'/3y=0,a+d+p'ad-p'bc=0.

#)(a b}_ as+/3c8)`c dl-(ya+&

as + bybl a 8

b+/3dyb+8da/3 + b8_YX-(c ) _

dl(y c/3+dSAnd so we get

as+/3c-as-by0 (Pr),ab+/3d-a/3-b80 (Pr),ya+8c-ca-dy0 (pr),yb+8d-c/3-d80 (P r),

where r = 2 if p = 2, and r = 1 if p 41- 2. This gives us

/3c-yb-0 (Pr)b(a-8)+/3(d-a)m0 (pr),y(a-d)+c(8-a)0 (pr).

'

547

First, suppose p 0 2. If a - 8 - 0 (mod p), then since a + 8 - 0 (mod p), weget

a - 8 - 0 (mod p).Thus, either /3 itt 0 (mod p), or y is 0 (mod p), and so we conclude thata - d . 0 (mod p), whence

a = 0 (modp),d 0 (mod p).

548 J. G. THOMPSON

If in addition, /3 - 0 (mod p), then we get b - 0 (mod p), and so X, Y arelinearly dependent (mod p). This is false, and so by symmetry, /3ybc i L5 0(mod p). But /3 / y - b/ c (mod p) and again X, Y are linearly dependent(mod p). We conclude that

a-8 0 (modp),and by symmetry

a - d m 0 (mod p).We are free to replace x, y by x", y" for any n, n' which are prime top, and sowe may assume that

a - 8 = 2 (mod p),

a - d - 2 (mod p).

Now our congruences become

b = /3 (mod p),

c = y (mod p).

But then we get

a+8+p`a8 -p`/3y-p'bc-a+d+pJad0 (mod p),

whence we get:

a + 8 0 (mod p), a + d 0 (mod p),

a - 6 = 2 (mod p), a - d = 2 (mod p),

a 1 (mod p), a I (mod p),

8 -1 (mod p), a -1 (mod p),

and X, Y are linearly dependent (mod p).We may now assume that p = 2. If a - 8 - 0 (mod 4), then we get a 8 -

0 (mod 2), whence either /3 z 0 (mod 2) or y 0 (mod 2), and we get a - d0 (mod 4), whence a = d - 0 (mod 2). Carrying out a similar argument to theone just completed for p 2, the proof of the lemma is easily completed.

LEMMA 13.2. Suppose A is an abelian subgroup of B" of type (p% p"2, p"s). Then

ni + n2 + n3 < 3n/2 + 9/2.

PROOF. Let xi, z2, X3 be a basis for A if p 2, and a basis for A2 if p = 2.Choose x, E zi. Then

xi E r(p"-,5.+e),, - i - 1, 2 ,3,

where e = 0 if p 2 and e = I if p = 2. Since the cyclic groups <x,>, <xj > aredisjoint if i j, Lemma 13.1 implies that

n <n-ni+e+n-nj +e+e,if 1 n,+nj -3e,

whence

3n > 2n, + 2n2 + 2n3 - 9e,so that

n1 + n2 + n3 < 3n/2 + 9E/2 < 3n/2 + 9/2,

as required.

LEMMA 13.3. Every abelian subgroup of C" has order at most p(3"+11)/2

PROOF. Since I'(p),,, and G,,," is of index 1 or p in I(p"),,, there is ahomomorphism 0: C" -* B" induced by the set-theoretic inclusion G,,,, C I'(p),,.Then

ker 0 =

is of order 1 or p. Since 4)(A) is abelian, for every abelian subgroup A of C, thislemma follows from Lemma 13.2.

REMARK. The 11/2 in Lemma 13.3 is distressing, but I do not see how tomake it go away. Its presence is an indication of the difficulty in dealing withp = 2.

LEMMA 13.4. If r E (2, 3) and x E G has order r, then the centralizer of x onC" has order at most p"+2

PROOF. I shall restrict myself to proving this lemma in the most difficult case:(1)r=2,(2) x is represented by X = (a d), a, b, c, d E Z,,, v(ad - bc) = 1,(3) plf.

(Of course, (2) (3), and (2) (1).)In this situation, we know that v(c) = 1, v(a) > 1, v(d) > 1, v(b) = 0, a + d

= 0. Conjugate X by (o al'), noting that a/c E Z,,, and get

1

0

a

c (ac

a a2 1 ac = 0 b+ c c =1 c -ac 0 1 c 0

Thus, we may assume at the outset that a = d = 0. Replacing X by -b-'X, wemay assume that b = -1, c = pu, u is a unit.

Now suppose y = (Y 6) E SL(2, Z,,) represents an element y of G,, such that[z, y] E G We then have

a - 8 - 0 (mod p),

/3 E p =- 0 (mod p).

Also

X-i= 10 1),c

(-c 0

550 J. G. THOMPSON

and

X 'YX c\-0c

0)(Y 8 C 0)

= 1 ( Y 8 )i0 -1)C -ca -c/3 c 0

- 1 ( 8c -y = 8 - Y

C -c2$ -caJ/I

-c/3a

Since a8 - 8Y = 1, we get

[Y,X]=8

-Y

C

-y8 - a/3c Y + a2C

-c/3 a

882 + c/32 Y

-/3 8 YC

This gives us the following congruences:(1) 82 + c/32 - 1 (mod p(2) y8 + a/3c - 0 (mod p"+' ),(3) y2/c + a2 - 1 (mod p").

Now(4)8=(1+8Y)/a(v(a)=0).

Using (4) in (1) gives(5) (1 + /3y)2 + ca2/32 - a2 (mod p').

Using (4) in (2) gives(6) y + 8Y2 + a2/3c - 0 (mod p

From (5), we get(7) a2 = (1 + 8Y)2/(l - c/32) (mod p").

Using (7) in (6) gives(8) Y + /3Y2 + /3c(1 + 8Y)2/(1 - c/32) = 0 (modpn+').

Clearing the denominator in (8) yields

Y - cf32Y /3Y2 -cR3y2 + /3c + 2c/32y + c/33y2 = 0 (mod p

which in turn gives

/3Y2 + (c/32 + 1)Y + /3c 0 (mod p

Define = y + c/3. We compute that

/3Y2 + (c/32 + l)Y + /3c = (/3 + 1 - c/32).Since v(/3 + 1 - c/32) = 0, we get = 0 (mod p"+'), which means that

(9) Y = -c/3 (modp"+i).If we use (9) in (7), we see that a2 (mod p") is determined by /3 (mod p").

Since a - 1 (modp), it follows that if p 2, then a, /3, y, 8 are uniquelydetermined (modp") by /3 (modp"), and there are pn-1 choices for /3, as

v(/3) > 0. If p = 2 and /3 is given, then there are 4 choices for a (mod p') (whenn > 3); if ao is one of them, the other 3 are -ao, a0(l + 211-'), -ao(l + 211-').The lemma follows.

The untreated cases of this lemma are easier, since the matrix X representingx may be taken to lie in GL(2, Zr). Thus, X acts on B" as well as C. Using thelatitude given in GL(2, Zr), we choose a handy conjugate of X, check that itsfixed point set on B" has order at most pand use the incluson G 1 C I'(p)to show that the fixed point set of x on C" has order at most p2.

We need some properties of p-groups.

LEMMA 13.5. Suppose p E (2, 3), and P is a p-group with the followingproperties :

(a) P has a metacyclic subgroup Q of index p,(b) P = <xIx E Q, x° = 1>.

Then the following hold:(i) Q has an abelian subgroup of index at most p,(ii) P has at most p3 conjugacy classes of subgroups of order p which are not

contained in Q.

PROOF. Let Q = Q/Q', and let t E P - Q be of order p. Then t acts on Qand if u E Q, then hypothesis (b) implies that

u u` . . u`°-1 = 1. (13.1)

Since a is generated by 2 elements, it follows that P/Q' has at mostp2 classes ifsubgroups of order p which are not contained in Q. Since Q' is cyclic, (ii)follows. As for (i), we break up the argument into cases.

Case 1.p=2. _Here t inverts Q, and so if H is a cyclic normal subgroup of Q with Q / H

cyclic, then H necessarily admits t. By (13.1) we see that

H = <h>, Q = <h, k>,

h' h", k`=k-',where a E Z is determined (mod 2"), and 2n = I H 1. Also of course,

k-'hk = hft (13.2)

for some /3, where /3 is also determined (mod 2"). Conjugation of (13.2) by tgives

kh'k-' = haft = h"#-'.

Since a is odd, we get /3 - /3 ' (mod 2"), whence <H, k2> is abelian and ofindex at most 2 in Q.

Case 2.p=3.Let a be the direct product of two cyclic groups of orders 3°, 3b with a > b.

By (13.1) we get a - b < 1.Let C be the centralizer in Q of Q'. Since Q is metacyclic, it follows that Q/C

is cyclic, and by (13.1) again, we have I Q / C I = 1 or 3, as Q / C admits t. _Next, let R/Q' be the set of elements of a of order 1 or 3, R/Q' = S21(Q). If

R/Q' is cyclic, so is Q, and so Q itself is cyclic, and (i) obviously holds. We mayassume that R/Q' is of type (3, 3). If [t, R] C Q', then (13.1) forces R = Q, and

552 J. G. THOMPSON

again (i) holds trivially. Thus we may assume that t does not act trivially onR/Q'. This implies that for each integer c, with 0 < c < a + b, Q has a uniquesubgroup of order 3` which admits t; call it Qc.

We may, of course, assume that Q' 1, and even that I Q'I > 3z, otherwise (i)clearly holds. Since IQ'I > 32, it follows that R C Q, and so R = Q2 C Q3 =S/ Q'.

Visibly, R is not cyclic, and so A = S21(R) is abelian of type (3, 3). Thus,<R, t> = R has the property that 9,(k) = <A, t> is of order 33. We will derivea contradiction from this.

By inspection, Q'A / Q' = Q1

and so [ Q3, t] = Qz = R / Q'. On the otherhand, t E S21(R) char R d <S, t>, the normality relation holding since IS : R I= 3, so that I<S, t>: RI = 3. Hence [S, t] C S21(R) n Q = A, so that [Q3, t] CQ1. This contradiction completes the proof of this cumbersome lemma.

LEMMA 13.6. Suppose p, r are distinct primes and r E (2, 3). Suppose furtherthat Q is a metacyclic p-group and that P = Q<t>, where Q d P and t has orderr. If P is generated by elements of order r, then one of the following holds:

(a) Q is abelian,(b) p = 2 and Q is a quaternion group of order 8.

PROOF. If p 2, this lemma is a consequence of Hilfssatz 8.5 of [3]. If p = 2,and Q' 1, it is not difficult to show that (b) holds. The proof is omitted.

Now we return to G and its open subgroups.We are now ready to converge on the final theorem.

THEOREM 13.1. If x E G has order r E (2, 3), therefor each e > 0, there areonly finitely many open subgroups K of G such that

X(K, E.

PROOF. By §11, we may assume that K E `K(l) for some l E (2, 3,p). Ifl r, then e(K, 0. Thus, we may assume that l = r, and that K = <K nG j, x'>, where x' is a of x.

Let n = n(K). We must, of course, bound n by a function of E. We assumewithout loss of generality that n > 10 so that, by Lemma 10.1, K n ismetacyclic.

Let Ko be the normal closure of x' in K and set

L = Ko G, ,,.

For each subset X of G,,, let X = XG,,,,,/GQJ,. By Lemmas 13.5(1) and 13.6, Ehas an abelian subgroup of index < pr since L = <x'>L n G, , and L n G,,,, ismetacyclic. Thus

ILI < pr -p"/2+11/2

Let (XI.... , xti) be a subset of L consisting of of x which isminimal with the property that every y of x which is in L satisfiesthe condition that

y is L-conjugate to a power of some x;.

By Lemma 13.5(ii), h < p3. For each i = 1, ... , h, let

C, = {k EKn Gi1l[ xi,k]

and let i be the set of E-conjugates of x;. Since k permutes9cl, it follows that

IK:GLI <p3, i = 1,2,...,h.Let

c = min{c1, . . . , ch}.

We conclude thatIKI < r p3+c+1+3n/2+1112

Furthermore, by Lemma 13.4,

c,<n+2, i= 1,2,...,h.Now we turn to Sx(K, We partition Sx(K, into subsets S1, ... , Sh

where

Si = { Kgl Kgx = Kg, gxg-1 is L-conjugate to x; } .

If Kg E Si, there is an element go E Kg such that goxgo' = x;. Now supposeKg1, Kg2 E Si, and

g1xg1' = gzxgz' = x;.This means that g1'gz = g satisfies

g, x] E G,.nLet H = { g E G,, 1[ g, x] E G,,,n}. Then, by Lemma 13.4,

IHI < IG,,: G,,11 pn+z,

and if Kgo E S;, then

si = (Kgohlh E H).We must unravel when it happens that Kgo E S,, h1, h2 E H, and

Kgoh 1 = Kgohz

This occurs if and only if

gohgo' E K, where h = hIh2'.

Now goxgo' = x;, and so

goh-'go' ' goxgo' ' gohgo' = goxgo' ,

whence

gohgo' E K gohgo' E C.

Conversely, if gohgo' E C;, then gohgo' E K. This tells us that

IS;1 < HI/pc < IGv: Gv1Ipn+2-c,

< IGn+z-c: 'P

554 J. G. THOMPSON

Hence, as h < p3, we get

IS.(K, Gj I < IG,,: G,,.1Ipn+s

By definition of eX(K, we get

e(K G)< IG,,:

But IG,,: KI = IG,,: KI, and so we get

IGL: Go.ll pn+5-ce .(K,X(K, G)

I GJ

pn+5-c.r.pc+3n/2+19/2p3(n-1)-1

as I G"I > I Ge,: G"'11 p3(n-')-'. This gives usp37/2 r 3 p37/2

fX(K, pn/2 pn 2

Since e < eX(K, G,), we find that

pn < 32.p37/E2

and so n is bounded by a function of E. The proof is complete.

14. An application. In this final section, I invoke the result that as f -* oo,g(ro(f)+) -* oo to show that

For each integer g, there are only finitely many discrete subgroups Kof PSL(2, R) such that:(a) K j ro(N) for some N.(b) (o 1) generates the stabilizer of the infinite cusp.(c) g(K) < g. (14.1)

By Helling's theorem, there is 0 E GL(2, Q) such that KO C ro(f)+ for somesquare free f. Since g(ro(f)+) < g(K") < g, f is bounded. Since ro(f)+ istransitive on cusps and contains (' ), we may assume that 0 = (0 9), whereh, q, l are integers, (h, q, 1) = 1, h > 0, 1 > 0, and 0 < q < h. Thus, KO is acongruence subgroup of ro(f)+, so by Theorem 4.1, there is an integer Mdepending only on g such that KO D r(M).

To complete the proof of (14.1), we must bound h and 1. Since

0(O M)0-' E K,

hypothesis (b) implies that Mh l-' E/ Z, so it suffices to bound h. Sincero(N)O C ro(f)+, we have

'(o1)o E ro(f)+,

whence hll. Let l = NO with 1o E Z. Since (h, q, 1) = 1, it follows that

(h, q) = I. (14.2)


As (, d) ranges over I'o(h2IN), a - d ranges over a set of integers whose g.c.d.divides 24 [4]. Since I'o(h2IN)' C ro(J)+, calculation of X in

* , ad - be = 1, c =- 0 (mod h2IN).1a )=(:X

forces h to divide q(a - d), whence by (14.2), h divides a - d, so h divides 24,and (14.1) follows.

It was (14.1) which prompted this paper, and the case g = 0 of (14.1) hasrelevance to the sporadic simple groups.

BIBLIOGRAPHY

1. H. Helling, Bestimmung der Kommensurabilitatsklasse der Hilberischen Modulgruppe, Math. Z.92 (1966), 269-280.

2. S. J. Patterson, On the cohomology of Fuchsian groups, Glasgow Math. J. 16 (1975), 123-140.3. B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin and New York, 1967.4. J. Lehner and M. Newman, Weierslrass points of ro(n), Ann. of Math. 79 (1964), 360-368.



CHARACTERS ARISING IN THEMONSTER-MODULAR CONNECTION

PAUL FONG1

This is a report on computing being done by A. O. L. Atkin, S. Smith, andmyself on a conjecture of Conway and Thompson in the monster-modularconnection. The observations which led to this conjecture are fully reported in[1], [3], [4].

The nontrivial irreducible characters of the monster M have degree at least196883. Under the assumption that the degree 196883 actually occurs, Fischer,Livingstone, and Thorne have computed the character table of M.

In the monster-modular connection a meromorphic function jc, called aHauptmodul, is attached to each conjugacy class C = cm of M. Herejc has theform

co

jc(q) = an(c)g`n--1

where the an(c) are integers, q = e2' , and T varies over the upper half-plane H.Thesejc arise as follows: The group PSL(2, R) acts in well-known fashion on H.We define the subgroups ro(n) for each n in N by

1'o(n) = j I a ):cO(modn)}/ { ± I }.

Of particular relevance are subgroups G of PSL(2, R) which contain some ro(n)as a subgroup of finite index. If the compactification HI G of the orbit spaceH/G has genus 0 as a Riemann surface, then any homeomorphismj: H/G -* Cis called a Hauptmodul for G. If j is normalized so that its pole is at ioo, itsresidue at ioo is 1, and its zero is at 0, then j is uniquely determined. The jcarising in the monster-modular connection are all of this form.

The assignment of Hauptmoduls to conjugacy classes of M was dictated bycertain considerations reported in [1]. On the basis of this empirical assignment,

1980 Mathematics Subject Classification. Primary 20D08, 20C15, 20-04; Secondary 10D05.'This research was supported by grants from the NSF.


557

558 PAUL FONG

Conway and Thompson observed that the class function of M

B,,:g- a.(g)is a character of M for n < 10. For example, B1 =1 + 196883, where 1 and196883 are the unique irreducible characters of M of degrees 1 and 196883 Theythen conjectured that B is a character of M for all n. There is a strategy due toThompson for establishing the conjecture in a finite number of computations byshowing

(A) the B are generalized characters for all n,(B) the generalized characters B are characters.

In (A) we shall see the curious situation where Brauer's Theorem on thecharacterization of characters is applied in a situation where the character tableis known.

We introduce the following notation: Let G be a finite group of order g. Weset 9 = Q(pg), R = Z[ µg], where µg is the set of gth roots of unity in C. Foreach prime integer p, let P be a prime ideal divisor of p in R, and let R. be thering of p-local integers in Q. Let Cl(G) be the set of class functions on G, and letX(G) be the character ring of G.

We fix a pair (p, s), where p is a prime integer and s is a p'-element of G. LetSs, be the p'-section of G determined by s. If P is a Sylow p-subgroup of CG(s),then Ss, is a union of classes C1, C2, ... , Ck of G, where each C. has arepresentative of the form st; for some t; in P. For each 0 in Cl(G) define thek-tuple 1(0) by

1(0) = (B(st1), B(st2), ... , O(stk)).

Let Mr(s) be the set of all k-tuples y = (y1, y2, ... , yk) with components y; inR. such that the dot product

x(O) . =_O (mod iPIR11)

for all 0 in X(G).

THOMPSON'S FORMULATION OF BRAUER'S THEOREM. Let 0 be an integer-valuedclass function on G. Then 0 is in X(G) if and only if for each Mr(s), the congruence

x(O) . =_O (mod IPIRI)

holds for ally in Mr(s).

In applying the theorem s need range only over representatives for thep'-classes of G. If the character table of G is known, then the Mr(s) can beeffectively constructed since the Mr(s) are finitely-generated free R,,-modules.The condition that 1(0) y - 0 (mod I P I R,) for all j in Mr(s) can then beexpressed as a set of k congruences on the components of x(0) modulo P1,where k is the Rc-rank of Mr(s).

In the case where G = M and 0 = 8,,, such a set of congruences involves thenth coefficients of jc,,jc...... jck' where C1, C2, ... , Ck are the classes in thep'-section of M corresponding to Mr(s). For a fixed Mr(s), these jc, can beexpressed as Laurent series in a fixed function such that the coefficients of theseLaurent expansions are p-adically convergent rational integers. Thus an integerN(p, s) exists such that x(0) y - 0 (mod IPIR,) for ally in Mr(s) and all

THE MONSTER-MODULAR CONNECTION 559

n > N(p, s). It follows that the proof of (A) can be reduced to a finite numberof computations. For the monster there are 195 different sets M,(s); the mostdifficult set of congruences occurs for M2(1), where k = 18 and IPA = 246. Thesesets of congruences have all been checked so (A) does in fact hold.

As for (B) it is known that o(a,,(1)) when c 1. Thus there exists aninteger N such that for n > N, the generalized characters B are necessarilycharacters. Estimates on N can be obtained by following the methods ofRademacher in [2], where expansion formulas for the are obtained. Anestimate by Thompson indicates that N = 1300 may suffice. To date it has beenchecked that the B for n < 400 are characters, and there is some evidence thatthis may be sufficient.

REFERENCES

1. J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979),308-339.

2. H. Rademacher, The Fourier coefficients of the modular invariant J(T), Amer. J. Math. 60 (1938),501-512.

3. J. G. Thompson, Some numerology between the Fischer-Griess monster and the elliptic modularfunction, Bull. London Math. Soc. 11 (1979), 352-353.

4. , Finite groups and modular functions, Bull. London Math. Soc. 11 (1979), 347-351.


MODULAR FUNCTIONS AND FINITESIMPLE GROUPS

LARISSA QUEEN

Let M be the Fischer-Griess new simple group, known as the Monster, oforder

246. 320.59. 76. 112. 133. 17 . 19.23.29.31 41 47.59.71= 8080, 17424, 79451, 28758, 86459, 90496, 17107, 57005, 75436, 80000, 00000.

The Monster has not yet been proved to exist but its character table has beencomputed [1].

It has recently been discovered that to every element m E M there corre-sponds a series, known as the Thompson series,

Tm(z) = q-1 + 0 + Hl(m)4 + H2(m)q2 + ,

where q = e2wiz and Hi(m) is a character of M evaluated at m [2]. All thefunctions computed in this way from the character table of the Monster can berecognized as well-known modular functions on various discrete subgroups ofPSL(2, R). In particular, evaluating this series at m = 1M one gets

J(z) = j(z) - 744,

the fundamental modular function on t = PSL(2, Z) [3].Following [2], we denote by F(m) the subgroup of PSL(2, R) consisting of all

elements fixing Tm, for m E M. We call F(m) the fixing group of Tm. It turnsout that for every m E M the genus of the fundamental region of the fixinggroup F(m), RF(m), is zero and T. plays the same role in F(m) as J plays in t,namely it generates the genus-zero function field invariant under F(m); in otherwords, T. is a Hauptmodul for F(m).

As usual, for N E Z + we define

r (N) - 1(c d) E I': a = d ° l (mod N), b = c 0 (mod N)

t°(N)- {(cdl EF:c=0(mod N)).

1980 Mathematics Subject Classification. Primary 20D08; Secondary 10D05, 20G30.


561

562 LARISSA QUEEN

We also define

a b/had-bcn=1},cnh d JJ J

which is a subgroup of PSL(2, R) and is conjugate by (o °) to 110(n). We notethat in order to have the same name for a Hauptmodul and the correspondingfixing group we have slightly changed the notation used in [2].

If h is the largest divisor of 24 such that h2IN and N = nh then the normalizerof ro(N) in PSL(2, R), 4SL(2R)(ro(N)), consists of all 2 X 2 matrices of theform [2]:

(ae b/h l: e divides n/h exactly and ade2 - bcn/h = e > 0J.cn de J

lW = `( ae b ) : e divides N exactly and ade2 - bcN = e > 0 }cN de J

is a single coset of ro(N). The We, known as the Atkin-Lehner involutions, forma subgroup of the full normalizer. Similarly,

we = (ae b/h l: e divides n/h exactly and ade2 - bcn/h = e > 0cn de J

is a single coset of ro(n/h: h). The we are called the Atkin-Lehner involutionsfor I'o(n/h: h).

For n and h as above,

9"PSL(2,R)(ro(N)) _ (ro(h : h), we: e divides n/h exactly).

Following the notation in [2], we write

/

I'o(n : h),,,. f... for <I'o(n : h), we, wf,... >,I'o(n : h)+ when all we for I'o(n : h) are present,I'o(n : h) when no we 1 is present.If F(m) is a subgroup of I'o(n : h)+e f .., of index h and

(Tm(z))h = K,

where K is a constant, we denote F(m) by ro(nhlh)+ej and T. byTMJh+eJ,... .

An element m E M determines a number N as the least N s.t. (N °) E F(m).For this N, we have

F(N) c ro(N) c F(m) c PSL(2, R).In fact, as long as we stay in the Monster, T. is a Hauptmodul for a groupbetween ro(N) and its normalizer in PSL(2, R).

We have examined other finite groups G, usually involved in some way in thecentralizers of elements of M. In particular, we discussed

.0, E, 3.2.Suz, 2.HJ, F, 2.A71 H, M12.

We still found that to every element g E G there corresponds a series

tg(z) = 4-' + ho(g) + hi(g)4 + h2(g)4Z + ... ,

MODULAR FUNCTIONS AND FINITE SIMPLE GROUPS 563

where q = ezw;z and h.(8) is a proper character of G if G is E, F, H or M12 and isi

a generalized character if G is .0, Suz, H-J or 2.A7. In the former case theconstant term is immaterial, as in M, while in the latter case the constant term issignificant. Moreover, tg is again a Hauptmodul for some discrete subgroup ofPSL(2, R) containing ro(N) for some N.

I would like to remark, in connection with recent suggestions that these"moonshine" properties of finite simple groups can be explained by certainMacDonald-Kac identities in the framework of the theory of Lie superalgebras,that this property is not shared by infinite Lie groups. We have computedmoonshine characters of EB for elements of order 1, 2, 3 and for the indicatedeigenvalues (blank = 1) these are

eigenvaluesfunction

t313t613

t9 t913t913+

where

and

2

1613 + t6I3 2z

1913 + (t)2913

t613(z) = 21(3z)e t913(z)=

Z)4

.q(6z)e71(9Z)4

71(z) = q'124(q - 1)(42 - l)(q3 - l)(q4 - 1) . . .

which is the Dedekind ,n-function. However,z

t613(z) + 32and t913(Z) +

\ 9 /1613(2Z) t913(Z)

are not Hauptmodul for any discrete subgroups of PSL(2, R), as can be easilyseen by considering their poles.

Again we have

F(N) c ro(N) C F(g) C PSL(2, R),where F(g) is the fixing group of g E G; however, ro(N) is not always normal inF(g). In fact it turns out that the possible fixing groups are all discretegenus-zero subgroups of PSL(2, R) containing ro(N) and such that the stabilizerof the cusp at z = ioo is generated by (o 1).

Using the Helling theorem [4], which says that every subgroup of PGL(2, C)commensurable with t is contained in ro(n)+for some square-free n, afterconjugation by a suitable element of PGL(2, C), we have proved:

THEOREM 1. All discrete maximal subgroups of PSL(2, R) containing ro(N)where N is not divisible by 4 or 9 are ro(n : h)+ for some n and h such that n issquare-free, nIN and hl(N/n).

564 LARISSA QUEEN

THEOREM 2. All discrete maximal subgroups of PSL(2, R) containing r°(N)where 4 or 9 divide N are r°(n : h)+ and r°(n(f/g) : h)+ for some n, h, g and fsuch that n is square-free, hI(N/n), gJ24, g2IN and (g, f) = 1, where

r°rn9 = to gf to gJ(we observe that (9 f

8)induces the map z ---> z + f/g)

r°(ng.+ -(0 h )ro(n g(

0h

Oi )

Our objective is to find all genus-zero discrete subgroups of PSL(2, R)containing r°(N) for some N and such that G. = <(o'I)) and to compute theirHauptmoduln. At the same time we are compiling a list of functions arisingfrom the character tables of finite simple groups as was done for M [2], hopingfor a nearly 1-1 correspondence between the functions and the Hauptmoduln.Unfortunately, there exist some Hauptmoduln (namely, 25Z, 49Z, 50Z [2]),known as the ghost elements, which do not happen in any finite simple group.

As an example, we have a look at r°(50). r°(2 : 5)+ is a discrete maximalsubgroup of PSL(2, R) containing r°(50). Transforming by (o °),

r°(2:5)+ r°(2)+,

r°(2:5) -* r°(2),

r°(SO) -* r(10:5) {(1gc d) : ad - 50bc = 11.

Reducing modulo 2,

r°(2)+ - PGL(2,5) = S5,

I'0(2) -* PSL(2, 5) = A5,

r(1o:5) - c2 = f (0?)' (2

r(10:5) A r(5) = r°(2) A r(5) -* 1.

Let K = z r°(50) be a subgroup of r°(2:5)+ that maps onto 1. Then K is anormal subgroup of r°(2:5)+ and

r°(2:5)+/K - r°(2)+/r°(2) n r(5) = S5.We are interested in the subgroups of S5 containing a given C2:

MODULAR FUNCTIONS AND FINITE SIMPLE GROUPS 565

Now we want to know which of these groups have genus zero, and to do thiswe use the Riemann-Hurwitz formula:

2(g + n - I)=2 ind IT,where g is the genus of the group G, n is the degree of the permutationrepresentation on the cosets of G, and if IT is a permutation with cycle shapeIT = SIS2 . . . S.

m

ind IT = (Si - I )i-1

(ind Ir is the minimal number of transpositions that generate 7r).As an example, we calculate the genus of A4 and S4. r0(2) has two parabolic

elements of order oo and an elliptic element of order 2, while F0(2)4- has oneparabolic element of order oo and two elliptic elements of order 2 and 3. Weconsider the permutation representations on cosets of A4 and S4, respectively:

order order cycle shape ind order order cycle shape ind(mod 5) of A5 on A 4 (mod 5) of S5 on S4

cosets cosets00 5 5 4 00 5 5 4

2 2 22.1 2 4 4 4.1 3

00 5 5 4 2 2 2.13 I

Hence we deduce that A4 has genus 1 and S4 has genus 0. All the groups belowthe line in the figure on page 564 have genus greater than zero.

The final table which follows gives all genus-zero subgroups between I'0(50)and I'0(2:5)4-and their Hauptmoduln. We note that the notation I'0(nlh)+has aslightly different meaning in the case when I'0(N) is not normal in the maximalsubgroup. (* denotes algebraic conjugation.)

Genus-zero corresponding arises as Thompson Atlas name ofsubgroup of S5 Hauptmodul function of the elementD10 ti0 M (Monster) 10E

H-J (Double cover + 2A

D 10 t 10(52)

of Hall-Janko)

As 12(5z)

F20 110+2 M IOCH-J -2A

Fzo t,0+2(5z)D8 150+ M 50A

H-J -10C, -10DF (Harada-Norton) 10F

D8 t10 10+[50-] H-J +10C, +10DD 110110+[50-1 F 10G, 10HD6 x 2 11010+[50+] H-J -5A, -5BD6 x 2 110 10+[50t] F 10D, 10EV4 150+50 ghost element 50ZS4 t1015+ F 10BS5 12+(5z)

566 LARISSA QUEEN

The details of these calculations appear in [5].

REFERENCES

1. B. Fischer, D. Livingstone and M. P. Thorne, The characters of the "Monster" simple group,Birmingham, 1978.


3. J. G. Thompson, Some numerology between the Fischer-Griess monster and the elliptic modularfunction, Bull. London Math. Soc. 11 (1979), 352-353.

4. H. Helling, On the commensurability class of rational modular group, J. London Math. Soc. (2) 2(1970), 67-72.

5. L. Queen, Some relations between finite groups, Lie groups and modular functions, Ph.D.Dissertation, University of Cambridge, April 1980 (submitted).



EUCLIDEAN LIE ALGEBRAS AND THEMODULAR FUNCTION j

J. LEPOWSKYI

1. Recent empirical calculations ([8], [11], [1]) have related the dimensions ofirreducible modules of the Fischer-Griess "monster" sporadic group, and of theLie group ES, with the coefficients of the modular functions j and j1"3, respec-tively. This suggests trying to construct infinite-dimensional graded vectorspaces, the dimensions of whose components equal these coefficients. In thisnote, we point out that the theory of Euclidean Kac-Moody Lie algebras and atheorem of Kac (generalizing a theorem and idea of Feingold and Lepowsky)provide such spaces, suggesting the striking possibility that there might be somerelation between the monster and Euclidean Lie algebras 2

I would like to thank Howard Garland for several stimulating conversations.

2. Let F be the monster, conjectured3 to exist and to have an irreduciblemodule of dimension d = 196883. Consider the modular function

j(z) = I unq', (1)n>-1

with q = e2"1Z, z in the upper half plane. The un are positive integers andu-1 = 1. McKay noticed the coincidence ul = d + 1, and Thompson has foundthat the first few un (n > 1) are linear combinations, with small positive integralcoefficients, of (conjectured) dimensions of irreducible F-modules [11]. This ledThompson to ask whether there is a natural graded vector space V = ® Yn withun = dim V, and if so, whether each Yn (n > 1) can be viewed as an F-module.

It is known that

j(z) = (2)

1980 Mathematics Subject Classification. Primary 05A17, 05A19, 10D05, 17B65, 20D08.'Partially supported by NSF grant MCS 78-02439.

2After circulating this work, I [earned from Kac that he also noticed that his theorem "explains"McKay's observation about E8 and j 1/3 (see §2). The version of [1] available to me when the presentwork was done was a preliminary version with no Postscript.

3Proved recently by Griess.


567

568 J. LEPOWSKY

where OE.(z) is the theta-function of the root lattice of the Lie algebra E. (withthe roots of E8 normalized to have norm 1) and 71(z) is Dedekind's eta-function(cf. [10]). Now

OE,(z)/'l(z)8 =q-1/3( E bngn),

n>O

where the bn are positive integers, and McKay has noticed that the bn areapparently linear combinations, with "small" positive integral coefficients, ofdimensions of irreducible E8-modules [8].

In the theory of Euclidean Lie algebras, there is a very natural gradedE8-module B = ®,,>0B,, such that

OE.(z)/'l(z)8 =q-1/3 E (dim (3)

n>0

by a theorem of Kac [3(c)] generalizing a theorem and idea of Feingold andLepowsky [2] (see below), "explaining" McKay's observation. Take C = B 0 B0 B with the natural tensor product grading

C ® C.n>O

Then (1), (2) and (3) imply that

un = dim C, -I for all n > -1,

giving what seems to be a reasonable answer to Thompson's first question. Thisraises a new question: Can ®,>2C,, or some related graded vector space withthe same dimensions, be made naturally into a graded F-module?

3. We sketch the relevant background on Euclidean Lie algebras (cf. also[3(d)], [5(d)]): The Kac-Moody Lie algebras ([3], [9]) are generalizations of thecomplex semisimple Lie algebras. Among them, the Euclidean Lie algebras([3(a)], [9(a)], [9(b)]), which are infinite-dimensional, were used by Kac andMoody ([3(b)], [9(c)]) to interpret Macdonald's eta-function identities [7]. Ingiving a new general proof of these identities, Kac also introduced the "stan-dard" modules (which are irreducible and are the natural generalizations of thefinite-dimensional irreducible modules for semisimple Lie algebras), and provedthe natural generalization of Weyl's character formula for them [3(b)].

For each complex simple Lie algebra g, Kac and Moody defined a certainEuclidean Lie algebra d, a central extension of g ®c qt, t-1] ([3(a)], [9(a)],[9(b)]). This Lie algebra has "simple roots" ao, . . . , a,, where 1 is the rank of gand al, . . . , a, are identified with the simple roots of g. Let R be the formalpower series ring on the analytically independent generators e-"°, ... , e. Fora standard a-module M, we define X(M) to be the element of R [6, p. 28]obtained by dividing the character of M [3(b)] by the exponential of the highestweight of M. Let f E R and let (so, ... , s,) be a sequence of nonnegativeintegers. If the formal power series in q obtained by specializing e-% to q'(i = 0, ... , 1) is defined, we call it the (so, . . . , s,)-specialization of f. (This ideawas introduced in [5(b)], [5(c)], [6, pp. 27-28].) When each s; = 1, it is definedand is called the principal specialization of f. This concept was introduced and

EUCLIDEAN LIE ALGEBRAS, THE MODULAR FUNCTION j 569

shown to be of interest for Euclidean Lie algebras in [5(b)], [5(c)] and forstandard modules in [6] and [2].

The standard modules were first studied "concretely" in [6] and [2].4 Thecatalyst was the surprising discovery that for certain standard modules M1, M2of X1(2, C)", the principal specializations of X(M1) are closely related to theproduct sides of the Rogers-Ramantrjan identities [6]. The resulting analysis ofstandard modules for X1(2, C)" led to the following unexpected findings connect-ing standard modules with combinatorial identities and automorphic functions[2]: Denote by C)) the "fundamental" standard module associated withthe simple root ao. (We now call this the basic X1(2, C)"-module.) Then theweight multiplicities for C)) are exactly the values of the classicalpartition function p, and

C))) = 2(e_"°);2(e a1)'(1+1))

\i

(2 p(k)(e__°)k(e-')k)(4)

k>O

[2, §4]. When the left-hand side of (4) is expressed by the character formula, theresult is a (two-variable) identity in R [2, Theorem 4.6], which upon principalspecialization becomes Gauss' identity

71(2z)2/71(Z) = ql/8 X gk(2k+1) (5)

keZ

[2, Corollary 4.7].Suppose g is complex simple with equal root lengths. The above module

C)) generalizes to the fundamental g-module associated with the simpleroot ao ([3(c), p. 129], [4]), called the basic g-module in [4]. We denote it by B(g).Kac has generalized (4) to B(g) [3(c), formula (3.37)], using a generalization of(5) which follows from an identity of Macdonald together with empiricalinformation from [5(b), Remark 2] (Proposition 3.7(h) of [3(c)]).

Now the (1, 0, . . . , 0)-specialization of X(B(g)) is defined, and when appliedto Kac's expression, as in [3(c), p. 131 ], it immediately yields

g1/240 (ZO(Z)r, (6)

where Bg(z) is the theta-function of the root lattice of g, with the roots of gnormalized to have norm 1.

On the other hand, B(g) has a natural grading such that eachB(g)_ is a finite-dimensional g-module and such that the (1, 0, . . . , 0)-speciali-zation of X(B(g)) is (Let A be the highest weight of B(g).Then B(g)_ is the sum of the weight spaces of B(g) for those weights µ such thatA - µ = nao + (a linear combination of a1, . . . , a1).) The case g = E. gives (3);the graded E8-module B in (3) is the basic ES-module B(E8). (Note that the(1, 0)-specialization of the right-hand side of (4) is ql/24(2rezq')/7j(z). It wasthis special case of (6) (for g = X1(2, C)) which drew my attention to Kac'sgeneralization.)

4We take this opportunity to correct two misprints on p. 279 of [2]. In line 6 of Proposition 2.3, afactor of n should be inserted into the exponent of u, and at the end of the previous line, a factor ofu raised to the same power as in line 6 should be inserted.

570 J. LEPOWSKY

The decomposition of each B(g)_ under g can be computed using Kostant'smultiplicity formula [unpublished] as stated and used in [5(a)]; it follows easilyin the present generality from the character formula.

REFERENCES


2. A. Feingold and J. Lepowsky, The Weyl-Kac character formula and power series identities, Adv.in Math. 29 (1978), 271-309.

3(a). V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR 32(1968), 1323-1367 = Math. USSR-Izv. 2 (1968), 1271-1311.

3(b). , Infinite-dimensional Lie algebras and Dedekind's 71-function, Funkcional. Anal. iPrilozen. 8 (1974), 77-78 = Functional Anal. Appl. 8 (1974), 68-70.

3(c). , Infinite-dimensional algebras, Dedekind's'q-function, classical Mobius function andthe very strange formula, Adv. in Math. 30 (1978), 85-136.

3(d). , Highest weight representations of infinite-dimensional Lie algebras, Proc. Internat.Congr. Math. Helsinki, 1978, part 1, Academia Scientiarum Fennica, Helsinki, 1980, pp. 299-304.

4. V. G. Kac, D. A. Kazhdan, J. Lepowsky and R. L. Wilson, Realization of the basicrepresentations of the Euclidean Lie algebras, Adv. in Math. (to appear).

5(a). J. Lepowsky, Multiplicity formulas for certain semisimple Lie groups, Bull. Amer. Math. Soc.77 (1971), 601-605.

5(b). Macdonald-type identities, Adv. in Math. 27 (1978), 230-234.5(c). , Generalized Verma modules, loop space cohomology and Macdonald-type identities,

Ann. Sci. Ecole Norm. Sup. 12 (1979), 169-234.5(d). , Lie algebras and combinatorics, Proc. Internat. Congr. Mathematicians, Helsinki,

1978, part 2, Academia Scientiarum Fennica, Helsinki, 1980, pp. 579-584.6. J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in

Math. 29 (1978), 15-59.7. I. G. Macdonald, Affine root systems and Dedekind's'q-function, Invent. Math. 15 (1972),

91-143.8. J. McKay, E. and the cube root of j(z) (preprint).9(a). R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211-230.9(b). , Euclidean Lie algebras, Canad. J. Math. 21 (1969), 1432-1454.9(c). , Macdonald identities and Euclidean Lie algebras, Proc. Amer. Math. Soc. 48 (1975),

43-52.10. J: P. Serre, A course in arithmetic, Graduate Texts in Math., vol. 7, Springer-Verlag, Berlin and

New York, 1973.11. J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular

function, Bull. London Math. Soc. 11 (1979), 352-353.

RUTGERS UNIVERSITY


EXPOSITION ON AN ARITHMETIC-GROUPTHEORETIC CONNECTION

VIA RIEMANN'S EXISTENCE THEOREM

M. FRIEDI

There is a triumph for the group theory that is applied to the examples of thispaper (Examples 1 and 2 of §1.A, § LB and §3). Starting with an infinitecollection of arithmetic questions parametrized (for example) by the degrees ofcertain polynomials involved in the phrasing of the problem, we apply theoremsof Burnside, Feit, Schur, Scott, Wagner, Wielandt, et al. to conclude a list ofsolutions that has a reasonable finitistic description. I selected these examplespartly to illustrate this remarkable circumstance; a circumstance commensurate(as the Schur problems will show) with the finitistic description in [Maz] of theQ-rational points on the complete set of modular curves { Y(n) } =1. Uncoveringa simple explanation for this surprising phenomenon, which occurs repeatedly inthe examples of [Fr, 0], remains one of the unsolved problems in this areajoining group theory to arithmetic.

This paper focuses on two examples from the motivational section [Fr, 0, §7]of my partially completed book. As examples of diophantine geometry problemsthey are archetypal of those that can be rephrased in terms of arithmeticmonodromy groups, and illustrative of those whose solutions apparently demandan open eye to rich connections with many diverse areas of mathematics. Informat we follow, for each of the problems, the outline of the 4 stages thatappear in § I.A. In scope we will be long on the first 3 stages: introduction ofRiemann's existence theorem; arithmetic monodromy interpretation; and il-lustration of the special type of permutation group theory that arises so naturallyin this area. The classical Riemann's existence theorem tells us a prescription(§O.A) for asserting the existence of ramified covers of the Riemann spherehaving degree n and explicit points of branching. The final list of covers resultsfrom the combinatorial expedient of computing some equivalence classes of

1980 Mathematics Subject Classification. Primary 14H25, 14H30.'The author was funded by NSF Grant No. MCS-78-02669. We would also like to thank Horst

Zimmer for his comments and careful reading of the manuscript.C American Mathematical society 1980

571

572 M. FRIED

homomorphisms of a group F (in this case, a free group on r generators moduloone relation; r being the number of branch points of the cover being considered)into the symmetric group on n letters, S,,.

There are problems that arise in applying Riemann's existence theorem. First,it is just an existence theorem; it does not explicitly produce algebraic equationsfor these covers. Secondly, it does not directly give us information aboutarithmetic monodromy groups of these covers. The examples of this papersuggest the direction that is taken in [Fr, 0, §§6, 9] to remedy these deficiences.In the end, much of this delicate arithmetic information can be obtained, ingeneralization to the original Riemann's existence theorem, by computing equiv-alence classes of permutation representations of certain finitely presentedgroups. Artin's braid group, the Hurwitz monodromy group, and many of theother classical combinatorial groups arise and present us with representationtheory problems. Suggested reading order: §1 (follow Schur's problem) -4 §O.A-3, §O.C -4 §3 -* rest of paper.

Table of Contents

§0 Directions on a Riemann surface§0.A Riemann's existence theorem§O.B Neighborhoods of a cover; Artin braid group; Hurwitz monodromy

group§0.C Modular curves.

§ 1 The four stages and elementary arithmetic monodromy§I.A Outline of the four stages and two examples of Stage I considerations§ 1.B The Cebotarev theorems and the Hilbert-Siegel theorem with applica-

tion to the Schur problem and Hilbert's theorem

§2 Group theory and Stage III considerations§2.A Group theory and geometric monodromy as applied to the reducibil-

ity of variables separated polynomials§2.B Newly reducible polynomial pairs; living up to an example of B.

Birch; and some theorems of Feit§2.C Double degree representations; theorems of Scott and Wielandt; and

the irreducible components of composite pairs

§3 Conclusion of the Schur problem and explicit aspects of Hilbert'stheorem

0. Directions on a Riemann surface.O.A. Riemann's existence theorem. We consider a compact Riemann surface 5

with a "sphere of reference", a situation that is not well represented by a picturein R3. We identify P' (projective 1-space over the complex numbers) with theone-point compactification C u {oo) of the complex plane. By a sphere ofreference we mean a surective complex analytic map S - P' presenting S as adegree n ramified cover of P'. Let

D(y)de 1

(branch points of 9); points z E P'such that Jyo-'(z)J is less than n}. Now consider the properties of such a map qp.

AN ARITHMETIC-GROUP THEORETIC CONNECTION 573

For each z E P' let Nr be a small "disc" neighborhood of z, and considerN, the restriction of cp over N.

For z (4 D(q)), N, can be selected so that qo naturally presents asisomorphic to n copies of N, For z E D(q)) we choose NZ so that

S IN, - {q)-'(z)) - NZ - (z) is an unramified cover. (0.1)

The fundamental group of a punctured disc is naturally isomorphic to Z bychoosing as a generator a "circle" about z in the counterclockwise direction.

DEFINITION 0.1. Let'' ,,, IX2, ')t be three connected manifolds and let

'Jl, - O,, i = 1, 2, be an unramified covering morphism. We say that 91, 6xand IX are equivalent (as covers of Ot,) if there exists a (not necessarilyunique) homeomorphism : )'1 ---> '2 such that 4,2 0 i = ¢,. From the theoryof the fundamental group we know that equivalence classes of connected coversof St are in one-one correspondence with conjugacy classes of subgroups ofer'(%, zo)d= the fundamental group generated by the homotopy classes of pathson % based at the point zo.

In expression (0.1) write SIN - {(p-'(z)) as a disjoint union of connectedcomponents U,_, M; where t is equal to Jpr-'(z)I. Then, up to equivalence (as acover of N. - z) M, is uniquely determined by the degree of the restriction of qDto M;. Indeed, if z = 0, and No is a small disc about the origin in C, then thecover of degree e is represented by

proj. on zM'_((w,z)ECXCIwe=z)IN0_(o) --" No-(0).

Thus, corresponding to 9) we have two pieces of data:(0.2) (a) the collection of points D((p) in P'; and

(b) for each z E D(q)) the collection of integers (some repeated) given bythe degrees of the connected components of S 1 N as covers of Nz (as above).

For each z E D(q)) let a(z) be a symbol of the form (sl)(s2) . . (s,) wheres,, . . . , s, are the integers associated to z by (0.2)(b). Since l;_, s, = n, it iscustomary to leave out of this symbol those integers s; for which si = 1. Wedefine the index of a(z) (denoted by ind(a(z))) to be the integer I',- 1(s, - 1). Interms of these quantities the Riemann-Hurwitz formula becomes

(0.3) 2(n + g - 1) = 2zED(q,) ind(a(z)) where g = g(S) is the genus of S (thenumber of handles in a description of S as a sphere with handles; the numberof linearly independent holomorphic differentials on S ; etc.).

Now let zo(q)) be a preselected base point, not in the support of D(q)) _(z,, ... , z,). Let { P,, ... , P } be a naming of the points of the fiber qq-'(zo(q))).The fundamental group a'(P' - (z,, . . . , z,), zo((p)) is a free group on rgenerators, which we denote by E1, . . . , 1 modulo the one relation 11 .. 1,= Id, where Id denotes the identity in this group. In Figure 1 the paths91, ... , 9, are representatives of the homotopy classes of paths that givegenerators E1, . . . , 2,.

574 M. FRIED

FIGURE 1. Generating paths on an r-punctured sphere

The paths 91, ... , 6, are (excluding their beginning and endpoints): nonintersecting; oriented, inorder, clockwise around zo(p); and'P; is homotopic to the path o B o I ' where B, the boundaryof a "small" disc neighborhood of z;, starts and ends at y.

Note that the path 15', c 92 o - - is homotopic to the identity by goingaround the "back" of the sphere.

For each degree n cover S IF P' with branch points among the set{z1, ... , z,) we associate to each Xi (as above) a permutation a; of the points{ P,, ... , in the fiber above zo(op) in the following way. The effect of a;applied to Pk is P, where: P, is the endpoint of the unique path on S lying over°, and starting at Pk. For convenience we change (P1, . . . , to( 1, 2, ... , n) so that we may regard ai as being contained in S,,. Thus we obtaina homomorphism a'(P' - {z,, ... , z,), zo) - S,,.

DEFINITION 0.2. We call the ordered r-tuple (a,, ... , a,) - a a description ofthe branch cycles of the cover S - P1 with respect to the base point zo(g)) and thecollection o f paths °P,, ... , amJ,. If we reorder the points P,, . . . , P by permut-ing the subscripts via T E S,,, a new computation of the branch cycles will yield(a...... a) where a; = T- a; - T, We say that or and a' areequivalent. If we write a; as a product of disjoint cycles in S,,, then the lengths ofthese disjoint cycles are the same as the integers that appear in the symbol a(z,)following expression (0.2).

Conversely, for a given homomorphism of ir'(P' - (z,, ... , z,), zo) into S,,we may consider the images of E,, ... , 1, (denoted a,, ... , a,) as the branchcycles of a cover S 10V computed with respect to the paths J,.Indeed, from fundamental group theory, we obtain a cover X- P' -( z, ... , Zr). We compactify X to give S _ S(9)) by making a relative com-pactification of (qg')-'(D°) where Dz is a disc neighborhood of z; on P' andDO = DZ - {z1). The upshot of all this is:

RIEMANN'S EXISTENCE THEOREM. Given paths 91, ... , Jr as described inFigure 1, there is a one-one correspondence between equivalence classes of

(0.4)(a) connected covers S (op) - P' of degree n; and(b) r-tuples (a,, ... , a,) E (S )r such that a, a, = Id, and a,, ... , a,

generate a transitive subgroup of S,,.


The group G(o) generated by a,, ... , a, is called the geometric monodromygroup of the cover S - P'. Let Aut(S , q)) be the group of analytic isomorphismsa: S -+ S for which qo - a = qv. The cover S _T_)-P' is said to be Galois if theorder of the group Aut(S , qp) is n (i.e., it is as big as it can be). For any cover

S -P' there exists a unique minimal Galois cover -'P' fitting in a commuta-tive diagram (the Galois closure diagram)

(0.5)(a)

The group Aut(S, 0) is isomorphic to G(o), but it is most canonically identifiedwith the elements of SN that centralize the image of G(o) in its right re ularrepresentation. Here N is the order of G(o). For any subgroup H of Aut(, 0)(or, by a slight abuse, of G(o)), the classical Galois correspondence produces forus a diagram

(0.5)(b)

where the cover - S. is Galois with group isomorphic to H.Let TH be the permutation representation of G(o) given by multiplication on

the right cosets of H in G(o). Then (TH(a,), ... , TH(a,)) gives us a descriptionof the branch cycles for the cover 55, P'

O.B. Neighborhoods of a cover; Artin braid group; Hurwitz monodromy group.

Let S(O' - P' be an n-sheeted cover of the Riemann sphere, as described in§O.A, where the branch points of qp() are z(O), ... , zr(O) Notationally we denotethe cover by the pair (i0), gpro>)

576 M. FRIED

DEFINITION 0.3. A neighborhood (9, c, P) of (S(°), op(°)) over 9 consists of thedata

(D pr,J-->9 xP'9prz

and a specified point p(°) E 9 where(a) c is a proper map of degree n;(b) J and P are complex connected manifolds;

P1

(c) for each p E'P, pre' b presents the fiber , el(pr1 o 0)-'(V) as ann-sheeted cover of P' having exactly r branch points; and

(d) the covers ,,., --* P' and S(0) . P' are equivalent as covers of P' (thenatural extension of Definition 0.1).

We consider a few comments on fiber products. Let X, Y, Z be three sets; f:X ---* Z, g: Y ---* Z any functions. Then the fiber product of X and Y over Z isthe set X xZ Y = {(x, y)lf(x) = g(y)} C X X Y. If X, Y, Z are complex (oralgebraic) sets, and the functions f and g are induced by complex (or algebraic)morphisms, then X X Z Y has the structure of a complex (or algebraic) set andthe natural maps prx: Y ---* X and pry: X x z Y ---* Y are induced by complex(or algebraic) morphisms.

Let A' and A' be two copies of affine r-space, and consider the natural map*r

AR ---* A' that sends (x1, . . . , x,) to the r-tuple of symmetric functionsC

(y1, ... , y,) = ( . , (-1)' I XxII ... XjoI.... ).

The subscripts R and C stand for (resp.) Roots and Coefficients. The cover4'r

AR ---* AC is called (at least by the author) the Noether cover; it is Galois withgroup S,. The variety A` can be regarded as an affine subset of both (P')' and ofPr: A' = P' - { oo } embeds (A')' in (P')'; and A' can be regarded as the subsetof P' represented by r + 1-tuples (yo, y1, . . . , y,) with yo = 1. Interestinglyenough (P')' and P' are joined in a commutative diagram

AR

I(P')'

A'C

I

- P'

where the vertical arrows are the respective identifications of A' with subsets of(P')' and P' given above.

The true nature of this diagram is best understood by considering the set ofnonzero polynomials in Z

Pr = l yi - Z'I(yo, . . . , y,) E C'+' - {(0, . . . , 0)} ,

i=°

modulo the action of C* that equivalences two polynomials if one is anonzeromultiple of the other. This set is then identified with P', and the map 'f, maps(x., .. , x,) to 11'_1(Z - xi) with the stipulation that if xi = oo, Z - x; isreplaced by the constant 1. Thus P' can be regarded as the quotient of (P')' byS,. Finally, let A, be the subset of AR consisting of the points having two ormore equal coordinates, and let D, (the discriminant locus of the Noether cover)be the image of Ar under 'l',. By abuse we also denote by A, (resp., D,) theclosure of A, (resp., D,) in (P')' (resp., P'). We regard P' - D, as the collectionof r unordered distinct points of P'.

For ('5, 4, fl a neighborhood of (5'0), (p(O)), there is a natural map 'J' :

":P - P' - D, which associates to l' E 9 the collection of branch points of thecover v -* P' that appears in expression (0.6)(c). Let q(0) = 49(p(0)) and forconvenience assume that q(0) E Ac. The neighborhoods of (V0), T(0)) are animportant consideration in problems in which Riemann's existence theorem isapplied; especially those neighborhoods (°T, D, °P) for which the map 'I'6 is afinite map (i.e., proper with fibers consisting of a finite number of points). Itwould be very valuable if we could find a neighborhood of (5t0>, ggt0)) for which

= P' - D but this is very rarely the case as we shall see.The fundamental group of A' - D, = A' - D,, denoted ¶1(A' - D,, q(0)) is

called the Artin Braid Group. Similarly, the fundamental group ir'(P' - D q10))is called the Hurwitz Monodromy Group. Let

G(E1, ... , 2,; 2:1 ... 2:,)ae' G(1)

denote the free group on the generators 2:1, ... , 2, modulo the one relationE1 ... 2, = Id. Let Aut(G(E)) (resp., Aut(G(E))/Inn(G(E))) be the group ofautomorphisms (resp., automorphisms modulo inner automorphisms) of thegroup G(E).

THEOREM ([Ar, E, 1], [Ar, E, 2], [Bo], [Nil, [Fr, 0; PROPOSITION 0.1 ]). Thefundamental group ir1(A' - D,, g101) is a subgroup of the automorphism group ofthe free group on E1, ... , s given by generators Q1, ... , Q,_, (see (0.8)) subjectonly to the relations: Qi U, = Q Qi for 1 <i <j <r- 1, j i + 1 or i - 1;and Qi ' Qi+ 1 ' Qi = Qi+ 1 . Qi ' Qi + v i= 1, . , r - 2. In addition, the naturalmap coming from the embedding of A' - D, in P' - D, in expression (0.7)induces a surjective map of ir'(A' - D,, q10>) onto ir'(P' - D q(0)). The groupIff'(P' - D q(0)) naturally maps to Aut(G(E))/Inn(G(E)), and the image group iscalled the Mapping Class Group.

Indeed, Qi acts on the r-tuple (2,, ... , s,) by:

(E)Q, is equal to

(21,...>21-191:1'li+i'21+ 1,2p...,s,), i= 1,...,r- 1.We also denote by the images of Q1, in7r'(P' - D,, qt°').

Let o °1 = (ai°l.... , a,°)) be a description of the branch cycles of the cover

(0) - P' (as in §0.A). Let Ni(c 0)) (the Nielsen classes associated to ot0)) be theequivalence classes of elements z E for which r = (T1, ... , z,) and there

578 M. FRIED

exist y E S and Q E S, for which:(0.9)(a) G(Y-' T Y) = G(a(01);

(b) T1 T, = Id; and(c) y-1 T(i)# y is conjugate to a;°) in G(o), i = 1, ... , r.

Through expression (0.8) the group 7r1(P' - D,, q(0)) acts on Ni(a(0)).DEFINITION 0.4. The Hurwitz number of o(0), denoted Hur(a(0)), is the number

of orbits of Ir1(P' - D q(0)) on Ni(a(0)). The Braid classes associated to a(0)consist of the elements in the orbit of o(0), denoted Br(a(0)), under the action of7T1(P' - D q(0)). From the theory of the fundamental group, the transitiverepresentation of ir1(P' - D,, q(0)) on Br(E(°)) corresponds to an equivalenceclass of unramified covers, denoted

'Y%: Mn, r; 5(0), c(°)) - P' - D (0.10)

where ¶;(n, r; 5(0), q)(0)) is called the Hurwitz parameter space associated to('5(°), T(0')

The space 'X (n, r; 50), c(0)) has a universal property. If (9, c, Jam) is anyneighborhood of (55(0), T(O)),',5,: P - P' - D, the associated map, then thereexists a commutative diagram

'?

Z*.'X(n, r; CS(o), T (0)

(0.11)

P'- D,

Thus (as explained in [Fr, 0, §3D we may use ¶;(n, r; 5(0), c(0)) as a guide intraversing the neighborhoods of the cover (55(o) q)(O))

When Aut(S(0), T(0)) consists of only a single element then there exists aunique neighborhood (9, c, 9C(n, r; 5(0), ,T(0))) of (55(0),'T(0)) inducing the mapof expression (0.10). In the general case the existence of such a neighborhood isa delicate problem interpreted in [Fr, 0, §4] as a problem about special represen-tations of the Hurwitz monodromy group. For the exposition of this paper we donot consider the neighborhoods of (V°), T(°)) but only the space¶;(n, r; 5(0), q)(0)). However, we should point out that we cannot do arithmeticwithout coordinates giving an algebraic structure on ¶;(n, r; 5(0), q)(0)) and someof the related neighborhoods of (V0), q)(0)). The problem of finding algebraicversions of Riemann's existence theorem is phrased in [Fr, 0, §6] in terms offinding explicit algebraic coordinates for these spaces. Of all the quantitiesintroduced in [Fr, 0] none is so crucial as the Hurwitz number, Hur(a(0)). Ingeneral Hur(a(0)) is 1. Indeed, the irreducibility of many of the foundationalspaces of classical algebraic geometry (e.g., the moduli space of curves of genusg) is a corollary of the Hurwitz number being one for special choices of a(0). Butit is not always one [Fr, 9, §3] and this possibility is the deepest challenge to thearithmetic-algebraic theory developed to date. The investigation of the Hurwitznumber of special a(0) is an important problem in combinatorial group theory.We conclude this section with an example of utmost importance to the sequelwhich may help the reader with the relation of these concepts to other classicalmoduli spaces.

O.C. Modular curves. Let L be a discrete (additive) subgroup of C such thatE = C/L is compact. There is an invariant j(C/L) = j(E) of E such that C/L,and C/L are analytically isomorphic if and only if j(C/LI) = j(C/L). Weregard the function j as a uniformizing function on the space of one-dimensionalcomplex tori. This space is represented as 2t /PSL(2, Z): the quotient of theupper half plane by the action of the group PSL(2, Z) of Mobius transformationswith integer coefficients.

Let p E E be a point of order n in the group Cl L; n a fixed integer. Then p isrepresented by one of the complex numbers a/n where a E L but a/m is not inL for m a divisor of n and m > 1. We let (p> denote the subgroup of Egenerated by p. From (p) we obtain a complex analytic map of groups:

E- EI = E/(p> = C/(L, a/n> (0.12)

where <L, a/n) denotes the group generated by L and a/n.Denote the diagram of (0.12) by (E, E,). Two such diagrams (E, 4, E)

and (E', t', E1) are equivalent if there exists a commutative diagram

E'

*1

E

1*1 (0.13)

E El

where ' and *I are analytic group isomorphisms.Let Yo(n) denote the equivalence classes of triples (E, 0, E,).Now, it is also true that [Ah, Chapter 7] every complex torus C/ L = E can be

presented as a 2-sheeted cover E- PI branched over 4 points of P', so that theorigin of E lies over the place at oo on P'; and the diagram (0.12) can beextended to

E

91P1

0El

110,

P1

(0.14)

Here P' is canonically presented as the quotient E/(-1 > where (- I > is thegroup (of order 2) generated by the automorphism of E induced by "multiplica-tion by -1" on C. Since <o> is invariant under this automorphism, the diagramis commutative.

We are interested in a description of the branch cycles of the cover(0.15) PI -, Pt, a cover of degree n having the same branch points,

,61{z1, z2, z3, z4), as does EI - * P'.

Indeed, the cover E -p P1 is a Galois cover, and Aut(E, 13, o 1) is generatedby translation by p and "multiplication by -1". We easily conclude that thecover of expression (0.15) has monodromy group isomorphic to the 2 x 2 matrixgroup ((o b) modulo n where a = ± 1, and b is any integer modulo n). Denotethis group by A((-1>, n). If we let a;°), i = 1, 2, 3, 4, be a description of the

580 M. FRIED

branch cycles of the cover in (0.15) then

a (0) = b0)

0 1(-1where 0,(0). ai" a3°" or (o °). Thus bi°) - b2(°) + b3°r - b4°) = 0 modulo nand b(°), b2°), b30) generate Z/(n).

Now consider the Hurwitz parameter space X (n, 4; P', q)(0') where P' _' P' isa choice of one of the covers given by expression (0.15). To every point of`JC(n, 4; P', 9,(0)) we associate a cover as in (0.15) with monodromy groupA((-I >, n) having a description of its branch cycles given by (a1, a2, a3, a4) asfollows:

o. _ (01 b'1

(b; - bj, I < i, j < 4) generates Z/(n) and b, - b2 + b3 - b4 = 0 modulo n.Let

a-l01)'

TO111

Then

and

= 1 b; - 2c0 1

1 2c - b.to 1

We easily conclude that the elements of the Nielsen classes associatedto o 0), are represented by the equivalence classes of such o; and representativesmay be chosen so that one of b1, b2, b3, b4 is 0 and another equal to 1.

For any cover P1 . P' with such a description a for its branch cycles, let-Wi P' P' be the associated Galois closure diagram (expression (0.5)(a)). By

using the formula (0.3), the genus of is computed to be 1, and thus may beregarded as a complex torus E. We recover an unramified map E - E, byletting E, be the quotient of E by the subgroup of A((-l>, n) generated by (o 1)-Thus we have a natural map

n3C(n, 4; P'. YO(n). (0.16)

There is, up to isomorphism, only one two-sheeted cover of P' having a given 4points of P' as branch points (Riemann's existence theorem). Therefore thediagram of (0.16) extends to a commutative diagram

.1C(n, 4: P', p(O)) Yo(n)

(0.17)

P4 - D4 4 ?l. /PSL(2, Z)

where O, - 'I' takes a point q of 1(, corresponding to the diagram (0.14) to thevalue j(E1).


Because of the importance of the modular curve Yo(n), this example shoulddispel any notion that the Hurwitz families are "simple" when we considercovers of P' by other copies of Pl. Even though the map'P% is unramified, thecover Yo(n) ---> q[ /PSL(2, Z) is ramified over the points of GlL/PSL(2, Z) thatcorrespond to the two isomorphism classes of elliptic curves that have nontrivialautomorphisms different from that induced by "multiplication by -1" on C (seethe Kummer-Ritt functions of §2.C). Let q E P4 - D4 be a point such that

is not one of these two special points on 21 /PSL(2, Z). Then the fiber of*X over q is mapped in a one-one way by A. Since A is a fiber preserving mapbetween complex manifold covers of a surjective map of complex manifolds, Ais an open map. The curve Yo(n) is irreducible, and therefore from the diagram(0.17) we easily deduce that the Hurwitz number is 1 in this case. The irreduci-bility of Yo(n) in this case follows from the description of Yo(n) as a homoge-neous space; Yo(n) - 21 /I'0(n) where F0(n) is the subgroup of PSL(2, Z) whoseelements are represented by matrices (c d) such that ad - be = 1 and c ° 0modulo n.

1. The four stages and elementary arithmetic monodromy.I.A. Outline of the four stages and two examples of Stage I considerations. The

remainder of this paper concentrates on applying the ideas and notations of thefour stages listed below to the Examples 1 and 2. In this subsection the examplesmerely illustrate the formulation of the type of problems that fit in our Stage Iformat. The examples are continued in §1.B, in accordance with Stage IIconsiderations, where they are rephrased entirely in terms of elementary arith-metic monodromy. Finally, utilizing the Stage III considerations of §2, theexamples are treated (with a complete exposition) as partially solved problemsby an analysis of the Stage IV considerations in §3.

Stage I. The Diophantine problem data. We start with a number field M (afinite extension of Q) with ring of integers R, and a diagram

(2'- (- p. (1.1)

The spaces C', C, and P are algebraic varieties; is regarded as a parameterspace; and for a point p E Jam, the fiber 0.' - C4, represents one algebraic curve,possibly a singular affine curve, covering another. Also, we are given a diophan-tine question D which can be asked of any one of the curve coverings C?y ---> (2, inthe family, for p an M-rational point of 'J' . The reader is welcome to takeM = Q, R = Z if that should ease the burden of these formulations.

EXAMPLE 1. The Schur problem for rational functions. Let J(y) E M(y) be arational function; f(y) = fl(y)/f2(y) where f1, f2 E R[y] are relatively primepolynomials. The degree of f is defined to be the maximum of the degrees of fland f2. For m a maximal prime ideal of R, the quotient R/m is a finite field. Solong as some of the coefficients of f2(x) do not lie in m, we may consider f as amapping on R/m u { oo }. We denote by f mod m the rational function obtainedby regarding the coefficients of f as being in R/m. Here {oo} designates thepoint at oo on the affine line; f(oo) = 0 if deg(f2 mod m) > deg(f, mod m);f(oo) = oo if deg(f, mod m) > deg(f2 mod m); and f(oo) = alb ifdeg(f2 mod m) = deg(ft mod m) and a and b are (resp.) the leading coefficients

582 M. FRIED

of f, mod m and f2 mod m. Quickly stated, the Schur problem is the problem offinding those rational functions f which give a one-one (and therefore onto) mapon R/m u ( oo ) for infinitely many primes m. In order to fit this into the Stage Iconsiderations we consider only those rational functions having a fixed degreeequal to some integer n.

Let C' consist of the collection of affine algebraic curves, in the two variablesx and y, of the form f,(y) - x - f2(y) = 0 where fl, f2 are as above withmax(deg(f,), deg(f2)) = n. For 6J' we take an open subset of affine 2 (n + 1)-space whose coordinates correspond to the coefficients of the pairs of polynomi-als represented by f, and f2. For V E Jam, the cover C2y C2y consists of the curve( (given by fl(y; p) - x - f2(y; p) = 0) mapped to the x-line (the curve Cry) viathe natural projection that takes a point (x, y) on C2y to its x-coordinate. Thediophantine question D: for an M-rational point p E 'J', do there exist infinitelymany prime ideals m of R for which the rational function fl(y; p) givesa one-one map on R/m u ( oo )? Equivalently, if Cry mod m (resp., (2b mod m) isthe completion in projective 2-space of the reduction of Cab (resp., E) modulo m,then we are asking that each R/m-rational point of i; mod m lies over exactlyone R/m-rational point of i mod m. Primarily we use the formulation that

r1(R/m U{oo})aer (xo

E R/m U {oo}I

f(yo) = xo for someyo E R/m U too))

is equal to R/m U ( oo ) for infinitely many primes m.EXAMPLE 2. Explicit aspects of Hilbert's Irreducibility Theorem. We continue

notations from above. Let f(x, y) E M[x, y] be an irreducible polynomial. Let6Af (R) be the set {x0 E RI f(xo, y) is reducible, as a polynomial in M[y]). Oneversion of Hilbert's irreducibility theorem states that the complement of 6Af (R)in R is infinite. In § 1.B we have an arithmetic monodromy tool that allows us toconsider the prospect of realizing groups as Galois groups over Q(x). We reserveour strongest inspection for the "simple" case when f(x, y) = f(y) - x for somepolynomial f E R[y]. Let `\J(R) = (x0 E Rlf(yo) = xp for some yo E M).Clearly `RJ(R) c 6Af (R). We seek here the "theory" of the set 6Af (R) - `VJ(R),denoted by S1(R); when is 51(R) a finite set? It is an understatement to say thatthe group theory involved in this problem (both solved and unsolved aspects)seems to be quite deep. We leave to the reader the analogous Stage I formula-tion, as it is quite similar to Example 1.

Stage II. Translation into elementary arithmetic monodromy data. If we arelucky, the diophantine question D of Stage I has an equivalent formulation interms of arithmetic monodromy groups. That is, for p E Jam, let My be the fieldgenerated by the coordinates of p over M, and let M,,(C',) (resp., be thefield of My rational on (resp., ). Then Mb((2b) is a field extensionof M,(4). Let My(C') be the Galois closure of the

---field extension

MM(Cy')/MM(Ly) and let Mb be the algebraic closure of My in Mb(Cb).Then the arithmetic monodromy (resp., geometric monodromy) group is

Gdet G(My(C'y)/Mb{y)),


the Galois group of M,(33P)/M,(4) (resp., G, = G(M( )/MJE ))). Again, ifwe are lucky, there is a statement DG about groups such that this statementholds for GP if and only if D holds for the cover C'" - OP.

After looking at the translation of Examples 1 and 2 into arithmetic mono-dromy in § LB, you might ask: how do we know which diophantine problemscan be translated into arithmetic monodromy statements? You might especiallyask this when you realize that this is the very step which allows us to bring tobear the powerful results on permutation groups without which there would befew definitive solutions to the types of problems presented in Examples 1 and 2.A proper answer would require a considerably more advanced exposition on theidea of decomposition groups (see § 1.B), the idea of a Galois stratification as in[FrS], and parts of the answer would still be conjectural.

Stage III. Analysis of the geometric data. Our analysis here switches to thegeometric monodromy group. We turn to group theory in order to attempt aclassification of the groups GP that satisfy the conditions resulting from Stage II.The conditions on the group G. are quite precise (see, for example, Proposition2.1) coming from use of the Riemann existence theorem. They are usuallyphrased as conditions on a permutation representation and certain selectedgenerators, the branch cycles of §O.A.

At this point we (mentally) carve the parameter space 9 of expression (1.1)into a union of disjoint pieces, 9 = U '_i X; whereby the geometric mono-dromy conditions are constant along each X;, i = 1, ... , t. Indeed, assumingthat each of the curves (2, in Stage I is of genus 0 (which we do assume for the

sake of simplicity) X; is equipped with a natural map X, - 'K; to one of theHurwitz parameter spaces of expression (0.10). The map *i attests to a descrip-tion of the branch cycles for the cover C'; - (2, (as in Stage I) for 1' E Xi.Assuming the success of our group theory considerations we may determine asubset S of { 1 , 2, ... , t} such that the cover C',' -3..4 satisfies the geometricmonodromy conditions for 17 E 9 if and only if 17 E U ; Es Xi.

Stage IV. Diophantine solution data. Now, continuing the notation from StageIII, we consider separately each of the Xi's for i E S. We change notation, sothat X; becomes X equipped with a map X 5C to one of the Hurwitzparameter spaces. We say that X provides a positive solution to the diophantineproblem D if there exists i E X such that 17 is M-rational and if the arithmeticmonodromy group GP corresponding to the fiber at 1' satisfies the conditiongiven by DG in Stage II. Explicitly deciding which of the irreducible componentsX;, i E S provide positive solutions to D is the most difficult part of the wholeprogram. The methods (still partly conjectural) by which this final step can beachieved include the results of [Fr, 1], [Fr, 2] and [Fr, 0, §9]. It is these that weregard as an arithmetic form of Riemann's existence theorem, and they proceedthrough a delicate analysis of the fields of definition of X and the variousneighborhoods attached to X.

One last point. The question of whether or not X provides a positive solutionto the diophantine problem D is sometimes overly subtle and outside theprovince of present day technique. For example, in considering the solution toExample 1 we will see that we return to the question of M-rational points on

584 M. FRIED

modular curves for part of the answer. In the case that M = Q we couldcomplete the answer to our original problem quite nicely since [Maz] shows thatmost modular curves have very few Q-rational points. However, there areseveral things wrong with concluding at this point. Most Hurwitz parameterspaces do not fit into diagrams related to modular curves; and secondly, if weconsider a field M Q we do not have [Maz] to call upon (yet, anyway; see[Frey]). A third point is this; sometimes we are not so very interested in a fixedfield M. Therefore there is a natural way out of this diophantine impasse in thecase where the statement D is first considered over a field M, but for whichthere is a natural interpretation of D over every finite extension L of M (denotesuch an interpretation by D(L)). We say that the pair (p, L) satisfies D if p Eis an L-rational point of 6Y, and if 0' _3.0 has the desired property over L.With X as above, we say that the solutions to D are arithmetically dense in X ifthe set XD

a=I{p E X I there exists L containing M with (p, L) satisfies D(L)}, is

Zariski dense in X.I.B. The Cebotarev theorems and the Hilbert-Siegel theorem with application to

the Schur problem and Hilbert's theorem. Ah, if only it were possible to exposethe tools of arithmetic monodromy in a complete way in a short space. Since thetheorems we state here are very generally applicable to diophantine problems,for the reader inexperienced with algebraic number theory, they are bestregarded as a machine whose readout is an arithmetic monodromy analysis ofthe diophantine properties of an irreducible curve f(x, y) E M[x, y] (see [Fr, 0,§8]).

We start with the arithmetic monodromy interpretation of the theorems of [S].Suppose that W is a projective curve, and W -* P' is a cover with W and opdefined over a number field M having ring of integers R. Suppose also that x isa uniformizing variable for P' and there exist infinitely many M-rational placesp E W for which x(op(p)) is in R. Then W is itself isomorphic to P', and if y is auniformizing variable for this copy of P', op is given by a rational functionf(y) = x for which there are at most two places (values of y) lying above theplace x = oo.

Thus, the branch cycle for the cover W - P' corresponding to the place at oois either an n-cycle or the product of two disjoint cycles of length s and n - s,respectively, where the degree of op is n. In addition, if R has only finitely manyunits (e.g., R = Z) then this branch cycle is either an n-cycle or a product of twodisjoint n/2-cycles.

Let f(x,y) E Z[x,y] be an absolutely irreducible polynomial over Q (i.e.,irreducible over Q. the algebraic closure of Q). Let Q f be the splitting fieldf(x, y) over the field of Q(x), and let G = G(S2f/Q(x)). For xo E Q let Of(..) bethe splitting field of f(xo, y) over Q. Then G(I f(XO)/Q) is naturally identified witha conjugacy class of subgroups of G, so long as xo is not one of the branchpoints x,, ... , x, = oo of the cover of the x-sphere coming from the projectionof (x, y) satisfying f(x, y) = 0 to the x-sphere. For H a subgroup of G weconsider 6A (H, Z) = {xo E ZIG(S2f(XO)/Q) is conjugate in G to H). We let TH:G - be the representation of G obtained from the action on the n(H) rightcosets of H. Let a,, ... , a, be a description of the branch cycles of the cover

above, corresponding, respectively, with the points x1, ... , x,. Then, generaliz-ing [S] we have

THEOREM 1.1 ([Fr, 0, §8.3]; use last comments of §O.A). A necessary conditionthat T.(H, Z) be infinite is that

(1.1)(a) J;-1 ind(T (a(i))) = 2(n(H) - 1), and(b) is either an n(H)-cycle or a product of two disjoint n(H)/2-

cycles.

The case when H = G is the theorem of [Hi]. The question of sufficiency inTheorem 1.1 is considered in [Fr, 0, §8.6].

Now we turn to the Cebotarev theorems. Let f(x, y) E R[x, y]; let m be aprime ideal of R; let x0 E RI m; and let SZRXO).m be the splitting field of f(xo, y)over R/m. Then, there exists an explicitly computable nonzero polynomialg(x) E R[x] (possibly a constant) such that if g(xo) 0 mod in, thenG(SZf(x jm/(R/m)) is identified with a conjugacy class of cyclic subgroups ofG(Slf/M(x)). Indeed, there is a canonical conjugacy class, denoted (a(m, x0)),in G(SZf/M(x)) for which the subgroups of G(S2f/M(x)) associated toG(Jftso),m/(R/m)) are generated, respectively, by the elements of (a(m, xo)>. Inaddition, for (in, x0) for which g(xo) izt 0 mod in,

(1.2) the number of points (xo, yo) with coordinates in R/m u too) is equal tothe number of disjoint cycles of length 1 in x0)) where H isG(SZ f/ M(x, y)).

Let M be the algebraic closure of M in Of. The restriction of <a(m, xo)>, itturns out, does not depend on x0; therefore we let (a(m)) be the resultingconjugacy class of G(M/M). Theorem 1.2 is an arithmetic monodromy combi-nation of the classical Cebotarev density theorem of [Ce] and the Riemannhypothesis for curves over finite fields (see [Bom] and [W]).

THEOREM 1.2 ([Fr, 5, PROPOSITION 2]). There is a constant C (dependent only onthe degree off in x and y) with the following property. Let JR/mi be larger than C,and let r be any element of G(SZf/M(x)) whose restriction to M is a(m). Then thereexist at least c, IR/ml values of x0 E R/m for which r is in the class of(a(m, x0)>, where c, > 0 is not dependent on in. In addition, for any z EG(M/M), there exist infinitely many prime ideals m for which z is in the class of(a(m)).

When we apply Theorems 1.1 and 1.2 to our examples, they present acommon feature: the arithmetic hypotheses of the example are translated into asearch for reducible members of a family of curves.

EXAMPLE I (continued). Arithmetic monodromy interpretation of the Schurproblem. Recall that we have fixed an integer n, and we seek the rationalfunctions of degree n for which

NR/m u [cc)) = R/m u (oo) for infinitely many prime ideals m of R.

(1.3)

By applying (1.2) to the case where (1.3) holds, we conclude that for eachx0 E R/m, for which g(xo) s2t 0 mod m (notation as above), a(m, x0) fixes

586 M. FRIED

exactly one of the zeros y1, ... , y" of f(y) - x = 0. By applying Theorem 1.2 weconclude that for IR/ml large and in satisfying (1.3), if T is an element ofG(M(y1, . . . , y")/M(x)) whose restriction to M is a(m), then the coset

(1.4) G(M(yl, ... , y,)/M(x)) T consists only of elements that fix exactlyone of y 1, ... , y".

If T itself fixesyl, by considering the elements of G(M(y1, ... , y")/,4(y1)) T

we easily conclude that expression (1.4) is equivalent to(1.5) each orbit of G(M(yl, ... , y")/M('_)(y1)) on ys, ... , y, breaks up into

strictly smaller orbits under the action of G(M(yl, . . . , y")/M(y1)) whereT = a(m) and M() is the fixed field of T in M.

By a piece of arithmetic magic [Fr, 4], (1.5) is actually equivalent to (1.3), evenif IR/ml is not large. We are not, of course, suggesting that (1.5) is a simplestatement: it merely leads to an accurate arithmetic monodromy interpretationthat does not involve a statement about an infinite number of (possiblyunknowable) primes.

Schur's original conjecture in [Sch, 1] (see [Fr, 3]) was that if f is a polynomialsatisfying expression (1.3) then f must be a functional composition of linearpolynomials and the classical polynomials given by

(1.6)(a) y" (nth degree cyclic polynomial); and(b) Tn(y) = 2-n-1((y + (y2 + 4)1/2)n + (Y - (Y2 + 4)1/2)n) (nth degree

Chebychev polynomial).We epitomize the ad hoc nature of the subject matter around the time of the

original Schur conjecture by comparing expression (1.5) with an elementaryproof that compositions of linear, cyclic, and Chebychev polynomials (of degreerelatively prime to 6) give one-one mappings on Z/(p) for infinitely manyprimes p. This argument [Fr, 3, Lemma 13] was worked out by Davenport andmyself the summer before he died. It has therefore a certain sentimental valueeven if it may not be entirely new.

If h(y) = y", h(y) clearly gives a one-one map on Z/(p) if (p - 1, n) = 1because the multiplicative group of nonzero elements of Z/(p) is a cyclic groupof order p - 1. Let Tn(y) be the nth Chebychev polynomial. There is an easilyderived, and far more useful, expression for T"(y). If we let 2z = y +(y2 - 4)1/2, then Tn(y) = (z" + z-")/2 where y = (z + z-1)/2. If y is an ele-ment of Z/(p) associate to y one of the solutions z (it makes no differencewhich) of y = (z + z-1)/2. All such z lie in the unique quadratic extension F(p)of Z/(p). If y1, y2 represent distinct elements of Z/(p) for which T"(y1)T"(y2), then either zi = zZ or zi = zZ" since we have zi + zi" = zZ + zZ". Themultiplicative group F(p) - (0) is cyclic and of order p2 - 1. If (n, ps - 1)1, then either z1 = zs or z1 = zsl. In either case y1 = ys contrary to ourassumption; and Tn(y) is one-one as a mapping on Z(p). Let h(x) =h1(h2(... (14(x)) ... )) be a composition of linear, cyclic and Chebychev poly-nomials of respective degrees n1, , .. , n,. for which (6, N) = I where n1 ... n, _N. Thus, from the above argument we have only to show that there existinfinitely many primes p for which (N, ps - 1) = 1. However, by Dirichlet'stheorem there are infinitely many primes in the arithmetic progression (jN + 21j E Z) because (N, 2) = 1. Also

(jN+2- 1)(jN+2+ 1)=(jN+ 1)(jN+3)

is relatively prime to N since (N, 3) = 1, and the argument is complete.Finally, we conclude from expression (1.5) that G(M(yl, . . . , y,)/M(y1)) is

not transitive on y21 ... , y,,, or(1.7) (fl(y) f2(z) - f1(z) f2(y))/(y - z) is a reducible polynomial in the

variables y and z where fl, f2 E C[y] are relatively prime polynomials for whichf = fl/f2.

EXAMPLE 2 (continued). Explicit aspects of Hilbert's theorem. For f(x, y) EQ[x,y] an irreducible polynomial, we are led to consider the behavior for largeN of the set R f(Z, N) = (xo E ZI f(xo, y) is reducible over M and Ixol < N ). Asa consequence of Theorem 1.1 we show that there exist constants c1, c2 > 0, andan integer I for which

(1.8)(a) C2N1/' < Af(Z, N)I < cl N 1/1, and I > 1; or(b) c2 (log(N))' < 16Af(Z, N)I < cl (log(N))'; or(c) I6Af(Z, N)I is bounded as a function of N.

Indeed, let Hi, i = 1, ... , t, run over the subgroups of G(2f/Q(x)) for whichH ; is not transitive ony1, ... , y,,, the zeros of f(x, y); the fixed field of H1 in Ofis of the form Q(t1) for an element t; E SIf; and there exists g1 E Q(z) for whichg;(t1) = x where either

(1.9)(a) g; is a polynomial; or(b) g1 is the ratio of two polynomials of equal degree with the denomina-

tor a power of an irreducible quadratic polynomial over Q.If 'Vg, (Z) _ (xo E ZI gi(z) = x0 has a solution in Q), then Theorem 1.1

concludes that(1.10) R1(Z) c (U ; 1 `fig (Z)) U V where V is a finite set.It is now easy to conclude expression (1.8) (see [Lev]) and that(1.11) f(g1(z), y) is reducible as a rational function in two variables, i =

1,...,t.We say that R f(Z) has exponential (resp., logarithmic) density if (1.8)(b) (resp.,

(1.8)(a)) holds.

2. Group theory and Stage III considerations. Literary motivation for thissection is contained in [Ca, 1], [Kan], [Sc, 1]-[Sc, 3] and many of the applica-tions and results are in [DLSc, 1], [DLSc, 2], [DSc], [E], [Fr, 3], [Fr, 5]-[Fr, 8],[FrSc], [FrSm], [Mc], and [Tv]. The problem: considering the reducibility ofh(z, y) E C[z, y] where h(z, y) has "variables separated". That is,

h(z,y) = hl(y)'g2(z) - gl(z)' h2(y) (2.1)

where hl, h2 (and g1, g2) are relatively prime pairs of polynomials in C[y] (resp.,C[Z]).

Of special importance is the case when h2(y) = 1 = g2(y), so that we areconsidering the reducibility of h(z,y) = h(y) - g(z). The impact of grouptheory appears in our use of [Bu], [F, 1], [F, 2], [Sco], [Sch], [Wa], and [Wie, 1].Each of the subsections contains explicit problems stated entirely in grouptheoretic terms.

2.A. Group theory and geometric monodromy as applied to the reducibility ofvariables separated polynomials. We start with an ordered pair of positive integers(n, m) and we let 6A(n, m) denote the ordered pairs of rational functions in C(y)of respective degrees n and m:

588 M. FRIED

IA(n, m) = ((h1(y), h2(y); g1(y), g2(y))I h1, h2, g1, g2 E C[y], andmax(deg(hl), deg(h2)) = n, max(deg(gl), deg(g2)) = m and h1 and h2 (resp., g1and g2) are relatively prime).

If (hl, h2; g1, g2) E 6,, then h1/h2 and g1/g2 represent rational functions ofrespective degree n and m. For our considerations a certain subset of %(n, m)should be removed.

DEFINITION 2.1. Given two rational functions h, g E C(y), g is said to becomposite with h if g = h(s(y)) for some s(y) E C(y). In the case that s(y) is alinear fractional transformation we say that g and h are linearly related. Suppose(hl, h2; g1, g2) E 6A(n, m) and there exists m(y) E C(y) with: deg(m(y)) > 1;and both h and g are composite with m(y). Write m(s(y)) = h(y) and m(s(y))= g(y) with sl(y)/s2(y) = i(y), and sl(y)/s2(y) = s(y) where sl, s2 (resp., sl,s2) are relatively prime polynomials. It is easily checked (use Gauss' lemma) thatii(y) S2(z) - S,(z) s2(y) is a factor of h(z,y) = hl(y) g2(z) - g1(z) h2(y)We say that (h h2; g1, g2) is composite with m(y). If there is no m(y) for whichdeg(m(y)) > 1 and (hl, h2; g1, g2) is composite with m(y), we say that(hl, h2; g1, g2) is not composite (or h(y) and g(y) are a noncomposite pair ofrational functions). We let 6JI(n, m)NC be the (hl, h2; gl, g2) in 6JI(n, m) which arenot composite.

It is clear that 1 (n, m)NC is naturally isomorphic is an open subset ofby using the coefficients of the polynomials involved as

variables.Now we introduce a new variable x, and we let P1 be a copy of P1 for which x

is a uniformizing variable. Similarly we let P1 (resp., P=) be a copy of P1 forwhich y (resp., z) is a uniformizing variable. For each (hl, h2; g1, g2) E,(n, m)NC we obtain maps: P1 (- )) Ps (i.e., y° E P'', B h(y°) = x°) and

1,P(g(=)) 1P= - P. For notational convenience we sometimes use the symbol (h, g) inplace of (hl, h2; 81, g2). Let 5 (h, g) be the fibered product P1 x r: P= (as in

tp(h, g) 1

§O.B). Then we have a natural map: ' (h, g) -+ P. The irreducible compo-nents of 5 (h, g) are in one-one correspondence with the irreducible factors of

h(z, y) = hl(y) g2(z) - gl(z) h2(y)

since the curve h(z, y) = 0 is easily identified with a Zariski open subset of5 (h, g)

For (h, g) E %(n, m)NC we let D(h, g) be the statement: 5 (h, g) is reducible.Our problem: explicitly describe the locus of points (h, g) E %(n, M)NC forwhich D(h, g) is true.

Now, in order to proceed to Stage III considerations, we need to interpreth

D(h, g) in terms of geometric monodromy groups. From the map P1 -) Ps(g)

(resp., P. Ps) we obtain an extension of fields, C(P')/C(Ps) (resp.,C(Pz)/C(P1)). By using the uniformizing variables, the extension C(P')/C(Ps)can be identified with C(y)/C(x). We let C(y) (resp., C(z)), as in § l.A, Stage II,be the Galois closure of the extension C(y)/C(x) (resp., C(z)/C(x)). In order tobe compatible with the literature, we use the following notation: Qih_x = C y);Qig_x = C(z); and 0(hg) = SZh_x Qg_s. The geometric monodromy group of the


cover (h, g) Ps is identified with G(SZ(h g)/C(x)). This group has two permuta-tion representations: Th: G(Q(hg)/C(x)) - S. obtained from the right cosets ofG(Q(h g)/C(y)); and Tg: G(Q(h g)/C(x)) - Sm obtained from the right cosets of

G(Q(h.g)/C(Z))'Consider the statement D(G, T1, T2), associated to triples (G, T1, T2) (where

Tl and Tz are representations of the group G): Tl and T2 are transitive,nonequivalent permutation representations, such that G(T1, 1) = (a E GI(1)Tl(a) = 1) is an intransitive group under the representation T2: G(T1, 1) -S.-

PROPOSITION 2.1. The statement D(h, g) (i.e., 5 (h, g) is reducible) is true if andonly if the statement D(G(St(h,g)/C(x)), Th, Tg) is true.

PROOF. Indeed, from the Galois correspondence, the irreducible componentsof 5 (h, g) are in one-one correspondence with the orbits of G(Th, 1) in therepresentation Tg.

Thus, in Proposition 2.1 we have completed the translation of statementD(h, g) to the geometric monodromy statement D(G, Th, Tg). We can expeditethe use of Riemann's existence theorem in the later stages by continuing withfurther Galois theoretic observations. For brevity we quote the appropriate,relatively easy, results from [Fr, 6] without proof.

DEFINITION 2.2. Let h(y) E C(y). We say that h(y) is decomposable if h(y) _h(')(h(Z)(y)) where deg(h(')(y)) > 1 for i = 1, 2. If such h(') and h(Z) do not existthen h(y) is indecomposable.

LEMMA 2.1 [Fr, 6, PROPOSITION 2]. Let (hl, h2; gl, g2) E 6A (n, m)Nc and as-sume that S (h, g) is reducible. Then there exist rational functions htl), h(2), g(l),

g(2) E C(y) with these properties:(2.2)(a) h(y) = h'1)(h(2)(y)), g(y) = g1l)(g(2)(y));

(b)'0'01_x = Qe)_x = SZ(ho,,J1)); and(c) the irreducible components of S (0), g(l)) are in one-one correspondence

with the irreducible components of 5 (h, g).

The effect of Lemma 2.1 is that, in considering the irreducibility of ' (h, g),we may restrict consideration to those h, g E C(y) for which Qh _ x = Qg _ x =Q(h g). In this case, the representations Th and Tg of G(Q(h,g)/C(x)) are bothfaithful representations. Now we quote Riemann's existence theorem in anappropriate form.

LEMMA 2.2 [Fr, 6, PROPOSITION 4]. Let G* be a finite group with two faithfultransitive representations T, and T2*. Suppose that G* is generated by elements

a(1) ... a(r) = Id. Let z1, ... , z, be distinct ele-ments of P. Then (2.3) and (2.4) are equivalent.

There exist rational functions h(y), g(y) E C(y) as follows:1 1(2.3)(a) Py -. Px and P1 - P1 are ramified only over z..... , z,;

(b) S2h_x = Qg_x = Q(h.g)'(c) G and Th (resp., Tg) is equivalent as a permutation

representation to Ti (resp., 72*); and

,p(h)

590 M. FRIED

(d) (Ti (a(1)`), ... , Ti (a(r)`)) (resp., (TT T (a(r)`))) is adescription of the branch cycles for the cover (Py, (p(h)) (resp., (P1, (p(g))).

The Riemann-Hurwitz formula (§O.A) takes the form.(2.4)(a) Ej _ 1 ind(Tl (a(j)*)) = 2(n - 1), and

(b) Ej-1 ind(TZ(a(j)`)) = 2(m - 1)where deg(T,) = n, deg(TT) = m.

In addition, if Zr = oo, then we may take h(y) (resp., g(y)) to be a polynomial ifand only if

(2.4)(c) n = m, and Ti (a(r)`) and Tz (a(r)`) are both n-cycles.

From these remarks, the consideration of pairs (h, g) E 6R(n, m)NC for which(h, g) is reducible is equivalent to the description of the Hurwitz parameter spaces

(of §O.B) `JC(n, r; Py, (p(h)) where Py"-L

P1 has a description of its branch cyclesgiven by a(r)*) = Q as in Lemma 2.2.

The case where h and g are polynomials and h(y) is indecomposable (as inDefinition 2.2) is of extraspecial concern. For the remainder of this subsectionwe concentrate on this case in order to focus our group theory and algebraicgeometry concerns.

DEFINITION 2.3. Let F be a finite ring. A set of distinct elements D =(d1, . . . , d,) form a difference set of multiplicity r if the differences (d, - dj fori z#j) run over all the values of F - (0) exactly r times. If I Fl = n we say thatwe have an (n, k, r) design. If F = Z/(n), then D is said to be a cyclic differenceset. In the latter case, an element a E Z/(n) is said to be a multiplier of thedifference set D if (a d1, ... , a dk) = (d1 + t, ... , dk + t) = D+ t. Thesets D, D + 1, . . . , D + t are the blocks of the design.

THEOREM 2.1. Let (h, g) E 6R(n, m)Nc where: h, g E C[y] (i.e., they arepolynomials); h is indecomposable; and 5 (h, g) is reducible. There exist polynomi-als gt1W, g(2) E C[y] such that: deg(h) = deg(g(1)); g(z) = g(1)(g(2)(z)); 'h-x =SZg,,,_x; and the irreducible components of 5 (h, g) are in one-one correspondencewith the irreducible components of 5 (h, g(u).

Furthermore,(2.5)(a) g(') is indecomposable;

(b) Th and Tg,u (representations of G(SZh_x/C(x))) are two doubly transitiverepresentations which are inequivalent as permutation representations, but equiva-lent as group representations;

(c) _S (h, g0)) has exactly two irreducible components;(d) if X1 and X2 are the irreducible components of S (h, g(1)) given in (c),

q(h.g`'))then the degree of the map X1 --> Pz, call this k, satisfies (n - 1)jk(k - 1);and

(e) there exists a difference set modulo n of cardinality k.Still further: for a* _ a description of the branch cycles of

hthe cover Py Pz we take a(r)* to be the branch cycle corresponding to co E P1.Then, since h is a polynomial, Th(a(r)*) and are both n-cycles (identi-fied as n-cycles of the design indicated in (2.5)(e)). In addition:

(2.6)(a) r is equal to 3 or 4; and(b) i-i Th(a(i)`) = n - I = ;_ Tg,(a(i)`).


OUTLINE OF PROOF [Fr, 6, §3 for details]. The first paragraph is a restatementof Lemma 2.1. From [Fr, 3, Lemma 9], since h is indecomposable, Th is doublytransitive unless h is a cyclic or Chebychev polynomial.

If h is a cyclic or Chebychev polynomial (expression (1.6)), [Fr, 3, Lemma 11]shows that h and g(1) are linearly related (Definition 2.1) contrary to ourassumption that (h, g) is in 6A(n, m)Nc

For the remainder of our comments we replace g(1) by g, so that deg(h) _deg(g). Let yi , ... , y- (resp., zi , . . . , z,`) be the zeros of h(y) - x (resp.,g(z) - x), so that C(Yi .....y,) _ Qh-x = Sag-x = C(z , . . . , z,`) where weidentify y,` with the integer i in the representation Th. Letyi ,y (2), ... , y (k) be inthe orbit of yi under the action of G(Q(hg)/C(z1*)). The linear representationassociated with a doubly transitive representation [H, Theorem 16.6.15, p. 284] isthe sum of the principal representation [H, Theorem 16.6.15, p. 284] is the sumof the principal representation and an irreducible linear representation. Thus,the subspace of relations (E a; y,` = b with a1, . . . , an, b E C) is of dimension1, generated by (E"_ 1 y,`) - c, for some constant c E C. In particular, y i + y (Z)+ +y (k) is not a constant: indeed, it is a z* + b for some a, b E C. Fromthis we immediately deduce that the representations Th and Tg (ofG(Q(h g)/C(x))) are equivalent as group representations, and Tg is a doublytransitive permutation representation. If g were decomposable, from Galoistheory it is easily deduced that Tg would be imprimitive (there would be aproper field between C(z) and C(x)). The double transitivity of Tg thereforegives (2.5)(a) and (b). It also gives (2.5)(c) by the following argument. Let m(z, y)be an irreducible factor of h(y) - g(z), and let y,,*(1), ... , be the zeros ofm(z*, y). Then, if the degree of m(z*, y) in y is k, the coefficient of yk is ofdegree 1 in zi , and we obtain the relation azi` + b = y$(,) + +y$(,) forsome a, b E C. From [H, loc. cit.], this is one relation too many, unless (2.5)(c)holds.

Let Th(a(r)*) = (1 2 . . n), and Tg(a(r)*) = (1 2 . . . n), an assumption thatwe can make with no loss by changing Th and Tg to equivalent permutationrepresentations. We easily see that the conjugates of zi over C(y;) are

zi ,(z1)(a(r)`)1-adz) ... , (zi)(a(r)`)1-a(k)

(2.7)

Thus, the conjugates of zi over C(y u.)) are of the form

(zI)(a(r)`)'O-a(;) i = 1, . . . , k. (2.8)

Now assume that z2 is a conjugate of z,* over C(y (j)) for t values of j. Since G isdoubly transitive on zi , . . . , z,!, each z. , with u 1, is conjugate to zi overC(y uj)) for exactly t of the y .*(j). Hence t (n - 1) = k (k - 1) and the set(1, a(2), . . . , a(k)) is a difference set modulo n. With this we conclude theresults of expression (2.5).

We give a proof of (2.6)(a) in the case that n - 1 = k (k - 1). Theremainder of expression (2.6) is given by Lemma 2.2.

Let (a(1), ... , a(k)) be the difference set determined above, and recall that

azj + b = yj+a(1) + . . +Yj+a(k), j = 1, .. , n.

592 M. FRIED

Define a finite projective plane 7r: (y i , ... , y,`) are the points and(zi, ... , z,*) are the lines; with y; "on" z provided i = a(u) + j (mod n) forsome u = 1, ... , k. Clearly G = G(Qh_x/C(x)) acts as a doubly transitivegroup of collineations on 7T. It follows from a theorem of Wagner [Wa] that ithas order n = q 2 + q + 1, where q is a power of a prime. Now, nonidentitycollineations of projective planes can never fix 4 noncollinear points. Hence,each a z# Id in G fixes at most q + 2 of the points (Y,*, . . . , y,*,). Thusind(a) > (q2 - 1)/2 and

r-1

q2 + q = n - 1 = ind(a(i)) > (r - 1)(q2 - 1)/2. (2.9)1

h

Therefore the number r - 1 of finite branch points of the cover P1 PX is atmost 2q/q - 1. For q = 2, the expression ind(a) > (qZ - 1)/2 used aboveshould be replaced by ind(a) > (q2 - 1)/2 + 1 (since q 2 - 1 is odd). Thus, inthis case we need an ad hoc argument that r - 1 < 3. For q > 2, 2q/q - 1 is atmost 3 (with equality only for q = 3, n = 13).

The general case of expression (2.6)(a) follows from a theorem of Feit [F, 1,Theorem 4] when combined with the method of proof given above.

Let 9 (n, m)Nc be the subset of 6A(n, m)NC consisting of the pairs (h, g) with hand g polynomials. From Lemma 2.2 we may, with no loss, assume that n = m ininvestigating the polynomial pairs (h, g) for which 5 (h, g) is reducible.

DEFINITION 2.4. Suppose that (h, g) E 'P (n, n)NC, and that 5 (h, g) is reduci-ble. We say that 5 (h, g) is newly reducible if S (0), g) is irreducible for ht1),h(2 E C[y] with V"(V'(y)) = h(y), and deg(h(1") < deg(h).

In §2.B we state the following problem entirely in terms of group theory.Problem 2.1. Describe explicitly the pairs (h, g) E 'P (n, n)NC for which:(a) (h, g) is newly reducible; or for which(b) (h, g) is reducible and h(y) is indecomposable (in particular, (h, g) is newly

reducible).2.B. Newly reducible polynomial pairs; living up to an example of B. Birch; and

some theorems of Feit. In this subsection we respond in depth to Problem 2.1.Let us start slowly by considering 'P(n, n)NC for low values of n. For n = 2:there are no reducible polynomial pairs (h, g) E 'P (2, 2)Nc, since ys - zZ - a isirreducible for a z# 0. The same holds for n = 3 by applying Lemma 2.2 to a

h

description of the branch cycles of P1 P. in the case deg(h) = 3, and notingthat S (h, g) reducible implies that g is composite with h. Our first real examplemust await n = 4. Indeed, for h(y) = y4 + 2y2, g(y) = -4y4 - 4y2 - 1,

h(y) - g(z) = (y2 + 2yz + 2z2 + 1)(y2 - 2yz + 2z2 + 1). (2.10)

In the notation of §2.A; there are generators a(l)', a(2)', a(3)' of G(Q(h,g)/C(x))for which Th(a(l)') = (1 3), Th(a(2)') = (4 3)(2 1), and Th(a(3)') = (1 2 3 4)-1(resp., (1 2)(3 4), (4 2), and (1 2 3 4)-1) is a

description of the branch cycles of P1-)

P1 (resp., Pz Ps).Now consider 9(5, 5)Nc. Since there is no difference set modulo 5, Theorem

2.1 implies that the set of (h, g) E 9(5, 5)Nc for which 5 (h, g) is reducible isempty. Similarly, for n = 6. If (h, g) E 'P (6, 6)Nc, and 5 (h, g) is reducible, then

h must be decomposable. The monodromy group of P, - Ps is easy to work


out in this case, and once again we are able to say that the set of (h, g) EP (6, 6)NC for which 5 (h, g) is reducible is empty.

When we come to the case n = 7, however, we come upon B. Birch's bruteforce calculation. Let

h(t,y) = h(y) =Y7 - 7Xt y' + (4 - X)t y4 + (14X - 35)t' -y'- (8X + 10)t2 ,2+[(3 - X)t2 +7(3X+2)t3] y -;t3

where t is a parameter, A = -'(1 - V-_7 ), and µ = (1 + ). Then, by takingg(z) = -h(z) (complex conjugation of the coefficients of h(z)), h(y) - g(z) _T1(y, z) - cp2(y, z) with

cp (y, z) z3- (31A t,

T2(y,z)z + 2(X -

(2.11)

All of the (h', g') E P (7, 7)Nc for which ' (h', g') is reducible can be obtainedfrom expression (2.11) by specializing t.

In [Fr, 0, §5.3] we show that there are two families of pairs of degree 13polynomials (h, g) for which S (h, g) is reducible where the families are definedover the field X13 + 13)

The main point is that the coefficients of the family explicitly written out byBirch lie in a genus zero function field. As n gets large, the amount of work incomputing such an example becomes catastrophic. Worse still, if the field ofcoefficients were not of genus zero, it is hard to image such a computation asthis, even for n = 7, being feasible. The example with n = 13 also has coeffi-cients in a genus zero field. This is shown through the theory of the Hurwitzparameter space.

Problem (Conjecture) 2.2. (See [Kan] for a complete discussion.) Consider triples(G, T1, T2) where T1 and T2 are distinct faithful, doubly transitive permutationrepresentations of G of degree n with these properties:

(2.12)(a) T1 and T2 are equivalent as representations of G; and there existsa E G such that

(b) T1(a) and T2(a) are both n-cycles.Must one of the following be true:(2.13)(a) G is a group of collineations of a finite projective geometry with TI the

representation of G on the points, T2 the representation of G on the hyperplanes; or(b) n = 11?

Now we concentrate on part (b) of Problem 2.1.

THEOREM 2.2. There exist polynomial pairs (h, g) E P (n, n)NC for which(2.14) h is indecomposable (Definition 2.2) and ' (h, g) is reducible in the case

that n = 7, 11, 13, 15, 21, and 31. If n - 1 = k(k - 1) where k is the degree ofone of the irreducible components of 5 (h, g) over Ps, then we must have n = 7, 13,or 21. If expression (2.12) implies expression (2.13), then the only possible pairs(h, g) for which expression (2.14) holds are those of degree n = 7, 11, 13, 15, 21,and 31. Thus, if we could answer Problem 2.2 affirmatively, we would have acomplete answer to Problem 2.1, part (b).

594 M. FRIED

DISCUSSION OF PROOF. From Lemma 2.2 and Theorem 2.1 it is sufficient toh

give a description of the branch cycles of the cover P1 Ps. This is a tediousexercise which we have undertaken for n = 7, 11, 13, 15, 21, and 31. The methodfor doing this appears in [Fr, 0, §5.3]. Feit (in [F, 2]) has also demonstrated theexistence of these branch cycles by another method.

Now assume that n - 1 = k (k - 1). We return to the proof of Theorem 2.1where we have seen that: r' = r - 1, the number of finite branch points, cannotexceed 3; and the group G(Q(h g)/C(x)) is a group of collineations on a projectiveplane. Let a(l)*, . . . , a(r)` be a description of the branch cycles of (Py, (o(h))where a(r)* is an n-cycle.

Suppose a branch cycle a is of order m > 3. Since G is a group ofcollineations on the projective plane ir; the element a carries lines to lines andhence both a` and (a*)2 fix at most q + 2 point. Therefore, in the expression ofa as a product of cycles all but q + 2 of the points lie in cycles of length at least3. Hence 3 3 (q2 - 1), with equality holding only if q + 2 points are leftfixed and a' is of order 3.

If r = 2 then h and g are both cyclic polynomials of the same degree andhence are linearly related, contrary to our assumptions [Fr, 3, p. 47].

If r = 3 and the branch cycles a(l)* and a(2)* are each of order 2, then h andg must be linearly related to a Chebychev polynomial [Fr, 3, p. 47]. Since h isindecomposable deg(h) is a prime p, and h and g are linearly related to theunique Chebychev polynomial of that prime degree having the same branch

points as the cover PT(h)

P >P.Y -Now suppose r = 3 and that a(l)* has order greater than 2. Then q 2 + q =

ind(a(2)') > 7(q2 - 1), whence q < 7. For q = 7, we must have2(q2 - 1) and ind(a(2)*) _ 1 (q2 - 1). Thus, a(1) and a(2)* both

fix a line and so both fix the common point; contrary to a(1) a(2) _(1 2 . . n)-i. The remaining possibilities q = 2, 3, and 4 give us the possibilitiesn = 7, 13, and 21. The case q = 5 (or q 2 + q + 1 = 31) does not yield anexample ([F, 2]).

If r = 4, then q2 + q > (q2 - 1) whence q < 3 (n = 7, 13) and each a(i)* isof order 2.

Details, and the general case (i.e., where we do not assume that n - 1 =k(k - 1)) can be found in [F, 2]. Feit has also pointed out that we do not needto know that Problem 2.2 has an affirmative answer in the case that n < 100 asthe calculations of [F, 2] handle this case.

2.C. Double degree representations; theorems of Scott and Welandt; the irre-ducible components of composite pairs. The notations of §2.B remain in force. In§2.B we investigated in great detail the irreducible components of 5 (h, g) for(h, g) E 'P (n, m)Nc and h an indecomposable polynomial; the case correspond-ing to many of the present applications of the theory of the irreducible factors ofpolynomials with separated variables. The success of that investigation providesmotivation for the further study of the irreducible components of 5 (h, g) where

(2.15) (h, g) E %(p, m)Nc and h is a polynomial of prime degree p.By the way, the case m = p is as easily handled as if g were a polynomial; and

so falls in §2.B considerations.


DEFINITION 2.5. A triple (G, T1, T2) is called a group with double degreerepresentation of degree n if these conditions holds:

(2.16)(a) T1 is a faithful doubly transitive representation of the group G ofdegree n;

(b) T2 is a faithful primitive (but not doubly transitive) representation ofthe group G of degree 2 n;

(c) there exists a E G such that T1(a) is an n-cycle and T2(a) is aproduct of two disjoint n-cycles; and

(d) the restriction of T2 to G(T1, 1) = (a E G1 (1)T1(a) = 1) is anintransitive group.

THEOREM 2.3. Let p be a prime. Let (h, g) E c (p, 2p)Nc where h E C[y] (as in

expression (2.15)), and 5 (h, g) is newly reducible, and the coverP1-(8)

P1 hasexactly two points lying over the point oo on P.

Then SZh_x = SZg_x and (G(SZ(h,g)/C(x)), Th, Tg) (as in Lemma 2.2) is a groupwith double degree representation of degree p. If Q = (a(1), ... , a(r)) is a descrip-

tion of the branch cycles of Py9,(h)

Ps, and a(r) corresponds to the place over oc,then

(2.17)(a) Th(a(r)) = (n) (i.e., an n-cycle) and Tg(a(r)) = (n)(n);(b) E;_1 ind(Th(a(i))) = 2(n - 1); and(c) Ei=1 ind(Tg(a(i))) = 2(2n - 1).

Conversely, if G is a group with double degree representation of degree p,(G, Th, T ), and generators Q satisfying expression (2.17), then there exists (h, g) E',(p, 2p)NC as above. If such a p exists, then 2p - 1 is a square, and we haveeither p = 5 or p > 333.

OUTLINE OF PROOF (see [Fr, 5, Corollaries 2 and 3]). Most of the proof followsimmediately, from Lemmas 2.1 and 2.2. However, in showing that Th is doublytransitive; and that Tg is primitive, but not doubly transitive there is some work.From [Fr, 3, Lemma 9] either Th is doubly transitive, or h is a cyclic orChebychev polynomial. If h is a cyclic polynomial we have C(SZh_s) = C(y),and if h is a Chebychev polynomial then [C(SZh_s): C(y)] = 2. Since g is ofdegree 2 p, and g is not composite with h, each of these cases is ruled out.

The representation Tg cannot be doubly transitive, for if it were, then T.would be a doubly transitive representation of G(SZ(h,g)/C(x)) of degree 2 phaving an intransitive subgroup, G(SZ(ti,g)/C(y)), of index p. Since p is less than2 p, it is (well) known that this is impossible. Now we show that Tg is aprimitive representation. _ _

If Tg is not primitive then there exists a group G with d properly betweenG(SZ(ti g)/C(x)) and G(SZ(ti g)/C(z)). From the fundamental theorem of Galoistheory we conclude that the fixed field of

G_

lies properly between C(x) andC(z), and g(z) = g"l)(g(2 (z)) where deg(g(')) > 1, i = 1, 2. Since deg(g) = 2p,either deg(g(n) or deg(g(2)) is 2. This is the first crucial place where we use the

assumption deg(h) is a prime. Since P1 P1 has 2 places lying over the placex = oo, we easily deduce there are only two possibilities:

(2.18)(a) g")(z) is a polynomial of degreep; or

596 M. FRIED

(b) g'2)(z) is a cyclic polynomial (of degree p) and the places 0 and oo lieq(9,.,)

over the place oo in the cover PZ ---> Px.In the case (2.18)(a), SZg,',_s C SZh-x; and since G(Slh_s/C(y)) has order

relatively prime top, this group cannot be transitive in the representationTit).

This contradicts our assumption that S (h, g) is newly reducible, and it is thesecond place where we use that deg(h) is a prime. In case (2.18)(b), sinces2h-x = 0g-x, the characterization of Chebychev polynomials in [Fr, 3, Step 3 ofLemma 9] shows that h is a Chebychev polynomial. We deduce that h and g area composite pair (as in Definition 2.1).

The fact that 2p-1 is a square, under the hypotheses of the theorem, is a resultof [Wie, 1]. The case p = 5 is treated in [Fr, 0, §5.4]. Finally, in [Sco] it is shownthat a group with double degree representation p does not exist for 5 <p < 333.

Problem 2.3. For what integers n do there exist double degree representations(Definition 2.5) of degree n?

From the beginning of this section we have removed from consideration theinspection of S (h, g) in the case that (h, g) is a composite pair of rationalfunctions: there exists m(y), h(2'(y), g(2)(y) E C(y) such that deg(m(y)) > 1,h(y) = m(h(2)(y)), and g(y) = m(g(2)(y)) (as in Definition 2.1). We did thisbecause we were investigating the irreducibility of S (h, g), and if (h, g) is acomposite pair, then 5 (h, g) is "trivially" reducible.

However, the archetypal example in this area actually involved the simplestcase of composite pairs: the case h = g. Therefore we conclude this subsectionwith an inspection of the irreducible components of S (h, h), for h E C(y). Since

S (h, h) is defined to be the fibered product P1 X p, PZ where P' P1 andPZ -41 Ps, , (h, h) contains a "trivial" irreducible component corresponding tothe diagonal A(y, z) C PY'' x r; P. An open subset of S (h, h) is given byht(z) . h2(y) - hl(Y) - h2(z) = 0 where h(y) = hl(y)/h2(y) with hl, h2 E C[Y]Then y - z = 0 corresponds to the locus of A(y, z). The number of irreduciblecomponents of S (h, h) is identified, via the fundamental theorem of Galoistheory, with the orbits of G(Slh_x/C(x)) in the representation Th coming fromthe right cosets of G(S2h_s/C(y)). If h = V)(ht2'(y)), then we automaticallyobtain "extra" components of S (h, h). For many applications we may con-centrate on the case that h is an indecomposable rational function (Definition2.2); or equivalently G(Slh_x/C(x)) is a primitive group. From Riemann'sexistence theorem, in a manner analogous to the previous examples of thissection, the search for indecomposable rational functions h(y) E C(y) for whichS (h, h) has 3 or more components is equivalent to the following problem.

Problem 2.4. Find explicitly the 4-tuples (n, G, T, Q) for which(2.19)(a) T is a primitive, but not doubly transitive, permutation representation of

degree n of the group G;(b) Q = (a(1), ... , a(r)) are generators of G for which a(l) . . a(r) _

Id;

(c) E;_1 ind(T(a(i))) = 2(n - 1).In the case that n is a prime, or when the search is for polynomials h(y) for

which S (h, h) has 3 or more components, Problem 2.4 is completely solved. Wenow describe these results, and remark only that beyond these cases we know

very little. The next lemma combines the famous theorems of Burnside [Bu] andSchur [Sch, 2].

LEMMA 2.3. Let T. G -* S. be a primitive, but not doubly transitive, faithfulrepresentation of the group G. If n is a prime p then G is a proper subgroup of thematrix group

G((Z/ (p))*, p) _ { (afla E (Z/ (p)`), b C Z/ (p) } .

If n is not a prime, then G cannot contain an n-cycle in the representation T.

THEOREM 2.4. Let h(y) C C(y) be such that deg(h) = p, a prime, and 5 (h, h)has at least 3 components. Then the monodromy group G(S2b_z/C(x)) of the cover

hpy, -L Ps is one of the following:

Z/ (p); or G(A, p) _ {(g b)1

aCA,bCZ/(p)}

where A is a subgroup of (Z/(p))'° of order 2, 3, 4, or 6. Further, a description ofh

the branch cycles of the cover Py -* Ps is given by a = (a(l), ... , a(r)) where(2.20)(a) r = 2, a(1) = (o i), a(2) = (o -i): or

(b) r = 3, a(i) _ (°(a b( with a(i) E (Z/(p))* is of order 3, i = 1, 2, 3; or(c) r = 3, a(i) = (a( b(ra) with a(1) of order 2, a(2) of order 3, and a(3) of

order 6; or(d) r = 3, a(i) = (%b(,) with a(1) of order 2, a(2) and a(3) of order 4; or(e) r = 3, a(i) _ oo b(i) with a(1) = a(2) _ -1, a(3) = 1; or(f)r=4,a(i)=(ob('i),i = 1,2,3,4.

OUTLINE OF PROOF. From Lemma 2.3, since (as noted above) the monodromygroup of the cover is a primitive, not doubly transitive group of prime degree,

hthe branch cycles of the cover Py Px are in G((Z/(p))', p). From expression(2.19)(c), 2(p - 1) _ ,_1 ind(a(i)). Let the order of a(i) be e(i). If e(i) is equaltop then ind(a(i)) = p - 1. Otherwise the index of a(i) is easily computed to be((p - 1)/e(i)) (e(i) - 1). If none of the a(i) is of order p, then

2= I,_i(e(i) - 1)/e(i).Combinatorics show that the possible values of e(1), . . . , e(r) correspond to thecases (2.20)(b), (c), (d), and (f). If just one of the o(i)'s is of order p, we have(2.20)(e).

The list given in expression (2.20) needs some further elaboration: details canbe found in [Fr, 2, §2]. The classical cyclic and Chebychev polynomials ofexpression (1.6) correspond to the branch cycles of expression (2.20)(a) and(2.20)(e) respectively.

Our next notation is compatible with that of §O.C. Let L be a discrete(additive) subgroup of C for which E = C/L is compact, and let p E C/L be apoint of order p. Let El = C/<L, a/p> where a E L, and a/p represents p inC/ L. We denote by B(E) a nontrivial subgroup of the analytic group isomor-phisms of E. For "most" E the maximal such group is of order 2, generated by"multiplication by -1" on C. The exceptional cases are E. (resp., En), the elliptic

598 M. FRIED

curve, determined up to isomorphism, with 4 (resp., 6) automorphisms of theanalytic group. We can form the quotient of E by 0(E), denoted E/0(E). Thenthe function field C(E/0(E)) is the fixed field of the action of the group 0(E) onC(E). We may describe C(E/0(E)) quite explicitly. An affine subset of E isrepresented by the equation (in C2) y2 = x' + ax + b. If 0(E) is of order 2,then C(E/0(E)) = C(x): if 0(E) is of order 4, then C(EQ/0(EQ)) = C(x2); if0(E) is of order 3, then C(E,9/0(E,9)) = C(y); and if 0(E) is of order 6, thenC(E,9/0(E,9)) = C(x'). Inn all cases E/0(E) - P', and in all cases we obtainfrom the natural map E - * El a commutative diagram

Epr(E),[,

IT>

E1

,[,pr(E,) (2.21)

E/9(E) - E1/9(E1)

where 0(E) and 0(E1) are taken to have the same order.By using the explicit generators listed above for C(E/0(E)) as uniformizing

variables for copies for P1, the map if is uniquely associated to a rationalfunction h4(y). In the literature, as far as we know, these rational functions havenever received a name. We hope that our next definition violates no traditions.

DEFINITION 2.6. If a rational function h(y) E C(y) corresponds to the map ifin a commutative diagram represented in expression (2.21), then we call h(y) a

1s(h(y))

1Kummer-Ritt function. The branch cycles for cover Py - Ps where h(y) is aKummer-Ritt function are given in (2.20)(b), (c), (d), and (f). Indeed, §O.C is thecomplete theory of Kummer-Ritt functions whose branch cycles are given byexpression (2.20)(f) [Fr, 2, Lemma 2.1].

3. Conclusion of the Schur problem and explicit aspects of Hilbert's theorem.Again, M is a number field, and R its ring of integers. In § I.B we left the Schurproblem at the point where we had discovered that if f E M(y) gives a one-onemap on R/m u (oo) for infinitely many primes in, then

(fl(y) .f2(z) - f1(z) -f2(y))/ (y - z)is a reducible polynomial in the variables y and z where f = fl(y)/f2(y). Fromthis point we assume that

(3.1) degree off is a prime q.

In Theorem 2.4 we have described the branch cycles of the cover P1 Psassociated to such an f. We conclude that f must be a Kummer-Ritt function(Definition 2.6) of degree equal to q. However, how (in the world!) are we todecide for which of these Kummer-Ritt functions we actually obtain pairs (M, f)for which expression (1.5) holds: how do we generalize the Davenport-Friedargument that works for cyclic and Chebychev polynomials? Here is the answer[Fr, 2, Theorem 2.2 for details]! We must find M for which f is defined over M,and M (the algebraic closure of M in M(y1, ... , Sif_s) is different fromM. In the case that the branch cycles are given by expression (2.20)(b) or 2.20(c)(resp., expression (2.20)(d)) we may take M = Q(om) (resp., M = Q(V-_I )),and the result follows immediately for q > 3 from the main theorem of Complex


Multiplication ([ShT, p. 135] or [Sw-D]), since M is generated by the coordinatesof q division points on the corresponding elliptic curves. In the case that thebranch cycles corresponding to the cover given by f correspond to expression(2.20)(f) then the pair (M, f) may be regarded as corresponding to a point on themodular curve of level q through the analysis of §O.C. These curves are definedover Q (a result that goes back to [FriKI] and that has been generalized in manydirections; see [Fr, 0, §9], [ShT], and [Sw-D]). Since by [Maz] these curves havevery few Q-rational points, for a given q > 3 the best result comes from theconcept of arithmetic density as described in Stage IV of §1.A. That is, let Yo(q)be the modular curve of level q. Consider the points P E Yo(q) for which P hascoordinates in M. and p corresponds to a rational function f, for which the pair(Mb, ft,) satisfies conditions above. Then we must show that this set of points P isinfinite. This argument is given in [Fr, 2, Lemma 2.2] as an application of thecurves used in [0] combined with Hilbert's irreducibility theorem.

Of course, there are Kummer-Ritt functions for other integers n (not justwhen n is a prime). If f(y) = f,(f2(y)) with fl, f2 E M(y), and if

`VAR/m U [co)) = R/m U (oo),

then

`Uf(R/m u* too)) = R/m u {oo), i = 1, 2.

Therefore, in our search for rational functions f satisfying condition (1.5) wemay assume that f is indecomposable over M (i.e., G(Of_s/M(x)) is a primitivegroup acting ony1, . . . , Apparently, it is possible thatf(y) may be indecom-posable over M but decomposable over k (i.e., G(Qf_s/M(x)) is not primitiveacting on y1 . yr). The geometric monodromy interpretation of this problemvia branch cycles (parallel to the examples of §2): describe the triples (G, d, Q)where

(3.2)(a) G is a normal subgroup of the primitive subgroup G of S.;(b) G is not primitive (but is transitive);(c) Q = (a(1), ... , a(r)) E is an r-tuple of elements that generates G

and satisfies a(l) a(r) = Id; and(d) _;_1 ind(a(i)) = 2(n - 1).

Condition (3.2)(d) is the Riemann-Hurwitz formula (expression (0.3)) and it

guarantees that there is a rational function f such that the cover P1 -L P's has adescription of its branch cycles given by Q.

Problem 3.1 (Primitivity problem). Describe the triples (G, d, Q) satisfyingexpression (3.2). Then for each such triple (G, G, Q) answer the question: does thereexist a pair (M, f) with f E M(y), Q a description of the branch cycles of the cover

Py - Ps, and G(Of-s/M(x)) ?

Problem 3.2 (Schur problem for rational functions of composite degree). Describethe triples (G, G, Q) with these properties: n is a composite integer;

(3.3)(a) G is a primitive but not doubly transitive subgroup of S.;(b) G is a normal subgroup of G C S. and G/ G is a cyclic group;(c) for G(1) (resp., G(1)) the stabilizer of 1 in G (resp., d) the orbits of d(l)

on 2, .... n break up into strictly smaller orbits under the action of G(1); and

600 M. FRIED

(d) the r-tuple Q E (S,,)' satisfies conditions (3.2)(c) and (d).Then for each such triple (G, G, Q) answer the question: does there exist a pair

J) 1

(M, f) with f E M(y), Q a description of the cycles of the cover P,''P(>

Ps, andG(Stf-s/M(x)) ^' G?

By the way, Problem 3.2 is interesting even with condition (3.3)(d) removed; itleads to the notion of a cover Y --> P' having the Diophantine Covering property[Fr, 2, Proposition 2.1 ].

Also, it is clear (to use a word that crops up a lot in [G]) that one of the goodthings about the prime degree case, given above, is the absence of sporadicgroups. In a manner compatible with the opening comments of this introduction,we are anticipating there being, at most, finitely many triples (G, G, Q) that donot fit into a readily recognizable pattern coming (preferably) from somegeometric situation. Of course there are already well-known rank 3 primitive,but not doubly transitive, groups that must be regarded as sporadic. Indeed, [G,p. 93], some of these figured in the production of sporadic simple groups. Thesemost certainly do not figure in Problem 3.2. In addition, the automorphismgroups of Grassmann varieties over finite fields have not yet been inspected for theireffect on Problem 3.2.

Finally we conclude with specific aspects of Hilbert's irreduciblity theoremapplied to the case where f(x, y) = h(y) - x with h(y) E Q[y] indecomposable(i.e., h cannot be written as a functional composition of two other polynomials,both of degree greater than 1)). From expression (1.10) we easily conclude that

'Ah-.(Z) = `Nh(Z) U (U ` g,(Z)) U

where g1, ... , g, are indecomposable rational functions satisfying expression(1.9)(b), and V is a finite set.

We conclude that g; in expression (3.4) cannot satisfy (1.9)(a) in consequenceof the hypothesis M = Q ([Fr, 6] or the first corollary to the main theorem of[Fr, 0, §9.1]).

defIn particular Rh_s(Z) - `Vh(Z) = Sh(Z) is of logarithmic density. By the way,

Sh(Z) may have exponential density if h is not indecomposable: take h(y) = y4+ y2 and use the factorization of expression (2.10). In addition we couldconsider Sh(R) for R the ring of integers of a general number field M. In thiscase Sh (R) may have exponential density even if h is indecomposable: take h ofdegree 7, 11, 13, 15, 21 or 31 for which we know that there are cases ofreducibility of h(y) - g(z) as in Theorem 2.2 (e.g., [Fr, 0, §5.3] or Birch's bruteforce calculation in expression (2.11)).

The problem of major concern: if h is indecomposable, is Sh(Z) finite?Theorem 2.3 applies immediately, and we conclude that Sh(Z) is finite if thedegree of h is a prime p for which either 2p - 1 is not a square or 5 <p < 333.We do have one case where Sh(Z) is not finite: h is of degree 5. Other than thiswe know of no other indecomposable polynomials h for which Sh(Z) is infinite.In order to go further we must know if there exist other examples of double degreerepresentations as in Problem 2.3.


Actually, if the classification of simple groups does go through as expected atthe Santa Cruz Conference, then there will be a complete classification of groupswith a doubly transitive representation containing an n-cycle (as told to me byFeit and Kantor). With this the last problem will be completely resolved. Again,this is fitting tribute to the role of the classification of simple groups in regard toapplications. More details on this will appear in [Fr, 0]. See also [F, 3, seeTheorem 1.1] and [Wie, 2].

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[Ah] L. Ahlfors, Complex analysis, McGraw-Hill, New York, 1966.[Ar, E, 1] E. Artin, Theorie der Zopfe, Abh. Math. Sem. Hamburg 4 (1925), 47-72.[Ar,E, 2] , Theory of braids, Ann. of Math. (2) 48 (1947), 101-126.[Bo] P. Bohnenbeust, The algebraical braid group, Ann. of Math. (2) 48 (1947), 127-136.[Bom] E. Bombieri, Counting points on curves over finite fields, Sem. Bourbaki 25 (1972/73), No.

430.[Bu] W. Burnside, On simply transitive groups of prime degree, Quart. J. Math. 37 (1906), 215-222.[Ca, 1] J. W. S. Cassels, Factorization of polynomials in several variables, Proc. 15th Scandinavian

Congress, Oslo, 1968, Lecture Notes in Math., no. 118, Springer-Verlag, Berlin and New York, 1970,pp. 1-17.

[Ca, 2] , Diophantine equations with special reference to elliptic curves, J. London Math. Soc.41 (1966), 193-291.

[CaFro] J. W. S. Cassels and A. Frohlich, Algebraic number theory, Thompson, Washington, D. C.,1967.

[Ce] N. Cebotarev, Bestimmung der dichtigkeiteiner menge von primzahlen, welche zu einer gegebensubstitutionsklasse gehoren, Math. Ann. 95 (1926), 191-228.

[Co] S. D. Cohen, Value sets of functions over finite fields, Acta Arith. (to appear).[DSc] H. Davenport and A. Schinzel, Two problems concerning polynomials, J. Refine Angew. Math.

214 (1964), 386-391.[DLSc, 1] H. Davenport, D. J. Lewis and A. Schinzel, Equations of the form f(x) - g(y), Quart. J.

Math. Oxford Set. 2 12 (1961), 304-312.[DISC, 2] , Polynomials of certain special types, Acta Arith. 9 (1964), 107-116.[E] L. Ehrenfucht, Kryterium absolutnez hierokladnosci wieminow, Prace Mat. 2 (1958), 167-169.[F, 1] W. Feit, Automorphisms of symmetric balanced incomplete block design, Math. Z. 118 (1970),

40-49.[F, 2] , On symmetric balanced incomplete block design with doubly transitive automorphism

groups, J. Combinatorial Theory 14 (1973), 221-247.[F, 31 , Some consequences of the classification of finite simple groups, these PROCEEDINGS, pp.

175-181.[Frey] G. Frey, Rational isogonies of prime degree and nonstandard arithmetic (preprint).[Fr, 0] M. Fried, Applications of Riemann's Existence Theorem to algebraic and arithmetic geometry

(in preparation).[Fr, 1] , Fields of definition of function fields and Hurwitz families and groups as Galois

groups, Comm. Algebra 5 (1977), 17-82.[Fr, 2] , Galois groups and complex multiplication, Trans. Amer. Math. Soc. 237 (1978),

141-162.[Fr, 3] On a conjecture of Schur, Michigan Math. J. 17 (1970), 41-55.[Fr, 4] , On a theorem of MacCluer, Acta Arith. 25 (1974), 122-127.[Fr, 5] , On Hilbert's irreducibility theorem, J. Number Theory 100 (1974), 211-232.[Fr, 6] , The field of definition of function fields and a problem in the reducibility of

polynomials, Illinois J. Math. 17 (1973), 128-146.[Fr, 7] , On the diophantine equation f(y) - x = 0, Acta Arith. 19 (1971), 79-87.[Fr, 8] , On a theorem of Ritt, J. Reine Angew. Math. 224 (1974), 40-55.[Fr, 9] (with R. Biggers), Relations between moduli spaces of covers of P' and representations

of the Hurwitz monodromy group (preprint).

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[FrS] M. Fried and G. Sacerdote, Solving Diophantine problems over: all residue class fields of anumber field, and all finite fields, Ann. of Math. (2) 104(1976),203-233.

[FrSc] M. Fried and A. Schinzel, Reducibility of quadrinomials, Acta Arith. 21(1972), 153-171.[FrSm] M. Fried and J. A. Smith, Primitive groups, Moore graphs and rational curves, Michigan

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Amer. Math. Soc. (N.S.) 1 (1979),43-199.[H] M. Hall, Jr., The theory of groups, MacMillan, New York, 1963.[HI] D. Hilbert, Uber die Irreduzibilitat ganzer rationaler Functionen mit ganz zahligen Koeffizien-

ten, J. Reine. Angew. Math. 110 (1892), 104-129.[Kan] W. Kantor, 2-transitive designs, Part 3: Combinatorial group theory, Proc. Advanced Study

on Combinatorics at Nijenrode Castle, Breukelen, Netherlands, July 8-20, 1974, MathematischCentrum, Amsterdam, 1974, pp. 44-98.

[FriK1] F. Klein and R. Fricke, Vorlesungen Uber die theorie der modulfunctionen II, Leipzig, 1892.[Lev] W. LeVeque, On the equation y' = f(x), Acta Arith 9 (1964), 209-219.[Me] C. MacCluer, On a conjecture of Davenport and Lewis concerning exceptional polynomials,

Acta Arith. 12 (1967), 289-299.[Maz] B. Mazur, Rational points on modular curves, Modular Functions of One Variable V, (Bonn,

1976), Lecture Notes in Math., vol. 601, Springer-Verlag, Berlin and New York, 1977, pp. 107-148.[NI] J. Nielsen, Untersuchungen zur Topologie der geschlossenen Zweiseitigen Flachen. I-III, Acta

Math. 50 (1927), 189-358; 53 (1927), 1-76; 58 (1931), 87-167.[0] A. P. Ogg, Rational points of finite order on elliptic curves, Invent. Math. 12 (1971), 105-111.[Sc, 1] A. Schinzel, Reducibility of polynomials of the form f(x) - g(y), Colloq. Math. 18 (1967),

213-218.[Sc, 2] , Some unsolved problems, Mat. Bibl. 25 (1963), 63-70.[Sc, 3] , Reducibility of polynomials, Actes Internat. Congr. Math. 1970, Vol. 1, pp.

491-496, Gauthier-Villars, Paris, 1971.[Sch, 1] I. Schur, Uber den zussamenhang zwischen einem problem der zahlentheorie and linear satz

Uber algebraische Functionen. S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. (1923), 123-134.[Sch, 2] , Zur Theorie der einfach transiliven Permutations Gruppen, S.-B. Preuss. Akad.

Wiss. Phys.-Math. Kl. (1933), 598-623.[Sco] L. Scott, Uniprimitive permutation group, Theory of Finite Groups, a symposium at Harvard

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Brook Sympos. Number Theory, Summer 1969, Amer. Math. Soc., Providence, R. I., 1971, pp. 1-52.[Tv] H. Tverberg, A study in the irreducibility of polynomials, Dept of Math., Univ. of Bergen, 1968.[Wa] Wagner, On collinealion groups of projective spaces. I, Math. Z. 76 (1961), 411-462.[W] A. Weil, Sur les courbes et les varietes qui s'endeduisent, Hermann, Paris, 1948.[Wie, 1] H. Wielandt, Primitive permutation gruppen Grad 2p. II, Math. Z. 63 (1956).[Wie, 2] , Permutation groups through invariant relations and invariant functions, Lectures

given at the Ohio State University, Columbus, Ohio, February 25-April 3, 1969.

UNIVERSITY OF CALIFORNIA, IRVINE


BURNSIDE RING OF A GALOIS GROUP ANDTHE RELATIONS BETWEEN

ZETA FUNCTIONS OF INTERMEDIATEFIELDS

D. HUSEMOLLER

Let K/F be a Galois extension of number fields with Galois group G =Gal(K/F). The question of the relation between the Zeta functions of inter-mediate fields between F and K can be solved as a purely group theoreticalproblem concerning the mapping of the Burnside ring to the representation ringof the finite group G. The purpose of this note is to survey this relation and givesome illustrations in particular cases. The method is an application of the resultsin the beautiful paper of E. Artin [3] where he introduced the nonabelianL-functions now called Artin L-functions. Zeta functions of intermediate fieldsassociate a holomorphic function to each element of the Burnside ring and ArtinL-functions associate a holomorphic function to each element of the representa-tion ring in such a way that it is compatible with the natural homomorphismfrom the Burnside ring into the representation ring. Since Artin showed that theL-functions associated with the set of irreducible representations of G =Gal(K/Q) are analytically independent, all analytic relations between Zetafunctions of intermediate fields are seen in the kernel of the natural map fromthe Burnside ring to the representation ring. We begin by outlining the basicproperties of Artin L-functions and then carry out the above program.

1. Artin L-functions. Let K/F be a Galois extension of global fields withGalois group G = Gal(K/F). To each prime v of F we have its absolute normNv which is a power of the residue class characteristic at the prime v. For aprime w of K over v or dividing v of F the decomposition group G. of w is thesubgroup of s in G such that ws = w and the inertia subgroup I. of G. consistsof all s with s(a) = a (mod MW). The Frobenius element Fr. is in G. anddefined modulo I. by the relation

Fr,(a) = a' (mod MW)

1980 Mathematics Subject Classification. Primary 12A70, 10H10, 14G10.C American Mathematical Society 1980

603

604 D. HUSEMOLLER

for any a in the integers of K. Any other prime w' over v is of the form w' = wsfor some s in G, see Lang [11, p. 18]. Then G.. = s - f GW s, IW = s-'7 s, andFr.. = s-1Fr,s for w' = ws.

Let p: G - GL(V) be a finite dimensional complex representation of Gdenoted by either V or p. Let Vi'- be the subvector space of V fixed by theinertia subgroup I., and let p(Fr,,)l V- be the restriction of the image of theFrobenius element to this subspace. Since I. is a normal subgroup of G. andFr. is well defined modulo IW, this linear transformation is well defined. Ifw' = ws is another prime of K dividing v, then the action of p(s) on the pair(Vi'-, p(Frj V'-) is an isomorphism

(Vi", p(Fr,) I V k) - (V k', p(Frw)l

and we denote this vector space and linear transformation up to isomorphism by(V'', p(Fr,,)). Further we can form a local Euler factor

det(I - (Nv)-5p(Fr,,))

which is (Nv)-' substituted into the polynomial det(I - tp(Fr,,)) of t where thedeterminant is of a linear transformation of the finite dimensional space V' intoitself. The product over all primes v of these local Euler factors is a holomorphicfunction in the right half plane Re(s) > 1.

(1.1) DEFINITION. The Artin L-function L(s, V; K/F), or for short simplyL(s, V), associated with the complex representation p is

L(s, V; K/F) = 11 det(I -v prime of F

If z1, ... , zf are the roots of the polynomial equation

0 = det(t -

then for the character X or Xv of V or the representation p, we havep(Frv)=zi"+... +zf

and

f- log det(1 - tp(Frv)) log(l - tzi)

i=1

f (zi)m », = X(FI')_ E E t tj=1 1<m m 1<m m

Thus we have the formula for the logarithm of an Artin L-function

log L(s, V) _ E X(Frv

Fr- m(Nv)'(1.2) THEOREM. Let V1, ... , V, denote the r distinct irreducible representations

of G with characters X1, ... , x, Then there are no multiplicative analytic relationsbetween the Artin L-functions L(s, V1), ... , L(s, V,) as holomorphic functions inthe right half plane Re(s) > 1.

BURNSIDE RING OF A GALOIS GROUP 605

PROOF. Form the product L(s, V1)', ... L(s, V,)`- where c1, ... , c, are com-plex numbers. If this product is 1, /then the logarithm is zero and

0 = ck log L(s, Vk) = E Ck XklFr'")

k=1 k-l Fr" m(NV)

E CkXk(Fr.) 1

Fr^ k-1 m(Nv)-Hence Ek

1ck7'(s) = 0 for each s in G by the Cebotarev's density theorem, see

Serre [16, pp. 1-7, 8]. By the orthogonality of the irreducible characters all thecoefficients ck = 0, see Serre [15 and deuxieme edition, p. 28]. This proves thetheorem.

(1.3) THEOREM. The Artin L -function L(s, V; K/F) has the following formalproperties.

(1) (Additivity) L(s, V1 (D V2; K/F) = L(s, V1; K/F). L(s, V2; K/F).(2) For the trivial representation 1 of G = Gal(K/ F)

L(s, 1; K/F) = F(s),the Dedekind Zeta function of the global field F.

(3) For K'/F a sub-Galois extension of K/F with Galois group G' _Gal(K'/F) a quotient group of G = Gal(K/F), a representation V' of G' definesa representation V of G on the same vector space and

L(s, V; K/F) = L(s, V'; K'/F).(4) Let L be the intermediate field of K over F fixed by the subgroup

H c G = Gal(K/F) so that H = Gal(K/L). If W is a representation of H, thenform the induced representation IndH( W) = V of G and

L(s, W; K/L) = L(s, IndH(W); K/F).PROOF. Statements (1), (2), and (3) follow directly from the definition of the

L-function. Assertion (4) is straightforward but it involves keeping careful trackof the relations between primes in the three fields F, L, and K, see Artin [3, Satz1, pp. 93-96] and Lang [11, Chapter XIII, §3].

(1.4) REMARKS. As special cases of induction (1.3)(4) we take W = 1. ThenIndH(1) = C[G/H] is the permutation representation, denoted rGIH, and

L(s) = L(s, 1; K/L)k

= L(s, rG,H; K/F) _ II L(s, V;; K/F)"")t-t

where rG1H = Vt (1) ® ... ®J/ ') is the decomposition of rG1H into irreduciblerepresentations of G.

Further take W = 1 and H = 1 or L = K. Then Indi(l) = C[G] is theregular representation rG of G and

OK(s) = L(s, 1; K/K) = L(s, rG; K/F)

= 5F(S) II L(s, X; K/F)d (xlx irreducible

where d°(x) = x(l) is the degree of X.

606 D. HUSEMOLLER

2. Meromorphic continuation of Artin L-functions and properties of complexrepresentations. For a 1-dimensional representation V of G the action of G isgiven by a linear character A: G -- C*. The Artin L-function L(s, V; K/ F), alsodenoted by L(s, A; K/ F), is equal to the abelian L-function associated with a(linear) character $ on the ideal class group where ¢ is related to A by thereciprocity mapping of class field theory. From this identification we deducethat for d°(a) = 1 the holomorphic function L(s, A; K/ F) defined on Re(s) > Iextends to a meromorphic function of C and satisfies a suitable functionalequation. Further, if A I. then L(s, A; K/F) is holomorphic on C and theZeta function F(s) = L(s, 1; K/ F) only has poles, see Hecke [10], Artin [4],and Lang [11, Chapter XI1, §2].

The attempt to extend these results to nonabelian L-functions by Artin andBrauer had a fundamental influence on the development of the general theory ofcomplex representations of a finite group.

(2.1) ARTIN's THEOREM. For every representation V of a finite group G thereexists one dimensional representations L,, ... , L, of cyclic subgroups C1, ... , Cro f G and integers m, n(l), ... , n(r) such that V"' and

®Ind° (L,)"(') are equal in R(G).

From this theorem with (1.3) we deduce that L(s, V; K/ F)' is a meromor-phic function L(s, L1; K/K1)"(l) ... L(s, Lr; K/Kr)"(') defined on C where Kj isthe subfield of elements of K fixed by Cj. This does not give the meromorphiccontinuation of the Artin L-function to the plane because it does not rule outthe possibility of multivalued continuation.

(2.2) BRAUER's THEOREM. For every representation V of a finite group G thereexist one dimensional representation L ... , L, of subgroups El, . . . , E, of G andintegers n(1), . . . , n(r) such that V and Ind (L1)"(l1 ® ®IndE(L)'(' areequal in R(G).

In this theorem, the subgroups E. can be chosen to be elementary, that is, aproduct of a cyclic group and a p-group. Applying (1.3) again as with Artin'stheorem, we can now deduce the meromorphic continuation of the ArtinL-functions, see Brauer [6].

(2.3) THEOREM. Let V be a representation of the Galois group G of a Galoisextension K/F of global fields. Then L(s, V; K/F) prolongs to a meromorphicfunction on the entire complex plane.

For a proof of the Artin and Brauer theorems, see Serre [15, deuxieme edition,§§9 and 10]. Now if the coefficients n(l), ... , n(r) in the Brauer theorem werealways positive, then we would be able to prove more about L(s, V; K/F)namely that it had a holomorphic continuation to the entire plane when the onedimensional representations L1, ... , L,. are nontrivial. This condition is equiva-lent to the assertion that V does no have the trivial representation as aconstituent by Frobenius reciprocity. This leads to one of the fundamentalconjectures in number theory.

(2.4) ARTIN CONJECTURE. Let K/ F be a Galois extension of global fields withGalois group G = Gal(K/F). If V is a finite dimensional complex irreducible


representation of G different from the trivial representation, then L(s, V; K/F)has a holomorphic continuation to the entire complex plane C.

The assertion of the conjecture could be stated equivalently for representa-tions not containing the trivial representation in its direct sum decomposition.

(2.5) REMARK. The Artin conjecture was established by Weil for the functionfield case, see [17]. Until recently, the conjecture was known in the number fieldcase only for obvious cases like representations induced from one dimensionalrepresentations. In [12] Langlands established the Artin conjecture for tetra-hedral and certain octahedral representations of dimension 2 over the rationalnumbers.

If the Artin conjecture is true, then for any extension of global fields L/F thequotient of Zeta functions is a holomorphic function on the entireplane. This is seen by embedding L into a global K such that K/ F is a Galoisextension with G = Gal(K/ F) and H = Gal(K/ L) c G. By (1.4) the inducedrepresentation rGIH = IndH(1) = V"0) ® ®V; (') where we add now thatthe trivial representation 1 = Vj for exactly( one j and n(j) = 1 in this case. Thus

L(s)/JF(s) = L(s, rG/H)/L(s, V) = II L(s, V,)n(r)

is a holomorphic function.An alternate argument gives, see also Serre [15, deuxieme edition, p. 89].

(2.6) THEOREM (ARAMATA [2]). For a Galois extension K/ F of global fields thequotient is a holomorphic function on the plane.

PROOF. Let G = Gal(K/ F) and n = # G = dimF K. The difference rG - 1 =IndG(l) - 1 in R(G) can be studied using the relation

n(rG Indc(Xc)C cyclic subgroup of G

where k = 4(c)rc - Oc in R(C). The natural number 0(c) is the Euler functionof c = # C; it equals the number of generators of C. The class function Oc isdefined by Oc(x) = # C if x generates C and zero otherwise. Using the lemman = >2c in G Indc(Oc), one can prove that 9c is in R(C) and that Xc is a characternot containing the trivial character in its decomposition.

From (1.3) we deduce that

( rK(s)/ rF(s))n = 11 L(s, Xc; K/ Kc)C

where Kc is the subfield of K fixed by C. Thus the nth power of the quotient isholomorphic, and hence, the meromorphic quotient is holomorphicon the plane.

3. The Burnside ring and representation ring. To define the Burnside ringA(G), we consider first the semiring of finite G-sets X up to isomorphism withaddition given by X + Y = X u Y the disjoint union or coproduct G-set andmultiplication XY = X X Y the product G-set. We form the associated ringA(G) of formal differences of isomorphism classes of G-sets.

(3.1) DEFINITION. The ring A(G) is the Burnside ring.

608 D. HUSEMOLLER

Note that A(G) is just the Grothendieck ring on the isomorphism classes ofcertain objects as is the representation ring R(G) the Grothendieck ring associa-ted with finite dimensional complex representations under direct sum and tensorproduct.

(3.2) REMARKS. The abelian group A(G) is free abelian with a basis of G-setsG/H as H runs over representatives of conjugacy classes of subgroups of G.Similarly R(G) is free abelian with a basis consisting of the irreducible represen-tations. There is a natural ring homomorphism

0: A(G) - R(G)defined by 0(class of G/H) = class of the representation C[G/H] = rGIH

Let M denote the multiplicative group of holomorphic functions defined forall s with Re(s) > 1 without zeros. For example, the Zeta functions andL-functions considered in the previous section are elements of this group.

(3.3) REMARKS. Let K/F be a Galois extension of global fields with G =Gal(K/ F). We have a commutative diagram

A(G)9

>R(G)

"M

where '(class of G/H) = L(s) where L = K" is the subfield of K fixed by thesubgroup H and X(class of V) = L(s, V; K/F). By (1.4) we have

'(class of G/H) = L(s) = L(s, rGlH; K/F) = X(O(class of G/H))

which establishes the commutativity of the diagram.Since the irreducible representations form a basis of R(G) as an abelian

group, the theorem on analytic independence (1.2) implies, in particular, that themorphism X is injective. Thus ker(O) = and this yields the followinginterpretation in terms of relations between Zeta functions.

(3.4) THEOREM. Let K/F be a Galois extension of global fields with G =Gal(K/ F). Let K,, .. . , K, be intermediate fields which are the fixed fieldsassociated to subgroups Hl, ... , H, of G. Then the relation

JK(S)n") ... (s)n(r) = 1

between Zeta functions holds if and only if the relation

n(l) IndH (1) + +n(r)IndH(1) = 0

holds in the representation ring R(G).

Now we consider examples of relations between Zeta functions. For a cyclicgroup G the natural map A(C) - R(C) is injective and these are the only finitegroups for which this map is injective.

(3.5) EXAMPLE. Let K/F be a Galois extension of degree 4 with G =Gal(K/F) equal to the 4 group. Then rk A(G) = 5 and rk R(G) = 4 and thekernel is of rank 1. Let H, H', and H" be the three subgroups of two elements inG fixing the quadratic extensions L, L', and L" of F. Then

rG/H + rG,H, + rG,H = rG + 2


in R(G) and (3.4) yields the relation between the five Zeta functions

(3.6) EXAMPLE. Let K/F be a Galois extension of degree 6 with G =Gal(K/ F) equal the symmetric group on 3 letters. Let H, H', and H" be thesubgroups of 2 elements and C the cyclic subgroup of order 3. Let L, L', and L"be the cubic extensions of F in K which are conjugate and let M be thequadratic extension of F in K fixed by C. Then G/H, G/H', and G/H" are allisomorphic G-sets, and A(G) has rank 4. The representation ring R(G) has rank3 generated by the identity representation 1, the sign representation sgn, and a 2dimensional rational representation W. The relation in R(G)

rGIC+rGIH+rGIH=(1 +sgn)+(1 + W)+(1+ W)=r.+2

yields by (3.4) the relation between Zeta functions

JM(S)JL(s)2 =

and because the subgroups H, H', and H" are conjugate so are the fixed fieldsand the Zeta functions are equal IL(s) = L,(s) _ L,(s).

To the exercises 6.3 and 6.4 on pp. 362-363 of Cassels and Frohlich [8] thereis an example of two nonisomorphic fields E and E' over the rational numberswith the same Zeta functions. This example is due to F. Gassmann [9], and theyare contained in a Galois extension of the rational numbers with the symmetricgroup on 6 letters as Galois group.

(3.7) EXAMPLE (PERLIS [14]). The simple group G of order 168 is the Galoisgroup of an extension L/Q. The group G is isomorphic to both PSL2(F7) andPSL3(F2) = GL3(F2). If H is the stabilizer subgroup of a point P: (a, b, c) inP2(F2), and if H' is the stabilizer subgroup of the line L: aw + bx + cy = 0 inP2(F2), then A HA is a bijection H - H'. The subgroups H and H' arenonconjugate, of index 7 in G, and IndH(1) = IndH,(l) since the transposeinterchanges conjugacy classes.

If K is the field generated by a root of f(X) = X7 - 7x + 3, and if K' is thefield generated by a root of f'(X) = X 7 + 14X4 - 42X2 -21X+9, then Lcan be chosen to be a normal closure of K and K' such that H (resp. H') is thesubgroup of all s in G fixing the field K (resp. K'). Now K and K' arenonisomorphic since H and H' are not conjugate, but K(s) = K,(s) since GI Hand G/H' are equal when mapped from A(G) into R(G).

REFERENCES

1. H. Aramata, Uber die Teilbarkeit der Zetafunktionen gewisser algebraischer Zahlkorper, Proc.Imp. Acad. Tokyo 7 (1931), 334-336.

2. , Uber die Teilbarkeit der Dedekindschen Zetafunktionen, Proc. Imp. Acad. Tokyo 9(1933), 31-34.

3. E. Artin, Uber eine neune Art von L-Reihen, Abh. Math. Sem. Univ. Hamburg (1923), 89-108.4. , Beweis des allgemeinen Reziprozitatsgesetzes, Abh. Math. Sem. Univ. Hamburg 5

(1927), 353-363.5. , Zur Theorie der L-Reihen mil allgemeinen Gruppencharakteren, Abh. Math. Sem. Univ.

Hamburg 7 (1929), 292-306.

610 D. HUSEMOLLER

6. R. Brauer, On the Zeta-functions of algebraic number fields, Amer. J. Math. 119 (1947),243-250.

7. , On Artin's L-series with general group character, Ann. of Math. (2) 48 (1947), 502-514.8. J. W. S. Cassels and A. Frohlich, Algebraic number theory, Thompson, Washington, D. C.,

1967.9. F. Gassmann, Uber Beziehunge zwischen den Primidealen eines algebraischen K&-pers and den

Substitutione seiner Gruppen, Math. Z. 25 (1926), 661-675.10. E. Hecke, Algebraische Zahlen, Akademische Verlagsgesellschaft M. B. H., Leipzig, 1923, and

Chelsea, New York, 1948.11. S. Lang, Algebraic number theory, Addison-Wesley, Reading, Mass., 1970.12. R. P. Langlands, Base change for GL(2), the theory of Saito-Shintani with applications, Notes,

Inst. for Advanced Study, Princeton, N. J., 1975.13. J. Martinet, Character theory and Artin L-functions, Symposium on L-Functions and Galois

Properties (Univ. of Durham, 1975), A. Frohlich (ed.), Academic Press, New York, 1977.14. R. Perlis, On the equation rK(s) _ rK'(s), J. Number Theory 9 (1977), 342-360.15. J.-P. Serre, Representations lineaires des groupes finis, Herman, Paris, 1967, deuxieme edition,

1971.16. , Abelian l-adic representations and elliptic culues, Benjamin, New York, 1968.17. A. Weil, Sur les courbes algebriques et les varietes qui s'en deduisent, Publ. Inst. Math.

Strasbourg (1945), Hermann, Paris, 1948.

HAVERPORD COLLEGE, PENNSYLVANIA


FINITE AUTOMORPHISM GROUPS OFALGEBRAIC VARIETIES

D. HUSEMOLLER

The study of automorphism groups of algebraic varieties lies on the interfacebetween algebraic geometry and several other branches of mathematics. Finite-ness theorems are proved using algebraic groups or analytic methods, bounds onthe order of automorphism groups of curves are obtained by topology, andexamples of curves with large automorphism groups come from the theory ofmodular varieties. The classical work on this subject goes back to the 19thcentury with Hurwitz's bound on the order of an automorphism group of acurve of genus at least 2. The finiteness of this group goes back to Poincare,Klein, and Weierstrass together with Hurwitz.

From the point of view of the theory of finite groups there has been anunderstanding that finite simple groups should arise as the automorphismgroups of algebraic varieties. The determination of the finite subgroups of thelinear groups in low dimensions is the principal application of group theory tothis study at this time.

PART I. FINITENESS THEOREM FOR AUTOMORPHISM GROUPS OF VARIETIES

Using methods from the theory of algebraic groups, we show that a smoothalgebraic variety of general type has a finite automorphism group. Included inthis class of varieties of general type are algebraic curves (or Riemann surfaces)of genus ) 2 and smooth hypersurfaces in Pr+1 of degree > r + 2. A complexKahler manifold version of this result is also sketched. The local cross sectiontheorem of Rosenlicht is the essential result from the theory of algebraic groupswhich is used together with the observation that an algebraic group or compactLie group with zero connected component of the identity is a finite group.

1. Birational transformations and the cross section theorem. Varieties aresubsets of projective space P(k) defined as the locus of zeros of homogeneouspolynomial equations together with the Zariski topology where the closedsubsets are exactly the subvarieties. We will only be interested in irreduciblevarieties V, that is, varieties V where V = V, U V2 is a union of two open sets

1980 Mathematics Subject Classification. Primary 14H20, 14H30, 14H45.e, American Mathematical Society 1980

611

612 D. HUSEMOLLER

only if V = Vi or V = V2. Equivalently, irreducible varieties have the definingproperty that two closed sets have empty intersection only if one set is alreadyempty.

With respect to the Zariski topology, each variety V has a structure sheaf (S ofrings where the stalk OX at x in V is the ring of germs of regular functions on Vnear x. This sheaf embeds in a sheaf of germs of rational functions 6x on Vwhere OR is the field of fractions of the ring (9s. The global sections H°(V, (9)= k, the constants, and H°(V, Oil,) = k(V) the ring of rational functions on thevariety V.

(1.1) REMARK. On an irreducible variety every nonempty open set is dense inthe whole variety. Also, the ring k(V) of rational functions is a field. Hence-forth, we will use the term variety to mean irreducible projective variety over kalgebraically closed.

(1.2) DEFINITION. A rational map f: V - W is a morphism V 3 V' W' CW defined on nonempty open subsets V' and W' of the respective varieties. Arational map f is called a birational map provided V' and W' can be chosensuch that f: V' - W' is an isomorphism.

Strictly speaking, a rational map f: V - W is an equivalence class of mor-phisms defined on nonempty open sets where two morphisms V'--+ W' andV" -p W" defining f are equivalent when equal on V' n V" - W' n W".Conditions for the composition of rational maps to be defined can be given andbirational maps are isomorphisms with respect to this composition. Compositionis always defined for birational maps. For a complete discussion see Demazure[3] where a rational map is called a pseudomorphism.

(1.3) REMARK. For a rational map f: V - W with dense image and a rationalfunction u in k(W) the composite of is defined on an open dense set of Vdefining an element, denoted f°(u) = uf, in k(V). In this way f determines ak-monomorphism

f °: k(W) --+ k(V)of fields defined over k. Conversely, each such k-monomorphism h: k(W) --->k(V) is of the form h = f° for a rational map with dense image. Further, f is abirational map if and only if h is an isomorphism. If Bir(V) denotes the group ofbirational maps V - V then we have a group isomorphism Bir(V) -Gal(k( V)/ k) given by f H f°.

For a variety V we use the notation Vns for the open subset of smooth(nonsingular or simple) points.

(1.4) REMARK. For two curves X and Y a birational f: X -f Y restricts to anisomorphism f: Xns- Yns. For a smooth curve X the group Bir(X) = Aut(X)the automorphism group of X. In general Aut(V) is a subgroup of Bir(V).

Using Rosenlicht's local cross section theorem, we have the following criterionfor a connected linear algebraic group to act nontrivially on a variety biration-ally equivalent to a given variety.

(1.5) PROPOSITION. The following are equivalent for a variety V.(1) V is birationally equivalent to a product variety W X Pl.(2) A connected 1-dimensional linear algebraic group T acts nontrivially on some

V' birationally equivalent to V.(3) A connected linear algebraic group G acts nontrivially on some V' biration-

ally equivalent to V.

AUTOMORPHISM GROUPS OF ALGEBRAIC VARIETIES 613

PROOF. Since PGL2 acts nontrivially on W X P1 statement (1) implies (2) andclearly (2) implies (3). To show that (3) implies (2), we choose a Borel subgroupB of G which acts nontrivially on V. This is possible because the conjugates ofany Borel subgroup cover G, and then we filter B by connected subgroups B.where dim(B) = i

0= B° C B 1 c B Z C C B,,, = B

and B; _ 1 is normal in B;. Let j have the property that Bj acts trivially on Vwhile Bi does not. Then T = B, / Bj _ 1 satisfies (2).

The implication (2) implies (1) is Rosenlicht's local cross section theorem, seeRosenlicht [15].

2. Canonical dimension of a variety.(2.1) DEFINITION. Let V be an r dimensional variety over an algebraically

closed field k. The canonical ring R( V) of a variety V is 14°,,, H°( V, wy®) wherethe ring structure comes from the tensor product of sections of powers of theline bundle wy = SE of differential r-forms. The canonical dimension K(V) of Vis the transcendence degree of R(V) over the ground field k minus 1.

(2.2) REMARKS. The inequality -1 = K(V) = dim(V) holds. If V and V" aretwo birationally equivalent varieties, then K( V) = K( V'). For further properties,see Bombieri and Husemoller, [1, pp. 360-363].

(2.3) EXAMPLE. If V is birationally equivalent to W X P1, then K(V) _ -1.From the Kiinneth formula

H°(X X Y, (S2z)m® ® (S2Y)m®) = H°(X, (S2'x)m®) ® H°(Y, ( y)m®),

it follows that H°(W X P1 (S2')m®) = 0 for all m > 1. This means that R(V) _k and K(V) = -1. In fact, for a variety V, the canonical dimension K(V) = -1 ifand only if H°(V, wy ®) = 0 for all m > 1, or in geometric language I mKvI = 0form> 1.

From (2.3) and (1.5) we deduce immediately.

(2.4) PROPOSITION. Let V be a nonsingular projective variety. If K( V) > 0, thenBir(V) contains no connected linear algebraic group, in the sense that there is anontrivial action of a connected algebraic group 1 on a birationally equivalentvariety.

For curves (or Riemann surfaces when over C) we have K(C) = -1 if and onlyif C = P1, K(C) = 0 if and only if C is an elliptic curve, and K(C) = 1 if andonly if the genus q(C) > 2, or equivalently, the Euler number e(C) < 0.

3. Finiteness of the birational automorphism group of a variety of general type.(3.1) DEFINITION. An r dimensional variety V is of general type provided

K(V) = r.A variety V is of general type if and only if there exists an n such that the map

associated with the linear system mKvJ defined V -p V CProj H°(V, wy®) is a birational map.

(3.2) THEOREM. Let V be a variety of general type. Then the group Bir( V) of allbirational transformations of V onto itself is a finite group.

614 D. HUSEMOLLER

PROOF. Choose n such that 01"KI: V---,. V C PH°(V, wy®). Then each sin Bir(V) defines s' in PGL(H°(V, wy®)) preserving V and inducing s' inBir( V"). The homomorphism s H s' is an injection. Now Bir( V") is an algebraicsubgroup of PGL, and by (2.4) its connected component of the identity Bir( V")'= 1. Thus Bir(V") and hence also Bir(V) are finite groups. This proves thetheorem.

(3.3) REMARK. The above theorem applies to curves of genus at least 2 andhypersurfaces of degree at least n + 1 in P. Surfaces X for which KX > 0 andKX.C > 0 for each curve C on X are exactly the surfaces of general type andthus, Bir(X) is finite, see Bombieri and Husemoller, [1, p. 372, (5.4)].

For further remarks see the paper of Matsumura [11].

4. Finiteness theorems by analytic methods. Finiteness of automorphismgroups in the transcendental case is proved by showing a group is both compact,when it preserves a norm, and discrete, when it preserves a lattice. We canintroduce a norm on 0 in H°(X, w;®) for X a compact Kahler manifold by

IIBIIZ= f(BAB)'

X

The image of Aut(X) in Aut(H°(X, w;®)) preserves this norm, and hence, hascompact closure.

Now assume that X has dimension r and that n = 1. Consider the morphismAut(X) - Aut(H'(X, Z)) given by the induced mapping on integral cohomol-ogy. The image is a discrete group in Aut(H'(X, R)) and in Aut(H'(X, Q.Using Hodge theory where H°(X, ox) is a summand of H'(X, C) since wX =SE , we can show that the image is both discrete and compact, hence finite. Thecase n > 1 is reduced to n = 1 by a covering space argument.

(4.1) REMARK. The question of when Aut(X) operates faithfully on H'(X, Z)for an r dimensional variety is an important problem which is settled for curvesand some surfaces. If X is a curve of genus at least 2, then an application of theLefschetz fixed point theorem shows that the action is faithful. For K3-surfacesBurns and Rapoport [2] and Enriques surfaces Ueno [19] have shown thatAut(X) acts faithfully on HZ(X, Z). Peters has studied surfaces of general type,in [13], where the action is faithful.

PART II. BOUNDS ON THE ORDERS OF AUTOMORPHISM GROUPS AND EXAMPLES

Using the classical Riemann-Hurwitz relation, we derive Hurwitz's bound#Aut(X) < 84(q(X) - 1) for the order of an automorphism group of a curve Xor Riemann surface. This bound holds only if the order of Aut(X) is prime tothe ground field characteristic. In characteristic p larger bounds for the order ofthe automorphism group are stated. We consider some examples where theHurwitz bound is best possible and some genera where the bound is not bestpossible by giving bounds on the order of the automorphism group of ahyperelliptic curve.

Finally the question of automorphism groups of surfaces is taken up. Unlikewith curves it is possible for the automorphism group to be an infinite discretegroup for a surface.

5. The Riemann-Hurwitz formula. We call vr: X - Y a ramified covering whenIT is either an analytic map of compact Riemann surfaces or a finite morphism


of algebraic curves. In either case the degree of is defined to be [k(X) : k(Y)]where k(X) is the field of rational functions, and for each x in X a ramificationindex ex where is locally of the form 7r(t) = I- in terms of a local parameter t.

(5.1) THEOREM (RIEMANN-HURwITZ). The following relation holds between theEuler characteristics for a ramified covering fr: X -p Y where n and each ex isprime to the characteristic of k

e(X) = n.e(Y) - (ex - 1).x in X

The last term is the formula makes sense since ex = 1 except for finitely manyx in X. The simplest proof of the formula comes by examining a triangulation ofY where the vertices include all y = vr(x) where ex > 1 and then lifting thetriangulation to X. Since e = V - E + F, a direct count of vertices, edges, andfaces on Y and on X gives the relation. A proof using coherent sheaf cohomol-ogy follows easily by comparing ff,((9x) and (9; and using e(X) = 2X((9x), seeHartshorne [5, Chapter IV, §2]. For further discussion and a guide to thehistorical background see Kleiman [9, p. 300].

(5.2) REMARK. A ramified covering ff: X -p Y is called regular provided ex isconstant on each fibre 7r-1(y), and ey is defined to be ex where fr(x) = y. In thiscase eyIn and the quotient n/ey is the number of points in the fibre ir-1(y). If n isprime to the ground field characteristic, then the Riemann-Hurwitz formulabecomes

e(X) = n.e(Y) - n. E (i_1).yin Y ey

The ramified covering resulting from a finite group action of G on X given bythe projection X -p G \ X is regular and ex is the order of the isotropy subgroupGx at x.

In (5.1), if X is the Riemann sphere with e(X) = 2, then e(Y) is strictlypositive or the right-hand side of the formula would be negative, and so Y isagain the Riemann sphere.

(5.3) EXAMPLE. Let G be a finite group of order n acting on X the Riemannsphere and the form the quotient G \ X = Y again isomorphic to the Riemannsphere. If e1, . . . , e, are the orders of ramification overy1, ... , y, in Y, then

2=2n-nf I1- 11+... +II - 111LL \ e l

Jl e,Jl

JJ

Dividing by n and reorganizing terms, we obtain

(1 +... + 11-r+2=2.e1 e, n

Since each I/ ei > 1, we have only the following solutions.r = 2: (e1, e2) = (n, n) which corresponds to G cyclic of order n,r = 3: (e1, e2, e3) = (2, 2, n) which corresponds to G dihedral of order 2n,(e1, e2, e3) _ (2, 3, 3) which corresponds to G = A4 of order 12,(e1, e2, e3) _ (2, 3, 4) which corresponds to G = S4 of order 24,(e1, e2, e3) _ (2, 3, 5) which corresponds to G = A5 of order 60.

616 D. HUSEMOLLER

The three exceptional groups are exactly the automorphism groups of the fiveregular solids and frequently referred to that way.

(5.4) REMARK. The Riemann-Hurwitz formula for a ramified coveringis X -p Y takes the following form in terms of the genus q of the curves.

(2q(X) - 2) = n. (2q(Y) - 2) -- n. (1 - 1).

yin' eY

If the covering is totally ramified over yo in Y, then the formula takes the form

(2q(X) - 1) = n. (2q(Y) - 1) - n. I (1 - _L).Y 41Yo

ey

6. Automorphisms of hyperelliptic curves. A hyperelliptic curve X is a curvewith an involution t such that the quotient X/t by the action of t is P1(k) wherek has characteristic different from 2. If f: X -p X/t is the projection and ifFix(t) is the set of fixed points under t, then f(Fix(t)) = S is the ramificationlocus. Applying (5.1), we obtain

e(X) = 2.e(P1) - (2 - 1)S

or e(X) = 4 - # S and the genus q(X) is given by # S = 2q(X) + 2.The hyperelliptic involution t commutes with all elements of Aut(X) and the

two element subgroup <t> generated by t is a central, so normal, subgroup ofAut(X). The quotient group G = Aut(X)/(t) acts on X/t = P1 and hence isone of the groups of the form determined in (5.3). Thus the automorphism groupof a hyperelliptic curve X is a central extension of a cyclic group, a dihedralgroup, the tetrahedral group, the octahedral group, or the icosahedral group.

Next consider the projection ff: X/t = P1 -p P1/G = P1 as in (5.3). Then theramification locus S is carried onto itself by the group G since Fix(t) is carriedonto itself by Aut(X). This means that S is a union of fibres 'ff -1(y) which withat most 3 exceptions have n elements where n = # G. This analysis gives:

(6.1) PROPOSITION (OGG). For a hyperelliptic curve X in characteristic zero wehave # Aut(X) < 4(q(X) + 1) with four exceptions: q(X) = 2 where # Aut(X)< 8(q(X) + 1) = 24, q(X) = 3 where # Aut(X) < 6(q(X) + 1) = 24, q(X) = 5where # Aut(X) < 10(q(X) + 1) = 60, and q(X) = 9 where # Aut(X) <6(q(X) + 1) = 60.

7. The Hurwitz bound on an automorphism group; results in positive character-istic.

(7.1) THEOREM (HuRwITz). Let G be an automorphism group of a curve Xwhere # G is prime to the characteristic of the ground field.

(1) If q(X) > 2, then 84(q(X) - 1) > # G.(2) If q(X) > 1, and if G fixes a point of X, then 6(2q(X) - 1) > # G.

PROOF. We use the relation from (5.2)

m

(l/n)(2q(X) - 2) = (2q(X/G) - 2) + (1 - 1/e;).r=1

For q(X/ G) > 0 we havem

(1/n)(2&) - 2) > E (1 - 1/e,) > 1 - 1/e, >i-l

and for q(X/G) = 0 we have

(1/n)(2q(X) - 2) > -2 + (1 - 1/e,) > 0;al

> 1 - (1/e1 + 1/e2 + 1/e3) > 0

> 1 i+ 13+7 = 1

T2

Hence the following 84(q(X) - 1) > n always holds. The second relation isproved by a similar analysis using the relation in (5.4)

(1/n)(2q(X) - 1) = (2q(X/G) - 1) + (1 - 1/e,).=1

This proves the theorem.(7.2) REMARK. In characteristic 0 the above bounds apply to the full automor-

phism group of X for q(X) > 2. The automorphism group of an elliptic curve E,which is the same as the automorphisms of the group E preserving the origin,has order at most 6. In general it is a group of order 2 with two exceptions whereit is cyclic of order 4 in one case and cyclic of order 6 in the other case.

In characteristic p we have the following bounds on automorphism groups.(7.3) REMARK. In Iwasawa and Tamagawa [8] the group of automorphisms G

leaving a point fixed on a curve X was shown to have a normal p Sylowsubgroup N where p is the ground field characteristic such that #N 6p2(2q - 1)2 and the factor group G/N is cyclic of order 6 6(2q(X) -1)where q(X) > 0.

From these bounds the finiteness of Aut(X) where q(X) > 2 was establishedin all characteristics for curves.

(7.4) REMARK. In Roquette [14] the Hurwitz bound #Aut(X) 6 84(q(X) - 1)was shown to hold in characteristic p for p > 2q(X) + 1 and for 2q(X) + 1 >p> q(X) + 1. The inequality is valid also for p = 2q(X) + 1 with one exceptionthe hyperelliptic curve y" - y = x2 when p > 5. In this exceptional case # G 68q(X)(q(X) + 1) andp = 2q(X) + 1.

(7.5) REMARK. In Singh [17] and in Stichtenoth [18] we find the inequality

#Aut(X) < 4pq(X)2 - (2q(X) + Ill 4pq(X)2 + 1p - 1 p

1

(p-1)2

This is under the assumption that p 6 q(X) + 1 and so q(X) > 2.These results are obtained by a careful analysis of wild ramification and a

study of Artin-Schreier extensions.

8. Examples where the Hurwitz bound is realized. Since every curve of genus 2is hyperelliptic, we have #Aut(X) 6 12 = 4(q + 1) instead of #Aut(X) 6 84= 84(q - 1). Hence the Hurwitz bound is never obtained in genus 2. Thus thefirst example is possible only for genus 3.

618 D. HUSEMOLLER

(8.1) EXAMPLE. The Klein curve is given by the equation

WX3+XY3+ YW3=0in the projective plane and is isomorphic to the modular curve X(7) for the level7 modular subgroup of SL2(Z). The group PSL2(F7) acts on X(7) as a group ofautomorphisms. Since q(X(7)) = 3 and #PSL2(F7) = 168, we see that G168 =PSL2(F7) must be the full group of automorphisms of X(7). For a furtherdiscussion of this example and related questions, see Gross and Rohrlich [4] andWeil [20].

(8.2) REMARKS. In Shimura [16, §3.19] there are other examples of curves with84(q - 1) automorphisms coming from more complicated modular construc-tions. Hirzebruch has also obtained a central extension of G,68 by Z/2 as theautomorphism group of a Hilbert modular surface for Q(V) and its con-gruence subgroup for a prime ideal with norm 7.

(8.3) EXAMPLES. If X is a curve with #Aut(X) = 84(q(X) - 1), then using thejacobian J(X) of X and multiplication N: J(X) -p J(X) on J(X) we can inducea covering XN of X for each natural number N.

XN -----> J(X )l 1", dim J(X) = 2q (q = q(X)).X - J(X)

Now Aut(X) acts on J(X) and (Z/N)2 = ker(N) the group of coveringtransformations of vr, under translation. Thus the semidirect product G =Aut(X) X (Z/ N)2 acts on J(X) and on XN. Now calculate the order of thissemidirect product

# G = #Aut(X).N2 = 84(q(X) - 1).N2 = 84(q(XN) - 1).From this we deduce that G = Aut(XN). By this method starting with such an X,for example X = X(7), we obtain infinitely many curves and genera for whichthe Hurwitz bound is realized by the order of the automorphism group of somecurve.

This question was also studied by Macbeath [10] using geometric methods.

REFERENCES

1. E. Bombieri and D. Husemoller, Classification and embeddings of surfaces, Proc. Sympos. PureMath., vol. 29, Amer. Math. Soc., Providence, R. I., 1975, pp. 329-418.

2. D. Bums and M. Rapoport, On the Torelli problem for kahlerian K3-surfaces, Ann. Sci. EcoleNorm. Sup. 8 (1975), 235-273.

3. M. Demazure, Sous-groupes algebriques de rang maximum du groupe de Cremona, Ann. Sci.Ecole Norm. Sup. 3 (1970), 507-588.

4. B. Gross and D. Rohrlich, Some results on the Mordell-Well group of the jacobian of the Fermatcurve, Invent. Math. 44 (1978), 201-224.

5. R. Hartshorne, Algebraic geometry, Springer-Verlag, Berlin and New York, 1977.6. F. Hirzebruch (to appear).7. A. Hurwitz, Analytische Gebilde mit eindeutigen Transformationen in sick, Math. Ann. 41 (1893),

403-422 (Werke I, 391-430).8. K. Iwasawa and T. Tamagawa, On the group of automorphisms of a function field, J. Math. Soc.

Japan 3 (1951), 137-147; 4 (1952), 100-101, 203-204.9. S. Kleiman, The enumerative theory of singularities, Real and Complex Singularities (Oslo,

1976), Sijthoff and Noordhoff, Groningen, 1977.


10. A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90-96.11. H. Matsumura, On algebraic groups of birational transformations, Lincei-Rend. Sci. Fis. Mat.

Nat. 34 (1963),151-155.12. A. Ogg, On the automorphism group of a hyperelliptic curve (preprint).13. C. A. M. Peters, Holomorphic automorphisms of compact Kdhler surfaces and their induced

actions in cohomology, Invent. Math. 52 (1979), 143-148.14. P. Roquette, Abschatzung der Automorphismenanzahl von Funktionenkorpen, Math. Z. 117 (170),

157-163.15. M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401-443.16. G. Shimura, Construction of class fields and zeta functions of algebraic curves, Ann. of Math. (2)

85 (1%7),58-159.17. B. Singh, On the group of auiomorphisms of a function field of genus at least two, J. Pure Appl.

Algebra 4 (1974), 205-229.18. H. Stichtenoth, Uber die Automorphismengruppe eines algebraischen Funklionenkorpers von

Primzahlcharakteristlk, Arch. Math. 24 (1973), 527-544.19. K. Ueno, A remark on automorphisms of Enriques surfaces, J. Fac. Sci. Univ. Tokyo 23 (1976),

149-165.20. A. Weil, Sur les periodes des integrales abeliennes, Comm. Pure Appl. Math. 29 (1976),

813-819.

HAVERPORD COLLEGE, PENNSYLVANIA


TRANSFORMATION GROUPS ANDREPRESENTATION THEORY

TED PETRIE

The subject of transformation groups is concerned with the study of homo-morphisms of a group G (which here is finite) into the group of homeomor-phisms of a topological space. Representation theory is concerned with homo-morphisms of G into an orthogonal group or a unitary group. The first disciplinedraws motivation and machinery from the second. Mostly topologists fix thegroup and vary the topological spaces. Here we vary the group as well and treataspects of the subject which reflect the style of representation theory. Webroaden the notion of a representation of a group, illustrate some properties ofthese representations and show how questions about the structure of groups areobtained by asking for a classification of those groups which have a representa-tion with a certain property. The style of the paper is casual with emphasis onideas and properties rather than precise detail which is available in the refer-ences.

By a linear representation V of G we mean that V is a vector space (real orcomplex) and a homomorphism G -* Aut(V) also called V where Aut(V) is0(n) or U(n) accordingly as V is R" or C'; so for us V is both a vector spaceand a homomorphism. Equivalence of linear representations is defined as usualby conjugating by an element of Aut(V). By a smooth homotopy sphere Y wemean a smooth manifold which is homotopy equivalent to the unit sphere S ineuclidean space and dimension Y = dimension S. (This is equivalent to requir-ing Y to be simply connected and have the same cohomology and dimension asS.) A smooth representation Y of G is a smooth homotopy sphere Y together witha homomorphism G - Diff(Y) also called Y. Here Diff(Y) is the group ofdiffeomorphisms of Y. We declare Y1 and Y2 are equivalent smooth representa-tions if there is a diffeomorphism x from Yl to Y2 such that Y2(g) = xY1(g)x-1for all g E G. A homotopy sphere Y is a cell complex (do not worry about it;just think of a topological space) which is homotopy equivalent to the unitsphere S in Euclidean space and dimension S = dimension Y. A homotopyrepresentation Y is a homotopy sphere together with a homomorphism of G to

1980 Mathematics Subject Classification. Primary 20C15, 20G05, 20G20.0 American Mathematical Society 1980

621

622 TED PETRIE

Homeo(Y)-the group of (cellular) homeomorphisms of Y. Two homotopyrepresentations Y1 and Y2 are equivalent if there are continuous maps x:Y1 Y2 and z: Y2- Y1 such that Y2(g)x = xY1(g), zY2(g) = Y1(g)z and zxand xz are G homotopic to the identity of Y1, resp. Y2. This means there is acontinuous map h,, 0 < t < 1, of Y1 to itself such that h,(gy) = gh,(y), g E G,y E Y1 and ho = identity, h1 = zx. The precise definitions of the equivalenceshere is not important. They are provided for feeling.

Before moving on to points of interest in the structure of groups, we give someillustrations of these definitions for cyclic groups. Here is a homotopy sphere ofdimension 1.

Y:

There is an obvious homomorphism of the cyclic group of order 3 intoHomeo(Y) which takes a generator into a rotation by 27ri/3. This defines Y as ahomotopy representation. This is not so interesting since it is equivalent to therepresentation on the circle with the same rotation. In fact the inclusion of thecircle in Y provides the equivalence. It does illustrate the notion of smoothabil-ity. The homotopy representation Y is smoothable if there is a smooth represen-tation which is equivalent to Y as a homotopy representation. We provide anonsmoothable representation later. We say a smooth representation is linear ifit is equivalent to the smooth representation defined by the unit sphere SV ofsome linear representation V of G. Note because our linear representations arehomomorphisms into Aut(V), the unit sphere SV is invariant by G i.e. Aut(V)C Diff SV; so in this way a linear representation is also a smooth representation.

Here is an example of a nonlinear smooth representation of the cyclic groupof order 2 Z2: Let

fzo,z1 z,)=z0+z1+zZ+zj+ +z,, n, q odd.

Let V be the complex representation of Z2 of dimension n + 1 defined by

T(ZO z 1 . . . z,) = (Z0, -Z 1 . . . -z, )if T generates Z2. Then f: V -p C is invariant i.e. f(z) for z E V. Set

I = {z EVif(z) = 0, jzjj = 1).

Then E is a smooth homotopy sphere and Z2 is represented in Diff(E) by notingE C V is G invariant because f is invariant. For n > 3 these smooth representa-tions of Z2 are not linear (see [1]).

One reason why topologists have avoided complicated groups is that somevery simple questions involving cyclic groups are not known. For example, it isunknown whether Diff(l) contain Z2 for an arbitrary smooth homotopy sphere1. It may seem strange that the huge group Diff(E) might not accommodate Z2as a subgroup. There are smooth manifolds with no finite subgroups in theirdiffeomorphism groups.

TRANSFORMATION GROUPS AND REPRESENTATION THEORY 623

Although we are primarily concerned with representations, we are occasion-ally forced to mention more general topological spaces with G action namely Gmanifolds. A G manifold M is a smooth manifold M together with a homomor-phism called M of G into Diff(M). We say G acts on M and write M(g)x = gxfor g E G and x E M. Some important invariants of M are:

(i) The H fixed point set Mr" = {x E Mlhx = x, h E H}. Here His asubgroup of G.

(ii) The isotropy representations TIM, x E MG . Here TIM is the representa-tion of G on the tangent space to M at x E MG which is induced by the actionof G on M.

(iii) dim MH-the dimension of MH as a topological space.If Ml and M2 are two G manifolds, a G map f: M1 -p M2 is a continuous map

f which satisfies f(gx) = gf(x) for g E G and x E M1. Suppose n = dim M1 =dim M2 and both are oriented manifolds. This means specific generators [M;],i = 1, 2, of the n dimensional homology of each is given. Under these circum-stances the degree off written degree f is defined as

f#[ M1] = degree f [ M2].

Here f, is the homomorphism of HH(M1) to H (M2) induced by f.If H is a subgroup of G and M is an H manifold there is an induced G

manifold indHM with this property: If N is a G manifold and f: M - N is an Hmap, there is a unique G map indHf: indHM - N which restricts to f onM c indHM. The precise definition of indHM is unimportant. As a manifold itis a union of copies of M indexed by the cosets of H. We do need that thedegree of indHf is I G I H I1 degree f.

A smooth or homotopy representation Y of G such that YH = 0 for H 1 iscalled a free representation. A linear representation V is called free if, theassociated smooth representation on SV is free. If (Y - YG)H = 0 we say Y isa semifree representation. Here is the first question which asks for the structureof those groups which have a representation with a given property.

Question 1. Which groups have a free representation?This question has a rich history. The answer varies according to the specific

type of representation sought as we shall see. For a linear representation Y to befree it is necessary and sufficient that the only matrix Y(g) for g c G which has1 as an eigenvalue occurs when g is the identity of G.

THEOREM 2 (WOLF [ 22]). If G has a free linear representation every subgroup oforder pq (p, q primes not necessarily distinct) is cyclic. Conversely if G is solvableand this condition is fulfilled, free linear representations exist. However, fornonsolvable G the only noncyclic composition factor allowed is the simple group oforder 60.

THEOREM 3 (CARTAN AND EILENBERG [3]). If G has a free homotopy representa-tion, then G has periodic cohomology.

Cartan and Eilenberg also relate the dimension of a free representation to theperiod of the group. The structure of those G with periodic cohomology issimple. Every Sylow subgroup is cyclic or generalized quaternion.

Milnor added a condition for smooth representations.

624 TED PETRIE

THEOREM 4 (MILNOR [11]). If G has a free smooth representation, then everyelement of order 2 in G lies in the center.

Swan provides the converse to 3. He also relates the minimum dimension of afree representation of G to the projective class group of G. This is a subtlerelationship and even today the minimum dimension of a free representation isunknown for some groups. For those groups with periodic cohomology of period4, this is related to the Poincare conjecture.

THEOREM 5 (SWAN [20]). G has a free homotopy representation iff G has periodiccohomology.

Combining 4 and 5, we obtain an example of a nonsmoothable homotopyrepresentation of G. Take any free homotopy representation of the symmetricgroup of degree three. This exists because this group has periodic cohomology ofperiod 4. Since there are noncentral elements of order two, Milnor's theoremimplies this homotopy representation is not smoothable.

The first free smooth representations of groups which have no free linearrepresentations were given independently by Petrie [13] and Lee (unpublished).

THEOREM 6 (PETRIE [13]). Any extension G of a cyclic group Z, by a cyclic groupZq of odd prime order q has a free smooth representation.

One obtains an approximation to a free smooth representation for thesegroups this way. Let X be a complex one dimensional free representation of Z,and q a complex one dimensional free representation of Zq. Induce X up to Gand add it to q viewed as a representation of G via the homomorphism of G onZq. This gives a q + 1 dimensional representation of G. The complex polynomial

f(z1,. . . , zq, zq+l) = zl + . . . +Zp + zq+l

defined on V is invariant by the action of G (f(g) = fO for z E V) provided 1is a power of q. For suitable positive numbers e and q

K= {z EVIf(z) = e, JJYJJ = q) (7)

is a smooth G invariant submanifold of V of dimension 2q - 1 and G acts freelyon K. The only nonzero homology of K occurs in dimensions 0, 2q - 1 where itis the integers and in dimension q - 1 where it is a q torsion module. Since K isalso simply connected, it would be homotopy equivalent to a sphere (and sodefine a smooth free representation) except for the homology in dimension q.This is removed by a process called G surgery (which is incorporated in (19))and a free representation is produced.

Question 1 was finally settled by

THEOREM 8 (MADSEN, THOMAS AND WALL [10]). A finite group G has a freesmooth representation iff for all primes p, all subgroups of order 2p and p2 arecyclic.

Artin observed [9] that there is a function c such that whenever V is a linearrepresentation of G

dim VG = 'G (dim V c I C cyclic)


so when G is not cyclic, the dimension of VG is determined by the dimensions offixed sets of proper subgroups. Borel showed an analogous statement for anelementary abelianp-group G and homotopy representation Y. In fact

dim Y - dim yG = Z (dim YH - dim YG).HEX

Here 'C is the set of subgroups of index p. See [2, p. 175]. So there is a functionG such that for every homotopy representation Y

dim yG = PG(dim YHIH m# G)

provided G is an elementary abelian group of rank greater than 1. (The formulasays nothing if G = Zr.) These two relations motivate the

Question 9. For which groups G is there a function G such that for every (2representation Y of G, dim yG =G(dim YHJH m# G)? (Here (2 abbreviateslinear, smooth or homotopy.)

The above discussion shows that the structure of the groups for which qGexists depends on the type of representation considered. The answer for linearrepresentations is that G must be noncyclic. For other representations this isanswered by

THEOREM 10 (DOVERMANN AND PETRIE [23]). There is a function tpG such thatdim yG = qJG(dim YHIH m# G) for every smooth (homotopy) representation Y iffG is a noncyclic group of prime power order.

The next two results are concerned with proving equality of real representa-tions in two geometric situations. The discussion is facilitated by starting withcomplex representations and proving their underlying real representations areequal. Let G be a cyclic group of order m with generator t. Then a complexrepresentation V of G has the form

V = Y. X4, X(t) = C = exp[27ri/m] (11)

for integers a; defined mod m. Here X is the complex one dimensional represen-tation which sends t to C as indicated. If V is a free representation, then all theintegers a; are prime to m.

THEOREM 12 (DE RHAM [4], [12]). If two real linear representations of a groupare equivalent as smooth representations, they are equivalent as real linear repre-sentations.

The proof of this theorem is based on the torsion invariant. Let I be the idealin the rational group ring of G generated by the sum of the group elements andlet N = QG/I. The units of this ring U(N) have a subgroup generated by ±gfor g E G. This is denoted by ± G. Let us suppose G is cyclic of order m. If Y isa free representation of G, its torsion invariant

A(Y) E U(N)/±Gis defined. If Y = SV where V is a free complex representation as in (11), then

A(SV) = 11 (t4 - 1) E U(N)/±G. (13)

626 TED PETRIE

The main ideas in the proof of de Rham's Theorem can be seen in the case Gis cyclic and V and W are free complex representations of G which are smoothlyequivalent. Then A(SV) = A(SW). The Franz Independence Lemma [12, p.406], asserts the units t' - 1 E U(N) where (r, m) = I and I '( r < m/2 do notsatisfy any multiplicative relations. From this and the formula for the torsioninvariant, it follows that V and W are equivalent real representations. (Note(t' - 1) = -t'(t-' - 1) = (t' - 1) in U(N)/ ± G and that X' and X' are equiv-alent real representations.)

THEOREM 14 (ATIYAH AND BoTT [1]). If Y is a semifree smooth representation ofG with YG consisting of two points p and q, then the isotropy representations T,Yand Tq Y are equal.

The proof of this theorem is based on the G signature invariant Sign(G, M)defined for any smooth G manifold M. This is a character of the group whosevalue at g E G is denoted by Sign(g, M). The essentials of the argument arecontained in the case G is cyclic of prime order m generated by t. TheAtiyah-Bott fixed point formula gives

Sign(t, M) = Y, v(x),xEMG

1 + ka;(x)

I - C4(I)

if TIM = EX4(x) (see (11)). This involves a choice of a complex representationwhose underlying real representation is TIM. In the case of a smooth represen-tation Y of G, Sign(G, Y) is zero because Y has zero cohomology in dimensionsdifferent from 0 and dim Y. If yG = p U q, then T. Y and Tq Y are freerepresentations of G because Y is semifree, so the integers a;(p) and a.(q) areprime to m; moreover v(p) = -v(q) because Sign(t, Y) = 0. Another applicationof the Franz independence lemma (see [1, p. 479]) shows {a;(p)} _ {±a1(q)}; soTo Y and Tq Y are equivalent as real representations.

Theorem 14 motivates theQuestion 15. For which groups G do there exist smooth representations Y such

that yG consists of two points p and q and the isotropy representations differ atthese points?

Observe that the Atiyah-Bott Theorem implies that Y cannot be semifree. Infact their argument shows that ResH T. Y and ResHTq V are equal as representa-tions of H for every subgroup H of G of odd prime power order. This ispertinent to the characterization of those elements in the representation ring ofG which occur as the difference ToY - TqY for some smooth representation Yof G with yG = p U q.

The following partial answer to 15 is far from complete but it provides thefirst examples where the isotropy representations can differ:

THEOREM 16 (PETRIE [16], [17]). Let G be an odd order abelian group with atleast four noncyclic Sylow subgroups. Then there is a smooth representation of Gwith yG =p U q and Tp Y # Tq Y.


This theorem requires as input a partial solution of a very old question ofMontgomery and Samelson:

Question 17. Which groups have a smooth representation Y with yG equal toone point?

The first example of a group with such a smooth representation was given byStein. The group is the simple group of order 60, namely PSL(2, Z5) (see [19]).The answer to 17 is not complete but there are some interesting groups.

THEOREM 18 (PETRIE [14], [15]). These groups have a smooth representation Ywith yG equal to one point: G = S3, SO3, SL(2, Fo), PL(2, Fo) (characteristic Foodd) and every odd order abelian group with at least 3 noncyclic Sylow subgroups.

In addition [14] provides a lot of information about the isotropy representa-tions T. Y at the unique fixed point of the representation Y. This is relevant toTheorem 16.

There are some common features to the Theorems 6, 8, 10, 16 and 18. Here isa brief discussion. The essential tool is the G surgery sequence ([8], [17]):

hSG(Y,X)-* NG(Y,X)--G> I(G,X) (19)

defined more generally but in particular for a G representation Y. When Y is afree G representation, this is due to Wall [21]. In the general case it is due toDovermann and Petrie [8]. A rough description of the sequence must suffice.The parameter A is a record of some of the invariants of a G manifold. Inparticular A records the dimensions of fixed sets of subgroups and isotropyrepresentations at fixed points. Elements of NG(Y, A) are equivalence classes ofA approximations to the homotopy representation Y. These are G manifolds Xwhose invariants are specified by A and which have a G map f from X to Y ofdegree 1. Elements of hSG(Y, A) are smooth representations X whose invariantsare specified by A and which support a G map to Y of degree 1. The set I(G, A)(for some A, it is a group) has a distinguished element zero. The sequence isexact in the sense that an approximation X with parameter A gives rise to asmooth representation X' with parameter A iff a(X) = 0. Then X = d(X') forsome X' in hSG(Y, A).

One useful relation between the G surgery sequence and representation theoryis a theorem of Dovermann [7] which asserts that if Y is a linear representationand yG # 0, then NG(Y, A) # 0 implies hSG(Y, A) # 0. This says the ex-istence of an approximation (with parameter A) to a representation implies theexistence of a smooth representation with parameter X.

In Theorems 8 and 10 the sequence is applied like this: Let H be a subgroupof G, S a smooth representation of H and f: S --+ Y an H map. Then indHf:indHS -- Y is a G map whose degree is degree f I G I I H I -' . Suppose that 'C is afamily of subgroups such that the integers { I G I H I -' I H E '3C } are relativelyprime. Suppose also for each H E 3C, SH is a smooth H representation and fH:SH --+ Y is an H map of degree 1. Choose integers aH such that

2 aHIGI IHI-' = 1.HEX

628 TED PETRIE

Define X to be the disjoint union

X = H11X aH indHSH (20)E

where aH indHSH means the disjoint union of aH copies of indHSH. (Some of theaH's will be negative. In that case the orientation of SH is reversed.) The symbolIIHEX means disjoint union over X. At this point X is far from a homotopysphere. In fact it is a disjoint union of a number of spheres permuted by G.However there is a G map f of X into Y where f is defined on each copy ofind ,SH as indHfH; so

degree f= alI G I I H I-degree fH = 1.HE`3C

This means X is an approximation to Y.For specific application to Theorems 6 and 8 take 'C to be a complete

collection S of representatives of the conjugacy classes of Sylow subgroups. Thischoice is natural as the condition for periodic cohomology is specified in termsof the Sylow subgroups. They are either cyclic or generalized quaternion. Fromthis it is easy to see that for any free homotopy representation Y of G and anyP E 5, there is a linear free representation VP of P which has a degree 1 P mapfp: S VP ---> Y.

Take Y to be a free homotopy representation of G provided by Theorem 5and take the approximation

X = II aH indHSVH (21)HES

and observe that G acts freely on X. Choose A to be the record of the invariantsof X. Then X represents an element of NG(Y, A). In this case I(G, A) is a group,in fact a Wall group LA(G), n = dimension Y. When I G I is odd this group iszero; so there is a homotopy representation X' with parameter X. Since Aspecifies the dimensions of fixed sets of X as well as X' and since X H = 0 forH 1, the same is true for X'; so X' is a free representation of G.

This establishes 8 for groups of odd order. For the remaining groups in thattheorem one needs to use additional information about the sequence and therepresentation theory of G. In particular, the conditions in 8 imply that every 2hyperelementary subgroup of G has a free linear representation.

The hard part of the proof of Theorem 10 is to show that if G does not haveprime power order, there is no function q'c which determines dim yG in terms ofdim Y", H G. To do this we construct two smooth representations Y, and Y2such that dim V' = dim Y2 iff H m# G. We again apply the construction in(20). In this case we take 'C to be the set of all proper subgroups of G. (This isused in (22) below.) Since G1 is not a prime power, the integers { I G I HI-'H E X) are relatively prime.

We choose a suitable complex representation V of G such that SVG 0 andset Y = SV. For each H E 'JC we let VH be the restriction of V to H. For the Hmap fH: SVH -* Y take the identity. Then the approximation X provided by (20)has

dim X" = dim Y", H G,

dim XG = -1 and dim yG > 1. (22)


Choose A to be the record of the invariants of X. Then X represents an elementof NG(Y, A); so this set is nonempty. Since Y is a linear representation withyG 0, Dovermann's theorem applies and there is a homotopy representation

X' with parameter X.' In particular dim X'H = dim X H for all H; so dim X'H =dim YH iff H G.

Not much can be said here about Theorem 18. However, we can make someremarks of how 18 is used in 16. Let G be as given in 16 and let V be arepresentation of G such that dimR VG = 1; so if Y = SV, then yG has twopoints p and q. Actually the choice of V is delicate and depends on 18. Nowtake two smooth representations Y, and Y2 of G having unique fixed points p1and p2 as provided in 18. We suppose To Y1 is V minus the one dimensionaltrivial representation. Then there is a degree 1 G map of Yl to Y. We choose Y2so that the isotropy representation To2Y2 is not equivalent to To YI and map Y2into the fixed point q in Y. Set

X = Y1 1T Y2.

Then there is a degree 1 G map of X to Y which restricts to the given maps onY,. Thus X is an approximation to Y. Let A be the record of its invariants; soX E NG(Y, A). Note that X G= P1 U P2 and

ToX-To2X= To Y1-TP2Y2 A0.

We are able to show aX = 0 so there is a smooth representation X' withparameter X. In particular X'G = Pt U P2 and ToX' To2X'.

There is a way of organizing homotopy representations similar to the waylinear representations are organized by the representation ring. The essentialpoint is that homotopy representations can be added to give a new homotopyrepresentation. (The set of smooth representations is not closed under addition.)The addition is defined by the join construction * used often in topology. In factif V1 and V2 are two linear representations of G, then S(V1 ® V2) =S( V,) * S(V2). For any two complex representations Y, and Y2, we defineY1®Y2=Y1*Y2.

In order to obtain a manageable situation, we restrict to semilinear homotopyrepresentations. These are the homotopy representations analogous to complexlinear representations. They are the homotopy representations Y for which YH isan odd dimensionsional homotopy sphere for all H and N(H)/H acts triviallyon the cohomology of YH for all H. Assert that two semilinear homotopyrepresentations Y, and Y2 are equivalent if there is a semilinear homotopyrepresentation Y such that YI ® Y is equivalent to Y2 ® Y as a homotopyrepresentation. Henceforth a homotopy representation means a semilinear ho-motopy representation. Define ([5], [6]) V(G) to be the Grothendieck group ofthe semigroup of equivalence classes of homotopy representations. This is theconstruction which gives the complex representation ring of G when applied tothe semigroup of complex representations of G. We make no distinction betweena homotopy representation and its class in V(G).

'Added in proof. There are some inaccuracies in Dovermann's Theorem; however an alternativeargument can be used for this conclusion.

630 TED PETRIE

If Y is a homotopy representation of G, define an integral valued functionDim Y on the set of conjugacy classes of subgroups of G by

Dim Y(H) = 1 dim(YH + 1). (23)

Note if V is a complex representation of G, Dim SV(H) = dimc V. We viewDim Y as a character. These characters determine the group V(G) rationally [6].In fact there are only a finite number of homotopy representations Z such thatDim Z = Dim Y for any fixed Y. The "orthogonality relations" i.e. relationsamong the character values Dim Y(H) as H ranges over (conjugacy classes of)subgroups determine V(G) 0 Q. This leads back to

Question 9. For which groups G is there a function tpG such that whenever Y isa semilinear representation of G, dim yG = YHI H # G)? To answerthis question let G' be the commutator subgroup of G.

THEOREM 24 (TOM DIECK AND PETRIE [6]). There is a function tpG such thatwhenever Y is a semilinear homotopy representation of G, Dim Y(G) _tpG(Dim Y(H)IH # G) iff GIG' is not cyclic.

COROLLARY 25 [6]. The rank of V(G) (as an abelian group) is the number ofconjugacy classes of subgroups of G with cyclic commutator quotients.

Note that V H SV defines a homomorphism of the complex representationring R(G) to V(G) whose image is called J(G).

THEOREM 26 [6]. J(G) = V(G) iff G is cyclic.

This means that there are homotopy representations which do not come fromlinear representations. It is thus interesting to know which homotopy representa-tions are smoothable. We have already treated this for free representations.

The determination of the torsion subgroup of V(G) is quite subtle. It dependson group cohomology and the projective class group (see [6]). Complete compu-tations of V(G) are available only for G cyclic, Z. X Zo, p a prime, andmetacyclic groups of order pq where p and q are prime.

REFERENCES

1. M. Atiyah and R. Bott, A Lefschitz fixed point formula II, Ann. Math. 87 (1968), 451-491.2. A. Borel et al., Seminar on transformation groups, Ann. of Math Studies, no. 46, Princeton

Univ. Press, Princeton, N. J., 1960.3. H. Cartan and S. Eilenberg, Homological algebra, Oxford Univ. Press, Cambridge, 1956.4. G. de Rham, Reidemeister's torsion invariant and rotations of S", Differential Analysis, Oxford

Univ. Press, Cambridge, 1964, pp. 27-36.5. T. tom Dieck and T. Petrie, The homotopy structure of finite group actions on spheres, Proc.

Waterloo Topology Conf., Lecture Notes in Math., vol. 741, Springer-Verlag, Berlin and New York,1979.

6. , Homotopy representations of finite groups (to appear).7. H. Dovermann, Addition of equivariant surgery obstructions, Lecture Notes in Math., vol. 741,

Springer-Verlag, Berlin and New York, 1979.8. H. Dovermann and T. Petrie, G. surgery II (to appear).9. W. Feit, Characters of finite groups, Benjamin, New York, 1967.

10. I. Madsen, C. Thomas and C. T. C. Wall, The topological space form problem, Topology 15(1976), 375-382.


11. J. Milnor, Groups which act on S" without fixed points, Amer. J. Math. 79 (1957), 612-623.12. , Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426.13. T. Petrie, Free metacyclic group actions on homotopy spheres, Ann. of Math. 94 (1971), 108-124.14. , Groups which act on a homotopy sphere with one fixed point (to appear).15. , Pseudoequivalences of G manifolds, Proc. Sympos. Pure Math., vol. 32, part 1, Amer.

Math. Soc., Providence, R. I., 1978, pp. 169-210.16. , Groups which act on a homotopy sphere with distinct isotropy representations (to appear).17. , G surgery I - A survey, Algebraic and Geometric Topology, Lecture Notes in Math.,

vol. 664, Springer-Verlag, Berlin and New York, 1978, pp. 196-234.18. , Three theorems in transformation groups, Proc. Aarhus Topology Conf. (1978),

Springer-Verlag, Berlin and New York, 1980, pp. 549-572.19. E. Stein, Surgery on products with finite fundamental groups, Topology 28 (1977), 16-25.20. R. Swan, Periodic resolutions of finite groups, Ann. of Math. 72 (1960), 267-291.21. C. T. C. Wall, Surgery on compact manifolds, Academic Press, New York, 1970.22. J. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967.23. H. Dovermann and T. Petrie, The Artin relation for smooth representations (to appear).

RUTGERS, NEW BRUNSWICK


LIE ALGEBRAS WITH NILPOTENTCENTRALIZERS

I. M. ISAACS

This lecture is a report on some joint work of myself and Georgia Benkart. Iwill keep this writeup brief, since full details can be found in our paper [1].

The point I wish to stress here is how analogy with finite groups can suggestproblems about Lie algebras, and how the overall structure of a solution to sucha problem can parallel the general outline of many arguments in group theory.Of course, the details of the solutions of group and Lie algebra problems areoften entirely unrelated.

The group theoretic problems which have the most natural analogs in Liealgebras are those which do not mention explicitly the "arithmetic" structure ofa group: Sylow subgroups, orders of elements, etc. The problem considered hereis that of classifying Lie algebras in which the centralizer of every nonzeroelement is nilpotent. The details of our solution are totally unrelated to those ofthe Feit-Hall-Thompson paper [2] in which the corresponding group theoryproblem is considered. The overall structure of our proof, however, should besomewhat familiar to the group theorist.

Let F be any algebraically closed field. There is a unique simple Lie algebraS(F) of dimension 3 over F. (If char(F) 2, then S(F) = s12(F).) If char(F) =p > 3, let W(F) denote the Witt algebra, which is simple of dimension p.Finally, let C(F) = s13(F)/F 1 if char(F) = 3 so that C(F) is simple ofdimension 7. It turns out that the only simple c.n. (= centralizers nilpotent) Liealgebras over an algebraically closed field F are S(F) and, depending on thecharacteristic, C(F) or W(F).

To prove this classification of simple c.n. algebras, we found it necessary tostudy c.n. algebras which are not necessarily simple, so that we could obtaininformation about subalgebras of our unknown algebra.

1980 Mathematics Subject Classification. Primary 17B50.® American Mathematical society 1980

633

634 I. M. ISAACS

THEOREM 1. Let L be a nonnilpotent c.n. Lie algebra over an algebraicallyclosed field. Then L has a unique maximal ideal N. Furthermore, N is nilpotent,L/N is a c.n. algebra and either

(a) dim(L/N) = l or(b) L/N is simple.

We also needed a "recognition theorem" for the algebras S(F), C(F) andW(F). This is provided by a result of Kaplansky [3] which characterizes these interms of roots and a Cartan subalgebra (of dimension 1).

The final ingredient in the argument is this. Suppose L is c.n. and M C L is anonsolvable subalgebra. Then L is an M-module and we consider an M-com-position series for L through M. Let N be the maximum ideal of M. Then Nannihilates each composition factor. Each factor thus becomes a module for thesimple c.n. algebra S = M/N, and all composition factors except M/N itselfsatisfy a certain technical property which makes them what we call specialmodules.

Now let L be any simple c.n. algebra. If it does not satisfy Kaplansky'scondition, we construct a nonsolvable subalgebra M such that M/N does satisfythe condition, where N is the maximal ideal of M. Since C(F) and W(F) havesubalgebras isomorphic to S(F), it is no loss to assume MIN = S(F). Themodule theory of S(F) is known, and so we can determine all of its specialmodules. The fact that the M-composition factors of L are (almost) all special,then gives us additional information which shows that L satisfies Kaplansky'scondition after all. This proves that L = S(F), C(F) or W(F).

We are now in a position to prove the following, which is the main result ofthe paper.

THEOREM 2. Let L be a nonsolvable c.n. Lie algebra over an algebraically closedfield F. Then L has an abelian ideal N and L/N = S(F), C(F) or W(F).

The only additional item here, is that the maximal ideal N of L is abelian. Toprove this, it suffices to consider the case where L/N = S(F) and again theresult follows using our knowledge of the special modules for S(F).

REFERENCES

1. G. M. Benkart and I. M. Isaacs, Lie algebras with nilpotent centralizers, Canad. J. Math. 31(1979), 929-941.

2. W. Feit, M. Hall and J. Thompson, Finite groups in which the centralizer of any nonidentityelement is nilpotent, Math. Z. 74 (1960), 1-17.

3. I. Kaplansky, Lie algebras of characteristic p, Trans. Amer. Math. Soc. 89 (1958), 149-183.

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