investigations in algebraic theory of combinatorial objects

Investigations in Algebraic Theory of Combinatorial Objects

Mathematics and lts Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

A. A. KIRILLOV, MGU, Moscow, Russia Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, Russia N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, Russia S. P. NOVIKOV, Landau Institute ofTheoretical Physics, Moscow, Russia Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, Russia

Volume 84

Investigations in Algebraic Theory of Combinatorial Objects

Edited by

I. A. Faradzev

A. A. Ivanov

M. H. Klin Institute for System Studies, Moscow, Russia

and

A. J. Woldar Villanova University, Villanova, Pennsylvania, U.S.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

Issledovanifa po algebralchesko1 teorii kambinatornykh ob"ektov. English.

Investigt1ons in algebraic theory of combinatorial Objects I by I.A. Faradzev, A.A. Ivanov, M.H. Kl in. and A.J. Waldar.

p. cm. -- <Mathematics and its applications. Soviet series 84>

Inc 1 udes i ndex. ISBN 978-90-481-4195-1 ISBN 978-94-017-1972-8 (eBook) DOI 10.1007/978-94-017-1972-8 1. Combinatorial analysis--Congresses. I. Faradzhev, I. A.

II. Title. III. Ser1es: Mathmematics and its appl ications <Kluwer Academ1c Publ ishers>. Soviet ser1es ; 84. QA164.I8713 1992 511 ·. 6--dc20

ISBN 978-90-481-4195-1

Part of this book is a revised and updated translation of HCCJ1E,LJ,OBAHH51 no AJlfEBPAH4ECKOA TEOPHH KOMBHHATOPHblX OBbEKTOB

© Institute for System Studies, Moscow, 1985

Printedon acid-free paper

All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in I 994

92-27720

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

TABLE OF CONTENTS

Preface to the English edition

Preface to the Russian edition

PART 1. CELLULAR RINGS

1.1 I.A. Faradrev, M.H. Klin, M.E. Muzichuk, Cellular rings and groups of

automorphisms of graphs

vii

ix

1.2 V.A. Ustimenko, Onp-local analysis ofpermutation groups 153

l.3 J a.Ju. Gol'fand, A. V. Ivanov, M.H. Klin, Amorphie cellular rings 167

1.4 M.E. Muzichuk, The subschemes ofthe Hamming scheme 187

1.5 Ja.Ju. Gol'fand, A description of subrings in V(SP, X sP, X ••• X Sp) 209

1.6 LA. Faradrev, Cellular subrings of the symmetric square of a cellular ring of rank 3 225

1.7 V.A. Ustimenko, The intersection numbers of the Hecke algebras H(PGL.(q),

BWß) 251

1.8 LA. Faradiev, A.V. Ivanov, Ranksand subdegrees of the symmetric groups acting

on partitions

1.9 A.A. Ivanov, Computation of lengths of orbits of a subgroup in a transitive

permutation group

PART 2. DISTANCE-TRANSITIVE GRAPHS

2.1 A.A. Ivanov, Distance-transitive graphs and their classification

2.2 A.V. Ivanov, On some local characteristics of distance-transitive graphs

2.3 LV. Chuvaeva, A.A. Ivanov, Action ofthe groupM12 on Hadamard matrices

2.4 F.L. Tchuda, Construction of an automorphic graph on 280 vertices using finite

geometries

PART 3. AMALGAMSAND DIAGRAM GEOMETRIES

3.1 A.A. Ivanov, S.V. Shpectorov, Applications of group amalgams to algebraic graph

265

275

283

379

395

409

theory 417

3.2 S.V. Shpectorov, A geometric characterization of the group M22 443

3.3 M.E. lofmova, A.A. Ivanov, Bi-primitive cubic graphs 459

3.4 V.A. Ustimenko, On some properties of geometries of Chevalley groups and their

generalizations 473

Subject index 507

PREFACE TO TUE ENGLISH EDITION

This volume arose through the initiative of Kluwer Academic Publishers in an attempt to intro

duce some areas of research in algebraic combinatorics which originally appeared in Russian to a

wider mathematical community. The authors of the papers in this volume belong to two scientific

groups. The first consists of the people associated with the Iabaratory of Discrete Mathematics at

the Institute of System Studies in Moscow. The other belongs to the Department of Algebra at the

Kiev State University. Besides translations of research papers from Russian to English, the

volume contains four survey papers written expressly for this edition. The surveys are located in

the opening sections of each of the parts of the volume: two surveys belong to the first part, one

to the second and one to the third.

The core of the volume is formed by the c:ollection of papers "Investigations in Algebraic

Theory of Combinatorial Objects" (M.H. Klin, I.A. Faradrev eds.), Moscow, Institute for System

Studies, 1985, referred below as IATC0-85. For the papers translated from IATC0-85 we

indicate the corresponding pages at the end of the papers. The present volume contains transla

tions of all papers from IATC0-85 excepting the first and the last ones. The content of the first

paper is covered in the survey "Cellular Rings and Groups of Automorphisms of Graphs" by LA.

Faradtev, M.H. Klin and M.E. Muzichuk. For this reason tlle references in the translated papers

to the first paper of IATC0-85 were changed to the references to the above mentioned survey. On

the other hand, the volume contains a translatiort of a paper by A.A. lvanov on the computation of

ranks and subdegrees in permutation groups, which appeared originally in a different collection.

The volume consists of three parts. The papers in the first part are devoted to investigations and

applications of cellular rings (adjacency algebras of coherent configurations). The first survey in

this part is about the subject in general. The second survey is on the technique of p-local analysis

in permutation groups. The second part of the volume contains papers on distance-regular and

distance-transitive graphs. In the third part some results in a relatively new direction, amalgams

and geometries, are presented. As was predicted in the preface to the Russian edition, the method

of amalgams has come to play an increasingly significant roJe in algebraic combinatorics.

Some changes were made to the papers from the Russian collection during translation. The

paper "On some properties of the geometries of the Chevalley groups and their generalizations"

by V.A. Ustimenko was revised considerably. The paper "Amorphie cellular rings" by Ja.Ju.

Gol'fand, A.V. Ivanov and M.H. Klin in the Russian editionwas divided into two 'parts', with

different sets of authors. Finally in some papers the order of the authors was changed to be

alphabetic in the Western version.

The Publisher wishes to draw the readers' attention to the special issue of the Kluwer joumal

Acta Applicandae Mathematicae Vol. 29/1-2 entitled "Interactions between Algebra and Com

binatorics", edited by LA. Faradrev, A.A. Ivanov, and M. H Klin. This issue can be seen as a

sequel to the present volume. It deals with cellular rings, distance-regular graphs and group

vii

viii PREFACE TO THE ENGLISH EDITlON

amalgams; the papers provide examples of new applications of permutation group theory and

association schemes in algebraic combinatorics.

We are very grateful to Professor M. Hazewinkel for his interest in our research and for

introducing the idea for the present volume, to Dr. D.J. Lamer for his help and patience during

the delayed preparation of the volume, and to Adrianka de Wit and Anneke Pot for their kind and

perfect technical assistance.

We want to thank J. Remmeterand F. Lazebnik for their excellent and conscientious transla

tiontagether with the fourth editor of the massive paper "Cellular Rings and Groups of Automorphisms of Graphs".

I.A. Faradzev

A.A. Ivanov

M.H. Klin

A. Waldar

PREFACE TO TUE RUSSIAN EDITION

In its development modern combinatorics synthesized methods from many diverse branches of

mathematics, especially from algebra, geometry and number theory. The main roJe in this

synthesis was played by an extant body of algebraic ideas, starting primarily with techniques

from linear algebra and group theory.

Over the last twenty years a new tendency developed toward the interplay of combinatorial and

algebraic (particularly group theoretic) methods .. This tendency was caused by a sharp increase in

the importance of discrete mathematical applications. It was discovered that certain problems

related to experimental design, chemical structure analysis, design of logical schemes and of

various devices, etc., shared a common mathematical formulation. The content of these problems

is the identification of certain combinatorial objects (primarily graphs and networks) and the

characterization of their automorphism groups.

In the mid 70s, a variety of results from 11nite group theory on automorphism groups of

combinatorial objects, tagether with combinatorics and computing theory, formed the basis for

the direction and content of a new mathematical subject. Subsequently, the algebraic theory of

combinatorial objects became an independent branch of mathematics. The main goal of this

theory is to study the relationship between a combinatorial object's local features, defined in

terms of incidence of its component parts (e.g. vertices and edges of a graph; points, bloclcs and

flags of a block design), and the global properties of the object's automorphism group. From the

group theoretic viewpoint, interest in such a relationship is justified by the compactness and

convenience by which one is able to defme certain classes of groups as automorphism groups of

graphs, two-graphs and other appropriate combinatorial objects. The use of such definitions

appears to have been extremely efficient in studying the sporndie simple groups, for example.

Exploiting the relationship between local and global properties enables one to establish

necessary (and sometimes sufficient) conditions for the automorphism group of an object from a

certain class to have such extremal properties as transitivity, primitivity, distance-transitivity, etc.

From the viewpoint of complex systems theory, the primary objective of the algebraic theory of

combinatorial objects is to approximate the global (i.e. algebraic) properlies of a system by its

local (i.e. combinatorial) features, that is, features which involve incidence between its com

ponent parts.

The present collection of papers deals with an area of the subject that became uni11ed only a

few years ago. This area is founded on the following three methodological bases, which arose

independently of one another: the method of invariant relations, the theory of cellular rings, and

constructive enumeration of combinatorial objects. The oldest among these is the method of

invariant relations in the theory of permutation groups. This method was proposed by M. Krasner

and I. Schur in the 30s and was advanced fundamentally in papers by H. Wielandt, R.

ix

X PREFACE TO THE RUSSIAN EDITION

Köchendorffer, L.A. Kalu:lnin and their students in the 50s and 60s. Nowadays the most deeply

developed is the theory of binary invariant relations and their combinatorial approximations.

These combinatorial approximations arose repeatedly during this century under various names

(Hecke algebras, centralizer rings, association schemes, coherent configurations, cellular rings,

etc.- see the first paper of the collection for details) andin various branches of mathematics, both

pure and applied. One of these approximations, the theory of cellular rings (cellular algebras),

was developed at the end of the 60s by B. Yu. Weisfeiler and A.A. Leman in the course of the

first serious attempt to study the complexity of the graph isomorphism problem, one of the central

problems in the modern theory of combinatorial algorithms. At roughly the same time G.M.

Adelson-Velskir, V.L. Arlazarov, I.A. Faradtev and their colleagues had developed a rather

efficient tool for the constructive enumeration of combinatorial objects based on the branch and

bound method. By means of this tool a number of "sports-like" results were obtained. Some of

these results are still unsurpassed.

Toward the end of the 70s it became clear that an extensive knowledge of finite group theoretic

techniques was desireable in order to obtain important new results by analyzing the enormaus

amount of data on graphs and incidence systems whose automorphism groups had interesting

properties. It was also desirable to have some powerful software available so that one could carry

out computations in permutation groups and cellular rings in order to collect this data.

The necessity to unite the efforts of specialists from different scientific schools led to the

organization of a seminar on the algebraic theory of combinatorial objects. This seminar started in

1980. Its kerne! consisted of mathematicians from the Labaratory of Discrete Mathematics at the

Institute for System Studies and from the algebraic school of L.A. Kalu:lnin at the Department of

Algebra and Logic at the Kiev State University named after T.G. Shevchenko. In addition the

seminarwas attended by mathematicians from other institutions: the Department of Algebra and

Geometry at the Kaluga Pedagogical Institute named after K.E. Tsiolkovskir. the Department of

Mathematics and Mechanics and the Department of Chemistry at the Moscow State University

named after M.V. Lomonosov, the Moscow Physical-Technical Institute, etc.

The results presented in this collection were obtained through the interaction of the participants

of the seminar. The papers are divided into two parts. Those in the first part are devoted to

various aspects of cellular rings: axiomatics, the description of cellular rings possessing certain

extremal properties, enumeration of cellular rings and computation of their structure constants.

The first part opens with a survey paper by M.H. Klin which contains almost no new results but

presents a history and methodology of the subject, as weil as an introduction to terminology

requisite for an understanding of subsequent papers in the first part. The second part is devoted to

the study of automorphism groups of certain combinatorial objects such as diagram geometries,

distance-regular graphs, edge- but not vertex-transitive graphs, Hadamard matrices and structure

formulas of chemical compounds. This part opens with a paper by S.V. Shpectorov, in which a

new method for the characterization of certain combinatorial objects (diagram geometries and

graphs) is proposed which is based on a consideration of rank 3 amalgams. The amalgam method

PREFACE TO THE RUSSIAN EDITION xi

also appears in the paper by A.A. Ivanov and M.E. Ioftnova in the collection. We believe that the amalgam method will perhaps form what is ultimately the most important methodological foundation for the algebraic theory of combinatorial objects.

Unfortunately, some very interesting papers had to be excluded from the collection because of

volume limitations. Nonetheless, we believe that the resulting condensed issue adequately represents the ideology, techniques and achievements of the seminar participants.

The numerous cross references among the papers of the collection gives but a partial indication of the close collaborative efforts and mutual scientific interests of the authors. We did not wish to convert the introductions of the papers to a list of arcs in an almost complete multigraph of acknowledgements for useful advise and discussions, more precise definitions, etc. So we

requested that the authors Iimit their expressions of gratitude to a single collective sentiment, which we echo at this point of the preface. Nevertheless, we would be remiss in not singling out

for mention the many contributions of A.A. Ivanov, who has deeply scrutinized all of the papers with a high degree of enthusiasm and competence, proposed many valuable comments, removed

a number of inaccuracies, and improved upon the overall exposition. In this manner he has greatly facilitated the work of the editors, and for this he has our heartfelt thanks.

M.H. Klin

I.A. Faradrev

Part 1. Cellular Rings CELLULAR RINGS AND GROUPS OF

AUTOMORPHISMS OF GRAPHS I.A. Faradiev, M.H. Klin, M.E. Muzichuk

Introduction

The present article serves as an introduction to the first part of the book. In particular, it contains the terminology necessary to understand all articles from the first part, and also several articles from the second and third parts of the book. In the Russian edition a similar function was performed by the paper [*Kl 9], where the author considered elementary concepts of the theory of association schemes and gave a thorough exposition of different axiomatic analogues and interpretations of this theory. The last paragraphs of [*Kl 9] were written in collaboration with A.A. Ivanov and were significantly influenced by a book of E. Bannai and T. Ito [Ba 5] which had not been translated into Russian at that time. At the present time there is no need for them, since the book [Ba 5] has become widely known and accepted as fundamental for the theory of association schemes. For this reason, D. Reidel Publishing Company suggested that we write a new introductory article, which is a survey of some results of Soviet mathematicians related to applications of association schemes. Similar newly written art'icles open the other two parts of this book.

It is worth mentioning that several other surveys have been published; see [*Ka 7], [*Kl 10], [Fa 2]. The Russian version [*Iv 7] of [Fa 2] appeared as an appendix to the translation [*Ba 2] of the book [Ba 5] in Russian. However, volume restrictions on those surveys permitted a detailed consideration of only certain papers, with others represented by a list of results with short comments. Also in the book [Kl 7] some of the important concepts are treated in an elementary fashion and are illustrated by examples.

This work is based on the results of participants of our seminar (more details are given in the introduction to the Russian version andin section 1.4). Here, for the first time, a detailed exposition of methods developed and used by us is given. In general the presentation is self-contained ( especially in Chapter 2). For this reason, this article can be used not only as a source of facts published in Russian and unfamiliar to the native English reader, but also as a textbook on some topics of algebraic combinatorics.

The concept of a cellular ring ( cellular algebra), introduced by B.Yu. Weisfeiler and A.A. Leman [*We 1] in 1968, is fundamental for the whole article. It is equivalent to the concept of adjacency algebra of a coherent configuration. Also the concept of a cellular subring is introdueed, and the Galois correspondence between

2 I. A. FARADZEV ET AL

permutation groups acting on a set n and cellular rings over n is established. Then we discuss a methodology of application of the correspondence to the studies of permutation groups and groups of automorphisms of graphs. All this, along with a discussion of notation, terminology, axiomatics of cellular rings and historical remarks, forms the content of the first chapter. Chapter 2 occupies the central place in this survey, where the techniques used in the solutions to the following problems are presented:

construction and description of the centralizer ring of a given permutation group, answering questions about the primitivity of a cellular ring;

enumerating the cellular subrings of a given cellular ring;

determining the automorphism group of a cellular ring ( under the condition that information about a subgroup of the automorphism group is known);

finding those cellular rings which are centralizer rings of a suitable permutation group;

computer implementation of developed algorithms for computing in cellular rings and methodology for utilizing these programs. The technique we describe is illustrated by numerous examples which, we think, are interesting in their own right.

Chapter 3 includes a survey of some results obtained by Soviet mathematicians who used the technique described above. These results are mainly devoted to the enumeration of cellular subrings of cellular rings which are BM-algebras of some classical association schemes (Johnson schemes, Hamming schemes and their q-analogues). The determination of the automorphism groups for some of these subrings is also considered. Salutions of these and some other problems are given with more detail than in our previous surveys: we give precise formulations of all necessary statements and supply short sketches of proofs. Chapter 3 concludes with brief comments on those results which are not discussed in the main part of the survey.

We assume that the reader is familiar with basic notions and facts about permutation groups, graphs and association schemes. In all cases where an ambiguity in terminology arises, one can consult any of the following books: [Ha 5], [Wi 2], [Ha 10], [Ba 5]. The survey [Fa 2] is also useful, since some questions are discussed there more thoroughly than here. All cases of specialized usage of terminology and notation are specifically pointed out, e.g., we use classical notation for the wreath product of permutation groups (G. P6lya, L.A. Kaluznin), where the active group (rough in P6lya's terminology) is written on the left and the passive one on the right (see section 3.3). All references from Chapters 1 and 3 are in the text, but

CELLULAR RINGSAND GROUPS OF AUTOMORPHISMS OF GRAPHS

those from Chapter 2 appear at the end of each section. This is dictated by the additional tutorialload which we imposed on Chapter 2; we did not want to distract the reader's attention from the main line of presentation. There is a very simple explanation to our seemingly pedantic approach in the references: there are many results in the text which can be considered as folklore as well as those which were obtained independently, and almost simultaneously, in the USSR and abroad. In these cases we wanted, if not to confirm our priority, then at least to avoid undeserved accusations of plagiarism. On the other hand, we apologize in advance for all cases where a publication relatecl to one of the questions cliscussed in the text escaped our knowleclge. In general we tried to include the results of foreign authors wherever it was suitable. Note also that references of articles which appear in Russian c:arry an asterisk when cited in the body of text, e.g. [* Acl 1]. The reader is thereby referred to a second bibliography carrying the heading of "Russian References." As usual, the so-called "MR( new )" system for transliteration of Cyrillic is used. A few exceptions go back to other previously establishecl Romanizations occurring in English literature.

Finally we mention that the beginning and end of a proof are marked by ..,... and ..,., respectively, ancl the end of an example or a statement given without proof is marked by •·

4 I. A. FARADZEV ET AL.

CHAPTER 1. CELLULAR RINGS (GENERAL FACTS)

1.1. Questions of axiomatics

The main object of research in this paper will be the notion of a cellular ring, introduced by B.Yu. Weisfeiler and A.A. Leman [*We 1 J in connection with the first serious attempt to consider aspects of the complexity of the graph isomorphism problem. This was thoroughly considered in the collective monograph [We 2].

A cellular ring W of degree n isaring of n x n integer matrices which satisfy the following axioms:

(W1) W has a basis A = {A1 , ... ,A,.} of {0, 1}-matrices;

r

(W3) LAi = J, where J is the matrix with all entries equal to 1; i=l

(W4) the nurober of ones in each nonzero row of Ai is the same and is equal to ki ' 1 s; i s; 1'.

One can show that axioms (W1)~(W4) define a basis of the cellular ring uniquely, up to the order of its elements. This basis is called standard, and the fact that cellular ring W has standard basis A = {A1 , ... , A,.} is written as W = (A1 , ... , A,.). The cardinality r of the basis of the cellular ring is called its rank. One can embed the cellular ring W in the matrix algebra generated by A over the field C of complex numbers. We refer to this algebra as a cellular algebra. We will usually use the same notation for the cellular ring and the cellular algebra having the same basis.

Let us fix an n-element set n. In what follows we will usually assume that n = { 1, 2, ... , n}. Then a basis matrix Ai can be interpreted as an adjacency matrix of a digraph fi = f(A;) whose set of vertices is n and whose set of arcs is R; = R(A;). The corresponding graphs f; and relations R; are called the basis graphs and basis relations of the cellular ring. According to axiom (W3), the collection of all basis graphs of a cellular ring W forms a partition of a complete digraph which can be considered as a complete colared digraph f(W), where an arc (i,j) is assignecl color s if and only if (i,j) E Rs, 1 s; i,j s; n, 1 ::=; s s; r. In studying cellular rings, we will use the languages of matrices, graphs and relations interchangably, feeling free to switch from one to another. In particular, the notations W = (f1 , ... ,f,.) and W = (R1 , ... , R,.) will be used. At the same time, we will use the names for the matrix operations of addition and multiplication when discussing operations in the ring, although they correspond to union and convolution of relations (more

CELLULAR RINGSAND GROUPS OF AUTOMORPHISMS OF GRAPHS 5

precisely, of multirelations ).

We would also like to point out that according to axioms (Wl) and (W3) a cellular ring is always closed under one more operation, namely the Hadamard product ( o) of matrices. In terms of relations the corresponding operation is intersection.

Fora fixed n, we denote by r the functor which associates a graph to a matrix or a relation, and by R, the functor which associates a relation to a matrix. The adjacency matrix of a graph r = (n, R) is denoted by either A(r) or A(R).

It is easy to show that axiom (W4) is not independent, that is, that it can be derived from axioms (Wl)-(W3). The independence of axioms (Wl) and (W3) is obvious. The independence of axiom (W2) may be shown by means of the following example. Let N = {1,2, ... ,n}, n?:: 4, andn = N2 = {(i,j): i,j E N,i #j}. Let i, j, k and l be pairwise distinct elements of N. Let us consider the following binary relations Oll r:

R1 === {((i,j), (i,j))},

R2 = {((i,j),(j,i))},

R3 = {((i,j), (i, k))} u {((i,j), (k, i))},

R4 === {((i,j), (j, k))} u {((i,j), (k,j))},

R5 = {((i,j), (k, l))}.

It is easy to checkthat the ring W = (R1, R2, R3, R4 , R5) satisfies axioms (Wl) and (W3) but does not satisfy (W2).

The axioms of a cellular ring do not imply that the ring has the multiplicative identity, i.e., the identity matrix I is not necessarily in W. For example, Iet moK1

be th'~ disjohlt union of m complete l-vertex graphs, moK1 be the complement of moK1, and ~ml = r(I) be a reflexive graph on ml vertices formed out of loops only. Then W = ( m° Kz + ~ml, m° Kz) is a cellular ring, but I (/_ W. Sometimes the consideration of cellular rings without the identity matrix Ieads to fruitful results. See, for example, [*Mu 8]. Nevertheless, in this paper in the future we eonsider only eellular rings with identity, without mentioning this eaeh time.

The multiplication in a eellular ring (All ... , Ar) is defined eompletely by a tensor pt, 1::::; i,j,k::::; r, suchthat

r

Ai 0 Aj = I>7jAk. (1.1.1) k=l

6 I. A. FARADZilV ET AL.

The nurobers p~j are called structure constants of the ring W. For given i, j, k

p~j can be interpreted as the nurober of triangles in the colored graph r(W) which have one fixed edge of color k, and in which the other two edges have colors i and j.

Structure constants of a cellular ring satisfy the following relations:

k k' Pij = Pj'i';

r r "'s l "'I s L...-Pij · Psk = L...-Pis · Pjk· s=l s=l

(1.1.2)

(1.1.3)

A cellular ring with all of its basis graphs regular is called a cell. In a cell with identity, the basis graph which consists entirely of loops will usually be denoted by r 1 . For every basis graph r; the nurober k;, defined by (W 4), denotes the valency of r;. For the structure constants of a cell the following additional relations hold (see [We 2]):

r r r

LPL=:LP;;=k;, Lk;=n; j=l j=l i=l

r

L Pij · ks = k; · kj; s=l

(1.1.4)

(1.1.5)

(1.1.6)

(1.1. 7)

(1.1.8)

Let W = (A1, ... ,Ar) be a cellular ring, and let a subring W' ~ W satisfy axioros (W1 )-(W4) and have standard basis (A~, ... , A~). Then W' is called a cellular subring of cellular ring W ( denoted W' s; W). It is easy to see tlmt each basis eleroent A~ of cellular subring W' can be obtained by suroroing soroe basis eleroents of cellular ring W, that is, the basis { A ~, ... , A~} can be considered as a partition of the basis { A1, ... , Ar}. More precisely, there exists a partition of the set { 1, 2, ... , r} and a one-to-one roapping that associates a dass Tx of the partition to every x E {1, ... , s} in such a way that A~ = L A;.

iET~

Constructions leading to particular classes of cellular rings have appeared in different branches of roatheroatics ( roainly in group theory and corobinatorics) nuroerous tiroes. (For roore details see section 1.4.)

CELLULAR RINGS AND GROUPS OF AUTOMORPH!SMS OF GRAPHS 7

I. Schur [Sc I] considered the centralizer ring of a permutation group having a regular subgroup, in connection with the study of B-groups. R. Kochendörffer and H. Wielandt (see [Wi 2]) generalized Schur's ideas to arbitrary permutation groups and made the centralizer ring (Vertauschungsring in H. Wielandt's terminology, or more briefly V -ring) a powerful tool in the study of permutation groups.

Let ( G, !1) be a permu tation group and M (g) be the permutation matrix corresponding to a permutation g E (G, !1). The ring of n x n integer matrices, where n = 1!11, which commute with M(g) for every g E (G, !1) is called the centralizer ring (or V-ring) of the permutation group (G, !1), and is denoted as V(G, 0). It is easy to see that V(G, !1) is a cellular ring with standard basis { A1, ... ,Ar} whose elements form the set 2-orb( G, n) of orbits of the induced action of ( G, !1) on the set n X n (2-orbits of the group (G,n)). If the action of Gon n is transitive, then V(G, !1) is a cell. In this case, the rank of the cell and the valencies ki of the basis graphs coincide, respectively, with the m1mber and cardinalities of orbits of the stabilizer Ga of a point o: E n (i.e., with the rank and subdegrees of the group (G, !1)).

It is not true that every cellular ring is the V-ring of a permutation group. The first examples of such cellular rings ( they are called non-Schurian rings) were given in [Sh 4], [Wi 2] and [*Ad 1].

The results of I. Schur, which he obtained by using centralizer rings, can be proven also by means of the traditional technique of computing with double cosets. This is due to the one-to-one correspondence between double cosets of G with respeet to a subgroup F and 2-orbits of the action of G on the right cosets of this subgroup.

Let F be a subgroup of G, let {Fg1F, ... , Fg8 F} be the set of double cosets of G with respect to F, and let n be the set ofright cosets of G with respect to F. It is easy to check that the set Ai = { ( o:, ß) : o:, ß E !1, ß = o:9_, g' 9 for some g E G, g' E

FgiF} represents a 2-orbit of the transitive permutation group (G, !1). Conversely, if o: E: n is the coset Fe = F, then the set {g : g E G, (o:, o:9) E Ai} is a double coset of G with respect to F. In particular, this implies that s is equal to the rank of V(G,n).

Let us consider the group algebra A(G) formed by the formal series 2:: a(g)g gEG

with the operations of addition, multiplication, ancl multiplication by a number. To a double coset FgiF we associate an element Qi E A(G), where

I. A. FARADZEV ET AL

Since the number of representations g = g'g" of an element g E FgiF (with g' E FgjF and g11 E FgkF) does not depend on the choice of g E FgiF, and is equal to a multiple of IFI, we have

Qi. Qj = 2:.>7jQk, (1.1.9) k

where tt are nonnegative integers.

Thus we see that the set of linear combinations of elements Qi, 1 ~ i ~ r, with integer coefficients, forms a ring, which is called the Hecke ring H(G, F). It turnsout that the intersection indices tfj of the Hecke ring H(G, F) and the structure constants pfj of V( G, D) are related by the equalities tfj = pji, i.e., the rings H(G,F) and V(G,D) are anti-isomorphic.

There is a description of a V-ring of a permutation group having a regular subgroup which is very elegant. The corresponding axiomatics suggested by I. Schur [Sc l]lead to the concept of an S-ring.

Let R be a group and A(R) be its group ring, i.e., the set of formal sums 2..:: a(g)g with integer coefficients. If T ~ R, then the element T_ = 2..:: g of the

gER gET

group ring is called a simple quantity. A subring ß of the group ring A(R) is called an S-ring over R if it satisfies the following axioms:

(SI) ß has a basis {T1 , ... , Tr} formed of simple quantities;

r

(S2) T1 = e, the identity element of R, and 2..:: Ti = E., i.e., {T1 , ... , Tr} is a i=l

partition of R;

Let us consider a permutation group ( G, R) containing a regular subgroup ( R, R). Let Ge be the stabilizer of the identity element e E R in ( G, R) and {Tl, ... , Tr} be the orbits of Ge acting on R, with T1 = { e }. The set of integer linear combinations of T1, •.• , Tr is called the transitivity module ß(G, R) of the permutation group (G, R) (sometimes the notation ß(Ge, R) is used). It is easy to check that ß( G, R) is closed with respect to multiplication and satisfies axioms ( S 1) (S3), i.e., tlmt it is an S-ring over R. It is also easy to establish the isomorphism of ß( G, R) with the Hecke ring H( G, Ge), and therefore the anti-isomorphism with the V-ring of the group (G, R), the cellular subring of the V-ring of the group (R, R).

It is not true that every S-ring over R is a transitivity module of a permutation group (G, R), i.e., there exist non-SchurianS-rings (e.g., see [Wi 2]). Nevertheless,


it turns out that there exists a one-to-one correspondence between S-rings over R and cellular subrings of V(R, R).

Let us now fix the terminology, introduced by D. Higman ([Hi 2], [Hi 3]), of the theory of coherent configurations.

Let n be a set of cardinality n and let 0 be a family of binary relations on n (subsets of the set fl X fl) that satisfies the following axioms:

(Cl) 0 is a partition of S1 x fl;

(C2) if f E 0 and f n I i= 0, where I= Io = {(x, x): x E fl}, then f CI;

(C3) if f E 0 then jl E 0, where jl = {(y,x): (x,y) E f};

(C4) for every f, g, h E 0 and (x, y) E h, the number p19 (x, y) of elements z E n suchthat (x, z) E f and (z, y) E g does not depend on x and y.

A pair (n, 0) satisfying axioms (CI )-(C4) is called a coherent configuration on n, and the family 0 is called a coherent collection of binary relations on n. A coherent configuration is called homogeneous if, in addition, the following condition holds:

(C5) I= I0 E 0.

The axioms for a coherent configuration represent a natural combinatorial abstraction of the properties enjoyed by the family of 2-orbits of an arbitrary permutation group acting on n. A pair (f2,2-orb( G, n) ), where ( G, fl) is a permutation group, is always a coherent configuration. It is homogeneaus if G acts transitively on n.

It is easy to see that the class of coherent configurations is equivalent to the class of cellular rings with identity. We note that the class of cellular rings is larger, since, as was noted before, there are cellular rings without identity.

Let us consider now the best known and most commonly used axiomatics for the theory of association schemes.

Let n be a set of cardinality n, and let ~, 7 = 0, 1, ... , d, be subsets of n X n which satisfy the following conditions:

(Al) Ro = {(x, x) : x E fl};

(A2) Ro U Rt U ... U Rd = n x n, ~ n Rj = e1 for i i= j;

(A3) R~ = Ri' for some i' E {0, 1, ... , d}, where R~ = {(y, x) : (x, y) E ~};


(A4) for every triple i,j,k E {0, 1, ... ,d}, and for given (x,y) ERb the number pfj of elementszEnsuch that (x,z) ER; and (z,y) E RJ does not in fact depend on the choice of the pair ( x, y).

Then M = (n, { R;}f=o) is called an association scheme on n with d classes, and the numbers pfj are called the intersection numbers of M. Thus the class of association schemes coincides with the dass of homogeneaus coherent configurations and therefore with the class of cells with identity.

If an association scheme satisfies the condition

(A5) i' = i for all i E {0, 1, ... , d}

then it is called symmetric or a Hose-Mesner type scheme.

For an association scheme M = (n, {R;}f=0 ), we denote by A; = A(Ri) the adjacency matrix of the graph ri = (n, R;) (the i-th adjacency matrix of the scheme M).

As follows from axioms (A1)-(A4), (Ao, ... , Ad) is a cellular ring. At the same time, if we consider all linear combinations of matrices Ao, ... , Ad with complex coefficients we will obtain the cellular algebra A(M), which is traditionally called the adjacency algebra of the scheme M, or its Hose-Mesner algebra (HMalgebra).

In what follows in this article and other articles of this collection, we will freely use two main languages, the ones of cellular rings and association schemes. In the latter case, the element of the basis of the BM-algebra which is the identity matrix is denoted by A0 .

Traditionally, the language of association schemes is used when M is a syrnmetric scheme. The syrnmetry of an association scheme implies the comrnutativity of its BM-algebra, and the theory of commutative BM-algebras has become in the last decade the most developed part of algebraic combinatorics.

In order to keep the presentation self-contained, we Iist below some of the most important facts about BM-algebras. (For more, see [Ba 5], Chapter 2.)

The following statement (Theorem 2.3 in [Ba 5]) gives an isomorphic representation of the adjacency algebra of a scheme M by matrices of order d + 1.

Theorem 1.1.1 Let M be a symmetric association scherne with d classes having Ao, ... , Ad as adjacency matrices and pfj as intersection nurnbers. Consider matrices B; of order d + 1 defined by (B;)jk = pfj, and Iet ß be the subalgebra of the matrix algebra over C, of order d + 1, which is generated by B0 , ... , Bd. Then the algebra


A(M) is isomorphic to the algebra ß, where the isomorphism maps the matrix A; to the matrix B;. In particular, the matrices A; and B; have the same minimal polynomial. •

Let A(M) be the adjacency algebra of a symmetric association scheme M. We denote by M the space of column vectors of length n over C. Then A(M) can be considered as an algebra of linear operators over M. lt is known (see, e.g., [Ho 2]) that M admits the decomposition

M =Mo ffi Mt ffi ... ffi Md

as the direct sum of common eigenspaces of the algebra A(M). ForA E A(M) we denote by Pi ( A) the eigenvalue of matrix A which corresponds to the subspace Mi. Then PJ(A' + A") = PJ(A') + PJ(A"), PJ(A' · A") = PJ(A')PJ(A") and PJ(o:A) = o:pj ( A), for o: E C and 0 ::; j ::; d. Let us denote by m; the dimension of the

d

subspace M;, so that 2:::: m; = n. A(M) has a basis Eo, ... , Ed consisting of minimal i=O

mutuaJly orthogonal idempotents characterized by the property PJ(E;) = Oij· Then E; · Ej = O;jE;, and one may assume that Eo = ~.ln, where .ln is the n x n matrix with all entries equal to one. Then

d

Ai= L>J(i)EJ, 0::; i :S: d. j=O

Here PJ(i) = PJ(A;) is the eigenvalue of A;, which is also the eigenvalue of B; (Theorem 1.1.1). Let

The matrices P = (Pj ( i)) and Q = ( qj ( i)) of order d + 1 are called the first eigenmatrix and the second eigenmatrix of the association scheme M. We remark that P · Q = Q · P = nld+l, where Jd+l is the identity matrix.

Since Pj ( i) is an eigenvalue of B;, the eigemnatrices P and Q are determined uniquely by the intersection numbers P7j.

Let us consider the operation of the Hadamard product o on the set of matrices of the algebra A(M). Wehave A; 0 Aj = O;jA;, therefore {A 0 , ... , Ad} is a basis of the algebra A.( M) consisting of minimal mutually orthogonal ( with respeet to the Hadamard product) idempotents.

12 L A. FARADZEV ET AL.

Let

The numbers qt are called Krein parameters. Theseparameters have no explicit combinatorial interpretation, but in a certain sense they are dual to the intersection numbers pfj· The following statement (Theorem 3.6 from [Ba 5]) supports this claim.

Lemma 1.1.2

k m;mj ~ 1 --qij = -n- L k2 pz(i) · pz(j) · pz(k),

l=O l

kk d 1 --Pk· = _i_i L - 2 qz(i) · qz(j) · qz(k).

'J n l=D mz

The bar here denotes the complex conjugate. • In particular, Lemma 1.1.2 implies that the Krein parameters are defined

uniquely by the intersection numbers.

An important property of the Krein parameters was proved by L. Scott [Sc 2] who used some results of M.G. Krein [*Kr 1]. It is stated in the following

Lemma 1.1.3 The Krein parameters arenonnegative numbers. • A simple and clear proof of this Iemma can be found in the book [Ba 5] (The

orem 3.8 and Lemma 3.9).

Throughout this article the bases {A0 , ... , Ad} and {E0 , ... , Ed} of a commutative BM-algebra will be called the first standard basis and the second standard basis, respectively. In all cases where we use the first standard basis only, we will refer to it just as the standard basis, as we did earlier.

1.2. Invariant relations of permutation groups

Let (G, !1) be a permutation group, i.e., a faithful action of Gon an n-element set !1. Let !lk be the k-th Cartesian power of !1, i.e., the set of all sequences of elements of !1 of length k. The elements of !lk will be called k-points. Bachsubset ~ nk is called a k-relation (a k-ary relation). A relation is called antireflexive if all the coordinates of each k-point of are distinct. A relation is called symmetric if, for every (a 1 , ... , ak) E and every permutation h from the Symmetrie group sk, ( al h' a2h' ... , akh) E . To any anti-reflexive k-relation <1>, one


can associate the collection of k-element subsets { {a1 , ... , ak}i(ab ... , ak) E «P} of n. In this c:ase we talk about a symmetrized k-relation.

We define an action of an arbitrary permutation g of the symmetric group Sn = S(n) Oll k-points of nk as follows: (al, ... , ak)g = ((al)Y, ... , (ak)Y). Let q, be a k-relation Oll n. Then q,g is defined as q,g == { aY : a E q, }. A relation q, is called invariant with respect to a permutation g E Sn if q,g = «P. We say that «P is invariant with respect to the group ( G, n) if it is invariant with respect to every permutation of the group.

By k-rel( G, 0) we denote the set of all k-ary relations invariant with respect to the group (G,n). Nonempty relations from k-rel(G,n) which areminimal with respect to inclusion are called k-orbits of the group ( G, n). The set k-orb( G, n) of all k-orbits of the gTOUp ( G) n) always forms a partition of the set nk, and every relation from k-rel( G, n) is a union of some elements from k-orb( G, 0).

For a permutation group ( G, n), we denote by lnv( G, n) the set of all its invariant relations:

Inv(G,n) = U k-rel(G,n). k=l

Usually one considers only the finite part U~=l k-rel( G, n) of the set Inv( G, n), since all the relations of lnv( G, n) can be uniquely reconstructed from the k-relations, k::; n.

For every set of relations {«P1, ... ,«Ps} on n, let Aut(«P 1, ... ,«Ps) denote the automorphism group of the set, where

Aut(«P1, ... ,«Ps) = nAut(«Pi), i=l

and Aut( «Pi) denotes the group of all permutations of Sn which map «Pi into itself.

Since {«P 1, ... ,«Ps} ~ lnv(Aut(«P 1, ... ,cP8 )) and (G,n) ~ Aut(Inv(G,n)), the pair of functors Inv and Aut establishes a Galois correspondence between the sets of permutationsOll n and the sets of relationsOll n. The Galois-closed objeets under this correspondence are all permutation groups as well a."i the so-called Krasner alge bras ( [*Bo 1], [*Ka 1]), which are sets of relations closed und er a particular collection of operations performed on the relations. These Operations obey the rules of formulae construction of the first order languages (FOL). The relations in this ca.se are interpreted as domains of truthfulness for the predicates (see [*Ro 2], [*Ka 1]). The term "Krasner algebra" is chosen in honor of M. Krasner, who was the first to eonsider the Galois correspondenee described above in [Kr 1].

If (H, n) and (G, n) are two permutation groups Oll n and H:::; G, then Gis called an overgroup of Hin the symmetric group So. The lattice of overgroups of a group ( G, n) is anti-isomorphic to the lattice of subalgebras of the Krasner algebra Inv(G, n). Therefore the study of the structure of the lattice of overgroups can often be reduced to the consideration of some combinatorial properties of Krasner algebras.

Let il>r, il>2, ... , il>s be arbitrary relationsOll n and let l]i be a k-relation Oll n. We will say that \If is formula-expressible in iJ>1, ... , iJ> 8 Oll D if there exists a formula F(i1> 1, ... ,il>8 ;zr, ... ,zk) in FOL, with free variables z1,z2, ... ,zk, whose domain of truthfulness is l]i. In this case we write

il>1, ... , il>s ~0 W.

We will use just the symbol ~ when n is clear from context.

Two families of relations iJ> = { il>;} and l]i = { l]i J} are called formulaequivalent if each relation w1 E l]i is formula-expressible in the relations of 

and vice versa.

Formula-equivalent relations generate the same Krasner algebra and therefore have the same automorphism group. Thus the Galois correspondence mentioned above gives a one-to-one map between permutation groups and their Krasner algebras: a given permutation group is mapped to its Krasner algebra and at the same time to its class of formula-equivalent relations. Any information about a permutation group ( G, n) can be obtained principally from information about an arbitrary set of relations which generate the algebra Inv( G, n). From the combinatorial point of view, those permutation groups which are completely defined by the information contained in a set of k-relations, where k is small, are of special interest.

Following H. Wielandt [Wi 3], we call two permutation groups (G1, D) and (G2 , D) k-equivalent (and denote this by G1 ~k G2 ) if k-rel(G1 , D) = k-rel(G2 , D). It is clear that two groups are k-equivalent if and only if they have the same set of k-orbits. All k-equivalent groups share many common properties (see [Wi 3]).

Among all k-equivalent permutation groups there is a unique one which is maximal with respect to inclusion, i.e., the one which contains all groups from the given class of k-equivalent groups. Set G(k) = Aut(k-rel(G, D)). The group G(k)

is called the k-closure of the group ( G, n). If G(kJ = G, then the group ( G, D) is said tobe k-closed. G(k) is exactly the maximal group in the class of k-equivalent permutation groups. For a group ( G, n) the property of being k-closed means its Krasner algebra is generated by the invariant k-ary relations. The following inclusions hold for k-closures:

G = c<nJ <;;;: c<n-l) <;;;: .•• <;;;: G(k) <;;;: ... <;;;: G(2l <;;;: G(ll <;;;: So.

For a given permutation group ( G, f2), we denote by a( G) the least value k suchthat (G, n) is k-closed. Some relationships between a(G) and other numerical invariants are studied in [*Ka 3].

In what follows the case considered will almost always be k = 2, and primary attention will be given to transitive groups. When binary relations are discussed, it is convenient and conventional to use the language of graph theory. We have used it already in section 1.1. Herewe just want to note that a permutation group (G,n) is 2-closed if and only if there exists a complete colored digraph r with vertex set n, suchthat Aut(r) = (G, n). The partition of the set of arcs of this graph induced by the colaring forms a set of relations which generates the Krasner algebra Inv( G, n).

Below we give a natural interpretation of the concept of a cellular ring in terms of invariant relations. In order to do this we need several more concepts.

Each function J : f2k -+ Z is called a k-function (or k-multirelation) on n. Following H. Wielandt [Wi 3] we say that fisinvariant with respect to a permutation g E Sn iffor every (xr, ... ,xk) E nk, f((xr) 9, ... ,(xk)9) = f(xr, ... ,xk)· If f is invariant with respect to all permutations from (G, n), then we say J is invariant with respect to ( G, f2). The set of all k-functions invariant with respect to (G, f2) is denoted by k-Rel(G,n). The automorphism group of a k-function can also be de

fined naturally. It is clear that Aut(J) = n!=r Aut(<l\), where { 1 , ... , 1} is the set of all distinct "level surfaces" of f (i.e., i = {(xr, ... ,xk) E nk: f(x 1, ... ,xk) = ci} where Ci E Z). We notice also an obvious inclusion k-rel(G, n) ~ k-Rel(G, n), which can be obtained from viewing the relations as functions with values 0 and 1. All the set theoretic operations of a Krasner algebra, in pa.rticular the operations of union, intersection, convolution and projection, can easily be extended to the k-functions in such a way that the set U~=r k-Rel( G, f2) turnsout tobe closed with respect to all these operations. (The projection operation will be considered in section 2.1.) It also turns out that the set k-Rel( G, f2) will be closed with respect to the same operations. Here we restriet ourselves to the operations of union, intersection and convolution.

Every 2-function with nonnegative values will be interpreted as a complete oriented multigraph with vertex set n and in which the multiplicity of each edge is equal to the value of the function. The union and intersection of functions f and g are obtained by taking the union and intersection of the multisets of their arcs. The c:onvolution f * g of the functions f and g is a function h such that

h(x, y) =o L f(x, z)g(;~, y). zE!!

Now let 2-orb(G, n) = {Ao, A1 , ... , Ad}, and let Ii be the function corresponding


to Ai. Then the following is obvious:

A(fi n fi) = A(li)oA(fi)

A(J; * fj) = A(fi)A(Jj)

Extending these operations by linearity on the set of all 2-functions, we arrive at an isomorphism between the algebra of 2-functions from 2-Rel( G, n), with Operations U, n, * and multiplication by a number, and the V-ring of the permutation group ( G, n). An isomorphism between an arbitrary set of 2-functions Oll n, closed with respect to these four operations, and the cellular ring on n can be obtained in precisely the same way.

From our point of view, the existence of this isomorphism explains the role of the theory of cellular rings (association schemes, coherent configurations) in modern algebraic combinatorics. These are the most natural among all algebras of multirelations which possess the essential core of information regarding the corresponding Krasner algebras. This statement will be made more precise in the next section.

1.3. Galois correspondence between cellular rings and permutation groups

If ( G', n) is an overgroup of a permutation group ( G, n), then obviously V( G', n) is a cellular subring of the V-ring V( G, n). On the other hand, if W' is a subring of a cellular ring W, then all elements of W' are formula-expressible in a basis of the ring W, so that Aut(W):::; Aut(W').

In addition, the following inclusions hold:

Aut(V(G)) 2: G;

V(Aut(W)) 2: W.

It follows from this (see [Fa 2] for details) that the pair of maps (Aut,V) defines a Galois correspondence between cellular rings and permutation groups. V-rings of permutation groups and 2-closed permutation groups are the Galois-closed objects of this correspondence.

A cellular ring which is a V-ring of some permutation group is called a Schurian or Schur type cellular ring ( for clarification of the terminology see section 1.4). Otherwise we speak about non-Schurian or non-Schur type cellular rings.


The Galois correspondence presented here is coarser than the one described in section 1.2:

distinct 2-equivalent permutation groups have the same V-ring but different Krasner algebras;

distinct cellular rings can have the same group of automorphisms and therefore generate the same Krasner algebra.

Despite this fact, cellular rings are quite suitable combinatorial objects to use to distinguish 2-closed permutation groups, because the Galois correspondence discussed above establishes a one-to-one correspondence between 2-closed permutation groups and Schurian cellular rings. This correspondence will be the subject of our further research. In comparison with the Galois correspondence from the previous section, this one has an important advantage: If the relation of formulaexpressibility in the Krasner algebra is hard to see, the relation of cellular computability can be checked in polynomial time.

Let A be a square matrix of order n. By (A) we derrote the matrix algebra generated by A, i.e., the set of all matrices of the form p(A), where p(x) runs over all polynomials with complex coefficients. Next, by [A] wederrote the cellular algebra generated by A, i.e., the minimal cellular algebra containing A. We note that if A is an integer matrix then the same notation [A] will be used for the cellular ring generated by A. It is clear that (A) c [A]. In general, the opposite inclusion may not hold. (The simplest example is the matrix A = A(n°Km), the adjacency matrix of the disconnected union of n copies of the complete graph on m vertices; less trivial examples can easily be constructed by using, for example, some results from [Be 1]). An explanation for this is that a cellular algebra, in contrast to the usual matrix algebra, must always have a basis formed by (0, 1)-matrices. Therefore it is closed under the complementation of a graph and under the operation of the Hadamard product, the use of which enables one to construct the first standard basis of the cellular algebra. An algorithm for stabilizing a matrix A of order n ( the WeisfeilerLeman algorithm, which first appeared in (*We 1]) allows the construction of [A] in polynomial time. It can be found in [We 2].

We can now introduce the following definitions. Let A and A' be two square matrkes of order n. The matrix A' is cellular computable ( or just computable) through A if A' E [A]. The matrix A' is polynomial computable through A if A' E (A). The matrices A and A' are cellular equivalent if each is cellular computable through the other. They are polynomial equivalent if they are polynomial computable through each other.

It is obvious that polynomial computability (equivalence) implies cellular com-

18 1. A. FARADZEV ET AL.

putability ( equivalence), and that this in turn implies formula-expressibility ( equivalence). The converse is not true in general. It is clear that A and A' are cellular equivalent if and only if [A] = [A']. Therefore cellular equivalence can be established in polynomial time. As an example of polynomial equivalence we consider, following [We 1], a regular graph r and its complement r, in the case when both graphs are connected. (Using methods of linear algebra, one can show that an adjacency matrix A(f) is polynomial computable through A(f) if r is connected.) We note that r and r for an arbitrary r are cellular equivalent.

The implication [A] = [A'] ===:;. Aut(f(A)) = Aut(f(A')) gives a convenient way to describe groups of automorphisms of graphs. Its simplest application is to show that the automorphism groups of r and r are the same. A more interesting example follows. But first we shall formulate a very useful lemma.

Lemma 1.3.1 Let W = (A1, ... ,Ar) be a cellular ring, B = (bij) E Wand C = (cij), where for some c

c·. - "J' { b· t]- 0,

Then CE W.

if bij = c; otherwise.

• This obvious fact, which goes back to I. Schur, is a slight generalization of

Proposition 22.1 from [Wi 2]. Here we shall use it in the classical situation of S-rings.

Example 1 Let (Zn, Zn) be the regular action of a cyclic group of order n on { 0, 1, ... , n - 1} generated by the permutation x f-+ x + 1 ( mod n) and let n =

2k. Further, let Cn = f(1,n- 1) be the graph corresponding to simple quantity 1, n- 1 from the S-ring over Zn. Cn isasimple cycle of length n. It is clear that Aut(Cn) = Dn, the dihedral group of degree n and order 2n. Following Guy and Harary [ Gu 1], by M2k we denote the Möbius ladder of order k, i.e., the graph f(l,k,2k-1) (denoted by W2k in [Be 1]). The group of automorphisrns Aut(M2k) of the Möbius ladder was studied in [Si 1] and [Wa 1], where sorne special tricks were applied. Let us show that Aut(M2k) = D2k for k > 3. In what follows the graph is identified with its adjacency matrix.

Since the graph C2k is a distance transitive graph (d.t.g.), then f(k) E [C2k] and therefore M 2k E [C2k]· Let us now carry out calculations in the S-ring over Zn. Wehave (1, k, 2k- 1)2 = 3 · Q + 2, 2k- 2 + 2 · k- 1, k + 1. From this, 2, 2k- 2 is obtained through 1, k, 2k- 1 and we can compute 1, k, 2k- 1 · 2, 2k- 2. Wehave

K"- k k k - { 1, 3, k- 2, k + 2, 2k- 3, 2k- 1, if k > 3, -1, ' 2 - 1 · 2, 2 - 2 - 2·1,k,2k-1, ifk=3.


If now we use the operation of the Hadamard product, we get Ko1, k, 2k- 1 = 1, 2k- 1 for k > 3. Therefore 1, 2k- 1 is computable in the S-ring through 1, k, 2f=. 1, i.e., Czk E [Mzk]· From here we have [Czk] = [Mzk] for k > 3, so that Aut(M2k) = Aut(Czk) = Dzk· •

The following scheme of investigation ( which we called the standard in [Fa 2]) offers broader opportunities for using the Galois correspondence in studying permutation groups.

Let a permutation group ( G, n) be given and suppose we are interested in all 2-closed overgroups of (G, !1) and/or all graphs which are invariant with respect to ( G, !1) and the automorphism groups of such graphs. The standard scheme of investigation contains the following steps:

1. Constructing the V-ring V( G, !1), i.e., describing the standard basis and the structure constants;

2. Finding the 2-closure of the group ( G, !1);

3. E:numerating all cellular subrings of the ring V( G, !1);

4. Discarding all non-Schurian subrings;

5. Finding automorphism groups of all cellular subrings (Schurian subrings).

Let f(A) be a graph obtained after the applieation of this scheme where A E

W = V(G, !1). Then Aut(f(A))= Aut([A])=Aut(W'), where W' is the minimal Schurian cellular subring in W containing [A] as a subring. (Such a subring is always unique.)

The general methodology of application of the standard scheme and some modifications are contained in [Fa 2]. Section 2. 7 of this survey describes a package of programs for computing in V-rings. In particular, they allow one to carry out the standard scheme. Finally, sections 2.1-2.6 present a technique for the realization of some steps of the scheme, both on computer and on the theoretical level.

If a cellular ring W has no nontrivial cellular subring then W is called a simple ring. The simplicity of V( G, !1) means that every overgroup of the group G(2) is 2-transitive, and that Aut(r(A)) = G(2) for A E V(G,n), A f. I,J,O,J- I. The property of simplicity of V-rings was often used in the proofs of maximality of permutation groups of degree n in the symmetric or alternating groups of degree n (see Chapter 3). When multi-parametric infinite series of permutation groups were considered, wespoke about the asymptotic simplicity, meaning the simplicity of V-rings from the series for sufficiently large values of one of the parameters.


To conclude this section, let us give one more interesting example of the use of the Galois correspondence.

Example 2 Let A be a square matrix and let p(x) be a polynomial. We say that p( ;r) distinguishes ( "respects" in another terminology) the spectrum of A if p(.Ai) =f. p(.Aj) for distinct eigenvalues Ai and Aj of A. The following conjecture from [We 1] is posed by Paul Weichsel.

Conjecture. Let r be a regular connected graph. Let p(x) be a polynomial such that p(r) is a connected graph. Furtherlet Aut(r)=Aut(p(r)). Then p(:r) distinguishes the spectrum of the matrix A(r).

Some counterexamples to this conjecture can easily be obtained from some of our results. For example, we showed that the adjacency algebra of the Johnson scheme J(ll, 4) has a cellular subring W' = (A0 , A 1 + A4 , A2 + A3), which is a non-Schurian subring. (See sections 2.5, 2.6 and 3.2.) Since the Johnson scheme is P-polynomial, A1 has five distinct eigenvalues and A1 +A4 is polynomial computable through A1 . Since J(ll, 4) is primitive, r(A1 +A4 ) is a connected graph. Since W' is a cellular subring of rank 3, the matrices A1 + A4 and A2 + A3 have 3 distinct eigenvalues. At the same time, Aut(f!)=Aut(J(11,4))=Aut(W')=Aut(r(A 1 + A 4 )),

since W' is a non-Schurian subring. Therefore we have a counterexample to the Weichsel conjecture. •

1.4. Historical notes

As we already mentioned in section 1.1, several versions of the axiomatic theory of cellular rings or particular cases (interpretations) of it are known. The reason for this is simple: All of these theories appeared independently at different times in the works of different authors in relation to different problems. Here we restriet ourselves to a sketch of those few lines which tie directly to our work and the works of others mentioned in this book. More about the history can be found in the hook [Ba 5] and our previous surveys [*Kl 9], [Fa 2].

Apparently, the first was the classical article of I. Schur [Sc 1] where, to each overgroup (G, R) of a regular permutation group (R, R) in the symmetric group S(R), he associated a subring of the group ring Z(R) generated by the orbits of the stabilizer Ge of the identity element e in the group R (transitivity module in the terminology of H. Wielandt, who was a student of Schur).

I. Schur axiomatized the concept of the transitivity module by introducing the structure which we now call S-ring (Schur ring). He managed to find sufficient conditions (in terms of S-rings) for the given group R to be a B-group, which means that each overgroup of (R, R) is either imprimitive or doubly transitive.


(This problern was first considered by Burnside.) Using these conditions Schur was able to prove that the cyclic group of order n ( n cornposite) is a B-group. (Burnside could show this result only in the case n = pm, p prime, m > 1.) Using the tedmique of their teacher, H. Wielandt and R. Kochendörffm expanded the dass of known B-groups (see [Wi 2]).

By the end of the 30's, W. Manning noticed (see [Ma 3]) that the rnain results of Schur and his followers could be obtained by using techniques which involve computing in the algebra of double cosets, which is traditional in group theory. The awakening interest in these objects was also due to the appearance of articles by Frame (see [Wi 2] for references), who applied the theory of characters to the double coset algebra. Let us here rernark that, following Bourbaki [Bo 4], the double coset algebra of a group G with respect to a subgroup H is called a Hecke algebra. The use of this terminology can be explained by the fact that Hecke hirnself considered analogaus algebras in his work concerning the theory of autornorphic functions. (See rnore details in [Ne 2], p.l07.)

Finally in the 50's, H. Wielandt, again following Schur's ideas, suggested consideration of the centralizer ring of the permutation group (G, n), i.e., the set of all integer matrices of degree n = lr21 which commute with all permutation matrices corresponding to elements of ( G, n). In the case where ( G, n) is transitive, its centralizer ring is anti-isomorphic: to the ring of integral linear combinations of the basis elernents frorn the Hecke algebra, but an application of the language of matrices allows one to consider these structures for intransitive groups as well. The main results from the theory of centralizer rings appear in [Wi 2], whose rotorprint version appeared several years earlier. In 1969 Wielandt (see [Wi 4]) identified the theory of centralizer rings, along with character theory and the method of invariant relations [Wi 3], as one of the three basic tools in the theory of permutation groups.

At the same time, a parallel and initially independent development of similar concepts was taking place, due to the needs of applied combinatorics. As early as 1939, R. Bose and K. Nair [Bo 1] suggested consideration of partially balanced block designs, i.e., incidence systems with the property that the multiplicities of all points (blocks) have the same value, but the multiplicities of pairs of points can attain d distinct values (in the siruplest case d = 2).

For the construction of such designs, special kinds of partitions of the cartesian square were considered. Due to Bose and Shimamoto [Bo 2], they came tobe called association schemes with d classes. The problErn of a systematic study of association schemes was posed. To each association scheme was associated a matrix algebra [Bo 3], which presently is called a Bose-Mesner algebra or BM-algebra. A strong new motivation for the study of association schemes was a connection, realized by Del'3arte, between some classes of association schemes and the theory of

22 LA. FARADZEV ET AL.

algebraic codes [De 1]. An explosion of interest in strongly regular graphs (s.r.g.), distance-regular graphs ( d.r.g.) and distance-transitive graphs ( d.t.g. ), related to the outstanding achievements in the theory of finite simple groups, placed the theory of association schemes at the center of modern algebraic combinatorics. This position was strengthened by the book of Bannai and Ito [Ba 5].

When Schur introduced the concept of S-ring as an axiomatization of the concept of module of transitivity, he thought that every S-ring was a module of transitivity ( this is the origin of "Schur type ring"). Despite the fact that this turned out to be false, Schur's method undoubtedly survived the test of time. In the 60's, due to the needs of computer science, the problern of developing algorithms for the identification of graphs and other combinatorial structures became quite prevalent. On a heuristic Ievel, the booming research activity (see the survey [*Ze 1]) satisfied some users of the algorithms who worked mostly with restricted classes of graphs, but on the theoretical Ievel the situation needed improvement. The first serious attempt at an algebraic analysis of the problern was made in 1968 by B. Yu. Weisfeiler and A. A. Leman in [We 1]. In principal, their initial position coincicled with the approach of Schur (up to the langnage used): a correspondence between a graph r and a matrix algebra containing the adjacency matrix of r as an element. Like Schur, they assumed the basis elements of the algebra were adjacency matrices of 2-orbits of the automorphism group Aut(f) of the graph r. To the credit of the authors [We 1], they soon found their own counterexample to this assumption [* Ad 1]. The concept they developed, namely that of a cellular algebra, is presented in full in [We 2].

The concept of cellular algebra (more precisely, cellular algebra with identity) coincides ( up to langnage) with the concept of coherent configuration, independently introducecl by D.G. Higman ([Hi 2], [Hi 3], etc. 1l) The coherent configuration is a cellular algebra with identity over C generated by T adjacency matrices of graphs with vertex set n, such that the edges of each graph are pairs belanging to one of the T classes of the configuration (adjacency algebra in Higman's terminology). The subset of a cellular algebra consisting of the integral linear combinations of its basis elements forms a ring which we call a cellular ring. Higman's work was mainly aimed toward extending certain results of group representation theory to the theory of coherent configurations. In contrast, in [We 2] we find an analysis of a wide range of questions basecl on the concept of a cellular algebra: identification of graphs, cletermination of their canonical forms, computation of their automorphism group, constructive enumeration of various classes of graphs, and so on. We note

!) After the work on this survey was completed, P.J. Cameron informed us that the lecture notes by D.G. Higman: Combinatorial Consideration about Permutation Groups, Math. Institute, Oxford, 1971, contain an axiomatic account of coherent configurations as Higman had developed it at that time (including some facts which are considered in our survey).


that in their very first publication [We 1], Weisfeiler and Leman make reference to the remarkable paper of Higman [Hi 1] on the theory of centralizer rings, but at that time there had been no article by Higman on axiomatic analogues of centralizer rings. Finally, it is a pleasure to remark that Higman hirnself [Hi 4] mentions the Weisfeiler-Leman algorithm from [We 1], [We 2] in one of bis latest papers. Therefore, the equivalence of the languages of coherent configurations and cellular rings can be considered as well known now.

One additional source for the ideas underlying the theory of cellular rings (in the form in which it is represented in this book) is the method of invariant relations (see sections 1.2, 1.3), which stems from the not- so- well known work of M. Krasner [Kr I] and is developed in the works of H. Wielandt [Wi 3] and L.A. Kaluznin and bis students [*Bo 1], [*Ka 1]. The method is thoroughly presented in the monograph [Po 3]. The idea of using the Galois correspondence, described in the previous section, originated within this approach. The first time the correspondence was actually used was in the work of M.H. Klin [*Kl 1] (see section 3.I for details), though the concept of the V-ring appears there only implicitly. In explicit form, the use of Galois correspondence is suggested in [*Kl.3] and employed in [*Ka 1]. We wish toremarkthat in [*KaI], and in several subsequent publications, the following terminological imprecision was consciously allowed: it was never emphasized that, in general, a subring of a V-ring could be a non-Schurian cellular ring. This is due to the fact that there had been no standard terminology in the theory of cellular rings prior to the publication of the book [We 2].

As is mentioned in the foreword, the methodology we are using has developed from the combined efforts of several groups of people. In this survey we use the terminolo,gy of cellular rings and V-rings, which is consistent with that used in almost all of our publications. We note that, along with the term "cellular subring", the equivalent "subscheme of an association scheme" is often used. As a rule we prefer the first term, since subscheme is sometimes used in the theory of association schemes to denote a completely different object (see [De I], section 5.3.3; [Ba 5], section 3.3).

Wehave already presented (in [Fa 2], §5) some arguments which support our statement that the not-so-well known concept of Galois correspondence is an elementary but, nevertheless, very effective method for studying combinatorial objects with rieb symmetry. In this connection, we find it appropriate at this point to direct the reader's attention to two more examples: the refutation of the conjecture of Weichsel given in section 1.3 and the discovery of a new infinite family of (P and Q)polynomial association schemes in [Iv I] (see also section 3.4). What unites these two examples is that both were obtained rather "cheaply"; the desired result follows in just a few lines by combining results obtained earlier. We also want to pointout


that a particular case of our methodology (see section 1.3) was rediscovered in [As 1].

CHAPTER 2. THE TECHNIQUE

2.1. Computation of rank, subdegrees and structure constants

In this section we consider various rnethods for cornputing the rnost irnportant characteristics of a V-ring of a transitive permutation group (G,D), narnely its rank, subdegrees and structure constants. Knowledge of the structure constants is suffkient for the solution of the problern central to this survey: the enumeration of all cellular subrings of a V-ring. But for solutions to certain cornbinatorial problerns, it often suffi.ces to know just the rank and subdegrees. For exarnple, it is possible to forrnulate suffi.cient conditions for the existence or nonexistence of distance-transitive representations of G in terms of subdegrees. Examples of such conditions and their applications are eonsidered in the survey article of A.A. Ivanov [*Iv 9] in this collection. Because of this we start our presentation with the problern of cornputing the rank and subdegrees.

2.1.1. Computation of rank and subdegrees

It is well known that knowledge of the characters of a transitive group ( G, n) is suffi.cient to determine its rank. Therefore the problern of cornputing rank will be considered not on its own but in eonjunction with the cornputation of subdegrees. On one hand, in order to use sorne rnethods for cornputing subdegrees, one has to first know the rank of the group. On the other ha:nd, if one rnanages to cornpute subdegrees, then the rank can be deterrnined autornatically as the eardinality of the rnultiset of subdegrees.

Let us also notiee that sornetirnes it is appropriate to consider the problern of determining subdegrees as a final step in solving the rnore general problern of deterrnining the 2-orbits of the perrnutation group. In this setting the problern actually becornes a eornbinatorial one. Therefore the rnethods for cornputing subdegrees of a transitive permutation group ( G, !1), where n is the set of cosets of G with respect to a su.bgroup H, ean be divided into two classes: algebraic rnethods and cornbinatorial rnethods. The algebraic rnethods use only information about the groups G and H. The cornbinatorial rnethods are ernployed when we use inforrnation ab out the eornbinatorial structure of the set n to cornpute subdegrees, and the problern of cornputing subdegrees is eonsidered, implicitly or explicitly, as apart of the problern of describing the 2-orbits.

2.1.1.1. Combinatorial methods for computing subdegrees

Let us eonsider a function <P : n X n -+ A which rnaps a pair of elernents frorn n to an elernent of a suitable set A, and which is invariant with respect to the action of (G,D), i.e., if (w1 ,w2 ) and (w~,w~) belong to the sarne 2-orbit, then

26 I. A. FARADZEN ET AL.

<P(w1,w2) = 1/J(w~,w~). When the converse is true, i.e., <P(w1,w2) = <P(w~,w~) implies that (w1,w2) and (wi,w~) belong to the same orbit of (G,D), we call the invariant <P exact ( complete in another terminology).

When Dis an orbit of the induced action of Gon a relation in a natural representation (G,D') ofthe group (see subsection 2.7.1 for details), then the invariants of ( G, D) on n X n are easy to construct. The use of the geometry of the natural representation allows one to describe 2-orbits of the induced action in terms of the values of the exact invariant and to calculate their cardinalities.

Example 1 The action of Sv on the set {,~} of the m-element subsets of a v-element set.

From the v-transitivity of the natural action of Sv, we conclude that the invariant 1/J, defined by the formula <P(A,B) = lA n BI, for A,B E {,:',}, is exact. Denoting by A; the 2-orbit of the group (Sv, { ,:',}) consisting of all pairs ( A, B) such

that lA n BI = m- i, it is easy to get an expression for subdegrees: k; = (';')(:-~), 0 :s; i :s; m. •

Example 2 The imprimitive action of the group G = PGL3 (q) on the flags of the desarguesian projective plane of order q.

We remind the reader that a flag of a projective plane is a pair ( A, a ), where A is a line and a is a point on A.

In order to describe the 2-orbits of the action of a group Gon the flags, we use a well-known property of collineation groups of finite projective Galois planes [Ha 5]. It turns out that altogether there are the six following 2-orbits:

A0 : A = B, a = b;

A1 : A = B, a =/= b;

A2: AnB = {a} = {b};

A3 : A n B = { a} =/= { b};

A4: An B = {b} =/= {a};

A5 : An B f. { a, b }.

Indeed A0 is a 2-orbit due to the flag transitive action of PGL3(q); A1 - due to the 2-transitivity of the stabilizer of the line on its points; A2 - due to duality considerations. In the cases of A3 , A4 , A5 , we fix a flag (A, a) and describe the corresponding suborbits using suitable elations of the projective plane. (In order

to show that two flags belong to the same suborbit, it suffices to use one elation in the case of A3 and A4 , and two consecutive elations in the case of As.)

Therefore each 2-orbit can be described by a configuration of the intersection of a pair of flags (A, a) and (B, b) which form a 2-point of the 2-orbit. From here the subdegrees can easily be obtained: k0 = 1, k1 == k2 = q, k3 = k4 = q2 , k5 = q3 •

• If <h' <h, ... , </Jk are invariants ofthe action of ( G, n) on n X n and <Pi : n X n -t

Ai, then the mapping <P = (<fJ1, ... ,</Jk): n X n -t (A1, ... ,Ak) is also an invariant of the action. It may even happen that </J is exaet though none of </J1, ... , </Jk is. The method of subsequent splitting consists of a sequential construction of invaria.nts </J1,</J2, ... ,</Jk until theinvariant </J = (q>1, ... ,</Jk) becomes exact. The exactness of an invariant is achieved when the number of its values on n X n coincides with the rank of ( G, n). The latter can be computed from the decomposition of its permutation character into irreducibles or by using Burnside's Lemma ( or more precisely the Cauchy-Frobenius-Burnside Lemma ( see section 3.1) ).

Example 3 The primitive action of the Mathieu group M12 on the set {132 } of a.ll

3-element subsets of a 12-element set.

It is evident that lA n BI, A,B E { 1n, is an invariant of the 2-orbits of the

group (M12 , e32 } ). From the 5-tra.nsitivity of M12 on 12 points it follows that for

lA n BI > 0 the correspondence between 2-orbits and lA n BI is one-to-one:

Ao: lA n BI= 3;

At : lA n BI = 2;

A2 : lA n BI = 1.

Since the natural action of M 12 is not 6-transitive, we cannot conclude that all the pairs ( A, B) with lA n BI = 0 belong to the same 2-orbit of ( M12 , { 1n ). It is well known that the stabilizer of three points in a natural representation of M12 is isomorphic to M 9 · S3 . From here, using Burnside's Lemma., one can easily calculate that the rank of (M12 , { 1

32 }) is equal to 5, i.e., the set of disjoint pairs from {1

32 }

breaks into two 2-orbits. Due to the fact that M12 is the automorphism group of the Steiner System S(5, 6, 12), we have that the predicate (AUB E S(5, 6, 12)) is an invariant of the 2-orbit of the group (M12 , { 1

32 } ), the one which distinguishes those

2-orbits having lA n BI = 0:

A3 : lA n BI = o, Au B E S( 5, 6, 12);

A.1 : lA n BI = o, Au B tf. S(5, 6, 12).


Having the description of the 2-orbits, it is not hard to compute the subdegrees: ko = 1, k1 = 27, k2 = 108, k3 = 12, k4 = 72. The numbers ko, k1, k2 and k3 + k4 are computed by the formula from example 1. The subdegree k3 can be computed in the following way: there are C52 ) I(~) = 132 blocks in the Steiner System S(5, 6, 12).

Each block contains (~) = 20 triples and the total number of triples in all blocks of

S( 5, 6, 12) is 2640. But I { 132 } I = C32) = 220, hence each triple from { 132 } is contained in 12 blocks of the system S(5, 6, 12). •

Example 4 The primitive action of the Mathieu group M 23 on the set { 233} of all 3-element subsets of a 23-element set.

Due to the 4-transitivity of the natural action of M 23 on 23 points, the invariant rPI ( A, B) = lA n BI' for A, B E { 233}' distinguishes precisely those 2-orbits of (M23, g3 }) for which lA n BI > 1, but it is insufficient to distinguish 2-orbits for which lA n BI s; 1. Using the fact that M23, in its natural representation, is the automorphism group of the Steiner System S( 4, 7, 23) ( which is obtained from the Stein er System S( 5, 8, 24) by removing a point x and all blocks which do not contain x), we construct two more invariants:

rP2(A., B) = max IBn Cl; CES(4,7 ,23)

ACC

rP3(A., B) = (3C E S(5, 8, 24) : (A. U B) c C).

Using the decomposition of the permutation character of this representation into irreducibles [Co 1], one can easily determine that i ts rank is 8. The description of 2-or bi ts of the group (Mn, { 233 }) by means of the exact invariant ( 4> 1 , rP2, 4>3) follows. The computation of subdegrees can be clone using simple combinatorial calculations similar to ones in the previous example.

2-orbit rP1 rP2 rP3 subdegree

Ao 3 3 Yes 1 A1 2 3 Yes 60 A2 1 3 Yes 90 A3 1 2 Yes 480 A4 0 3 Yes 20 A5 0 2 No 480 A6 0 1 Yes 160 A1 0 1 No 480

•


When the rank is large one has to use many invariants. This makes the method of combinatorial splitting more difficult to use and the description of 2-orbits becomes too cumbersome. In this case one can sometimes find a one-to-one correspondence between 2-orbits and the isomorphism classes of a suitable set of combinatorial objects. Then, in order to compute the rank and the subdegrees, it suffices to compile the complete list of mutually nonisomorphic objects from the set. (This can be clone by the methods of constructive enumeration [*Fa 2].) An example of this approach was given in [*Fa 5]. Different approaches to the problern of computing subdegrees were discussed there as well.

When the group we study is obtained from other groups by means of certain operations, the methods for computing rank and suhdegrees are also considered to be combinatorial. In particular, the description of the 2-orbits of the direct product, wreath product and symmetric exponentiation of permutation groups can be easily obtained from information about the 2-orbits of the groups involved. An example of this was considered in [*Fa 4].

2.1.1.2. Algebraic methods for computing subdegrees

The first example of an algebraie method is the direct computation of subdegrees of a permutation group ( G, n) as cardinalities of orbits of the stabilizer of a point. Let H = (g1 , ... , gk) be the stabilizer of a point w E n in the group ( G, ü). Consider a graph whose vertex set is n with vertices w and w' adjacent if and only if wg; = w' for some generator g; of H. Obviously, the connected components of this graph are the orbits of (H, ü). Applications of this approach are restricted by the great difficulty of obtaining a generating system for the group (H, ü) from a given g;enerating system for the group ( G, r!). (The generating system of permutations for the transitive action of group G on its cosets with respect to subgroup H can be obtained by the Todd-Coxeter Algorithm. Another way, named "inducing", is discussed in Section 2.7.1.) The best of the known algorithms for obtaining a generating set for a stabilizer of a point has complexity of order lül3 s, where s is the cardinality of a set of generators for ( G, ü). This method was used for the computation of ranks and subdegrees of primitive representations of nonabelian groups of order less than 106 [*Iv 2] (see also section 3.5).

Another algebraic method for the computation of subdegrees is the method of Burnside marks, which is closely related to the computation of the number of fixed points of the set n under the action of all subgroups of the stabilizer in two different ways.

The Burnside mark m(K,H,G) is defined as the number of fixed points of the group K ~ G in the action ( G, n) of the group G on the cosets of the subgroup H. Since the stabilizer of an arbitrary point in the group ( G, n) is conjugate in G

30 l. A. FARADZEV ET AL

with H, then m(K, H, G) is different from zero if and only if K is conjugate in G to a subgroup of the group H.

If ß 0 , ß 1 , ... , ßr_ 1 is the complete set of orbits under the action ofthe stabilizer of the point H on the set n, then the action of H on each orbit is equivalent to its action on the set of cosets of some subgroup K ~ H, and the length of the orbit is equal to the index of K in H. Let JC = (H = K 0 , K 1 , ... K 8 ) be the complete list of subgroups of H which are mutually nonconjugate in H, and let JC be ordered in such a way that if K; is conjugate in H with a proper subgroup of the subgroup Kj, then i > j. By Q; we denote the number of orbits of the stabilizer of the point H on which H acts similarly to its action on the cosets of its subgroup K;.

Theorem 2.1.1

' L m(K;, Kj, H)Qj = m(K;, H, G), for 0::; i ::; s. j=O

..,. The number of fixed points of the set n under the action of the subgroup K; is equal to m(K;, H, G). On the other hand, it is equal to the sum of the numbers of fixed points under the action of K; taken over all orbits of H such that H acts on Q1 of them similarly to its action on the cosets of its subgroup Kj, j ::; i ...,..

Therefore we obtain a system of s + 1 linear equations in the s + 1 unknowns Q i. That the system has a unique solution follows from the fact that it is triangular with diagonal entries m( K;, K;, H) > 0. Hence, in order to compute the subdegrees of the group (G,D), it is sufficient to find marks m(K;,H,G) for each subgroup K; ~ H (up to conjugacy in H) and the marks m(K;, Kj, H) for all pairs K;, Kj E JC for which 0 ::; j ::; i. The method is treated with more detail in [*Iv 4], where a more general version is considered as well.

The following theorem can be used to compute marks. The proof is omitted.

Theorem 2.1.2 Let ( G, f!) be a transitive permutation group, H the stabilizer of a point wEn, and L :S: H. Let {L 1 , ••. Lm} be the complete list of subgroups of H which are mutually nonconjugate in H but which are conjugate to L in G. Then

m

m(L,H,G) = ~_)NG(L;): NH(L;)]. i=1

• In the case where L is conjugate in H with all subgroups of H which are

conjugate with L in G, the formula for the mark takes on the simpler form

m(L,H,G) = [NG(L): NH(L)].


The marks m(K;, Kj, H) for K;, Kj EH are usually easy to compute by using

Theorem 2.1.2. On the other hand, the computation of INa(K;)I for some K; C

H and establishing conjugacy in G of two isomorphic subgroups of H which are nonconjugate in H can create substantial difficulties. Sometimes, here, the use of computers can be helpful.

Example 5 The primitive action of the Mathieu group M 24 on the cosets with respect to its maximal subgroup L2(7).

The lattice of subgroups of L2(7) is given in Figure 2.1.1 (in this example, as in (Co 1], D 2 n denotes the dihedral group of order 2n).

Figure 2.1.1

The marks of the form m(K;, Kj, L2(7)), for K; ~ Kj ~ L2 (7), can be found easily if we apply Theorem 2.1.2. For example, A4 contains one conjugacy dass of Za, and these subgroups are self-normalizing, but NL2 ( 7)(Z3 ) = D6 • Therefore m(Za, A4 , L2(1)) = 2. There are two conjugacy dasses of involutions in S4 , whose respective normalizers are isomorphic to D 8 and .E4 • These dasses of involutions are fused in L2 (7), and NL2 ( 7)(Z2 ) = D8 • This implies m(Z2 , S4 , L2 (7)) = 3. The marks of the form m(Z;, L2 (7), M 24 ), i = 2, 3, 4, 7, are easily computed by identifying the conjugacy dass of M 24 containing non-trivial elements of Z;. The following trick can be used to compute orders of the normalizers of the subgroups

S4,S~,~,A~,Ds,D6,E4 and E~ in M24.

It is easy to establish that all these subgroups act semiregularly in the natural representation of M24 , and the complete lists of elements of these subgroups (as permutations of degree 24) can be compiled. By using the fact that the centralizer in the symmetric group of the left regular representation is equal to that of the right regular representation, one can determine all elements of the centralizers of


these groups in S24 . Finally, from the obtained centralizers in S24 , one can easily decide ( using computer) which of those elements belong to M 24 . In order to do this one uses the fact that such elements preserve the blocks of the Steiner System S(5, 8, 24). The computations described above show, in particular, that the orders of centralizers in M 24 of each pair of subgroups S4 and S~, A4 and A~, E4 and E~ are distinct, which implies that the subgroups in each such pair are not conjugate in M 24 . From the solution of a system of equations, it is established that the representation in question of M 24 on 1457280 points has rank 8858 and set of subdegrees (1 7 85 142 21 2 286 4213 5673 84220 1688535 ) where the superscript indicates the

' ' ' ' ' ' ' ' ' ' multiplicity of the corresponding subdegree. •

Because the direct method of computation of subdegrees is naturally applicable to permutation groups of small degree, the Burnside mark method allows a relatively simple computation of the subdegrees of permutation groups in which the lattice of subgroups of the stabilizer of a point is not too rich.

In other cases the methods described above are not applicable due either to complexity or lack of sufficiently detailed information. In this case, we usually apply the method of exhaustion. This technique amounts to a generation of all possible partitions of the number n = IDI into sums of addends, where each acidend can potentially serve as a subdegree of the group ( G, !1), and a rejection of those which fail to satisfy certain available necessary conditions expressed in terms of the limited information on ( G, !1). If more than one partition passes through this sieve, then, in order to determine which one is the set of subdegrees, we need a more thorough analysis which addresses more subtle characteristics of the group under consideration.

Below we are going to present some necessary conditions on both the subdegrees and the decomposition of the degree of the group into a sum of subdegrees. They will make use of the following easily available information regarding the group: its permutation character (see Section 3.1 for the definition), its maximal subgroups and the normal subgroups of the stabilizer of a point.

Let (G, n) be an action of G Oll the cosets of a subgroup H, X its permutation character, r the rank of the group and 1 = k0 ::; k1 ::; k2 ::; ••• ::; kr_ 1 its subdegrees corresponding to the orbits ~0 , ~ 1 , ... , ~r-l of the stabilizer of a point.

Lemma 2.1.3

(1) k; divides IHI, 0::; i::; r- 1;

(2) Let {K1 , ... ,K8 } be the complete list of maximal subgroups of H. Then, for any k; -#1, there exists j suchthat [H: K 1] divides k;;

CELLULAR RINGS AND GROUPS OF AUTOMORPHISMS OF GRAPHS 33

(3) Let M <l H. Then, for every 0::; i ::; r- 1, there exists a subgroup K ~ M, suchthat [M: K] divides k; and k;/[M: K] divides [H: M] .

.,.. (1) and (2) follow from the fact that the action of H on an orbit 6.; is similar to its action on the cosets of some subgroup K. For (3), the action of Mon any orbit 6.; partitions the orbit into orbits of the same size [M : K]. But a group of order [H: M] acts on them.,..

Lemma 2.1.4 For every g E G with x(g) > 1, there exists i 2:: 1, suchthat J(g)J divides \H\/k; .

.,.. A proof follows from the fact that, for x(g) > 1, there is an orbit of length k;, i 2:: 1., such that (g) fixes at least one point of it, i.e., H acts on it similarly to its action on the cosets of a subgroup of order \H\/k; which contains (g) .,..

Lemma 2.1.5 Let g E G, \g\ = p'\ p prime, x(g) > 0. Let t; be the remainder ··-1

upon division of k; by p. Then the quantity x(g)- 2::: t; is nonnegative and divisible ,:=o

by p .

.,.. Since g fixes at least one point, there exists a permutation g' E H which belongs to the same conjugacy class of G as g. But g' stabilizes every orbit 6.;, and its number of fixed points on an orbit of length k; is congruent to t; modulo p .,..

Lemma 2.1.6 Let M <l H, M = Ep". Then

.,.. All orbits of the action of M on the orbit ß; have the same length, which does not exceed GCD(k;,po:). Therefore the number of such orbits is at least k;/GCD(k;,po:). Denoting by cp; the character of the action of Mon the orbit 6.;, and using Bumside's Lemma, we obtain:

L cp;(g) 2:: Po:. GCDk(~- o:)" gEM ,,p

The statement of the lemma is obtained by summing this inequality over all i rauging from 0 to r - 1 .,..

In the case where ( G, f!) is primitive, the following strong necessary conditions on the subdegrees hold.


Lemma 2.1. 7 [Si 3] lf ( G, D) is primitive, then

I) k1 is greater than or equal to the greatest prime divisor of IHI;

2) if k1 is prime, then ki does not divide IHI;

3) if kr-1 >I, then GCD(k;, kr-d =/=I for I :Si :Sr- I;

4) k; :S k1k;_ 1 for I :Si :Sr -I; if the number of subdegrees equal to k1 is odd, then k; :S (k1 - I)k;_1 for 2 :Si :Sr -I;

5) if ~k; :S p :S k; for a prime p which divides IHI, then there exists kj > k; which divides k;(k;- I);

6) nk; is even for I :S i :S r - 1. • Lemma 2.1.8 Let r be a graph with vertices .6..1,.6..2, ... ,.6..r-I· Two vertices .6..; and tlj are adjacent if k; · a = kj · b for some nonnegative integers a, b which are less than or equal to k1 - 1. lf ( G, D) is primitive, then the graph r is connected .

.,.. All anti-reflexive 2-orbits of a primitive group are connected. The connectedness of r is a necessary condition for the connectedness of the 2-orbit 6.1 ~

An algorithm for finding subdegrees of a primitive group ( G, n) employing the lemmas given above is the following. First we use Lemmas 2.1.3 and 2.1.7(I,2) in order to find a set N of positive integers which can serve as the subdegrees of ( G, D). Next we construct all possible collections I = k0 :S k1 :S ... :S kr-l such

r-1

that k; E N for I :S i :S r - I and 2.::: k; = n, and for each such collection we i=O

verify the conditions given in the other lemmas. If only one such collection passes all tests, then this corresponds to the set of subdegrees of ( G, n). Otherwise, more subtle information about ( G, n) has to be used.

As a first illustration of the algorithm, we consider the following well known example.

Example 6 The primitive representation of the Tits group 2 F4 (2)' of degree I755.

The stabilizer of a point in this representation has order 211 · 5 and it is the normalizer of an involution 2A in the Tits group. From the decomposition of the permutation character into irreducibles: X = x1 + X6 + X9 + XI4 + XIs, it follows that the rank of this representation is equal to 5 and x(2A) = 91. It turns out that this information, borrowed from [Co I], is sufficient to establish that the only collection of possible subdegrees satisfying the described necessary condition is (I, IO, 80,640, I024). In this case, knowledge of the subdegrees is sufficient to prove


that the 2-orbit A1 with k1 = 10 determines a distance-transitive graph (for details, see [*Kl 10]). •

Example 7 The primitive representation of the Conway group Co3 of degree 11178.

The stabilizer of a point in this representation is isomorphic to the HigmanSims group H S. From the known decomposition o:f the permutation character into irreducibles: X = XI+ Xz + Xs + Xs +XIs, and from the list of maximal subgroups of H S, we eonclude that the only two collections which satisfy the necessary conditions are (1,100,352,1100,9625) and (1,352,1200,3850,571'5). •

Example 8 The primitive representation of Janko's group J4 of degree 173067389.

Using ad hoc methods, it was shown in [*Si 1] that the collection of subdegrees of this representation must be one of the following four:

1. (1,15180,28336,2040192,6800640,81607680,82575360);

2. (1,15180,28336,2040192,27202560,61205760,82575360);

3. ( 1 '15180,28336 ,27202560,30602280,32643072,825 75360 );

4. (1,15180,28336,3400320,32643072,54405120,82575360).

Since the stabilizer of a point is a normal subgroup of order 211 and the value of the character of the representation on every involution of J4 is at most 52349, we conclude by Lemma 2.1.6 that the fourth colledion represents the collection of subdegrees. •

References

Examples 1 and 2 are well known. Examples :3-5 were considered by LV. Chuvaeva as part of the problern of the analysis of V-rings of primitive representations of the Mathieu groups. The complete results are given in [*Chu 1], [*Iv 8], [*Fa 6], [*Chu 3]. The method of Burnside markswas first suggested in [*Iv 3]. In [*Iv 4] (the translation of this paper is included in this book), the method is considered in the more general context of computing the lengths of orbits of an arbitrary subgroup F ~ G in the permutation group ( G, n). This article also contains several examples of the computation of ranks and subdegrees of particular permutation groups (including intransitive ones) and of some series of groups. Subdegrees of the primitive representations of PSL2 (q) were computed by L. Tchuda in [*Tchu 1], [*Tchu 2], who used the same method. The method of Burnside marks was also used in [Di 2] for the computation of ranks of normalizers in the symmetric group of the primitive permutation groups of degree :::; 1000 which have an abelian socle.


The enumeration algorithm for computing possible collections of subdegrees was used in [*Iv 8] in one of the steps of the solution to the problern of constructing all distance-transitive representations of an abstract group by using its table of characters. Examples 6-8 were extracted from that work. Example 8 was recently considered in detail in [Si 2], in which it was also noted that this example had been earlier studied in [No 2].

2.1.2. Computations of structure constants

The computation of structure constants plays a most important part in the study of cellular rings. Knowledge of the structure constants of a cellular ring allows one to determine its primitivity, to enumerate its cellular subrings, to find its eigenvalues and their multiplicities, to determine its Krein parameters (when the cellular ring is commutative), and so on.

We are going to consider some of the most important methods for computing the structure constants of cellular rings.

Let W = (A0 , A1, ... , Ar-1) be a cellular ring on the set D, A; = A(f(A;)), and let

r-1 A; 0 Aj = LPfjAk.

k=O

Then, obviously,

pfi = J{c E D: (a, c) E A; and (c, b) E Aj}J (2.1.1)

where (a, b) is an arbitrary pair from Ak.

If a cellular ring of degree n and rank r is given by a colored graph, then direct application of (2.1.1) allows one to compute all its structure constants (for more details see section 2. 7).

In those cases where there is a combinatorial description of the elements of a basis for the cellular ring ( as was the case in examples 1-4 of V -rings of permutation groups), implementation of (2.1.1) is reduced to a sequence of simple combinatorial exercises. We shall illustrate the technique on the permutation groups whose 2-orbits were described in examples 1 and 2.

Example 1 ( continued) Let us fixapair ( A, B) of m-element subsets of a v-element set N such that JA n BJ = m- k. Then the set N becomes partitioned into four parts: N1 = AnB, N2 = A \B, N3 = B\A and N4 = N\(AUB) with cardinalities JN1J = m- k, JNzJ = JN3J = k, JN4J = v- m- k. Let us count the number of subsets CE {~} suchthat JA n CJ = m- i and JB n CJ = m- j. If JN1 n CJ = t,


then IN2 n Cl = m - i- t, IN3 n Cl = m- j - t and IN4 n Cl = i + j + t- m. This implies that, for a fixed t, there are

( m - k) ( k ) ( k ) ( v - m - k ) t m-i-t m- j- t \i + j + t- m

choices of c. Since 0 ~INs n Cl~ INs I for 1 ~ s ~ 4, t ranges from

tmin =max(m-k-i,m-k-j,m-i-j,O)

to tmax = min(m- k,m- i,m- j,v- k- i- j).

Summing over all t, we obtain the formula for the structure constants

k ~~ (m - k) ( k ) ( k ) ( v - m - k ) Pij = L.t t m - i - t m - j -- t i + j + t - m ·

trrun

• Example 2 ( continued) In this example we fail to obtain a single formula for all structure constants. Therefore each constant must be computed by using individual (though quite simple and uniform) considerations. We give several illustrations of such reasoning.

Let us fix fl.ags (A, a) and (B, b) suchthat ((A, a ), (B, b)) E Ak, and let us count the number of fl.ags (C,c) suchthat ((A,a),(C,c)) E A; and ((C,c),(B,b)) E Aj. For example,

Pit = I{(C,c): C = A,c #- a,C == B,c #- b}l.

Since .A #- B for ((A,a),(B,b)) E Az, we get pj1 = 0. Next,

P!3 = I{(C,c): C #- A,c #- a,c E A, C #- B,c -:f. b,c E B}l

for ((A,a),(B,b)) E A1 , i.e., for A = B,a -:f. b. There are q -1 ways to choose a point c on line A and q ways of choosing a line C through c. Therefore p!3 = q( q-1 ). Further,

p~5 = I{(C, c) : C #- A, c #- a, a tt C, c tt A, C #- B, c #- b, c tt B, b tt C}l

for ((A., a), (B, b)) E A5, i.e., for A #- B, a #- b, a tt B, b tt A. Let An B { d}. Then there are q - 1 ways to choose a line C which passes through d and q2 + q + 1- (2q + 1)- (q -1) = q2 - 2q + 1 ways to ehoose C not passing through d. A point c can be chosen on the line C in q ways in the first case and in q- 1 ways in the second case. In all, we obtainp~5 = (q-1)q+(q2 -2q+1)(q-1) = q3 -2q2 +2q-l. In a similar way all other structure constants can be computed, and they turn out to be polynomials in q of degree at most 3. The computed values are presented in Table 2.1.1, where the values of P?j,PL, ... ,pfj are located (reading from top down)

in the ith row and /" column.

38 LA. FARADZEV ET AL

i \ j 0 1 2 3 4 5

1 0 0 0 0 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

0 q 0 0 0 0 1 q- 1 0 0 0 0 0 0 0 q 0 0

1 0 0 1 q-1 0 0 0 0 0 0 0 q

0 0 0 0 1 q- 1

0 0 q 0 0 0 0 0 0 0 q 0 1 0 q- 1 0 0 0

2 0 0 0 0 0 q 0 1 0 0 q-1 0 0 0 0 1 0 q- 1

0 0 0 0 q2 0 I 0 0 q 0 q2- q 0 I 0 0 0 0 0 q2 !

3 1 0 q-1 0 0 q2- q

0 0 0 q 0 q2- q

0 1 0 q-1 q- 1 q2 - 2q + 1

0 0 0 q2 0 0 0 0 0 0 0 q2

0 q 0 q2- q 0 0 4 0 0 0 0 q q2- q

1 q- 1 0 0 0 q2- q

0 0 1 q-1 q- 1 q2 - 2q + 1

0 0 0 0 0 q3

0 0 0 q2 0 q3- q2

0 0 0 0 q2 q3- q2

5 0 q 0 q2- q q2- q q3- 2q2 + q

0 0 q q2- q q2- q q3- 2q2 + q

1 q-1 q-1 q2 - 2q + 1 q2 - 2q + 1 q3 - 2q2 + 2q - 1

Table 2.1.1


Let us remark that only the axioms for a projective plane were used in our computations of the structure constants. This means that the partition of the pairs of flags of a projective plane (not necessarily Desarguesian) of order q into the six classes described above generat es a cellular ring ( which is not necessarily a Schurian ring). •

Sometimes a group-theoretic interpretation of the structure constants of a Vring tums out to be useful in their computation.

Definition 2.1.1 Let Ok,i be a k-orbit and Ok+l,i a (k + 1)-orbit of the group (G,!l). Let (x1, ... ,xk) E Ok,i· The number

(2.1.2)

is called the coeflicient of projection of the orbit Ok+1,1 onto the orbit Ok,i·

The following statement is easy to prove.

Proposition 2.1.9 The coeffi.cient of projection of Ok+l,i onto Ok,i does not depend Oll the choice of (xl, ... , Xk) E Ok,i, and also q(Ok+l,j, ok,i) = [Ga,, ... ,ak :

Ga,, ... ,ak>z] where, for ( ai' ... , ak) E Ok,i, z is chosensuchthat ( ai' ... , ak, z) E ok+I,j·

• If q(Ok+I,j, Ok,i) -:/= 0, then we shall say that the orbit Ok,i is a projection

prl, ... ,k(Ok+I,j) of the orbit ok+I,j with respect to the first k Coordinates. Analogously, the projection with respect to any subset of k coordinates is defined.

Now we can give an interpretation of the structure constants Pf,j of the permutation group ( G, n) in terms of indices of subgroups of the group G.

Proposition 2.1.10 Let A;,Aj,Ak E 2-orb(G,!l), and let r 1 , ... ,r. be all temary orbits of (G, !1) for which pr1,2h) = Ak, pr1,3h) = A;, pr2,3h) = Aj' for 1:::; t:::; s. Then

• P~j = L q( Tt, Ak)-

t=I

~ We use formula (2.1.1) for fixed a, b suchthat ( a,, b) E Ak. Let us construct orbits of the subgroup Ga,b on the set {c E n: (a,c) E A; and (c,b) E Aj}. Then, for each of these orbits, there is a t E {1, ... , s} and a 3-orbit Tt which corresponds to it, with the correspondence being one-to-one. The contribution of this orbit to Pf,j is the coeffi.cient of the projection q(rt, Ak ), according to Proposition 2.1.9 ..,..

Proposition 2.1.10 is especially convenient in those cases where the sets 2-orb( G, !1) and 3-orb( G, !1) are described ( either for particular permutation groups


or for infinite families of permutation groups ). This, essentially, was the manner in which the structure constants of V-rings of the groups (Sv,{,:',}) was computed in example 1.

The problern of computing the structure constants of cellular rings abtairred from operations on other cellular rings (including the case of V -rings of permutation groups) deserves special attention. The case of the symmetrized cartesian power ( corresponding to exponentiation of a permutation group by a symmetric group) was considered thoroughly in [*Fa 4]. Same other group-theoretic operations are treated in [We 2].

References

Example 1 is well known as folklore and was published many times ( e.g. [*Ka 1 J, [De 1]). The computations which appear in Table 2.1.1 were clone by A .N. Kosminin. Proposition 2.1.1 0 was proved in [*Kl 5]. Different aspects of the usage of coefficients of projection appeared in [*La 1], [*Za 2]. The method of projections for the classical cases of the Johnson and Harnming schemes was applied in [*Kl 5]. The difficulties which arise during applications of the method of pro jections are discussed in [*Us 11].

2.2. The enumeration of subrings

Let W = (A0 , A 1 , ••• , Ad) be a cellular ring and let I= {0, 1, ... , d}. Fora partition {TI, Tz, ... , Tr} of the set I to generate a cellular subring W' of a cellular ring W, the following condition is necessary and sufficient: Va., ß, 1 E {1, 2, ... , r }, Vk, l E T'Y,

(2.2.1)

Condition (2.2.1) is equivalent to the fact that the matrices Ca = I: A;, 1 ::; a. ::; iETa

r, form the first standard basis of a cellular subring W' in W. We remark that if W is a cell it suffices to consider partitions of the set {1, ... , d}, since A0 is a basis element of an arbitrary subring.

Condition (2.2.1) shows that , in order to enumerate cellular subrings of a given cellular ring W, it suffices to know only the structure constants of W.

Let us illustrate this assertion, first on the example oftheinfinite one-parameter family of cells considered in section 2.1 ( example 2). We derrote the cells of this family by W(q).

Proposition 2.2.1 For q > 1 the cellular rings W(q) have the following subrings:

a) for all values of q, (Ao, Ar +A2+A3+A4+As}, (Ao, Ar +A2, A3+A4, As}, (Ao, Ar, A2+ A3 + A4 + As), (Ao, A2, Ar + A3 + A4 + As);

b) for q = 2, (Ao,Ar + A3 + A4,A2 + As}, (Ao,Ar + As,A2 + A3 + A4};

c) for q = 4, (Ao, Ar + A2 + As, A:1 + A4).

41

~ W(q) = (Ao,Ar,A2,A3,A4,As}, and A4 = A~. Let W' < W(q). Then W' = (Ao,Cr, ... ,Cr), where r < 5 and, for 1 ~ a ~ r, C,,. = I: A;. (T = {Tr,T2, ... ,Tr}

iET" is a partition of {1,2,3,4,5}.) Since 4 = 31, the elements 3 and 4 either belong to the same class of the partition or form two one-element classes. Therefore, the following nineteen partitions define the candidates for the bases of cellular subrings:

Tr = {{1},{2,5},{3},{4}};

T2 == { {1, 2}, {5}, {3}, {4}};

73 = {{1,5},{2},{3},{4}};

1"4 = {{1,2,5},{3},{4}};

Ts = {{1},{2},{3,4},{5}};

T6 = {{1},{2,5},{3,4}};

T7 = {{1,2},{5},{3,4}};

T 8 = {{1,5},{2},{3,4}};

Tg = {{1, 2, 5}, {3, 4 }};

Tro = {{1,3,4},{2},{5}};

~~r = {{2,3,4},{1},{5}};

T12 = {{3,4,5},{1},{2}};

Tr3 = {{1,3,4},{2,5}};

T14 = {{2, 3, 4}, {1, 5}};

Trs = {{3,4, 5}, {1, 2} };

~6 = {{2,3,4,5},{1}};

~7 = {{1,3,4,5},{2}};

Trs = {{1,2,3,4},{5}};

42 I. A. FARADZEN ET AL.

719 = {{1,2,3,4,5}}.

Foreach of the candidates we have to checkthat it satisfies relations (2.2.1).

For some of the partitions the existence of the subring is obvious. Partition 7 19 generates the cellular subring (Ao, A1 + A2 + A3 + A4 + A5 ) - the V-ring of the symmetric group S(q+l)(q2+q+l). Partition 7 7 generates the cellular subring (A0 , A 1 + A2 , A3 + A4 , A5 ) which is the V-ring of the primitive action of the group PfL3(q). The group PfL3(q) is obtained from the complete collineation group of the plane S2 ,q by adjoining the contragredient automorphism. Partitions 7 16

and 717 generate the cellular subrings < A 0 , A 1 , A2 + A3 + A4 + A5 > and < A0 ,A2,A1 + A3 + A4 + A 5 > which are the V-rings of the wreath product of Sq+l

and Sq2+q+l· Incidentally, the existence of these subrings could be checked directly by showing that the relations (2.2.1) become identities for each of the partitions.

For the remairring fifteen partitions, verifications of (2.2.1) show that either it is not satisfied for any value of q, orthat it leads to the solutions given in the statement of the theorem. As an illustration, we consider the following two partitions.

Partition 7 6 Let a = 1, ß = 1, 1 = 2, k = 2, l = 5. Then we obtain the relation Pi 1 = p~ 1 or 0 = 0, which is an identity. Let a = 3, ß = 3, 1 = 2, k = 2, l = 5. Then P~3 + P~4 + Pl3 + Pl4 = P~3 + P~4 + P!3 + P!4 or q2 - q = 4(q- 1), which has solutions q~ = 1, q~ = 4. Let a = 2, ß = 2, 1 = 2, k = 2, l = 5. Then we arrive at the equation q- 1 + q3 - q2 = 2(q- 1) + q3 - 2q2 + 2q- 1, having solutions q~' = 1, q~ = 2. Therefore the relations (2.2.1) do not hold for 7 6 when q>l.

Partition 7g Here (2.2.1) produces only two independent relations. For example, for a = 2, ß = 2, 1 = 1 and for a = 2, ß = 2, 1 = 2, we obtain the following system of equations:

q2 - 5q +4 = 0

q=q

which has as its only nontrivial solution q = 4 <1111

Remark 1 As follows from the proof, the subrings which exist for all values of q are of Schur type. The two subrings which exist for q = 2 define a strongly regular graph with parameters v = 21, k = 10, l = 10, ,\ = 5, 11 = 4. It is well known that such a strongly regular graph is unique up to isomorphism: it is the triangular graph T(7). Therefore the two subrings which appear for q = 2 are isomorphic and of Schur type. The subring appearing for q = 4 also defines a strongly regular graph ( corresponding to the matrix A3 + A4 ) with parameters V = 105, k = 32, l = 72, ,\ = 4, f1 = 12. This graph r is well known (see the


survey [Hu 1]), and the cellular ring it defines is not of Schur type (see section 2.6). Since the subring (A0 , A1 + A2 + A 5 , A3 + A4 ) isasubring of the Schurian subring (A0 ,A1 + A2 ,A3 + A4 ,A5 ), then Aut(r) is a group of rank 4. It is easy to show that this group coincides with Pr L3 ( 4) and its order is 28 · 33 · 5 · 7. (To see this one considers the graph r(A1 + A2 ), the line graph of the bipartite Levi graph for the projective plane of order 4.) The claim that the subring for q = 4 is of non-Schur type will be justified in section 2.6.

Remark 2 It has been noticed that the cellular ring W(q) exists whenever there exists a projective plane of order q. Foreach such case we have described all cellular subrings of W(q). If the corresponding projective plane is Desarguesian, then the cellular subring (A0 , A1 + A2 , A3 + A4 , A5 ) is always of Schur type. In other cases the question requires special consideration.

The approach based on relations (2.2.1) seems tobe a "naive" way to enumerate subrings. But with all its drawbacks, t.his method can be potentially applied to every cel.lular ring ( see section 2. 7, where algorithmic aspects of this approach are discussed ).

In cases where W is a commutative ring, it possesses a dual basis of orthogonal idempotents, the use of which can facilitate the sea.rch for cellular subrings.

We remind the reader that, for A E W where W is a cellular ring, [A] denotes the minimal cellular subring of W which contains .A.

Proposition 2.2.2 Let W be a commutative cellular ring of rank d and, for A E lV, let Pj(A) be the eigenvalue of A corresponding to the j-th eigenspace of the algebra W. Then the conditions

p;(A)-:/:- Pi(A) for all 0:::; i,j:::; d, i-:/:- j (2.2.2)

imply that [A] = W .

.,.. Since all d + 1 numbers p0(A),p1 (A), ... ,pd(A) are mutually distinct, one can find polynomials J;( z), 0 :::; i :::; d, with complex coefficients and degree at most d such that J;(pj(A)) = D;j, where 8;j is the Kronecker symbol. It follows that Pi(f;(A)) = J;(pj(A)) = 8ij· Therefore Pj(A) = E;;, where Ej is the j-th primitive idempotent. Therefore Ei E [A] for all j, 0:::; j :::; d, and [A] = W ..,..

It tums out that computations in the dual basis can be successfully applied in the exhaustive search for cellular subrings.

Let T = {T0 , Tb ... , Tm} be a partition of the set I = {0, 1, ... , d}. We call the number m + 1 the rank of the partition and we denote it by rank( T). For an


arbitrary (d+ 1)-dimensional vector x = (x 0,x1, ... ,xd), we define an equivalence relation Px on the set {1, ... , d} by:

Let T1, ... , Tm be the equivalence classes of the relation Px· Then we define a partition Tx of the set I by Tx = {{O},T1, ... ,Tm}. Next, for an arbitrary F C {0, 1, ... , d}, we denote by ~F the characteristic vector of the subset F, i.e., ~F = ((~F)o,(~F)l, ... ,(~F)d) where

{ 0 if i rt F; (~F ); = '

1, if i E F.

Finally, for an arbitrary F C {0, 1, ... , d}, we denote by AF (respectively EF) the element I: A; (respectively I: E;) of W. Here {A0 , ... ,Ad} and {E0 , ... ,Ed}

iEF iEF represent the first and second standard bases of W. Notice that ~F ·Pis the vector obtained by adding component-wise those rows of the matrix P which are labeled by the numbers from F.

Proposition 2.2.3 Let U be a cellular subring of the commutative cell W, let P and Q be the first and second eigenmatrices for W and, for some F C {0, 1, ... , d}, let AF E U ( dually EF EU). Then U contains the subspace (Er)rEr~p·P (dually (Ar)rEr~F q)· In particular, rank(U) ~ rank(T~p·P) (rank (T~p·Q)) .

..,.Let rank(T(F'P)=m+1, T(p·P={T0 = {O},T1, ... ,Tm}. Then, using the definition of Tf.p·P, we obtain

where the numbers ,\ 1 , ... , Am are all distinct. This, tagether with the fact that U being a subcell satisfies the conditions of Lemma 1.3.1, implies that Er. E U for all k = 0, 1, ... , m. Since the elements Er0 , Er1 , ... , Erm are linearly independent ({Ta, T1, ... , Tm} is a partition of the set I), we obtain that rank(U) ~ rank( Tf.F .p ).

The dual statement is obtained by replacing A by E and P by Q in all arguments presented above .,...

In what follows, for F C {0, 1, .... , d}, we will use the notation pp(j) = L p;(j). iEF

Let T] and T2 be two partitions of the set {0, 1, ... , d}, T] = {T1;}o<i<., T2 = {T2j }o:s;j:s;r· We define a partition Tjl\ T2 as Tj/\ T2 == {TI i nT2j }o:s;;:s;.,o:s;j~r~ For any partition a = {~o, ... , ~m} of the set {0, 1, ... , d}, we define the partition ap ( dually aQ) suchthat Te}Jo .pi\Te}J, .pl\ ... 1\Te}Jm .p ( dually T{JJo .qi\TeE, .ql\ ... 1\TeE·Q ). In other words, the partition ap is defined by an equivalence relation {( i, j) : Yk(pr,k ( i) = pr,k(j))}. We notice that for all F ~ {0,1, ... ,d}, TeF"P = {F,I\F}P (since the sum of elements in each column of the matrix P, except for the column labeled by zero, is equal to zero).

Theorem 2.2.4 Let W be a commutative cell of rank d + 1, all basis matrices of which are symmetric. Let a = {~o = {0}, ~~, ... , ~m} be a partition of the set {0, 1, ... , d}. Then the elements Ar, 0 , ••• , Ar,m ( duallly Er, 0 , ••• , Er,m) form a basis of a subcell if and only if rank( a) = rank( ap) ( correspondingly rank ( a) = rank( aQ) ) .

..,.. We present an outline of the proof for the direct statement. The dual statement can be proved in a similar way.

Vve introduce a subspace [ with basis { Ar,Jo<i<m. [ is a subcell of the cell W if and only if it is closed with respect to the opera~-i~ns t, "·", "o", and contains the elements A0 and E0 • Therefore it is necessary to show that the condition rank(a) = rank(ap) is equivalent to the operation of matrix multiplication being closed on [.

Let rank( a) = rank( aP). Then we show that [ ~ (Erro, Err" ... , Errm), where {7ro,7ri,···,7rm} = aP. Since rank(C) = m + 1, we get [ = (Err0 , ••• ,Errm). This, together with Err, · Err; = D;jErr;, implies that [ is closed under "·".

Let us now assume that [ is a subcell. Let v; =TeE, .p. By definition, {0} E v; for all i, 0 :::; i :::; m. In addition, EN;; E [ for each N;j E v;, which follows from Proposition 2.2.3. Let N;j, E v; for i = 0, 1, ... , m. It follows from the closure of [ with respect to "·" that [ contains EN0; 0 • EN1;, • ••• • ENmim = EN0; 0 nN1;, n .. nNmim ·

Notice that each set nk=oNkik is either empty or coincides with a term of the partition ap. Therefore V( 7r; E aP)Err, E [. Now we use the fact that each Ar,, is a linear combination of the idempotents { EN;;} N;; Ev;, and so of the idempotents {ErrJrr,EcrP· Therefore rank(C) = rank(aP). From here we obtain the desired equality rank (a) = rank(ap) .,.

Remark If W is a BM-algebra of an arbitrary commutative association scheme (not necessarily symmetric), then it is easy to see that the statement of Theorem 2.2.4 should contain the additional condition at == a, where at is a partition of the

set {0, 1, ... , d} of the form

where ~; = {jt : j E ~;}.

Let U be a subcell of the commutative cell W. Then U has a basis defined by the partition T = { {0}, T1, ... , Tm} of the set {0, 1, ... , d}, i.e., U = (Ao, Ar" ... , Ar=). The algebra (U, ·)also has a basis formed by the primitive idempotents 10 ,11 , ... , Im, and there is a partition 1r = {{0},7r1 , ... ,7rm} of the set {0,1, ... ,d} suchthat h = E.,.,k, k = 0, 1, ... , m. The proof of Theorem 2.2.4 shows that 7r = TP and T = 1rQ.

The partitions r and 1r will be called the first and second partitions associated with the subcell U. The sets {O},T1 , ... ,Tm will be called the first basis sets of the cell U, and the sets {0},7r1 , ... ,7rm, the second basis sets ofthe cell U. We say that A; and Aj ( dually E; and Ej) are merged in U if i and j belong to the same first ( second) basis set.

Proposition 2.2.5 Let U be a subcell of Wandlet E;, Ej (A;, Aj) be merged in U. Then the following equalities hold for any first (second) basis set F of U:

L)Pk(i)- Pk(j)) = 0 kEF

( 2) qk( i)- qk(j)) = 0). (2.2.3) kEF

.,.. Let T and 7r be the first and second partitions associated with the cell W. We have shown that 1r = TP. Since E; and Ej are merged in U, then i and j lie in the same set of the partition TP. By definition this means that, for every 7l'J E 1r = TP, p.,.,1 ( i) = p.,.,1 (j ). To finish the proof of the first equality, we simply note that p.,.,1 ( s) = L Pk( s ). The proof of the second equality is similar ~

kErrt

As a corollary, we obtain the following useful

Proposition 2.2.6 Let W be a commutative cellofrank d+1, and let A 0 , A 1 , ... , Ad and E0 , E 1 , ... , Ed be its first and second standard bases. Suppose there is a pair of indices (k, m), with 0 < k, m::; d, suchthat the following inequalities are satisfied:

\fj f. 0, m, \fi f. 0, k, (Pj(k) > Pi(i))

(dually (qj(k) > qj(i))).

Then an arbitrary nontrivial subcell U of the cell W contains Ek ( dually Ak)-

~ We are going to prove the direct statement. The proof of the dual is analogous. Let U be a nontrivial subcell of the cell W, and let T and 1r be the first and second partitions associated with U. Since rank(U) 2:: 3, there exists T; E T such that O,m 1- T;. Let 7l"j E 1r and k E 7l"j. We want to show that 7l"j = {k}. Let us assume the contrary, i.e., that there exists s E 7l"j with s f=. k. Then, according to Proposition 2.2.5,

PTJk)- PT,(s) = 0.

On the other hand, PT;(k)- PTJs) = 2.:: (PI(k)- PI(s)), s f=. O,k, l f=. O,m. By lET;

assumption, Pt( k) -Pt( s) > 0 for alll f=. 0, m, s f=. 0, k. Therefore PT, ( k) -PT; ( s) = 2.:: (Pt( k) -Pt( s)) > 0, and we arrive at a contradiction <01111

lET;

Let us consider an example illustrating how the proven statements can be used in the search for subcells.

Let J( v, n) denote the Johnson association scheme and also its corresponding BM-algebra. We let m = v- n, and we restriet ourselves to the case n = 4. Hence we may assume m 2:: 4.

Proposition 2.2. 7 The subrings given below represent a complete list of the nontrivial subrings of the cellular ring J( v, 4) for v 2:: 8:

~ First we use Proposition 2.2.5, which reduces our problern to the consideration of only finitely many cases. In Table 2.2.1 we present the matrix P (see, e.g. [De 1]).

Let U be a nontrivial proper subcell of the cell J( v, 4) for which we are searching. A 1 1- U and E1 1- U, since J ( v, n) is a P- and Q-polynomial scheme. Therefore E 1 is merged with Ek for some k E {2, 3, 4 }, and U contains at least one of the elements A 2 ,A3 ,A4 ,A1 + A2 ,A1 + A.3 ,.41 + A.4 . Let us apply Proposition 2.2.5. First we find the difference between the first and second columns, then we sum, component-wise, the rows labeled by the elements of F and equate the obtained sums to zero. Here FE {{2},{3},{4},{1,2},{1,3},{1,4}}. We next do the same with the first and third columns, and also with the first and fourth columns. The results of these computations are given in Table 2.2.2 where, at the intersection of


the row corresponding to Fand the column (1, k), we have the set

{ m E N: m 2: 4 and :L>j(1) = L Pi(k)}, jEF jEF

keeping in mind that Pi(i) depends on m.

A; \Ei Eo Et E2 Ea E4

Ao 1 1 1 1 1

A1 4m 3(m- 1)- 1 2(m-2)-2 m-6 -4

A2 6( ';') 3(m;- 1)- 3(m- 1) (m;2)-4(m-2)-1 -3(m- 4) 6

A3 4{';) (m;t) _ 3{m;l) -2(m;- 2) + 2(m- 2) 3(m-3)-1 -4

A4 ( ';) -(m;t) (m;2) -(m- 3) 1

Table 2.2.1

F\{1,k} {1,2} {1,3} {1,4}

{2} 0 {4} {5}

{3} {9} 0 0

{4} 0 {4} 0

{1,2} 0 0 0

{ 1, 3} {8} {4,12} {7, 8}

{1,4} {5} {7} {7}

Table 2.2.2

It follows from the table given above that, for a nontrivial cell U, the following are the only possible cases:

(1) U s;;; (Ao,At + A3,A2,A4), m = 4;

(2) U s;;; (A0 ,A1 + A4,A2,A3), m = 5, Az EU or At+ A4 EU;

(3) U s;;; (Ao, At+ A4, A2 + A3), m = 7;

(4) U s;;; (Ao,At +A3,Az +A4), m = 7;

(5) U s;;; (A0 ,A1 + A3,A2 + A4), m = 8;

(6) U s;;; (Ao, At+ Az + A4, A3), m = 9;

(7) U s;;; (Ao,At + A3,A2 + A4), m = 12.

49

Since rank(U) E {3, 4}, we are left with the following possibilities. In cases (3)-(7) the inclusion sign can be replaced by equality. The remaining cases are presented below:

(1.1) U = (Ao, At+ A3, A2, A4), m = 4;

(1.2) U = (Ao,At + A2 + A3,A4), m = 4;

(1.3) U = (Ao, At+ A3 + A4, A2), m = 4;

(1.4) U = (Ao, At+ A3, Az + A4), m = 4;

(2.1) U = (Ao, At+ A3 + A4, A2), m = 5;

(2.2) U=(Ao,At+A4,A2+A3), m=5.

It is left to check which of the remaining 11 subspaces are really cells. We will apply Theorem 2.2.4. ForT= {To, Tt, ... , Tt} a partition of {0, 1,2,3,4}, we denote by Pr the (f + 1) X 5 matrix whose entries are defined by Pr(i,j) = I: Pk(j),

kET;

and by TP the dual partition ofthe set {0,1,2,3,4}. Now, for each ofthe eleven cases, we construct the corresponding matrix and the dual partition and check for equality of the ranks of the dual and direct partit.ions. A portion of the required computations for the cases (1.1), (5) and (7) appear in Table 2.2.3.

After carrying out all computations, we condude that the only possibilities that actually lead to subcells are (1.1), (1.2), (2.2), (3) and (5) <1111


Case m T pr ~ Existence

1 1 1 1 1

1.1 4 {0}, {2}, {4}, {1, 3} 36 0 -6 0 6 {0}, {1,3}, {2}, {4} 1 -1 1 -1 1 +

32 0 4 0 -8

1 1 1 1 1 5 8 {0}, {1, 3}, {2, 4} 256 -8 -8 16 -8 {0}, {3}, {1, 2,4} +

238 7 7 -17 7

1 1 1 1 1 7 12 {0},{1,3},{2,4} 928 32 -52 32 -8 {0}, {1,3}, {2}, {4} -

891 -33 51 -33 7

Table 2.2.3

Remark In a more general setting, the problern of enumerating the cellular subrings of the BM-algebra of the association scheme J( v, n) is considered in section 3.2. There, using examples of these cellular rings, we will also consider the problern of asymptotic simplicity of cellular rings, which from our point of view has independent theoretical interest.

References

The problern of enumeration of cellular subrings was first considered in [*Ka 1]. The "naive" approach to enumeration of cellular rings stems from [*Kl 5] and [*Kl 7]. Proposition 2.2.1 was first proved by A.N. Kosminin with the aid of a computer. Proposition 2.2.2 was proved in [*Kl 9], where it was mistakenly formulated as a necessary and sufficient condition. From this, using the Krein parameters qfj, the system of quadratic equations describing the ( o )-idempotents of W which generate proper subrings of W was derived. But, in practice, this system can be used only as an auxilary tool, for example, when one wants to show the nonexistence of nontrivial cellular subrings in W. Propositions 2.2.3-2.2.6 were proved by M.E. Muzichuk in [*Mu 7]. Proposition 2.2.7 was first proved in [*Kl 5] (see also [*Kl 8]), by using the "naive" approach for finding subrings. The condition for the existence of a cellular subring in a commutative cell, formulated in terms of the second standard basis and analogaus to Theorem 2.2.4, was recently and independently discovered by E. Bannai, see [Ba 6].


2.3. Primitive cells

In this section we consider cells only. Since the concept of a connected graph is used in the definition of a primitive cell, we re1nind the reader that a digraph r = (V, R) is called connected ( or weakly connected) if, for every x, y E V, x =/:- y, there exists a semipath in r from x to y. Here, semipath refers to an alternating sequence v0 , Xt, v1 , x 2 , ... , Xn, Vn of vertices v; E V and arcs x; E R suchthat each arc x; coincides either with (v;_1 ,vi) or (v;,v;_t).

Definition 2.3.1 The cell W = (Ao,A~, ... ,Ad), where Ao is the identity matrix, is called primitive if all anti-reflexive basis graphs r( At), r( A2 ), .•• , f( Ad) are connected. Otherwise the cell is called imprimitive.

Such a definition of primitivity is quite natural.

Proposition 2.3.1 The permutation group ( G, Sl) is primitive if and only if the V-ring V( G, Sl) is primitive. •

Corollary 2.3.2 Let W be a cellular ring. If Aut(W) is a primitive permutation group, then W is also primitive. •

The converse statement is false. There are many examples of primitive cells whose automorphism groups are imprimitive, or even intransitive.

In the study of the properties of primitive cellular rings, one can emphasize three main aspects: graph theoretic, algebraic and arithmetic. We start with the graph theoretical properties.

In order to claim primitivity of a given cellular ring W = (Ao, At, ... , Ad) on n points, it is sufficient to verify the connectedness of the basis graphs r( A1 ), ... , f( Aa). U sing standard algorithms for determining connectedness of graphs, in which the complexity is linear with respect to the number of arcs (see [Re 3]), this can be clone in O(n2 ) operations, with space requirements being O(nd). In most applications, d is much smaller than n. In this case one can suggest a faster algorithm which doesn't use the adjacency matrices of the basis graphs, but instead uses the structure constants pfj as input. In order to justify this algorithm, one can use either the isomorphism of a cell of rank d with the matrix algebra of rank d generated by the matrices B; formed by the structure constants (see Theorem 1.1.1), or direct combinatorial arguments.

Remark In the case when the cell is noncommutative, Theorem 1.1.1 holds, but the isomorphism is given by the map A; ,_..tß; (see [Ba 5], §2.2).

Definition 2.3.2 Having a given cell W = (Ao, ... , Ad), we define a family of digraphs Ft, ... , Fd, each of whose vertex sets is the set {1, 2, ... , d}, where vertices


j and k are adjacent in F; if and only if P7j f= 0.

Lemma 2.3.3 The basis graph r(Ai) is connected if and only if F; is. •

If some of the matrices A1 , ... , Ad are antisymmetric, then one can work with matrices of smaller orders for solving the question about the primitivity of a cellular ring. Let d = r 8 + 2r a, where r 8 is the number of symmetric matrices and r a the number of pairs of antisymmetric ones. Let r = r 8 + ra. Then we define graphs F; with vertex set {1, 2, ... , r }, where j and l are joined by an arc if and only if at least one of the structure constants with the index set { a, ß, 1} is different from zero, where a E {i,i'}, ß E {j,j'} and 1 E {l,l'}.

Proposition 2.3.4 The graph F; is connected if and only if the graph F; is. •

Therefore the most efficient algorithm for verification of the primitivity of the cellular ring works with the matrices of order r. Information about the complexity of the algorithm and its performance characteristics is given in section 2. 7.

Let us now switch to an algebraic interpretation of the primitivity property of c~llular rings. For matrices A, B E Mn(C), we denote ~tr(A · tf3) by (A, B), where tB is the transpose of the complex conjugate of B. Obviously, (A, B) is a positive definite Hermitian form. Let lAI = (A, In)· The following is obvious.

Proposition 2.3.5 The mapping A >--+ lAI from the cell W = (Ao, ... , Ad) to Cis d

an epimorphism of the algebra (W, ·) onto C. Moreover, the image of A = L; a;A; i=O

d

in W is lAI = L; aik;, where k; = lA; I= (Ai, In) are the subdegrees of the cell TV. i=O •

Definition 2.3.3 An element A E Mn(C) is called a cellular idempotent if

and A·A=(A,A)A (2.3.2)

It is clear that every cell contains the two idempotents I and I, which we will call trivial. (Here I is the identity matrix and J is the matrix all of whose entries are 1.)

Proposition 2.3.6 Let A E Mn(C) be a cellular idempotent satisfying the equality

(2.3.3)

Then R(A) is an equivalence relation of {1, 2, ... , n} whose classes have equal cardinalities.


~Since Ais a o-idempotent, then Ais a (0,1)-matrix. Let r = f(A) and m = (A, A). Then the relations A · A = mA and A · ln = mln (via (A, A) = (A, ln)) imply that R(A) isatransitive relation and that f(A) is a regular graph (in general, a digraph) of valence m. Let us prove the reflexivity of A. Let x be an arbitrary vertex of the graph f and let f( X) ( respectively, r-l (X)) be the set of vertices which are terminal points (respectively, initial points) of arcs initiated at ( respectively, terminated at) X. Let y E r(x). From the transitivity of R(A) it follows that f(y) c r(x). From the regularity of f(A) we conclude that lf(x)l = lr(y)l. Therefore f(y) = f(x), and so y E f(y ). Then the relation t A · ln = (A, ln)ln implies that for every vertex y we have y E f(x) for some vertex x. Thus R(A) is reflexive. Let us assume agairr that y E r(x). The reflexivity of R(A) implies that XE r(x). Sinee f(x) = r(y), as shown above, we have x E f(y) and thus R(A) is symmetrie. Therefore R(A) is an equivalence relation. The equality of the cardinalities of its classes follows from the regularity of the graph f(A) ...,..

By using Proposition 2.3.6, it is easy to prove

Theorem 2.3. 7 The cell W = (Ao, A1 , ... , Ad) is im primitive if and only if it eontains a nontrivial cellular idempotent. •

In some cases the following sufficient condition on the imprimitivity of a cell, formulated in algebraic terms, ean be used. We omit the proof.

Lemma 2.3.8 Let W be a cell with standard basis A = {A0 , A1 , .•• , Ad}· Suppose that A; · 2: Aj = k; 2: Aj for some i E {1, 2, ... , d} and F C {0, 1, ... , d}. Then W

jEF jEF is imprimitive. •

Now we present an imprimitivity condition for a cellular ring which is expressed in algebraic terms and which will be the most convenient for us.

Theorem 2.3.9 Let W be a cell with the standard basis A 0 ,A1 , ••• ,Ad. Suppose that there exist F C {0, 1, ... , d} and i E {0, 1, ... , d} suchthat the subspace .CF = (Aj)jEF is invariant with respect to the action of A; from the left, i.e., A;.CF t:;:: .CF. Then W is imprimitive.

~ Let MF = 2: Aj. To prove the imprimitivity of W it is enough to show, by jEF

Lemma 2.3.8, that A; · MF = k;MF.

Since the subspace .CF is invariant with respect to A; and MF = 2: Ai E .CF, jEF

we have A;. MF = 2: /jAj. This implies that IA;I. IMFI = I 2: rjAjl· Using jEF jEF

the equality IMFI = 2: kj, we obtain k, 2: kj = 2: /jkj, or 2: Cri- k;)kj = 0. jEF jEF jEF jEF

54 l. A. FARADZEV ET AL.

Using kj > 0 and /j ::=:; k;, we obtain /j = k; for all j E F, i.e., A; · MF = k;MF ..,..

Let us now consider the arithmetic properties of cellular rings. Many necessary conditions for the existence of primitive permutation groups, formulated in terms of subdegrees and structure constants of the V-ring, are well known. (See, e.g., [Si 3].) It turns out that many of these conditions are satisfied not only by Vrings of primitive permutation groups, but also by any primitive cell. Several such properties are presented in [We 2], pp. 71-75. Below we give several new conditions for the primitivity of a cell.

Theorem 2.3.10 Let W = (A0 , A1, ... , Ad) be a primitive cell of degree n and 1 = ko ::=:; k1 ::=:; ... ::=:; kd, d 2:: 2. Then

(i) if A;' · A; = k;A0 + pLA~o then PL ::=:; k;/2;

(ii) if d 2::3, then \fi\fl(k; 2::3 and l =f 0 ==? PL < k; -1);

(iii) if GCD(k;, kj) = 1 and k1 2:: 3, then there exists an l suchthat k1 > max(k;, kj) and k;kj = 0 (mod ki);

(iv) for every prime p from the set

1r(W) = {p: p is prime and 3i ( k; :::::: 0 (mod p) },

the inequality p ::=:; k1 holds.

~ (i) Let us assume the contrary, that is, for some i and l the following is true:

A;' · A; = k;Ao + pLA1 and p~'i > k;/2.

Consider the relation R = R( A 0 + A1). By the definition, R is reflexive, and it is easy to check that it is symmetric. Let us show that R is transitive. In order to do so, it is sufficient to show the transitivity of R(A1). Let (a, b), (b, c) E R(A1). Set S1 = {x E {1,2, ... ,n}: (a,x) E R(A;') and (x,b) E R(A;)}, S2 = {x E {1,2, ... ,n}: (b,x) E R(A;,) and (x,c) E R(A;)}. Due to our assumption, IS1I > k;/2 and I S21 > k;/2, and both SI and s2 are subsets of the k;-element set {X E { 1' 2, ... , n} : (x, b) E R(A;)}. Therefore, S1 n S2 =f 0, so let y E S1 n S2. Then (a, y) E R(A;,) and (y,c) E R(A;). Thus (a,c) E R(A1). Hence R is an equivalence relation and, since {0, l} =f {0, 1, ... , d}, W is imprimitive. The contradiction obtained proves (i).

(ii) Let us assume the contrary, i.e., that for some l, PL 2:: k; - 1. Then Pl'; E { k; -1, k;}. The case p:,; = k; implies P:,l = k1 and can easily be discarded, due to Lemma 2.3.8. In the case PL = k;-1, we can prove that A;,A; = k;A0 +(k;-1)AI· Now we can use the inconsistency of the inequalities k; 2:: 3 and k; - 1 ::=:; k;/2, the latter of which is obtained from part (i).


(iii) From [We 2], p. 27, property es, it immecliately follows that pLkl O(mod LCM(k;,kj)) for alll E {0,1, ... ,d}. Since k; and ki are relatively prime,

d

we have pLk1 :::::: O(mod k;kj)· On the other hand, 2.:: Pljkl = k;kj. Therefore k=O

there exists a unique l such that Plj :j. 0, or A; · Aj = PljAI. This implies that k;kj:::::: O(mod k1). It is left to show that k1 > max(k;,kj)· It follows from [We 2], p.28, property C9, that k1 ~ max(k;, kj)· We assume the contrary, i.e., that k1 = max(k;,kj)· Then from Pljkl = k;kj follows that PL = min(k;,kj)· Let us assume

for definiteness that k; ::; kj. Then Plj = k;. Now we use an equality which holds for every cell:

d d

LPl1 iP~j,kl = L(PL?kl 1=0 1=0

which follows from the equalities

(A ·A- A-·A-)=(tA-·A 4.··tA)=(A·,·A· A··A·,) I J> I J I I> •· J J I I> J J •

From (2.3.1) we have d

LPl1 iP~j,kl = (pL) 2 kl = kfkj. 1=0

On the other hand,

d d

LPl'iP~j,kl = k;kj + LPl'iP~j,kl::; k;kj + (kt- k;)max P~j'' 1=0 1=1

where the maximum is over 1 ::; l ::; d. Therefore,

(kt- k;)kj::; (kt- k;)maxp~i''

kj ::; max P~j'·

(2.3.1)

Since k1 ~ 3, then kj ~ 3, and for d ~ 3 we arrive at a contradiction to statement (ii), which we have already proved. For d = 2, the condition GCD( k;, kj) = 1 contradicts the primitivity of the cell (see [He 3]).

(iv) Let us assume the contrary, i.e., let p be a prime divisor of some kj and p > k1. Let F = {i: k;:::::: O(mod p)}. According to our assumption F :j. 0, and F :j. {0, 1, ... , d} since 1 t/. F. Accorcling to Theorem 2.3.9, the space .CF= (A;}iEF

cannot be invariant with respect to multiplication on the left by A1 . Therefore, there exist i E F and l rf. F for which p~ i of. 0. The number p~ i is not greater than k1, so it is mutually prime with p. Due to the choice of l, k1 and p are relatively prime also. Therefore, pLkt cj:. 0 (mod p). On the other hand, pi;kt is divisible by k1 and so also by p. We have reached a contradiction <1111

References

Graph theoretical and algebraic aspects of primitivity were considered in detail in [Ca 4] and in §2.9 of [Ba 5]. They were included in this survey mainly to make the exposition self-contained. Propositions 2.3.1-2.3.4 are used in the testing of the primitivity of cellular rings (see section 2.7). Propositions 2.3.5-2.3.9 are given in the same form as in [*Mu 7], where they were used for enumeration of primitive subcells. Theorem 2.3.10 was proved by M.E. Muzichuk in [*Mu 7].

2.4. S-Systems in S-rings

The problem of enumeration of cellular subrings in the V-ring of an arbitrary transitive permutation group ( G, D) has been considered in section 2.2. Here we return to a particular case of this problem, namely when Gis a regular permutation group, i.e., G acts on the set of its elements by right translation. It has been mentioned in section 1.1 that in this case the V--ring V ( G) is isomorphic to the group ring Z( G), and each cellular subring is isomorphic to an S-ring over G.

In this particular case the process of enumeration of all cellular subrings (Srings) can be greatly reduced by employing information about the automorphism groups of possible S-rings. This information allows one to obtain necessary conditions on partitions of the set of basis elements of the original S-ring, where the partitions correspond to basis elements of possible S-subrings. In some cases ( see section 3. 7) these conditions turn out to be extremely effective, to the extent that the enumeration of S-rings can be clone without use of a computer.

We will use the notation from section 1.1. Let R be a group with identity element e and let U = (T0 ... , Tm) be an S-ring with To = { e} and T/ = {x 9 : x E T;}, where g is a permutation from the symmetric group S( R) of the set R.

Definition 2.4.1 We say that the permutation group (A, R) preserves basis quantities of the S-ring U = (T0 , .•• , Tm) if, for all g E A and Ti E {T0 , ... , Tm}, TlEU. - -

It is easy to see that in this case Tl is also a basis element of U. Since the

permutation i : x 1---+ x- 1 preserves tl~basis quantities ( due to the definition of S-ring), we assume i E A in what follows.


If a concrete S-ring U is given, A can be taken as the direct product of symmetric groups, acting on the sets T0 , T1, ... , Tm. But actually the given definition is of non-trivial interest only if the group (A, R) is known a priori for allS-rings over R or for a wide dass of such S-rings. We give the two most interesting examples of such groups (A, R).

Proposition 2.4.1 Let (R, +) be an abelian group of order n. Let Z~ be the multiplicative group of residues mutually prime with n. For r E R and fl E Z~, let Tfl = r + r + ... + r(fl times). Then (Z~, R) preserves all basis quantities of any S-ring over R. •

Proposition 2.4.2 Let U = (T0 , T1, ... , Tm) be an S-ring over a group R, let X be the set of one-element basi;:;-;ets"" of U 0bviously X forms a subgroup of the group R). Let A = S2 1 X, and define an action of the group A on R in the following way: For r ER, [€;x1,xz] E A, x1,xz EX, and € E {-1,1},

Then the action of A on R preserves the basis quantities of U.

~ Let T; be an arbitrary basis set of U. Then T;-l is also a basis set due to the definition of S-ring. Since x 1 and x 2 aresimple basis quantities, then x}1T;'x2 E U. Therefore, [€; x1 , x2 ] preserves thebasis quantities of U ...,..

Let us now mention several useful elementary properties of pennutation groups preserving basis quantities of an S-ring.

By 0 0 , 0 1 , ... , Ot we denote the orbits of the permutation group (A, R). By Oo we always denote the orbit containing e E R. We define the binary relation Gu on the set of orbits by:

Proposition 2.4.3

(i) eu is an equivalence relation;

(ii) (00 , 0;) (j_ Gu for all i :f- 0.

~ (i) The reflexivity and symmetry of Gu are obvious. Let us show its transitivity. Let (0;, Oj) E 8u and (Oj, Ok) E 8u. Then there exist basis quantities Tq and

Tr with Tq n 0; :f- 0, Tq n Oi :f- 0, Tr n Oj :f- 0 and Tr n Ok :f- 0. Since Oj is an orbit of the group (A, R), there exists a: E Tq n Oj and a E A such that


aa E Tr n Oi. Since aa E Tqa, we have that T; = Tr. Since Oi is invariant with respect to A (as is any orbit of (A,R)), we have

Having Tr n Oi =f. 0 and Tr n Ok =f. 0, we get (Oi, Ok) E 8u.

(ii) Every a E 0 0 is obviously a basis set of the S-ring U. Therefore each basis set of U intersecting Oo is a one-element set and is contained in Oo. This implies that (Oo, Oi) t/. 8u for all i =f. 0 ~

Corollary 2.4.4 U Or.

(0,,0; )EE>u

Let Ti be a basis quantity of U and Tin 01 =f. 0. Then Ti ~

• Now we can introduce the concept of an S-system of the S-ring U with respect

to the group (A, R), which is the central to this section.

Definition 2.4.2 The set PA(U) = {A1 , ... ,Am;8u} is called the S-system of the S-ring U = (Ta, ... , Tm) with respect to the group (A, R) if the following conditions are satisfied:

(i) A; is the stabilizer of Ti in A, i.e., A; = { a E A : Tt = Ti};

(ii) 8u is the equivalence relation on the set of orbits of (A, R) described above.

Lemma 2.4.5 Let P A(U) = {A1 , ... , Am; 8u} be the S-system of the S-ring U with respect to the group (A, R). Then

(i) for all i E {1, ... , m} and a E A, a- 1 A;a E P A(U)

(ii) for all i E {1, ... , m} the family of subsets (Tin Oi)'j=1 forms a system of imprimitivity under the action (A, Oi);

(iii) if Ts n Oi =f. 0 and Ts n Oi =f. 0, then

~ (i) Let g E Ai, so that Tl= Ti. Then (Tt)a-'ga = (Tl)a = Tt and a- 1ga E Aj for some j, hence Tt = Tj.

(ii) Let us consider all non-empty sets ofthe form T1 noi, jE {l, ... ,m}. Clearly they form a partition of 0; on which A acts transitively. Since every element of A

either stablizes Tj or maps it to Tk, for some k :/= j, this partition is a system of imprimitivity for (A, 0;).

(iii) Let us consider all images of the set T8 under the action of permutations from A. It is clear that there are [A : A 8 ] different sets among them. The intersections of these sets with 0; and oj form systems of imprimitivity under the actions of (A,O;) and (A,Oj) correspondingly. The number ofblocks in each system is equal to the number of distinct images of Ts <1111

Let us introduce one more important concept.

Proposition 2.4.6 Let (R, +) be an abelian group of order n and (Z~, R) be a permutation group preserving all S~rings over R. Let 0 0 = 0, 0 1 , ... , Ot be all distinct orbits of (Z~, R). Then (00 , ... , Ot) is an S~ring over R. •

Definition 2.4.3 The S~ring (00 ,01, ... ,0t) and all its subrings are called Srings of traces over R.

Proposition 2.4. 7 Let U = (To, ... , Trn) be an S--ring over the abelian group R, 8u be the equivalence relation on { 00~~., Ot} defined above, r0 , r1, ... , Tt be the equivalence classes of 8u and Pz:, (U) be the S~system U with respect to Z~. Then

(i) if for some k E {1, ... , m}, T; n Ok :/= 0 and Tin Ok :/= 0, then A; =Ai;

(ii) U = (To, Q1, ... , Qt) is an S~ring of traces over R, where 8u = 8u and Q; = UTj' with the union taken over all Tj suchthat Tj n ok t= 0 and ok E Tj .

• In the case when A = Z~, Proposition 2.4. 7 allows a more compact definition

of S~system.

Definition 2.4.4 The S~system for the S~ring U over the abelian group R is the set P(U) = {A1, ...... ,At;8u} where A; is the stabilizer of all basis sets having nonempty intersection with 0;.

Below we consider an example which illustrates how the concepts and propositions of this section can be used to ease the task of searching for subrings.

Example Enumerate all S~rings over Z8 .

Here z; = {1,3,5, 7}, Oo = {0}, 01 = {1,3,5, 7}, 0 2 = {2,6}, 0 3 = {4}. According to Proposition 2.4.6, ( Oo, 01, 02, 03) is an S~ring. It is easy to check that (00 , 0 1 + 0 2 + 0 3), (00 , 0 1 ,02 + O;}and (00 , 0 1 + 0 2, 0 3) arealso S~rings. Moreover,they are S~rings of trac~ A~rther possibi~S~ring of traces is (00 , 0 1 + 03, 02). Let us check its existence. Since ( 02 )2 = 2 Oo + 203 and ( 02 )z;;-( 01 + -- - - - - -


03) = 03, this subspace doesn't form an S-ring, and the only possible equivalence relations in the S-systems are the following four: 8 0 = { {0}, {1}, {2}, {3} }, 8 1 = {{0},{1,2,3}}, 82 = {{0},{1},{2,3}} and 8 3 = {{0},{1,2},{3}} (here we show only sets of indices of orbits which form equivalence classes).

Let us enumerate the remaining S-rings. First of all we notice that there are five different subgroups of z;: F1 = z;, F2 = (5}, F3 = (3}, F4 = (7} and F5 = (1}. Now it is always the case that A3 = F1 and A2 = F1 or F2 . Therefore S-rings of traces are the only S-rings with 8u = 8 1 , 8 2 . Let 8u = 8 3 . Then A1 = A2 and there are two possibilities for these groups. If 8u = 8 0 , then all possibilities for A1

must be checked. In all cases except A 1 = F5 and A1 = A2 = F2, a "candidate" for being an S-ring can be immediately reconstructed from the S-system (See Table 2.4.1). In the case A 1 = F5 , one should also use Lemma 2.4.5(ii).

There is only one potential S-system which admits two distinct "candidates" for S-rings. A simple verification shows that there are indeed S-rings which correspond to rows 2, 3, 4, 5, 6 (for rows 1, 7, 8, 9,10 we have already seen this). The rejection of the last five "candidates" can be clone by easy computations, e.g., in order to reject line 13 it is enough to notice that 1°1 = 2. but 2. doesn't belong to the subspace. Therefore there are in all ten S-rings over Z 8 (including the two trivial ones ).

Other applications of this technique are discussed in section 3. 7. • References

Propositions 2.4.1, 2.4.6 and 2.4.7(ii), as well as the eoncept of S-ring, are due to I. Schur. Proofs of the statements can be found in [Wi 2], [Ko 1]. The concept of S-system was originally introduced and used for the enumeration of S-rings by P. Pöschel [Po 2]. It is eonsidered in Chapter 8 of the book [Po 3] from a more general perspective (the chapter was written with the participation of M.H. Klin). Propositions 2.4.2-2.4.5 are due to M.E. Muzichuk [*Mu 7], most of them being direct generalizations of corresponding results for cyclic groups from [Po 2]. Let us note that [*Mu 7] contains several more results useful for the enumeration of S-rings. The example was first published in [Kl 6] ( a eomputer was used for the computations). See also [Go 1], where the example is used to illustrate some of the obtained theoretical results. The concept of S--ring of traces was introduced in [Br 1] in a different context and using different terminology.

# Gu A1 A2 A3 Basis Quantities Existence of Ring

1 Go F1 F1 F1 Q,1,3,5, 7,2,6,1 + 2 Go F1 F2 F1 Q,1,3,5, 7,_2,§,1 + 3 Go F2 F1 F1 Q,.!_J'i_,~,2,6,1 + 4 Go F2 F2 F1 Q,.!_J'i_,~,_2,§,1 + 5 Go F3 F1 F1 Q,l~'~'~'1 + 6 Go F4 F1 F1 Q,.h1_,3,5,~,1 + 7 Go F5 F1 F1 Q,l,~,Q,1,.2,§,1 + 8 G1 Ft Ft F1 Q,1,3,5,7,2,6,1 + 9 G2 F1 F1 F1 Q,1,3,5,7,2,6,4 + 10 G3 F1 Ft F1 Q,1,3,5,7,2,6,4 + 11 Go F3 F2 F1 Q,~,5,7,_2,§,1 -

12 Go F4 F2 F1 Q,.h1_,~,_2,§,1 -

13 Go F5 F2 Fl Q,l,~,Q,1,2,6,1 -

14 G1 F2 F2 F1 Q, 1,5,2,3, 7,6,1 -

15 G1 F2 F2 Fl Q, 1,5,6,3, 7,2,1 -

Table 2.4.1

2.5. 2-Closure

Herewe consider the problern of determining the automorphism group Aut(W) of a Schur type cellular ring W. In this case the permutation group ( G, n) with V( G, n) = W is known. Therefore Aut(W) = G<2l, where G<2l is the 2-closure of G (see section 1.2.)

Fora given permutation group (G, n), the problern offinding G( 2l can be solved with the aid of a computer, provided n = 1n1 is not too large (see section 2.7). Here we consider several special cases of the solution of the problern at the theoretical ( computer-free) level, which can be applied when we have an infinite family of permutation groups, usually depending on several parameters. For one such method, we assume that the problern has been solved for the first several members of the family. Altogether we consider three such methods, and we illustrate them on some simple examples. Other possible applications of these methods are discussed in Chapter 3. Let us remark right here that these methods are not universal, and in some cases an ad hoc technique has to be developed in order to find the 2-closure of a particular group (see, e.g., section 3.7).

2.5.1. Using automorphism groups of partially ordered sets

Let a permutation group ( G, N) be given via some "natural" action, and sup-


pose we know that the group is 2-closed. We want to compute the 2-closure ( G, Q)(2 )

for another permutation representation of the group G on a certain set n. With this aim, we consider a family of induced actions (in the sense of Section 2. 7.1) of G of the same type Oll the sets L;, 1 ::; i ::; m. We suppose that n = Lm. Here, being of the same type means that there exists a unified combinatorial interpretation for the elements of L;'s in terms of 2-orbits of ( G, N) and points from L; and Lj, i =/= j, differ only in values of certain parameters. We assume it is possible to introduce a partial order 2: on L = U~ 1 L; which is preserved under the action of G. Then knowledge of the automorphism group of (L, 2:) can be used to compute the 2-closures ( G, L;)(2 ), provided that the set L satisfies conditions which we present below.

In what follows we restriet ourselves to the computation of ( G, Lm)( 2) ( the value i = m will be especially emphasized in the mentioned conditions ), but the technique we develop can also be used for all i, 1 ::; i ::; m.

Thus we let L be a finite partially ordered set with order ::;=:. Let V (respectively, 1\) denote the operation of taking the least upper ( respectively, greatest lower) bound.

We will need the following definitions. (More about semilattices can be found in [Bi 2].)

Definition 2.5.1 A semilattice with minimal element 0 satisfies the JordanDedekind condition if, for every x E L, allmaximal chains between x and 0 have the same length r( x ). In this case, the function r( x) is called the rank function Oll L.

Definition 2.5.2 Let r be a rank function on a semilattice L. A three element set { a, b, c} C L is called a triangle if:

(i) the elements a, b, c, a 1\ b, a 1\ c, b 1\ c are pairwise distinct;

(ii) r(a) = r(b) = r(c) = r(a 1\ b) + 1 = r(a 1\ c) + 1 = r(b 1\ c) + 1.

In what follows we assume that L satisfies the following conditions.

11. L is a 1\-semilattice with minimal element 0.

L2. L satisfies the Jordan-Dedekind condition, and r(x) is the corresponding rank function on L.

13. For x::; y the interval [x,y] is a chain if and only if r(y) = r(x) + 1.

L4. Let m = max r(x). Then, for every a, b E L with r(a) = r(b) = m- 1, there exist c, d E L such that c > a, d > b and c 1\ d = b 1\ a.

L5. For every triangle { a, b, c }, the least upper bound aVbV c exists and r( aV bVc) = r(a) + 1.

The number m = max r( x) will be called th~::; rank of the A-semilattice L. Let a E L, 0 ::::; k ::::; m. Then Lk,a = {x E L : r(x) = k and (a 2: x or x 2: a)}, Lk = Lk,O· Let G be the automorphism group of the lattice L. It is clear that the sets Lkl 0 ::::; k ::::; m, areinvariant blocks of the permutation group ( G, L ); therefore

the actions ( G, Lk) are defined. Our goal is to prove that ( G, Lm) is 2-closed.

Lemma 2.5.1 For every k ::::; m -1 and arbitrary a 1 , a2 E Lk, there exist c1 , c2 E Lm such that c1 > a1, Cz > a2 and c1 1\ Cz = a1 1\ az.

~ We will use induction with respect to m- k -1. The basis of induction ( k = m -1)

coincides with L4. Induction step: We assume that the lemma is proved for all k 2: n + 1 and we show that it is valid for k = n. Let a 1 , a2 E Ln. Then in the above layer Ln+I there are distinct elements b1 and b2 such that b; > a; (here and later i = 1, 2). By the inductive hypothesis, there are c1, c2 E Lm such that c; > b; and c1 1\ c2 = b1 1\ b2 • Since r( a;) = n ::::; m - 2, t.hen, according to L3, there are

b~ E [ai,ci] and b~ E [a2,c2] with bi 'I- b;. Since b~ 1\ b~ ::::; c1 1\ Cz = b1 1\ bz, we obtain

Again, by the inductive hypothesis, there are c~, c~ E Lm with ci > bi and c~ 1\ c~ = b~ 1\ b~. From this we conclude that c~ 1\ c~ = a 1 1\ a2 <01111

Lemma 2.5.2 For every k < m -1 and a E Lk, the set Lk+I,a has no least upper bound in L.

~Let us assume the contrary, i.e., that a E L with r(a) = k < m- 1, and the set

Lk+l,a = {x E Lk+I :X > a} has the least upper bound a. Let y E Lk+2, y > a. According to L3, there are x1, Xz E Lk+I,a suchthat a < x; < y, XJ 'I- X2. Then y =X] V Xz. Since x; E Lk+I,a, then x; ::::; a, hence y =X] V Xz :::; a. Repeating the argument as many times as necessary, we obtain that an arbitrary y E L which is greater than a is less than a. On the other hand, according to L4, Lm contains at least two elements greater than a which do not have an upper bound. This contradiction finishes the proof <01111

Lemma 2.5.3 Let AC Lk and for all a1,a2 E A, a1 'I- az, assume r(a1 1\ az) =

k- 1. Then at least one of the following two equalities holds: r(VaEAa) = k + 1 or

r(AaEAa) = k- 1.

~ Let us fix two arbitrary distinct elements a 1 , a2 E A and consider two sets:

Av = {a E A: a < a1 V az} and A" = {a E A: a > a11\ az} (Av = 0 if a1 Vaz does

not exist ). In order to prove the lemma, it is enough to show that at least one of the sets Av, AA coincides with A.

Suppose this is not the case, i.e., there are two elements a~, a~ E A (not necessarily distinct) such that a~ -/. a 1 V a2, a~ f a 1 1\ a2. Then we show that a 1 , a2, a~ form a triangle which has least the upper bound a 1 V a 2 V a~ of rank k + 1. This implies the existence of a1 V a2 and that a~ < a1 V a2. Next we observe that a1, a;, a2 are pairwise distinct, but {aha;, az} is not a triangle. This implies that a~ > a1 1\ az. Finally we obtain

(The inequality is strict since a~ < a1/\a2.) From here we conclude that r( a1/\a2) > r( a; 1\a~), which contradicts the definition of the set A in the statement of the lemma ~

Now we prove the theorem on which the first method of computation of 2-closure is based.

Theorem 2.5.4 Let G = Aut(L, ~) be the automorphism group of the /\semilattice L of rank m, which satisfies the conditions L1-L5. Then the permutation group (GILm,Lm) is 2-closed .

..,.. Let us define the binary relations cPk,l on Lk as cPI,k = { ( x, y) E Lk x Lk : r( x 1\ y) = k -l}, l = 0, 1, ... , k. lt is clear that all the relations cPm,l areinvariant with respect to (GILm,Lm)· Let 7r be an arbitrary permutation from (G1Lm,Lm)( 2l. Then 7r

preserves the relations cPm,l, 0 :::; l :::; m. In order to finish the proof, it is enough to extend 1r to an automorphism of the /\-semilattice L.

We are going to exhibit just such a construction for the extension of 1r : Lm -+

Lm to the permutation 1r1 : Lm U Lm-1 -+ Lm U Lm-1 such that 1r1 preserves the relations ~ and cPm-1,/, 0 :::; l :::; m - 1. "Layer-by-layer" continuation of 1r1 up to the action on the layer L 1 can be done in an entirely similar manner and, as a result, 7r will be extended to an automorphism of the whole lattice L.

First we consider the subsets Lm,a of the set Lm, where a E Lm-1· Since a = 1\xELm,.x, the mapping a ~--* Lm,a is injective. The permutation 7r preserves cPm,l and hence, for every x,y E Lm,a, r(x". /1. y".) = r(x /1. y) = m -1. This shows that the set L":r, a satisfies the conditions of Lemma 2.5.3. Using the fact that L":r, a

contains at least two maximal elements of L, one can easily show that L":n,a = Lm,~, where b = 1\xEL~ x. Therefore 7r permutes the sets of the form Lm a·

m,a '

Now we define an extension 1r1 of the permutation 7r on Lm U Lm-l by the rule: for a E Lm-1, a".1 = b where L":n,a = Lm,b· By definition, 1r1 preserves the relation

~ on the set Lm U Lm-l· Using Lemma 2.5.1, we see that 1r1 also preserves the

relations <Pm-1,/, 0 ~ l ~ m- 1 <1111

As we mentioned above, it is possible to prove that the groups ( GIL.; Lk) are

2-closed for 1 ~ k ~ m, by slightly modifying the arguments of Theorem 2.5.4.

Now we consider two examples of applications of Theorem 2.5.4. In each case we

just give a description of the /\-semilattice L and its automorphism group Aut(L ). Verification of conditions L1-L5 is left to the reader. The first example, which is

of folklore type, is given just to illustrate the method; the fact that the groups are

2-closed can be easily established without the use of Theorem 2.5.4.

Example 1 Prove that the exponentiation Sm j Sn of symmetric groups of degrees

m and n acting on mn elements is 2-closed.

Let M and N be finite sets, IMI = m, INI = n. Let L be the set of partial

functions from N to M, i.e., L = {(A, f): AC N and f: A--+ M}. We introduce

a partial order on L by:

The rank of L is equal to n and the set of maximal elements Ln coincides with the

set of all functions from N to M. We show that Aut(L, ~) coincides with the action

of the wreath product of symmetric groups S(N) I S(M) on L which is defined by

The inclusion Aut(L, ~) ;:;? S(N) I S(M) is obvious. On the other hand, the restric

tion of Aut(L) to L1 = {( {i},!) : i E N, f : {i} --+ M} is exact and it preserves

the equivalence relation ß with equivalence classes {( { i}, f)}, i E N. Therefore

Aut(L, 2) ~ Aut(ß)= S(N) I S(M). From here we conclude that Aut(L, ~)=

(S(N) I S(M); L). The action of S(N) I S(M) on Ln is exact and equivalent to the

exponentiation S(M) j S(N). •

Example 2 Prove the symmetric group S(N), where INI = mk, m > 1, k > 2, is

2-closed in its induced action on the unordered partitions of the set N into k parts

of equal cardinality.

Let {~}={Ac N: lAI = m}. Let

L = {{A1 , ... ,A;} :Ap E {:} for 1 ~p ~ i, and Ap n At= 0 for p #land i E {1, 2, ... , k- 2, k }}.

Let {AI, ... , A;} 2 {B1 , ... , Bj} {o} Vs (B. E {A1 , ... , A;} ). Here

{ i, for i ::=; k - 2, r({AJ, ... ,A;})= k 1 r ·-k

- , 10r Z- .

We prove that Aut(L, 2) = (S(N), L). The inclusion S(N) <;;; Aut(L) is obvious. Conversely, the restriction of Aut(L) to L1 = {~} is exact and preserves the relation Rm = {(A, B) : AnB = 0} on L1 . The relation Rm is a 2-orbit of the induced symmetric group (S(N), {~}) and for n > 2m+ 1 it generates the V-ring of this group. Therefore Aut(L) <;;; (S(N); {~} )( 2). For n -=f. 2m, the group (S(N); {~}) is 2-closed (see section 3.2). Hence Aut(L, 2)= (S(N), L). The action of S(N) on Lk-l, where Lk-I = { {A1 , ... , Ak}: A; E {~} and A; n Aj = 0 for i -=f. j}, is exact and it coincides with the induced permutation group under consideration. •

2.5.2. Using suborbits of 2-closed permutation groups

Let ( G, D) be a transitive permutation group, and let 2-orb( G, D) = { 0 , 1 , ... ,

r-d, with as usual o = {(o:,o:): o: E !1}. Let <l>;(o:) = {ß E D: (o:,ß) E ;}. We remind the reader that ;( o:) is usually called the suborbit of the permutation group ( G, n) with respect to the point o:. The method for computing the 2-closure which we are going to discuss in this subsection is based on the study of the action of the group G);l on the suborbit with respect to o:. More precisely, we will give sufficient conditions for the action of G);l on such a suborbit to be exact and coincide with the action of Ga.

Let X c n, ; be a relationOll n and ;Ix denote the restriction of ; to X, 1.e. ;Ix = {(a,ß) EX x X: (o:,ß) E ;}.

Definition 2.5.3 Let ß, 5, 'Y E !1. We say that ß separates the pair ('Y, 5) with respect to 2-orb(G,D) ifthe pairs (ß, 5) and (ß,'Y) lie in different 2-orbits of (G,D).

Definition 2.5.4 Let X c n. We say that X separates 2-orb( G, D) if, for every 2-element subset { 'Y, 5} of the set !1 \X, there exists a point ß E X which separates the pair {'Y, 5}.

Definition 2.5.5 We say that a relation ; E 2-orb ( G, D) separates points of n if, for some a E n, the set ;( a) U { a} separates 2-orbits of the group ( G, D).

Remark Due to the transitivity of the group (G,D), Definition 2.5.5 doesn't depend on the choice of point o:.

Obviously, a point a E D separates every pair of points lying in different orbits of Ga. Therefore the relation ; separates points of n if and only if every pair of points from one orbit of the group Ga is separated by an element from ;(o:).

Lemma 2.5.5 Let ( G, !1) be a transitive permutation group and let <P1 be a 2-orbit which separates points of no Then the restriction Ga Oll <Pl ( a) is faithful.

.,.. Let us assume the contrary, namely that the kernel of the restriction of Ga on <P1 ( a) contains a nontrivial permutation 1r. Then there exists a point x E n \ ( <Pl ( a) u { a}) such that X 71" =F X 0 Since <Pl separates points of n, there exists z E <P 1 ( a) U { a} such that the pairs ( x, z) and ( x 71", z) lie in different 2-orbits of the group (G,f!)o On the other hand, the action of 1r on the set <P 1(a) U {a} is trivial; therefore (x,z)11" = (x1r,z), ioeo, the pairs (x,z) and (x1r,z) belong to the same 2-orbit of the group ( G, f!)o The resulting contradiction proves the lemma ..,..

Lemma 2.5.6 Let ( G, f!) be a transitive permutation group, and let <P1 E 2-orb(G, f!)o Then (Gai<P,(a))(Z) can be embedded in the group

.,.. The group Gai<P,(a) leaves invariant the restriction of every 2-orbit of the group (G,f!) to <P1(a), and therefore so does the 2-closure of the group Gai<P,(a) ..,..

Theorem 2.5. 7 Let ( G, !1) be a transitive permutation group, and let <P 1 be

a 2-orbit which separates points of flo Then ( G~), <P1 ( a)) embeds in the group

(H, <Pt(a)) = Aut( {<P;I<P,(a)}) o If (H, <P1(a)) =(Ga, <P1(a)), then G(Z) =Go

.,.. According to Lemma 20505, the restriction of G~) to <P1 ( a) is exact, ioeo,

Now the first part of the theorem follows from Lemma 205060 In order to prove the second part, it suffices to notice that the restrictions of 2-orbits of the groups G" and (G( 2))a on <P1(a) coincideo Hence (Gai<P,(a))( 2 ) = (H, <P1(a))::) (G(2l)a and

Since IGI :S IG(z) I is always true, we obtain IGI = IG(Z) I and G = G(z) ..,..

Remark In some cases the arguments used in the proof of Theorem 2050 7 allow us to obtain a stronger result: Namely, if one can show that, for every <P; E 2-orb( G, f!) with <P;I<P,(a) =F 0, the relation <P; is formula-expressible through the relation <P1 (in terms of section 1.2), then G = Aut(<Pt)o

We will illustrate both Theorem 2050 7 and the foregoing remark on the simple example of the induced action of S5 on 30 pointso More interesting applications are discussed in Chapter 30


Let N = {1, 2, 3, 4, 5}, and let i, j, k, l, m be pairwise distinct elements of N. Let D = {({i},{j,k})} be the set of all possible ordered pairs of nonintersecting subsets of N, where the first subset contains only one element and the second is a 2-element subset. It is clear that IDI = 30 and (55 , D) is a transitive imprimitive permutation group which is similar to the action of S5 on its cosets with respect to the subgroup z2 X z2. Using the Burnside Lemma, it is easy to determine that the rank of ( S5 , D) is 11. The 2-or bi ts of the grou p ( S5 , D) can be easily descri bed in terms of intersections of subsets forming the pairs from D. In particular, there is a 2-orbit 1>1, where 1>1 = {({i},{j,k}),({j}, {i,l})}. In what follows, we call the graph r = (D, 1>1) the Balaban-Randic graph.

Example 3 Prove that the automorphism group of the Balaban-Randic graph is isomorphic to Ss.

Let a = ({1}, {2,3}). Then

1>1 (a) = {({2},{1,4}),({2},{1,5}),({3},{1,4}),({3},{1,5})}.

Let 1>2 = {({i},{j,k}),({l},{j,k})},1>3 = {({i},{j,k}),({i},{j,l})} and 1>4

{({i},{j,k}),({l},{j,m})}. It is clear that 1> 2 ,1>3 and 1>4 arealso 2-orbits of (Ss, D). Let us apply Theorem 2.5.7 to this group.

First we verify that { a} U 1>1 ( a) seperates points of D. We assume the contrary, i.e., that there is a pair of distinct points of n, ß1 = ( { ii}, {j1' kd) and ß2 = ( { i 2}, {h, k2}) which is not separated by { a} U 1>1 ( a). This means that both { i 1} and { i 2} ( repectively, {j1, k1} and {iz, k2}) have the same cardinalities of intersections with all subsets forming the points from { a} U 1>1 ( a ). This implies that i1 = i2, {]I, kd = {h, kz}. We arrive at a contradiction since ß1 ::/:- ß2.

N ext we notice that the restrictions of the relations 1>2, 1> 3 , 1> 4 are all nonempty relations from {1>;lq, 1 (a), 1>; E 2-orb(G,D)}. The group

obviously coincides with the regular action of z2 X z2' that is, ( H, 1>1 ( Q')) = (Ga,1>1(a)), where Ga is the stabilizer of a point a in (S5 ,D), which is isomorphic to z2 X Zz. Thus we get that (Ss,D) is a 2-closed permutation group.

Let us show that (S5 , D) = Aut(f). According to the remark following Theorem 2.5.7, it suffices to show that 1>1 f= 1>2 , 1>3, 1>4 (actually it is enough to show that 1>1 f= 1>2, 1'4). In order to do this, we notice that (I', 8) E 1>2 if and only if the points "'( and 8 belong to a simple 4-cycle in r and are of distance 2. The relation 1>4 can be similarly described with the replacement of length of the cycle from 4 to 5. •

2.5.3. 2-Closedness of imprimitive Frobenius groups

Here we will prove the 2-closedness of the groups mentioned in the title; this result is used in the study of automorphism groups of cyclic graphs (see the references in section 3. 7).

We remind the reader ( see [Wi 2]) that a transitive permutation group ( G, n) is called a Frobenius group if, for every a E n, the stabilizer Ga acts semiregularly on n \ { a}. A transitive group ( G, n) is called 3 /2-transitive if all its subdegrees, except n1, have the same value. It is clear that each Frobenius group is 3/2-transitive.

Theorem 2.5.8 Each imprimitive Frobenius group is 2-closed.

~ Let ( G, 0) be an imprimitive Frobenius group. Then its 2-closure ( G< 2l, 0) is also imprimitive. Since G(z) i'::jz G, we see that G(z) has the same subdegrees as G. Therefore ( G<2l, 0) is also a 3/2-transitive permutation group. According to a theorem from [Wi 2], we conclude that an imprimitive 3/2-transitive group ( G<2l, 0) is a Frobenius group. Now let a, ß E 0, a f- ß, and let R; be the 2-orbit of the group (G,O) containing (a,ß). Then IRd = [G: Ga,,a] = IGI, since GisaFrobenius group. Similarly, we get IR;I = IG(2ll. Thus IGI = IG(2ll. Since G s:;; G(2l, we have G = G<2l ..,..

Remark For primitive Frobenius groups Theorem 2.5.8 is generally false. As an example, we consider two primitive permutation groups G1 and G2 of degree 16 from the list of Sims [Si 3]. Here G1 == E16 : Z 5 , G2 = E16 : Ds, where A : B denotes the semidirect product of the group A by the group B. The group G1 is obviously a Frobenius group. The stabilizer of point 16 in Gz is generated by permutations a15 and ei6 from the list given in (Si 3]. It is easy to see that the group (ats, ei6/ (as well as the subgroup Z5 of Gt) has 4 orbits on the set {1, 2, ... , 16}. From here we conclude that G1 i'::jz Gz, and G1 is not 2-closed.

References

Theorems 2.5.4 and 2.5.7 were proved by M.E. Muzichuk. The result given in example 2 was also obtained by V.A. Vyshenski} and S.D. Parashchuk (*Vy 2] using another, more complicated, technique. The Balaban-Randic graph was first considered in (Ba 3] in connection with problems from organic chemistry; it is the so-called "reaction graph of adegenerate rearrangement" (see more details in (Ba 3], [Jo 2]). In [Ra 1] M. Randic, by means of some special arguments, determined the automorphism group of this graph. U sing a list of examples, it was demonstrated in (Jo 2] that the application of the theory of centralizer rings to the study of automorphism groups of such chemical graphs seems to be quite natural. Theorem 2.5.8 was proved in [*Kl 5]. We also want to note that substantial progress has


been made very recently in the solution to the problern of computing 2-closures of primitive permutation groups (see [Li 1], [Li 2]). Using some results from the classification of simple groups, the authors managed to determine all cases where the 2-closure of a primitive quasisimple permutation group has a large socle.

2.6. The t-vertex condition

Several methods for the computation of Aut(W), when W is a Schurian ring, were presented in the previous section. Let now W be a proper cellular subring of V( G, D), where ( G, D) is a 2-closed group, and let us assume either that there are no intermediate proper cellular subrings between W and V ( G) or that all such subrings are of non-Schurian type. In this case, if W is non-Schurian, then we get Aut(W) = (G, D) (see section 1.3). In order to prove that W is non-Schurian, it is enough to show that at least one of the basis graphs in W can be "combinatorially destroyed". Let us formulate this suffi.cient condition for being non-Schurian in more rigorous terms.

Let A;" A;" ... , A;, ( s 2': 2) be elements of the first basis of V( G) such that A = A;, + ... + A;, belongs to the first basis of the subring W. Let r(A) be the corresponding graph for W. Let ep and eq be distinct edges from r(A) with ep an edge in r(A;v) and eq an edge in r(A;.), p f. q. Let a be a numerical combinatorial parameter of the edges of r(A) which is invariant with respect to Aut(W) (invariant of the 2-orbit in the terminology of section 2.1), i.e.,

Vg E Aut(W), Ve E E(r(A)), a(e) = a(eY).

Then using the propositions given in section 1.2, one can easily prove

Lemma 2.6.1 The condition a(ep) f. a(eq) implies that W is non-Schurian. Therefore Aut(W) =Aut(V(G)) = Q(Z) = G. •

Here we show a systematic way of obtaining edge invariants.

Let r be a colored graph with vertex set D and E : D2 -+ {0, 1, ... , d} be a mapping which assigns to each ordered pair (x,y) of vertices the value E(x,y), called the color of the arc from X to y. Let ]{ c n. Then ]{ generates a colored subgraph r(K) of r. Let J{l' Kz c n and X and y be two (not necessarily distinct) vertices of D which belong to both K1 and K 2 • We say that subgraphs r(K1 )

and r(Kz) are of the same type with respect to the pair (x, y) if there is an isomorphism from the subgraph r(KI) to r(Kz) which maps X to X and y to y.

We say that a colored graph r satisfies the t-condition on the arcs of color i, (0 ::; i ::; d), if for every k, 2 ::; k ::; t, the number of k-vertex subgraphs of each


fixed type, with respect to an ordered pair of vertices (x, y) joined by an arc of color i, is the same for all arcs of color i.

We call a colored graph r, which satisfies the t-condition on the arcs of all its colors, a graph with the t-vertex condition, or just the t-condition (in another terminology, a graph with depth t). The following proposition is obvious.

Proposition 2.6.2 Let r be a colored graph with n-element vertex set n.

a) r is a graph with the 2-condition if and only if all nonisolated vertices in each of its one colored subgraphs have the same valency;

b) r is a graph with the 3-condition if and only if r is a colored graph of a cellular ring over n;

c) r is a graph wi th the n-condi tion if and only if i t is a colored graph of a Schurian cellular ring over n. •

Let us consider a particular case, when f is a simple graph. Proposition 2.6.2 can be restated in this case as

Proposition 2.6.3 Let r be a simple graph with n-element vertex set n.

a) r is a graph with the 2-condition if and only if r is a regular graph;

b) r is a graph with the 3-condition if and only if r is a strongly regular graph;

c) r is a graph with the n-condition if and only if r is a graph of rank 3. •

As a consequence of Propositions 2.6.2 and 2.6.3, in order to show that a cellular ring w on n is non-Schurian, or that a strongly regular graph r Oll n is not of rank 3, it is sufficient to show that the t-condition fails for some t with 4 :::; t :::; lr!l.

Let us consider the case t = 4 more carefully.

Theorem 2.6.4 Let f be a strongly regular graph.

a) If the number of 4-vertex subgraphs of r of some particular type with respect to the edge { x, y} does not depend on the choice of { x, y}, then the number of 4-vertex subgraphs of any other type does not depend on the choice of edge {x, y}.

b) If the number of 4-vertex subgraphs of r of some particular type with respect to a non-edge { x, y} doesn't depend on the choice of { x, y}, then the number of 4-vertex subgraphs of any other type also does not depend on the choice of non-edge {x,y}. •


Let r be a graph with the 4-condition. By a (resp. ß) we denote the number of complete 4-vertex subgraphs (resp., subgraphs with 5 edges) containing a given edge ( resp.' non-edge) {X' y}. Theorem 2.6.4 implies that a strongly regular graph r is a graph with the 4-vertex condition if and only if { a, ß} is its system of invariants. It turns out that there is a relation between these two invariants.

Proposition 2.6.5 Let f be a strongly regular graph with basic parameters v, k, l, .\, p, and suppose that it satisfies the 4-condition. Then

k(A(.A- 1)/2- a) = lß.

• A consequence of Proposition 2.6.3 fort = 4, together with Theorem 2.6.4 and

Proposition 2.6.5, have been used many times by the authors and their colleagues. Let us illustrate different schemes of applications by several examples.

The simplest scheme is the following. If r is merged from s basis graphs of a cellular ring, then we calculate the value of a for the s edges which represent sets of edges for these basis graphs. If all s values are the same, then a is an invariant of the edges of the graph. Otherwise r is not a graph with the 4-condition. Similarly, we work with the set of non-edges of r, using ß instead of a.

Example 1 In Proposition 2.2.7 and section 3.2, existence of several strongly regular graphs corresponding to cellular subrings of rank 3 in the BM-algebra of the J ohnson scheme J ( v, k) is proved. In particular, these strongly regular graphs appear when k = 4 for v = 9, 11, 12. Results of verification of the 4-condition for these graphs are given in Table 2.6.1.

Therefore the first and third graphs satisfy the 4-condition and the second doesn't. In fact, both the first and third graphs are actually graphs of rank 3. (See section 3.2 for more details.) •

Kind of Parameters Values of a and ß edges of s.r.g. on graphs A;

V s.r.g. r II k I A I' A, A2 Aa A.

9 A, +A• 126 25 100 8 4 "'= 12 ß=4 ß=4 <>= 12 11 A1 +A4 330 63 226 24 9 Q' = 153 ß = 34 ß = 24 "'= 156 12 A2 + A• 495 238 256 109 119 ß = 3213 "'= 2430 ß = 3213 Q = 2430

Table 2.6.1

One similarly shows that the 4-condition is not satisfied by two additional strongly regular graphs, having parameters

v = 120, k = 63, A = 30, p = 36 (for 1(10, 3) );

v = 1716, k = 833, A = 400, p = 408 (for 1(13, 6) ).

(See section 3.2 for more details.)

Sometimes it turns out that a (resp., ß) is indeed an invariant of edges (resp., non-edges) of a strongly regular graph r, but the parameter ß (resp. a) calculated from Proposition 2.6.5 is not a nonnegative integer, hence is not an invariant.

Example 2 Let r be a strongly regular graph with parameters (105,32,72,4,12), the existence of which was proved in Proposition 2.2.1. Let { x, y} be an edge of this graph, say x = (a, L), y = (b, L), where a and b are points and L is a line of the projective plane of order 4. Then all flags adjacent in r to both X and y have the form ( c, L ), where c E L, c (/. { a, b}. This implies that a = 3 for the edge { x, y}. Similarly we can show that a == 3 for the other types of edges. Assuming now that r is a graph with the 4-condition, and using Proposition 2.6.5, we obtain the equation

32(4(4- 1)/2- a) = 72ß.


As a result, we obtain 3ß = 4, which has no integer solution. Therefore r does not satisfy the 4-condition, and hence is not a graph of rank 3. •

Until recently, all of the strongly regular graphs investigated by the authors and their colleagues which satisfied the 4-condition turned out tobe graphs of rank 3. The following problern appeared to be interesting in this connection.

Problem 1 Are there strongly regular graphs satisfying the 4-condition which are not of rank 3?

An affirmative answer to this questionwas obtained in 1987 by A.V. Ivanov, who constructed a strongly regular graph with parameters (256,120,135,56,56) satisfying the 4-condition and having a = 784, ß = 672. Aut(r) has order 220 · 32 · 5 and is a transitive group of rank 4 ( there are two orbits on non-edges). The graphs r 1

and r 2 of neighbors and non-neighbors, respectively, of an arbitrary vertex of r are also graphs with the 4-condition which are not graphs of rank 3. f 1 has parameters (120, 56, 63, 28, 24), a = 216, ß = 144 and r2 has parameters (135, 64, 70, 28, 32), a = 168, ß = 192. Several different descriptions of these graphs were found. Here we give one such for the graph f 1.

Consider the cellular subring (A0 , A 1 +A2 +A5 +A6 , A3 +A4 ) in the BM-algebra of the Hamming scheme H(6, 2). Then t; = f(A 1 + A2 + As + A 6 ) is a strongly regular graph with parameters (64,28,35,12,12) and Aut(t;) S=' E64: Sg. The group Aut(t;) acts as a group of rank 3 on the set X of vertices oft;, and Aut(t;) acts transitively on the set of 240 8-vertex completely disconnected subgraphs oft;, This set of completely disconnected subgraphs is split into two orbits of the same cardinality under the action of the group E64 : A8 , which is a subgroup of Aut(t;) of index 2. Let X be one of these orbits. Let .i be the graph with vertex set X, where two completely disconnected subgraphs are joined by an edge if and only if they intersect in two vertices from X. Graph .i is isomorphic to the graph r I -

one of the graphs found by A.V. Ivanov which satisfies the 4-condition but is not of rank 3.

It is interesting that a graph r Oll 256 vertices found by A.V. Ivanov is also a graph with the 5-condition ' but not the 6-condition (graphs rl and r2 do not satisfy the 5-condition). In this connection, we now believe the following problern is very interesting.

Problem 2 Are there strongly regular graphs with the 6-condition which are not graphs of rank 3?

One of the authors (M.H. Klin) conjectures that there exists a sufficiently large t 0 ( t 0 2: 6) such that any strongly regular graph satisfying the t 0 -condition is of rank 3.


Prior to the examples of A.V. Ivanov, the authors knew of no n-vertex graph ( colared or simple) having the 4-condition but not the n-condition.

In 1988 another example of such a graph, on 15 vertices, was constructed by D.V. Pasechnik. This digraph is the first in an infinite family of antisymmetric association schemes with 2 classes. Colored graphs of these schemes satisfy the 4-condition, but their BM-algebras are of non-Schur type.

Theorem 2.6.6 Let {R0 ,Rt,R2 } be an antisymmetric association scheme on q points and having 2 classes which satisfies the 4-condition, and let { A0 , A1 , A2 } be a basis of its BM-algebra. We denote by l (resp., 0) the row vector of length q consisting of all ones (resp., zeroes). Let

and let Ao be the identity matrix of order 2q + 1. Then

i) (Ao, A1, Az) is a BM-algebra of an antisymmetric association scheme on 2q + 1 points;

ii) the BM-algebra (A0 ,A1 ,A2 } also satisfies the 4-condition;

iii) for q ~ 7, the cellular ring (Ao, A1 , A2 ) is of non- Schur type. • Remark It is well known that the conditions of Theorem 2.6.6 imply q :::::: 3 ( mod 4). By considering the Paley scheme on q points as an example of {R0 ,R1 ,R2 }, we obtain the

Corollary If q isapower of a prime, q:::::: 3 (mod 4), then there is an antisymmetric association scheme with 2 classes on 2q + 1 points which satisfies the 4-condition.

References

The concept of the t-vertex condition for strongly regular graphs was considered, and systematically investigated for t = 4, in [He 3]. In the case of colored graphs the concept was considered, in a different context, in [We 2], where the term "a graph of depth t" was used. Lemma 2.6.1 and Propositions 2.6.2, 2.6.3 are folklore. Theorem 2.6.4 and Proposition 2.6.5 were proved in [He 3], where Proposition 2.6.5 was attributed to C. Sims. The computations of o: and ß in example 1 were


clone by M.H. Klin (see [*Kl 5] and [*Kl 8]). We corrected here some inaccuracies in [*Kl 8]. The fact that the graph r from example 2 is not of rank 3 was mentioned in the survey by Hubaut [Hu 1]. Another justification of the fact that r does not satisfy the 4-condition was given in [*Go 1]. Problem 1 was raised many times by M.H. Klin in the 70's at various conferences and workshops on graph theory. It was published in [*Kl 10].

The examples of graphs with the 4-condition found by A.V. Ivanov were announced in [Iv 2] and [*Iv 10], and a complete description of the results obtained can be found in [Iv 3]. The description of f 1 was obtained by M.H. Klin. Using results from [Ca 1], A.E. Brouwer noticed that the examples found by A.V. Ivanov are the first members of an infinite family of graphs with the 4-condition (see [Br 4]). Theorem 2.6.6 was announced in [Fa 1]. A proof of the existence of an association scheme with the same parameters as those of the scheme given in the first part of Theorem 2.6.6 can be obtained by combining the remark of section 2.4 [De 1] with Lemma 14.1.6 of [Ha 6]. Recently, Theorem 2.6.6 was proved by D.V. Pasechnik in more general form.

2.7. A package of programs for computing in V -rings of permutation groups

In this section we are going to describe the algorithms realized in a package of programs which allows one to study transitive permutation groups according to the standard scheme presented in section 1.3. The package also contains some additional programs especially designed for the solution of some concrete problems (see sections 3.5, 3.6).

The package consists of the following programs:

(1) Inducing. Given a generating system of permutations for a transitive permutation group, we construct a generating system of permutations for its induced action on an orbit of a relation.

(2) Construction of a V-ring. Given a generating system of permutations of a transitive group, the colored graph of the V-ring of the group is constructed.

(3) Computation of structure constants. Given the colored graph of a cell, we calculate its structure constants.

( 4) Primitivity testing. Given the structure constants of a cell, we determine the connectedness of its basis graphs.

(5) Computing the lattice of subrings. Given the structure constants of a cell, we compute the lattice of its cellular subrings.


(6) lsomorphism of cells. Given the colored graphs and structure constants of two cells, it is determined whether they are isomorphic.

(7) Computation of automorphism groups. Given the colored graph of a cell and the lattice of its subrings, strong generating systems for the automorphism groups of all elements of the lattice are constructed.

(8) Enumeration of graphs. Given the colored graph of a cell and the lattice of its subrings, a list of all nonisomorphic graphs belonging to the cell is constructed.

Input and output formatting of the programs from the package is designed in a way which allows one to carry out the main task, that of investigating transitive permutation groups, without any additional programming. At the same time all programs of the package can be made compatible with other programs, including ones from available packages for computations in groups and computations with combinatorial objects, by the use of interface routines. For example, sometimes the generating permutations of the group can be obtained more easily, not by inducing the group, but by using a program which realizes the Todd-Coxeter algorithm [Ca 6]. Colored graphs of some permutation groups (such as symmetric groups induced on subsets, exponentiation of symmetric groups, actions of the classical groups on maximal isotropic subspaces) and their structure constants are better obtained by means of fast customized programs.

All programs from the package are written in Fortran-4, the only exception being the module dealing with packed rows of colored graphs written in Assembler /370. The package was run on EC-1040, EC-1045 and BASF 7/65 machines (the speeds are about 250000, 600000, and 1500000 operations/second, respectively ).

Let us now pass to a deseription of the algorithms and of the technical characteristics of programs from the package.

2.7.1. Inducing

Let the group ( G, N), N = {1, 2, ... , n }, be given by a generating set of permutations (91, ... , g 8 ). Let H be a subgroup of G. Suppose we are asked to construct a transitive action of G on the cosets with respect to H, and we want to find a system of permutations ?h, ?h, ... , g., where g; is the action of g; on the cosets of G with respect to H, 1 ::::; i ::::; s. The process of construction of such permutations !h, fh, ... , Ys we call inducing. We assume that we are given a relation U on N which is invariant with respect to H, and for which Aut(U) n G = H. Then it is easy to see that the action of G on the cosets with respect to H is similar ( as a permutation group) to the action of G on the set U of all images of U with respect to permutations of G.


To the orbit U of the relation U on N we associate a colored oriented multigraph II with vertex set U and arcs of the form (U', U") of color i (1 ::::; i ::::; 8) if U'Y• = U". Obviously, each vertex has one arc entering it, and one leaving it. Therefore, each graph II; consisting of arcs of color i represents a union of simple cycles corresponding to the permutation action g; of generating element g; on U. The set of all such permutations forms a generating system of permutations (g1, ... , g.) of the induced group (G,U).

The algorithm for the construction of the graph II consists of subsequent applications of each generating permutation g1, g2 , . .. , g8 to the set of already constructed vertices of the graph (originally this set contained relation U only ), and of the identification of the obtained relations with those previously constructed. The complexity of the algorithm is mainly related to the necessity for numerous verifications of isomorphisms of relations. The programs of the package use inducing only on orbits of relations of three special kinds, the choices of which were motivated on the one hand by their frequent appearance and on the other hand by the existence of effective algorithms to find canonical forms. They are:

(a) The unary relation or symmetrized k-point U = { ( iJ), ... , (ik) };

(b) The binary relation U = {( i, i + j mod n) : j E J} or cyclic graph Cu( J), where J C Zn= {0, 1, ... , n- 1}, [J[ = k;

(c) The binary relation U = {(i,i'): i,i' E N.}, where {N.} is a partition of the set N into n/k classes of the same cardinality k.

For these three types of relations the complexity of obtaining canonical representations is O(klogk), O(nk) and O(nlogk), respectively. After a canonical form is obtained, the identification is clone by hashing [Kn 1]. Therefore the complexity of the described inducing algorithm is O(t8nl}, where n 1 = [U[, and t is the complexity of the algorithm for obtaining canonical representatives. Runlling times for the inducing algorithm Oll certain groups Oll the EC-1040 are givell in Tables 2.7.1-2.7.3.

Group 8 k n n1 Time (sec.) PSL(3,4) 2 7 21 120 2 PSL(3,3) 2 4 13 234 2 Sz(8) 2 13 65 560 5 Jl 2 38 266 1540 24 ]2 3 6 100 10080 61

Table 2. 7.1. lllducing Oll k-points.


Group s k n nl Time (sec.) PSL(3,3) 2 3 13 144 2 PSL(3,5) 3 3 31 4000 150

Table 2.7.2. Inducing on cyclic graphs.

Group s k n nt Time (sec.) PSU(4,2) 2 3 27 40 1 PSU(3,5) 2 5 :so 126 3 Ml2 2 4 12 495 3

Table 2. 7 .3. Inducing on partitions.

Applications of the described inducing algoritlun are restricted mainly by the internal memory required to hold n1 relations. The program can be used in practice for groups of degree up to 105 .

2.7.2. Construction of a V-ring

In this subsection andin those which follow, we consider a transitive permutation group action on the n~element set Q = { 1, 2, ... , n}. The construction of the colored graph r( G) of the V ~ring of a transitive group ( G, Q) (i.e., partition of the set !12 into 2~orbits Qi ofthe group (G,!l)) can be donein two steps. First, using a generating system X = {g1 , ... , g.}, we construct partition r 1 ( G) of the set n into the orbits of the stabilizer of the point 1 in (G,n) (the first row in the adjacency matrix of the graph r( G)), and we determine a system of representatives h2 , .•• , hn of the cosets of G with respect to its subgroup G1 : 1 h; = i. Next we act by the permutations hj ' 2 ::; i ::; n, Oll r 1 ( G) and obta.in other rows of the adjacency matrix of the graph r( G). This method allows one to construct the matrix for r( G) row by row, writing the rows already obtained in the external memory, which saves a considerable amount of internal memory for large n.

Having a generating system X = {g1 , ... , g s} of the group ( G, n), we construct a colored oriented multigraph A( X) with vertex set n and arcs ( i, j) of color r if i 9 r = j. Let T(X) be a spanning tree of A(X) rooted at vertex 1. It is obvious that the product of generating permutations which correspond to the colors of arcs along the path in the tree T(X) from the root to t.he vertex i is the representative hi of the coset of the group G with respect to the subgroup G 1 . For every arc (i,j) of color r from A(X) which is not in T(X), the permutation h;grhj1 belongs to the stabilizer of point 1 in G, and the set of such permutations over all arcs of A(X)\T(X) gives a generating system for G1 . This method of obtaining generators of a point stabilizer was suggested by Schreier ( see [Si 4]).

The algorithm realizing the method just described consists of the sequential


generation of arcs of A(X), where an obtained arc is either included in T(X), or used as a generator of the stabilizer of point 1 which is used immediately in the computation of r 1 ( G). The sequence for generating arcs is such that arcs leaving vertex i f=. 1 are constructed only after at least one arc entering i has been obtained. At the same time the first arc entering j is included in T(X) and generates a representative of the coset hj, but all other arcs entering j are used to construct elements of the stabilizer of the point 1. This is possible since, at the moment of generation of arc ( i, j), the representatives of cosets h; and hj arealready determined. The sequence of generators of the stabilizer of point 1 generates itself a nested sequence of subgroups of G1: E =Ho ~ H1 ~ ... ~ Hm = G1, where m = s(n- 1) + 1 is the number of arcs in A(X)\T(X), and the subgroup Hi+1 is obtained from H; by adjoining the next generator of G 1 • The computation of r 1 ( G) is clone by subsequently finding the orbits of groups H;. This can easily be accomplished since joining a permutation f to H; leads to a union of orbits of H; containing j and k, if jf = k.

The first row of the obtained adjacency matrix of the colored graph allows one to determine the rank and subdegrees of the group ( G, n), and the comparison of its first row with its first column establishes a correspondence between the 2~orbit of color i and its inverse 2~orbit of color i'.

The described algorithm has complexity 0( sn3 ) and requires 0( n) internal memory. Table 2. 7.4 gives running times for the corresponding program appliecl to certain groups on the EC~1040 computer.

Group s n Time (sec) PSU(3,3) 2 63 3 Mu 5 66 8 PSL(3,3) 2 144 16 Mu 5 165 36 M22 2 330 66 J2 5 315 160 Sz(8) 3 560 250 Ag 2 840 420 J2 5 840 1100 M12 3 1320 1600 h 5 1800 5400

Table 2.7.4.

The time requirecl to construct the adjacency matrix of a colorecl graph is greatly reduced if ( G, n) is given by a generating system {91 , ... , 9n 9r+ 1, ... , 9,.} such that permutations 91, ... , 9r generate the stabilizer of point 1, because in this


case there is no necessity to find generators of G1 . The running times for certain groups of quite !arge degree defined in this manner are given in Table 2. 7.5 ( on the EC-1040 computer).

Group s n Time (sec) Sz(8) 3 1456 100 Jl 5 2926 360 PSL(3,5) 3 4000 660 ]z 3 10080 4200

Table 2.7.5.

The use of the program is limited not only by its complexity, but also by direct access external memory of volume O(n2 logr), where r is the rank of the group, needed for storage of the adjacency matrix of the colored graph. In practice the program can be used on groups whose rank is not too !arge (r :S 255) and whose degrees are up to 104 .

2.7.3. Computation of structure constants

In the computation of structure constants of a cell we use the interpretation of pfj as the number of oriented triangles whose sides have colors i and J and whose

common base has color k: pfj = l{t: (q,t) E Q; and (t,s) E Qj}l for (q,s) E Qk. Due to the regularity of the basis graphs of a cell, any point can be chosen as q, e.g., point 1. Since the number of triangles with common base and sides of colors i and J depends only on the color of the base, we can assume that s is chosen from the set S, ISI = r, consisting of the endpoints of arcs of different colors leaving vertex 1. (Here r is the rank of the group ( G, Q).) Finally, in order to construct all triangles needed for the computation of structure constants, every point of n has to assume the role oft.

With each triple (i,J, k) we associate a counter, setting it to zero at the beginning of the algorithm. We then move through the points of S, and for a fixed s E S, through all points t E H. The sth and tth elements of the first row of the adjacency matrix of the colared graph give us colors i and k, and the tth element of the sth row of the matrix gives color J. Foreach triple (i,J, k) obtained as a result of the described enumeration, we increase the value of the corresponding counter by 1. At the end ofthe enumeration, the counter associated with the triple (i,J, k) will have value pfj· We notice that in the implementation of the algorithm it is enough to hold only two rows of the adjacency matrix of the colared graph in the internal memory of the computer at the same time: the pt and the sth.

One can reduce the number of computed structure constants by using some of the relations which they satisfy. In particular, it is not necessary to compute


p~i' pf0 and P~j· The relation pfi = pj;i, makes the computation of Pfj redundant if the triple ( i, j, k) occurs lexicographically before the triple (j', i', k'). In reality, the number of nonzero structure constants is significantly smaller than r 3 . Therefore, only nonzero counters are kept, which results in an economy of memory. For the purpose of their storage and fast retrieval, the method of hashing is used [Kn 1]. After the computations are completed, the structure constants are retrieved from hash-memory and are put into lexicographic order of triples ( i, j, k) by the method of separated counting [Kn 1].

The complexity of the above algorithm is O(rn + L), where L is the number of nonzero structure constants computed. The running times for the computations on the EC-1040 for certain groups of not-too-small rank are given in Table 2. 7.6.

Group r n L Time (sec.) PSL(3,3) 18 234 948 8 Sz(8) 17 560 1495 15 JJ 22 1463 3215 53 PSU(3,4) 48 1600 22933 160 PSL(3,5) 50 3100 25552 280 PSL(3,5) 54 4000 45550 470 Sz(8) 165 2080 152101 820

Table 2.7.6.

2. 7.4. Primitivity testing

In order to determine the connectedness of each of the anti-reflexive basis graphs Q1, ... , Qr-l of a cell, it is enough to check the connectedness of nonoriented graphs which are either symmetric 2-orbits, or obtained by taking the union of a pair of antisymmetric 2-orbits. Let r = 1 + r s + 2r a, where r s is the number of symmetric 2-orbits and ra is the number of pairs of antisymmetric 2-orbits. Then in order to establish the primitivity of a cell, it is sufficient to check the connectedness of r' = r s + r a graphs. Let R be the set of indices of both the anti-reflexive symmetric and pairs of antisymmetric 2-orbits. Then Proposition 2.3.4 allows one to replace the connectedness test for the graph Q;, i E R, by a test for the connectedness of the graph Fi whose vertex set is R, and in which vertices j, k form an edge if at least one of the structure constants with the set of indices { i,j, k} is different from zero. The complexity of this algorithm is at most O(r'3 )

and it requires internal memory of volume O(r'2 ). As a by-product of running the program, we obtain the connected components of the F;'s, which are used for the computation of the lattice of primitive subrings. Running times for certain groups on the EC-1040 computer are given in Table 2.7.7.


-Group r' t n Time (sec.)

PSL(3,3) 12 234 7 Sz(8) 13 560 8 Jl 17 1463 11 PSU(3,4) 26 1600 61 PSL(3,5) 31 3100 76 PSL(3,5) 35 3875 110 Sz(8) 100 2080 610

Table 2.7.7.

2. 7.5. Computing the lattice of subrings

We construct all cellular subrings of a cell with basis graphs Q0 , Q1, ... , Qr-l in two steps. First we choose every proper subset T of the set {1, 2, ... , r -1} with ITI > 1 and suchthat the union Qr = U;ErQ; of basis graphs having indices in T satisfies some necessary conditions for the existence of a cell with basis graph Qr. Then we Iook for the partitions of the set of indices of the basis graphs such that all of their classes having more than one element were chosen during the first step of the algorithm, and for each such partition we check that it satisfies the axioms of a cellular ring.

In order for a subset T C { 1, ... , r - 1} to be chosen during the first step of the algorithm ( a "good" subset), it has to satisfy two conditions:

(a) Qr should be either symmetric (T contains only indices of symmetric and pairs of antisymmetric basis graphs), or antisymmetric (T contains only indices of antisymmetric pairwise non-inverse basis graphs);

(b) the coefficients in the decomposition of the ith power ( i = 2, 3, ... , k) of the graph Qr with respect to the basis graphs should be the same for all basis graphs with indices from T.

It is easy to see that if an antisymmetric subset satisfies condition (b ), then the subset formed by the inverse graphs is also antisymmetric and satisfies condition (b ). From this, it is easy to see that it is necessary to construct m = 2r,+r. + (3r•-

1 )/2- r"- r,- 2 subsets satisfying condition ( a) and then test each of them against condition (b ).

The generation of the necessary subsets is carried out in lexicographic order separately for the symmetric and antisymmetric cases. Condition (b) is checked for k = 3 (i.e., when Qr is raised only to the second and third powers), since experiments have shown that the time spent in verifying condition (b) for higher powers is not sufficient compensation for the slight reduction in the number of


"good" sets chosen.

For the second and the third powers, condition (b ), expressed m terms of structure constants, takes the form:

(bz) iET jET iET jET

r-1 r-1

(b3) 'Vk, k' E T, L L L L P~sPL = L L L L P~~Plj· t=l sET iET jET t=l sET iET jET

Thus, suppose a set T = {T} of "good" subsets of the set {1, ... , r - 1} was obtained in the first step of the procedure. The sets I: = { T1, ... , T.} ~ T, satisfying the conditions below, are considered during the second step:

( a) The elements T1 , ••• , Ts do not intersect;

(b) for every antisymmetric subset T = { i 1, ... , it} E I:, the subset T' { i~, ... , i~} also belongs to I:.

s

By adding to I; all one-element subsets of I: = { 1, ... , r - 1} \ U T;, we obtain i=l

a partition of {1, ... , r- 1 }:

{T1, ... , T., {ji}, ... , {jt}}

In order for this partition to generate a cellular subring, it 1s necessary and sufficient that

( c) In the decomposition of the product of all graphs which are generated by two partition classes, the coefficients of the basis graphs coming from the same partition dass are the same.

Forthose products of the form Qro · Qrß' Qra · Qj and Qj · Qj', condition (c) can be represented in terms of structure constants as


(c2) Va,ßE{1, ... ,s}, VjE~, Vk,k 1 ETo, LP~j= LP~;; iETß iETß

· ·I -,- I k k' Va E {1, ... ,s}, VJ,J E ~., Vk,k E To, Pjj' =Pjj'.

If condition ( a) fails to hold for a subset I; C T, then it will clearly fail to hold for every subset ofT containing I;, which allows some reduction of the enumeration process.

After all subrings have been found, they are arranged in decreasing order according to their ranks ( the rank of a subring is r = 1 + s + t), and for each subring its embeddings into subrings of smaller rank are determined.

The program has an option for the construction of primitive cellular subrings only, in case the cell itself is imprimitive. To achieve this, during the first step we select only those "good" sets in which the union of their basis graphs is connected. (This can easily be clone by using the connected components of the Fi 's which are obtained in the process of primitivity testing.) Then, in the second step, we consider only those subsets I; = {T1 , ... , T8 } for which all basis graphs with indices from { 1, ... , r - 1} \ U T; are connected.

The complexity of the first step of the algorithm can be estimated by O(r2 m), where m is the number of subsets of {1, ... , r - 1} considered. The complexity of the second step is usually less, except in those cases when almost all sets considered in the first step are "good" and almost all partitions of the set of basis graphs of the original ring generate its cellular subrings. Our experience with this algorithm has shown that it can be used for the computation of subrings of cells whose ranks are not too large (:::; 30) and which do not have too many subrings (:::; 100). The running times of the program on the EC-1040 computer for certain groups are given in Table 2.7.8.


Group r n Time (sec.) PSL(3,3) 18 = 1 + 7 + 2. 5 234 23 Sz(8) 17 = 1 + 10 + 2. 3 560 43 ]z 18 = 1 + 11 + 2 . 3 2016 76 ]1 19 = 1 + 14 + 2 . 2 1596 300 ]1 22 = 1 + 13 + 2 . 4 1463 780 Ml2 29 = 1 + 14 + 2. 7 1320 20000

Table 2.7.8.

2.7.6. Testing for cell isomorphisms

Cellular rings w = (Qo, ... 'Qr-1) and W' = (Q~, ... Q~-1) Oll n are isomorphic if there exists a permutation of the set n mapping { Qo, ... 'Qr-d to {Q~, ... , Q~-d· Obviously, in order for cellular rings to be isomorphic, the isomorphism of their tensors of structure constants P = IIPfjll and P' = !I(Pfj)'ll is necessary, as well as equality of their ranks and of their collections of valencies of basis digraphs.

lsomorphism testing of cells is carried out in two steps. First the isomorphism of their tensors of structure constants is verified and the group SAut(P) of automorphisms of the tensor of structure constants of the cell W, acting on the set of indices of the basis graphs of the ring W, is constructed. The action of each element g of this group can be extended to the colored graph f(W) by a suitable recoloring of f(W). The new colored graph obtained in this way is denoted by gf(W). During the second step, the isomorphism of colored graphs gf(W) and r(W') is established for all g E SAut(P).

Testing for isomorphism of tensors of structure constants of cells, as well as finding their automorphism groups, doesn't represent an essential difficulty for cells of small rank and low subdegree multiplicity. lt can be clone by a complete enumeration algorithm and we do not discuss it here.

The isomorphism of r with f' is shown by the method of iterative dassification [*Ze 1].

Let h = ( v1, ... , Vk) and I~ = ( v~, ... , vU be sequences of distinct vertices in the colored graphs r and r' such that the induced subgraphs of r and r' generated by h and I~ are isomorphic, with the isomorphisms carrying v; to v: for 1 :::; i :::; k. Sequences h and I~ partition other vertices of the graphs r and f' into dasses such that vertices of the same dass are connected to the vertices in h or I~ by arcs forming the same sequence of colors. Let us call these partitions K(h) and K(IU. The sequence of colors of arcs defining a dass is called the label of the dass. A necessary condition for the existence of an extension of the isomorphism

CELLULAR RINGSAND GROUPS OF AUTOMORPHISMS OFGRAPHS 87

of subgraphs generated by h and I~ to an isomorphism of the graphs r and r' is the coincidence of the collections of labels of the classes of partitions K(h) and K(IU and also of the cardinalities of dasses with the same label. If these conditians are satisfied we can extend h and I~ by joining ta the first sequence an arbitrary remaining vertex Vk+I af the graph r, a.nd by jaining to the second sequence a vertex v~+I af the graph r' belanging ta the dass of K(IU with the same label as the dass of K(h) cantaining vk+I· Obviously, the subgraphs induced by Ik+ 1 and I~+I are isamorphic. The pa.rtitians K(Ik+1 ) and K(I~+ 1 ) can be easily obtained by subpartitioning K(h) and K(IU, and this pracedure can be carried further. If the isomorphism of subgraphs induced by h and I~ cannot be extended ta an isamarphism of graphs r a.nd r' (i.e., if the necessary conditions above are not satisfied), we repla.ce v~ by a. vertex v~ from the sa.me dass of the partition K(I~_ 1 ) as v~. If all vertices of this dass of the partition are exhausted, we shorten the sequences h and I~ and replace vertex v~_ 1 • As a result of the application of the described algorithm we either find sequences In and I~, shawing the isomorphism of graphs r and r', or arrive at empty sequences I 0 , I~, proving that graphs r and r' are not isomorphic. Heuristics show that on the average the greatest reductian in the enumeration is achieved by choosing vk+ 1 (in order to extend h) from the dass of K( Ik) of minimum cardinality.

If graphs r and r' are transitive, then the enumeration can be reduced by fixing v1 = V~ = 1. If, in addition, r and r' are colored graphs of V -rings of some permutation groups, then further reduction in enumeration is obtained by fixing vertices v2 and v~ suchthat arcs (l,v2 ) and (l,v~) have the samecolor and belang to basis graphs of maximum valency.

The thearetical upper bound for the complexity of the described algorithm of cell isomorphism is exponential in the degree of the cells. Unfortunately, we do not have enough material to make statistical eondusions on its effectiveness in practice. Running times for some V -rings of primitive permutation groups are given in Table 2.7.9, where the upper portion of the table refers to pairs af isomorphic cells, and the lower to pairs of non-isomorphic cells.


Group n Subdegrees Time (sec.) M12 66 1,20,45 3 PSp(4,4) 85 1,20,64 4 M12 220 1,12,27,72,108 18 PSL(3,4) 280 1 ,9,18,36, 721+2 ·1 26 M12 495 1,32, 70,168,224 80 PSU(3,5) 525 1,20,48,96,1203 110 PSU(4,2) 40 1,12,27 5 PSU(3,3) 63 1,6,24,32 8 M12 495 1 ,6,16,24,64,962 ,192 430

Table 2.7.9.

2.7.7. Computation of automorphism groups

Let I = ( i 11 ... , im) be a base of the group ( G, D) [Si 4], i.e., a sequence of points of D suchthat the (pointwise) stabilizer G;,, ... ,im = E, where Eis the identity subgroup. We introduce a partial order on the elements of ( G, D) by saying that each element from the stabilizer of points i 1 , i 2 , ... , i j precedes all elements of the stabilizer of points i 1 , ... , i j-l which do not fix ij. Let us consider an arbitrary linear order on the elements of ( G, D) which extends the partial order just introduced. We will screen the group elements one by one and choose those which do not belang to the subgroup generated by previously chosen elements. It was shown in [*Za 1] that the chosenelementsform a strong generating system (in the language of Sims [Si 4]) of the group (G,D) with respeet to the base I, and the cardinality of the system doesn't exceed n - 1.

The algorithm described below constructs precisely such a strong generating system for the automorphism group of a colared graph r with vertex set n.

First we find a base of the automorphism group. Let h = ( v1 , ... , Vk) be a sequence of vertices of the graph r and let K(h) be a partition of the vertices of n \ h into classes, labeled by sequences of colors of arcs directed from the points of these classes to the vertices of h. If some of the classes of this partition have more than one element, we extend the sequence by joining to it a vertex Vk+l from the class K( vk+1 ) of the partition K(h). If all classes of the partition K(h) consist of one element, then h is a base of the automorphism group of r. Moreover, the subsequence of this sequence obtained by removing those vertices which were chosen from one-element partition classes, is a base of the automorphism group too. In order to obtain a base of minimum length we apply the following heuristic: If the partition K(h) contains any one-element classes, then we extend h by using vertices from these classes. Otherwise, we use a vertex from a class of maximal cardinality, in cantrast to what we did in the isomorphism problern (!) in subsection


2. 7.6. lt is easy to see that if I= ( v; 1 , ••• , v;m) is a base of the automorphism group obtained in the described manner from the base h = ( v1, ... , Vk ), then the orbit of the vertex v;, in the stabilizer of the points v;,, ... , v;,_ 1 in the automorphism group is completely contained in the dass K( v;,) of the partition K(I;, _I).

After the base I= ( v1 , ... , vm) of the group Aut(f) has been constructed, and candidates K( v 8 ) in the orbit of vertex Vs in the stabilizer of points v~, ... , Vs-1

in the automorphism group has been found, 1 :::; s :::; m, one can find a strong generating system by the method of iterative classification [*Ze 1]. Suppose a generating system for the stabilizer of vertices v1 , ... , Vs in Aut(f) has been found and, for each vertex v;, 1 :::; i:::; s, the partition of K(v;) into orbits of the action of the stabilizer has been obtained. Let us choose a vertex v~ from K( v.), which is distinct from v s, and let us try to extend the isomorphism of subgraphs generated by the sets VJ' ... 'Vs-1' v. and VJ' ... 'Vs-b V~ to an automorphism of the graph r. In case we fail, we would replace vertex v~ by v~' from K(v.), but from another orbit of the subgroup generated by elemeats already found. Otherwise, if an automorphism g from the stabilizer of points v1, ... , Vs-1 is found, we adjoin it to the already constructed system of generators and enumerate all partitions of the sets K( v;) into orbits, 1 :::; i :::; s. This is not hard to do, since joining g go the existing system of generators leads to a fusion of orbits containing vertices v and v', such that v9 = v1• Then we continue taking vertices from K(v 8 ) one by one, choosing each time a vertex from an orbit containing neither v., nor vertices already considered. After the set K ( v 8 ) is exhausted, we obtain a system of generators for the stabilizer of points v1 , ... , V 8 _ 1 in Aut(f), and the described process can be continued. After the last element of the set K ( vi) is used, we obtain a strong generating system of the group Aut(f) with respect to the base I = ( v1 , ... , Vm ), and orbits of every vertex v; of the base in the stabilizer of the points v1 , ... , v;_1 . The product of the cardinalities of these orbits gives IAut(f)l.

The program based on the algorithm allows one to compute not only Aut(f), but also the stabilizer of a set of vertices of r. In particular, if we are interested only in the order of a transitive group of automorphisms of a cell, it is sufficient to compute the stabilizer of the point 1 in the automorphism group. In the case when Aut(f) acts transitively on arcs of the same color, i.e., the cell is a V-ring of a permutation group, the computing time can be reduced even more by fixing a vertex joined to vertex 1 by an arc contained in a basis graph of maximum valency.

Despite the fact that the theoretic upper bound for the complexity of the algorithm is exponential with respect to the number of vertices of the colared graph, its behavior in numerous applications for computing automorphism groups of primitive cells has demonstrated at most quadratic dependence on the number of vertices. The running times for computing orders of automorphism groups of V -rings for certain


primitive permutation groups are given in Table 2.7.10.

Group n Time (sec.) A1 21 3 PSL(3,4) 56 7 Ag 84 15 PSp(4,4) 136 54 lv!n 330 180 M22 616 350 h 840 630 h 1596 650

Table 2.7.10

The following fact allows a considerable cut in running time when automorphism groups of cellular subrings of a cell are computed: The automorphism group of a subring W contains the autornorphism group of every subring W' in which W is embedded. Let r and f' be colored graphs of the cellular rings W and W', W' :=l W, and let I= ( v1 , ... , vk) be a base of Aut(f), such that I' = ( v1 , ... , vk' ), k' :::; k is a base of Aut(f') for which a strong generating system X' of the group Aut(f') C Aut(r) has already been constructed. For every vertex V 8 , 1 :::; s :::; k', the action of the stabilizer of vertices v1 , ... , v 8 _ 1 in the group Aut(f') partitions into orbits the set K ( v 8 ) of all candidates for the orbit of v 8 in the stabilizer of v1, ... , v 8 _ 1

in Aut(r). Applying the described algorithrn further, we find automorphisms of the graph r which, tagether with automorphisms of X I' give a strong generating system of Aut(f). The scheme is applied to all elements of the lattice of subrings in decreasing order of their ranks. When the automorphism group of a subring W is computed, we choose a subring W', among all subrings in which W is embedded, with automorphisrn group of maximum order. Running tirnes for the cornputation of the autornorphism groups of all m elements in the lattice of subrings of V--rings of certain primitive permutation groups are given in Table 2. 7 .11.

Group n m Time (sec.) PSU(3,3) 63 3 20 PSL(3,4) 280 13 960 PSU(3,4) 416 5 540 PSU(3,5) 525 8 1500

h 1008 2 1200 h 1800 2 2800

Table 2. 7.11

Our experience with this program shows that it allows us to compute automorphism groups of primitive cells of degree up to 104 . But it works far worse for


imprimitive cells. For example, we failed to compute the automorphism group of the lattice of subrings of the V-ring of the group S2 j S 9 of degree 512 in reasonable time ( several hours on the EC-1 045).

2.7.8. Enumeration of graphs

Fora given cell W = (Q0, Q1, ... , Qr-1}, we construct all graphs with valencies k :=::; (n -1)/2 of the form rJ = uiEJQ;, where I c {1, 0 0 0 'r -1}. Foreach of these graphs we find the smallest subring to which it belongs, and we select all pairwise non-isomorphic graphs from among them. Moreover, for k = (n- 1)/2, we check also for isomorphism of the complementary graphs. Of course, graphs which have different valencies or which generate non-isomorphic: subrings need not be tested for isomorphism. The isomorphism problern is answen~d by using the algorithm from subsection 2. 7 .6.

The program is restricted to enumeration of simple graphs ( unians of symmetric and pairs of antisymmetric basis graphs), and also those which generate only primitive subrings of the given cell.

The complexity of the described algorithm is linear with respect to the number L of constructed graphs ( L depends exponentially on the rank of the cell) and quadratically on the number L 1 of non-isomorphic graphs. Despite the fact that the theoretic upper bound of the complexity of the algorithm for the isomorphism problern described in 2.7.6 is exponential in n, its use for graphs with primitive automorphism groups demonstrates, at most, quadratic dependence on n. Running times on the BASF 7/65 computer for enumeration of primitive graphs invariant with respect to certain permutation groups are given in Table 2.7.12.

Group n r L L1 Time (sec.) A6 36 6 7 5 0.4 A6 45 9 15 11 0.7 PSL(2,11) 55 9 31 23 4.4 PSL(2,11) 66 10 31 31 13 PSL(2,13) 78 9 63 47 38 PSL(2,13) 91 12 255 159 815

Table 2.7.12

Restrietions on the running time and space necessary to store non-isomorphic graphs allow one to apply the program only for cells of small rank (:S:: 15) and small degree (:S:: 200).


References

The first version of some programs from the package were developed by V.A. Zaichenko in his thesis [*Za 2], also see [*Za 4]. The version of the package presented here was developed by LA. Faradzev. It was announced in [*Fa 3] and the description first appeared in [*Iv 5]. In this section we also presented some additional programs which were absent from [*Iv 5]. A new version of a sequence of programs (C language for IBM/PC) was created by LA. Faradzev in 1990. This version deals with intransitive permutation groups and arbitrary cellular rings. A brief preliminary description can be found in [Fa 3].

CELLULAR RINGS AND GROUPS OF A UTOMORPHISMS OF GRAPHS 93

CHAPTER 3. COMPLETED PROJECTS

3.1. Graphs with a given automorphism group

An algorithm to compute the number of graphs with a given automorphism group is briefly described in this section. As was mentioned in section 1.4, this result of ours was the first in which the idea of Galois correspondence was actually used. We believe that the material presented below is interesting, not only for its mathematical content, but also for the amazing coincidence of circumstances which tied several events of the last 100 years into one intriguing plot.

In 1969 two fresh unsolved problems attracted the attention of one of the authors:

(a) Findall graphs whose automorphism group contains a given permutation group (G,n) as a subgroup (V.G. Vising [*Vi 1]);

(b) Compute the number of graphs with a given automorphism group (G,n) (F. Harary [Ha 9]).

It is clear that problern (a) has a trivial solution: One has to find all (0,1)matrices from the V-ring V(G, n). In order to do this it is sufficient to know all 2-orbits of the permutation group (G, n). Therefore the solution to this problern had a purely methodological character and was reduced mainly to a suitable choice of language. This allowed one to clarify, for example, solutions of (a) for some permutation groups given in [*Da 1] and [*Da 2].

Problem (b) turned out to be more difficult. The first attempt at a solution was made, in [Sh 1] and [Sh 2], by J. Sheehan, who introduced the very important idea of using Burnside marks. (See the definition in section 2.1.) But Sheehan succeeded in solving (b) for multigraphs only. (Harary's formulation of the problern dealt with simple graphs, and with digraphs with no multiple arcs.) A scheme for solving problern (b) (in the form given by Harary) was described by M.H. Klin in [*Kl 1], and our present exposition follows that paper closely. We consider the cases of graphs and digraphs with n vertices simultaneously. In order to solve the problern for a given permutation group G of degree n, we assume as known the lattice of all overgroups of Gin the symmetric group Sn (actually only the 2-closed overgroups are needed).

Since we are interested in all permutation groups, including also the identity group, we assume that the list G1, ... , G8 of all pairwise non-conjugate subgroups of Sn is known. Let us introduce the following notations for 1 :<:::: i :<:::: s: By /;(x) (resp., j;( x)) we denote the generating function for the number of labeled simple graphs (resp., digraphs) which areinvariant with respect to the permutation group ( G;, n),


where n = {1, 2, ... , n}; by g;(x) (resp., g;(x)) we denote the generating function for the number of pairwise nonisomorphic ( or abstract, as they are sometimes called) unlabeled graphs (resp., digraphs) with automorphism group (G;, n).

Let us recall the concept of generating function. Consider a family of graphs, each having n vertices. Then the generating function for this family is a polynomial

n(n-1)/2

a( x) = I: akxk, where ak is the number of graphs of the family having exactly k=l

n(n-1)/2

k edges. For example, J;(x) = 2: f;Ix 1, where f;I is the number of labeled 1=1

graphs having l edges andinvariant with respect to the group G;. Polynomial g;(x) is defined similarly; for polynomials ]; ( x) and g; ( x) the summation is extended from l = 1 to l = n(n -1).

In what follows, we will use Burnside's Lemma. We want to emphasize (because it will matter in our presentation) that this well-known name is not quite appropriate. P. Neumann, in [Ne 3], was the first to show that this lemma was known to A. Cauchy and G. Frobenius (although in less general form). An important remark was made in [Wr 1]. It turns out that the first edition of the book [Bu 1], which appeared in 1897, had a reference (p. 119) to the work of Frobenius [Fr 1]. But in the second edition, the corresponding section was changed a lot, and for some reason the reference to [Fr 1] disappeared. Our point of view (which is justified in [Kl 7] with more detail) isthat the lemma should be called the Cauchy-Frobenius-Burnside Lemma or CFB Lemma.

Let (Sn, G}) be the action of the symmetric group Sn on 2-element subsets of

the set n, and (Sn, fi 2 ) be the action of Sn on the set of ordered pairs of distinct elements of n. We would like to note that the first (resp., second) action is similar to the action of Sn on the cosets ofthe subgroup Sn-2 X S2 (resp., Sn-2)· The problern of computing the number 9il of abstract graphs having l edges (resp., the number gil of digraphs with l arcs) with automorphism group G; can be reformulated as the problern of computing the number of such symmetrized I-orbits of the transitive group (Sn, G}) (resp., (Sn, D2 )) in which the stabilizer of the symmetrized I-point coincides with G;.

But we will be using a different approach. Let us consider the permutation group ( G;, n). Let d;I be the number of its symmetrized antireflexive 2-orbits containing l edges, and Jil be the number of antireflexive 2-orbits having l arcs. These numbers can easily be determined from a knowledge of the V-ring of the group ( G;, n). Knowing them allows us to find the generating functions f;(x) and ];(X).


Lemma 3.1.1 n(n-1)/2

J;( x) = 11 (1 + xl)d;z' 1=1

n(n--1)

J;(x)= 11 (1+xl)'7". 1=1

• Suppose the generating functions J;(x) and .f;(x) are known. Then clearly

n(n-1)/2 _ n(n-1) _

one can compute J;(1) = 2:: h1 and J;(1) = 2:: !;1, the numbers of all 1=1 1=1

labeled graphs and digraphs, respectively, which are invariant with respect to the permutation group (G;,n). It turnsout that the numbers J;(l) and .f;(1) can be computed directly, i.e., without an intermediate computation of the numbers

n(n-1)/2

d;1 and Jil· In order to see this, we introduce the numbers d; = 2:: d;1 and 1=1

n(n-1) _

J; = 2:: dil, standing for the number of symmetrized 2-orbits and 2-orbits, re-1=1

spectively, of ( G;, n).

To compute the numbers d; and J; we use the CFB Lemma. By x(g) we denote the value of the permutation character X on the permutation g E Sn, i.e., the number of elements from n fixed by g.

Lemma 3.1.2

d; = /~;! L [(x(g)) 2 - 2x(g) + x(l)J/2, gEG;

- 1 "" d; = TG;T ~ x(g)(x(g)- 1). _gEG,

• Proposition 3.1.3

- . ä. j;(l) = 2 i'

where d; and J; are the numbers defined above. • In order to pass from labeled graphs to abstract ones, we use the following

obvious

Lemma 3.1.4 Let r be a labeled graph, G = Aut(f). Let N(G) be the normalizer of the group G in the symmetric group Sn. Then there exist exactly [N( G) : GJ distinct labeled graphs isomorphic to r whose automorphism group coincides with

G. •

In what follows, we assume that the groups G1 , G2 , ... , G. are numbered in such a way that G;::::; g-1Gjg, g E Sn, implies that i;::: j. Let 9f = {g- 1Gjg: g E Sn and g-1Gjg ;::: G;}, i.e., 9f is the set of subgroups of Sn conjugate to Gj and

containing G;. Let kf = 19{1.

Theorem 3.1.5

- ( ) I G i I (!-( ) "' kj IN ( G j) I - ( )) g; X = IN(G;)I i X - L.....-<. i IGjl gj X 0

J l

~ We prove the statement for the case of undirected graphs. First we fix l and consider all graphs invariant with respect to G; having l edges. There are fit such graphs altogether. Some of these graphs have automorphism group greater than G;. First we consider allsuch graphs having automorphism group conjugate to Gj; there are [N(Gj) : Gj] · gjl such graphs (use Lemma 3.1.4). Then we consider all

such graphs for all groups from 9f for all j < i. Subtracting the obtained quantity from J;t, we get the number of labeled graphs with automorphism group G;. Then Lemma 3.1.4 is applied again. Repeating the same computations for alll, we obtain the first formula of the theorem. The case of directed graphs can be handled in an entirely similar manner ~

Now we transform the proven formula into a more compact form. In order to do this, we use Burnside marks. Let m{ = m{ ( n) be the number of fixed points of the subgroup G; in the action of Sn on the cosets with respect to the subgroup Gj. The following statement can be easily proved (see [*Iv 4]).

Lemma 3.1.6

• From Lemma 3.1.6 and Theorem 3.1.5 we immediately obtain:

Theorem 3.1. 7 1 "' . g;(x) = -;(/;(x)- L.....-mi gj(x)),

m l j<i


• If we are not interested in generating functions, but only in the total number

of graphs with a given group, then we consider the following relations which can be derived from Theorem 3.1. 7. We present them in the form of a system of linear equations given below (we assume m{ = 0 for j > i).

Corollary 3.1.8

Lmi 9j(l) == 2d', 1:::; i:::; s. j=l

l:m{ gj(l) == 2J', 1:::; i:::; s. j=l

• Thus we have described a scheme which, in principle, allows one to solve prob

lern (b) for all groups G; when the lattice of overgroups of G; in Sn is known. Nevertheless, the expressions we obtained for g;(x) and g;(x) cannot be considered "closed form" expressions, since we specified neither formulae for the numbers dil, dil, nor the Burnside marks. In [*Kl 1], [*Kl 5] these numbers were found by means of direct computations. In particular, generating functions g;( x) for all permutation groups of degree 5 were computed in [*Kl 5]. The obtained results coincide with the ones presented in [Ha 11] with reference to the thesis [St 3]. (This thesis is still unavailable to the authors.) We remark that a slight error in the statement of Lemma 3.1.2 from [*Kl 1] was corrected in [*Kl 5].

We did not consider the problern of getting closed formulae for d;1 and dil for an arbitrary permutation group (G;, Sl). In the case of a transitive group, the method of marks can again be used (but in this case for subgroups of G;); see section 2.1.

An explicit formula for the computation of marks (see section 2.1) is given in [*Iv 3], where it was obtained by using results from [Al 1]. A short direct proof of the formula is given in [*Iv 4]. Also we note that Corollary 3.1.8 immediately follows from Theorem 1 of [*Iv 4]. The same Theorem 1 can also be used for a justification of the whole computational scheme.

Startingin the mid-70's many publications have appeared in which the methodology for computing the number of combinatorial objects with a given automorphism group through the use of Burnside marks was presented from several different viewpoints. Many of the authors referred to the following works as the source of this


methodology: [Fo 2], [Sh 1], [St 3]. Herewelist several more papers but make no attempt to be complete: [Wh 1], [Wh 2], [Wh 3], [Kl 1], [Pl 1], [Ke 1]. Shortly afterwards, several works in mathematical chemistry were published in which either interesting applications of Burnside marks were pointed out or similar methods were developed. Methodologically, the results are related to the double coset approach; see [Ha 13], [Br 2], [Me 1].

A detailed comparison of different approaches using Burnside marks, and particularly an analysis of the capabilities of the scheme described above., is intrinsically interesting and probably deserves a separate publication. However, the priority in this question belongs not to any of the authors previously mentioned, but to the American mathematician, J. H. Redfield. The first mention of this matter can be traced to the sensational article of Sheehan [Sh 3], where he wrote about the unpublished article of Redfield which had been discovered by the English mathematician Lloyd. One year later a special issue of Journal of Graph Theory, which was dedicated to Redfield's memory, was published. It included the history of the unpublished Redfield articles [Ha 12], his biography [111], the Redfield article [Re 2], which was submitted in 1940 to the American Mathematical Journal and rejected by the editors, and finally its exposition [Ha 4] using modern terminology.

Another interesting survey of Redfield's works (in the context of problems arising in chemistry) is in [Ll 2]. It is clear now that Redfield twice managed to get 15-20 years ahead of his time: First, when the only article published during his lifetime [Re 1] appeared in 1927, where the main ideas ofmodern enumeration theory (P6lya, de Bruijn, Read) were foreseen, and the second time, in 1940, when for the first time Burnside marks were applied to the solution of problems of enumeration ( although the term Burnside marks did not appear explicitly in the article.)

Here the theme of two editions of the book of Burnside reappears. It is pointed out in [Ha 4] that in [Re 2] Redfield referred to the first edition of the book (1897); the second one [Bu 1] was probably unknown to him. But the method of Burnside marks was developed in the second edition. Therefore, Redfield had disadvantages in comparison with the mathematicians of the 60's and 70's for whom the reprinted edition of [Bu 1] was a handbook. Redfield had to rediscover the method and to apply it in a nontrivial way ([Re 2], in particular, contains tables of marks for S4 , A5

and PSL3 (2)). This shows us the grandeur of the scientific results of John Howard Redfield ( 1879-1944 ). At the same time, this shows us the tragedy of his situation: having no opportunity to use the simple, convenient, precise language developed by Burnside, Redfield (being no expert in group theory) had to develop his own language, which turned out to be more complicated and difficult to understand, and this doomed his work to be not understood by his contemporaries. We close this somewhat atypical survey section with a feeling of deep admiration for this


fascinating man.

3.2 The induced symmetric group

As was pointed out in section 1.3, the problern of enumeration of cellular subrings was considered for the first time in [*Ka 1], where the asymptotic simplicity (i.e., simplicity for values of v sufficiently large in comparison with n) of the BMalgebra of the association scheme J( v, n) was proposed.

Let IV\ = v and { ~} denote the set of all n-element subsets of the set V. lt is clear that \{:}1 = (~). Let Sv= S(V) be a symmetric group of degree v. Then its natural exact action on the set { ~} is defined, and the associated permutation group (Sv, { ~}) is called an induced symmetric group. The following theoremwas proved in [*Ka 1].

Theorem 3.2.1 There exists a function d( n) suchthat for any n _:::: 2 and v > d( n ), the group (Sv, {:}) is a maximal subgroup of either the symmetric or the alternating

group of degree (:) ( depending on the parity of c:=i) ). • The proof of the theorem followed the scheme given below.

1. It was shown that there existed a function c( n) such that for v > c( n) the group (Sv,{:}) is 2~closed.

2. It was shown that there existed a function b( n) such that for v > b( n) the BM -algebra of the association scheme J ( v, n) is simple.

3. It was shown that for v _=::: 7n + 2 the group (Sv, {:}) is not contained in any nontrivial multiply transitive permutation group of degree (:).

From here the validity ofTheorem 3.2.1 followed for d(n) = max{c(n), b(n), 7n+ 2}. All proofs were quite elementary and were basedonsimple combinatorial arguments. The original estimate for c( n) was a polynomial one, and the original proof of 2~closedness was quite cumbersome (although it had intrinsic interest, since it was based on an estimate of cardinalities of cliques in some basis graphs of the Johnson scheme). A particular case of the theorem was considered in [*Kl 2] for n = 2.

The proof of 2~closedness was simplified in [*Kl 5] (see also [*Kl 8]) and the exact value of c( n) = 2n + 1 was given. It was also shown that the 2~closure of the permutation group (S2n, e:}) is isomorphic to the group S2n X (T), where T is the permutation on { 2nn} which maps each n~element subset V to its complement. In another context, the statement about the 2~closedness of (Sv, {:}) was proved in [En 1] for v _:::: 2n + 1.


There was no estimate given for b( n) in [*Ka 1]. In [*Kl 8] a polynomial bound b( n) = 1

56 n 4 - ~n3 + 2n was established. We note that the scheme for the proof of

the existence of b( n) reappeared in [Fa 2], [Kl 7].

The very last step of the proof made use of Bochert-Manning bounds on the minimal degree (dass) of multiply transitive permutation groups; for references see [Wi 2]. We want to stress the fact that the proof of these bounds was obtained in an elementary way too.

Further study of the induced symmetric group has taken two directions: group theoretic and combinatorial.

The group theoretic direction

M. Krasner suggested the problern of the maximality of (Sv,{~}) to a former student of his, E. Halberstadt. Halberstadt announced his results in [Ha 1], [Ha 2] and partially published them in [Ha 3]. He restricted hirnself to the case v ~ 2n + 1 and noticed that for v = 2n the problern becomes too hard. For v ~ 2n + 1, he described all cases when (Sv, { ~}) has a nontrivial overgroup in S( { ~}) and proved that in all other cases the induced symmetric group is maximal. V.A. Ustimenko-Bakumovskif (= V.A. Ustimenko), a student of L.A. Kaluznin, studied Halberstadt's works and developed a more subtle technique, which he called p-local analysis on invariant relations of a permutation group ( see more in [*Us 13]). The first serious application of this technique was a complete description of the lattice of overgroups of the permutation group (Sv, { ~}) in S( { ~} ), including the case v = 2n.

Theorem 3.2.2

( a) If v =/:- 2n + 1, v =/:- 2n and ( v, n) =/:- ( 6, 2), (8, 2), (10, 3), (12, 4), then the permutation group (Sv, { ~}) is maximal in the symmetric (resp., alternating) group

of the set { ~} if ( ~=;) is odd (resp., even).

(b) If n 2 3 and v = 2n + 1, then the group (Sv,{~}) has only one nontrivial overgroup, and it is isomorphic to Sv+l·

(c) If (v,n) = (6,2),(8,2),(10,3),(12,4), then the group (Sv,{~}) has only one nontrivial overgroup, and it is isomorphic to PGL4 (2), Sp6 (2), Sp8 (2), 0]0 (2), respectively.

( d) If n = 2 and v = 5, then there are two nontrivial overgroups of Ss in S1o, and they are isomorphic to S6 andAut 56 •

( e) If v = 2n, then every nontrivial overgroup of Sv in S( { 2;:}) is im primitive.


(f) Let t:l' = e:_::-n. An arbitrary imprimitive overgroup of the permutation group

(S2 n, { 2:}) is contained in the wreath product S01 I S2 and can be represented as a semidirect product H: G, where His a subgroup (subdirect product) of the group M = (Z2 ) 01 and Gis an overgroup of the group (S2 n, { 2:_::-II }). The group His either M, M', ö or E, where M' is the subgroup of M of index 2 consisting of all vectors with an even number of nonzero coordinates, ö is the subgroup of order 2 consisting of vectors with all equal coordinates, and E is the identity subgroup. •

Theorem 3.2.2 was announced in [*Us 1] and the complete proof was given in [*Us 5]. The author did not publish the complete proof as aseparate article. Since this proof did not become known in the West, C. Sims suggested to J. Skalba that she reobtain the results from [*Us 1], which led to the paper [Sk 1].

The combinatorial direction

As we have already mentioned, Theorem 3.2.1 was proved using elementary tools, i.e., without the use of subtle group theoretic methods. Therefore, the following question is of some interest. Is it possible, at least for small n, to get a complete solution of the problem, i.e., to describe the lattice of overgroups of (Sv, { ~}) in S( ~ l? Some results in this direction were obtained by M. H. Klin and presented in his thesis [*Kl 5]. First, using a "naive approaeh", all cellular subrings in the BM-algebra of the scheme J( v, n ), n :::; 6, were enumerated. For n = 3, 4, the computations were clone by hand, and for n = 5, 6, the "Mir-1" computerwas used ( the computer computations were clone in collaboration with L. L. Verbitski]). As a result, all 2-closed overgroups for n ::=; 6 were found, as well as several non-Schurian cellular subrings of rank 3, i.e., strongly regular graphs. In all cases the property of being non-Schurian was established by using the 4-condition: for pairs of distinct ~~-orbits whose fusion was a basis graph, the number of complete 4-vertex subgraphs containing the given pair was calculated ( see section 2.6). Hereis the list of allnontrivial s.r.g. found in this way (v > 2n + 1). We give only one basis graph of the cellular subring and parameters of the s.r.g.: (v, k, l, .\, p,). Here v denotes a parameter of the Johnson scheme and v denotes the number of vertices of the s.r.g ..

V= 10 n=3 AI +A3 (120, 56, 63, 28, 24) V= 11 n=4 AI+ A4 (i130, 63, 266, 24, 9) V= 12 n=4 Az -tA4 (495, 238, 256, 109, 119) V == 1~~ n=6 A3 + As + A6 (1716, 833, 882, 400, 408).

The subring on 495 points is of Schur type and its automorphism group is isomorphic to Oi"Q(2); the other subrings are of non-Schur type. This list was also presented in [*Kl 8]. Let us remark that all these subrings, except for the last, were found independently by R. Mathon (the result was not published, see references in [Ca 3]

and [Br 3]). Another proof of the existence of subrings for n = 4 was given in section 2.2.

All existing nontrivial multiply transitive overgroups of the group (Sv, { ~}) were also found in [*Kl 5]. The search for overgroups was based on computations in the Krasner algebras: 3-orbits of the group (Sv, { ~}) were described, the coefficients of projection of these 3-orbits onto 2-orbits were computed as were the coefficients of the convolution of 3-orbits with respect to two arguments. Then we studied those fusions of 3-orbits which satisfied the necessary conditions for the existence of nontrivial 2-transitive overgroups. As a result, the lattice of overgroups for n = 2 and the lattice of 2-transitive, but not 3-transitive, overgroups for n = 3 were completely described. Therefore, a description of all nontrivial overgroups of the group (Sv, { ~ } ) , v ~ 2n + 1 first appeared in [*Kl 5] (also see the reference in [Ha 1 ]). Unfortunately, the aforementioned technique of computing with 3-orbits was never presented in aseparate publication; it was presented in part and used by F.G. Lazebnik [*La 1] in the solution of another maximality problem.

Finally, we mention that the problern of enumerating cellular subrings in the BM-algebra of the association scheme J( v, n) is interesting in its own right. It was considered even after the lattice of overgroups of (Sv, { ~}) were completely described by V.A. Ustimenko [*Us 3]. He proved that there are no other cellular subrings of rank 3 different from those found by M.H. Klin for 2 ::; v ::; 60, 2 ::; n ::; 20. M.E. Muzichuk, using his technique for the enumeration of cellular subrings ( see section 2.2), proved in [*Mu 7] that a BM-algebra of J( v, n) can contain cellular subrings only if v < 3n+4. But the conjecture that allsubrings of J( v, n) are among those which are presently known has not yet been proved.

3.3. Exponentiation of symmetric groups

Let us first recall the definition of wreath product of perrnutation groups. Let (G,M), (H,N) be two permutation groups, with JMJ = m and JNI = n. The wreath product (GI H, Mx N) of permutation groups (G, M) and (H, N) is the group of all possible maps of the form (x,y) r--+ (x',y') of the cartesian product MX N into itself such that g : x r--+ x' is a permutation on M, belanging to G, and h : y r--+ y' is a permutation in H which depends on x, i.e., h( x) E H M. Therefore a perrnutation from GI H can be defined by a table [g; h(x )] = [g; h1, hz, ... , hmJ, where g E G and h; = h( i) E H. The action of the table on elernents of the set M X N is defined by the rule

(x,y)[g;h(x)] = (xg,yh(x)).

It is easy to see that each table defines a permutation on M x N, and the product of tables defined as


corresponds to the product of permutations from GI H. Hence the set of all permutations from GI H indeed forms a group of order IGI·IHim. Sometimes the abstract group GI H itself is called a table wreath product.

The operation of wreath product of permutation groups was actually used by C. Jordan [Jo 4], p. 27. The first time the definition of wreath product appears explicitly, and in full generality, is in the paper [Po 1] of D. P6lya. Systematic applications of the wreath product in group theory started with the works of L.A. Kaluznin [Ka 2:1, [Ka 3]. More details on the history of such applications can be found in [*Ka 8]. Let us remark that, following L.A. Kaluilnin, in the notation for wreath product we write the "active" group G on the left and the "passive" group H on the right (which is different from [Ha 5], for example).

Besides the wreath product of permutation groups, another permutation representation of the table wreath product - exponentiation of permutation groups -is widely used in combinatorics and graph theory. The exponentiation ( H, N) i ( G, M) ( or simply H i G) of the permutation group ( H, N) by the pemmtation group (G,M) is a permutation group (GI H,NM), where the action of any table [g; h(x)] E GI H on an arbitrary map f E NM is defined by the rule

The operation of exponentiation was introduced by F. Harary in [Ha 8]; notice that Harary hirnself uses the notation [H]G instead of H i G. The permutational properties of the wreath product and exponentiation are very different, which explains why exponentiation is considered as an independent operation on permutation groups, though it can be thought of as an indueed action of the wreath product. In particular, the wreath product is always imprimitive ( or even intransitive), while exponentiation ean give rise to a primitive permutation group.

Theorem 3.3.1 The exponentiation (H,N) i (G,M) of permutation groups (H, N) and ( G, M) is primitive if and only if His a primitive non-cyclie pemmtation group and ( G, M) is a transitive permutation group. •

Apparently, this theorem should be considered to be of "folklore" type. Its statement and proofs were published in [*Kl 4], [*Ka 2], [*Ka 5], [Ca 5]. (P. J. Cameron has informed the authors that the version of this theorem which appeared in [Ca 5] was taken from talks by O'Nan and Scott at the Santa Cruz eonferenee.)

In particular, Theorem 3.3.1 implies that the exponentiation Sn i Sm is primitive if and only if n =/:- 2 ( which, by the way, is easy to prove directly ). In the beginning of the 70's, when interest in the study of maximal subgroups of permutation groups was growing (see section 1.3), the exponentiation of symmetrie groups


turned out to be one of the first candidates for being maximal. In particular, after proving Theorem 3.2.1 on the asymptotic maximality of the induced symmetric groups, it was expected that exponentiation of symmetric groups might have an analogous property. The first attempt to prove asymptotic maximality of the group Sn I Sm according to the scheme in [*Ka 1] was made by B.A. Romov in [*Ro 3]. (Here and in what follows we shall always assume m :::0: 2.) But the statement on asymptotic maximality for n f. 2 given in [*Ro 3] turned out to be false. One of the reasons may have been due to the lack of explicit formulae for the structure constants of the V-ring of the group.

M.H. Klin [*Kl 4] found formulae for the structure constants of V(Sn I Sm) and refuted the statement from [*Ro 3] Oll the maximality of s3 I Sm and s4 I Sm for sufficiently large m. Soon after this (tobe discussed below), V.A. Ustimenko, using p-local analysis, succeeded in proving the maximality of Sn I Sm for n > 4. Therefore the problern of giving an elementary justification for the asymptotic maximality of Sn I Sm for n > 4 lost its urgency, and we never returned to it. Further studies on Sn I Sm proceeded in two different directions, quite like the case of the induced symmetric group.

The combinatorial direction

G.A. Jones and K.D. Soomro, who probably did not know about [*Ro 3], [*Kl4], and subsequent publications, proved in [Jo 3] (see also [Jo 1]) the following result. (To do this, they proceeded along the lines of the scheme used in [*Ka 1] and described in section 3.2.)

Theorem 3.3.2 For n > 4 there exists a function f( n) such that for m :::0: f( n) the group Sn I Sm is maximal in the alternating or symmetric group on the set NM ( depending on the presence of odd permutations in Sn I Sm)· •

Let us note that a portion of the results from [Jo 3] appears in the thesis [So 1]. Also we note that, for n = 5, 6, the authors of [Jo 1] had to use some non-elementary methods, since the Bochert-Manning bounds failed to help. One ofthe steps in their proof was a demonstration of the asymptotic simplicity of the BM-algebra of the scheme H(m,n), n > 4.

A stronger result on the simplicity of the V-ring of the group Sn I Sm (which is known to coincide with the BM-algebra of the Hamming scheme H(m,n)) was obtained by M.E. Muzichuk.

Theorem 3.3.3 ([*Mu 7]) For n > 4 the V-ring of the group (Sn I Sm, NM) is simple, i.e., it doesn't contain proper cellular subrings. •

For n = 2, 3, 4, V(Sn I Sm) has proper subrings. Their study originated with [*Kl 4], where it was shown that Wm = (Ao, At+ A3 + ... , Az + A4 + ... ) is a subring of rank 3 in V(S4 I Sm), with one of its basis graphs being an s.r .. g. with parameters

if m is even, and

if m is odd. It was announced in [*Mu 5] that W m is the only proper cellular subring of V(S4 I Sm)·

The case n = 3 was considered by V.A. Ustimenko in [*Us 2], [*Us 6] (see also [*Us 5]). It was shown that, form= 2h, V(S3 -~ Sm) always contains a cellular subring of rank 4: Uh = (Ao,<Po,<P1 ,<P2 ), where 'Pi= I:i=oj(rnod 3lA;. The basis graphs f(<Pj), j = 0, 1,2, are strongly regular with parameters:

k = 32h-l + 2 0 3h-1 0 ( -:L)h- 1,

). =32h-Z+ 2. 3h-1. (-l)h _ 2,

/1> = 3Zh-Z + 3h-1 . ( --1)h

when 2h =/= j(mod 3), and

k = 32h-1- 3h-1 0 ( -1)\

). =32h-Z'

/1> = 3Zh-Z + 3h-1. ( --1)h

when 2h = j(mod 3). It was also shown there that the graph f( <Pi) is a graph of rank 3 (resp., rank 4) if j =/= 2h(mod 3) (resp., j == 2h (mod 3)). For h = 2,3,4,5, it was shown by computer in [*Za 5] that, in those cases where the s.r.g. r( <Pi) is a graph of rank 4, it does not satisfy the 4-condition.

M.E. Muzichuk and V.A. Ustimenko announced that, form= 2h, the cellular ring U(h) and its subrings exhaust allproper subrings of V(S3 I Sm), but a complete text of the proof is not yet prepared.


The case n = 2 was considered also in the aforementioned papers of Ustimenko, where some subrings of V(S2 I Sm) were found. Then V.A. Zaichenko [*Za 4], through the use of a computer, enumerated all cellular subrings of V(Sz I Sm) for m :::; 16. Analysis of these results allowed Muzichuk to formulate a conjecture about the structure of allsubrings in V(S2 I Sm) for all m, which was proven in [*Mu 4]. Also it was shown in [*Us 2] that a cellular subring of rank 3 in V(Sn I Sm), having one of its basis graphs an s.r.g. with ,\ = p,, can exist only for n = 2 or n = 4.

The results in [As 1], where the nonexistence of uniprimitive overgroups of Sn I Sm is proved for n ::=: 5 and 2 :::; m :::; 5, should be viewed as combinatorial also.

Therefore one may say that, at the present time, the problern of enumerating all cellular subrings in the BM-algebra of the Hamming scheme H( m, n) is close to completion. We remark that some examples of cellular subrings of these BMalgebras were found independently in [Hu 2] and [Ka 1].

The group theoretic direction

V.A. Ustimenko, using his technique of p-local analysis on invariant relations of permutation groups, but without using any results from the classification of simple groups, described all cases when the permutation group Sn I Sm is maximal.

Theorem 3.3.4 Let n > 4. The permutation group Sn I Sm is maximal in the alternating group on the set NM only if n is even and m > 2 or n = 4k and m = 2. Otherwise, it is maximal in the symmetric group on the set NM.

This result was announced in (*Ka 4], [*Ka 5], and published in its entirety in [*Us 5], [*Us 10]. The proof is presented also in the paper [*Us 13], which appears in this book.

For n = 2, 3, 4, the group Sn I Sm is not maximal, since it is a subgroup of the affine group AGLm(2), AGLm(3), AGL2 m(2), respectively. In these cases it is interesting to study the lattice of overgroups of Sn I Sm in S(NM). It seems reasonable to solve this problern in accordance with the following scheme:

(a) all cellular subrings in Sn I Sm are described;

(b) using the method presented in section 2.5 for computing 2-closures, the automorphism groups of the described subrings are determined;

( c) the remaining overgroups ( ones which are 2-transitive, or areunitransitive but not 2-closed) are described by the method of p-local analysis.


At the present time, stages ( a) and (b) have been clone by M.E. Muzichuk. The results related to (a), as we have mentioned, are partially contained in [*Mu 4] and [*Mu !5]. Some results related to (b) were am;tounced in [*Us 7], but have not as yet been published in their entirety.

Therefore, the problern of describing all 2-closed overgroups of the exponentiation Sn j Sm in the symmetric group Snm can be considered to be completely solved at the present time, and its solution does not use results from the classification of simple groups.

3.4. q-Analogues of the Johnson scheme and the Hamming binary scheme

Among all possible classes of cellular rings, the BM-algebras of (P and Q)polynomial association schemes are of the greatest interest in modern combinatorics. The property of being P-polynomial has a natural combinatorial interpretation (one of the basis graphs is a d.r.g.). The property of being Q-polynomial doesn't have so obvious a combinatorial interpretation, but Q-polynomial schemes are used extensively in coding theory. A thorough exposition of the current state of the theory of (P and Q)-polynomial schemes is contained in the book [Ba 5]. Appearing there is also a list of all known (P and Q)-polynomial schemes of sufficiently large diameter d, and the conjecture that every (P and (J)-polynomial scheme either

(a) has parameters which coincide with the parameters of a scheme from the list;

or

(b) is an antipodal cover or a bipartite extension of a scheme from the list.

A solution to the following problern which appears in the survey [Fa 2] could lead to indirect confirmation (refutation and/or modification) of the above conjecture. It is: Describe all cellular subrings in the BM-algebras of known (P and Q)-polynomial schemes. We notice that this problern can be solved on two levels:

(i) Give an explicit description of cellular subrings and their automorphism groups;

(ii) Give a clescription of all possible partitions of the set {l,···,d} lea.ding to cellular subrings, and the parameters of these subrings.

In the second case the problern can be solved even if we do not have a complete list of sehemes with given parameters, since it is sufficient here to know only the structure consta.nts p~i' which can be uniquely determined by the parameter set (intersection array) of the P-polynomial scheme.

Actually we have alreacly considerecl this problern in the two previous sections

for the two best known (P and Q)--polynomial schemes: the Johnson scheme and the Hamming scheme. But the main emphasis there was focused on the group theoretic motivation of the problem. For these schemes the problern turned out to be most difficult, and for Johnson schemes it has not been solved completely. Regardless of the presence of a relatively large number of cellular subrings in the V-rings ofthe groups (Sv,{~}) and Sn j Sm (the BM-algebras ofthe Johnson and Hamming schemes), the aforementioned conjecture for these classes of schemes has been confirmed at the present time: All families of cellular subrings which have been found either corresponded to (P and Q)-polynomial schemes from the list, or led to schemes which were not (P and Q)-polynomial.

In this section the stated problern is considered for several other classes of schemes, joined by the common name of q-analogues of Johnson and Hamming schemes. Most of the results we consider were obtained by M.E. Muzichuk.

3.4.1. q-Analogues of Johnson schemes

Let Vm(Fq) be a vector space of dimension n over the field of q elements, and let p be the characteristic of the field. By G~ = G~ ( q) we denote the set of subspaces of the space V m ( Fq) of dimension n ( sometimes this set is called the Grassmanian of index n- 1 of the space Vm(Fq)). In what follows, we will assume 2 < n::::; T· The family of relations c/Jo, cp1 , · · · , rPn, where c/J; = {( A, B) : dim( A n B) = n - i} forms an association scheme over the set G~, which is called the q-analogue of the Johnson scheme and which is (P and Q)-polynomial [De 1]. We denote this assocation scheme by lq(n,m), and its BM-algebra by Uq(n,m). We notice that relations c/J; are 2-orbits of the natural permutational representation of the group PGLm(q) on G~. It was proven in [*Us 4] (see also [*Us 5], [*Us 3]), that the permutation group (PrLm(q), G~) is maximal for n f. 2m, and for n = 2m it has the unique proper overgroup obtained as an extension of Pr Lm ( q) by means of the contragredient automorphism (here Pr Lm ( q) is the group of all semilinear transformations of the projective space PVm(q)).

Here we are interested in the problern of describing of the cellular subrings in Uq(n,m). The firsttime this problern was considered was in [*Us 8] (see also [*Us 12]), where it was shown that, for q > max{b(n),(~)}, the cellular ring Uq( n, m) is simple. Here b( n) is the same function which was present in the statement of Theorem 3.2.1 for the induced symmetric group. M.E. Muzichuk ([*Mu 3], [*Mu 7]) showed that the result holds also if the restrictions on q are removed.

We will need the following numerical values for some elements of the eigenmatrix P ofthe cell Ug(n,m) [De 2]:

(3.4.1)


[k + 1];Pk+l(i) = Pl(i)pk(i)- [k]q(q[rn- n]q + q[n]q- (q + 1)[k]q)Pk(i)

- q2k- 1[m- n- k -1]q[n- k- 1]qPk-1(i), (3.4.2)

where i = 0, ... , n, [x]q = 1 + ... + qx-l, x E N, IO]q = 0.

Lemma 3.4.1 Let W be a proper subcell of the cell Uq( n, rn ). Then A1 and A 2

are merged in W.

~ Let us consider the eigenmatrix P of the cell Uq( n, rn ). The elements Pk( i) are algebraic integers. On the other hand, formulae (:3.4.1) and (3.4.2) imply that the Pk(i) arerational numbers. Therefore the Pk(i) arerational integers, and matrix P, formed by residues ofpk(i) modulo q2 , ean be defined. The relation (3.4.1) implies:

- ( ") - { -q - 1 ' if i ::::: 2, PI l - -1 "f . = 1

' I l .

Then, using (3.4.2), one obtains that

Finally, using the relation

we get:

- ( . ) - { q' if i ::::: :~, P2 z - 0 "f . = "l

' I l . '

p3 (i) = 0 for all i :2: 1.

Pk(i) = 0 for all k :2: 3.

Let us now assume, that A1 and A2 are not merged in W. Then, by using Proposition 2.2.6 and the incongruence -1 =/'- -1 - q (mod q2 ), we conclude that E 1 E W. Since Uq(n, rn) is Q-polynomial, we conclude that W = Uq(n, rn). We have arrived at a contradiction ..,..

The following lemma, which is of interest in its own right, can be easily proved by means of a weil known recurrence for the intersection numbers of a P-polynomial association scheme ([De 1], relation 5.15) which we write as

PI(i)pj(i)/kj = PL+IPHl(i)/k~Hl + P{,jPi(i)/kj + PL-IPj-I(i)/kj-1,

and use for j = 1.

Lemma 3.4.2 Let P be the first eigenmatrix of a P-polynomial association scheme with d dasses and PL be its intersection numbers. Then PI (0) > PI ( i) > -1 implies PI(i)/kl > P2(i)/k2, where kt = Pt(O). •

Now we state and prove the main result of this subsection.

Theorem 3.4.3 For all n -::; m/2, q 2: 2, the cell Uq( n, m) is simple .

.,._Let W be a proper subcell of the cell Uq(n, m). Then, according to Lemma 3.4.1, A1 and A2 are merged in W. Let us use Proposition 2.2.5, according to which, for an arbitrary basis set F from the second partition of the subcell W, the following equality holds:

I:cqj(1)- qj(2)) = o jEF

By using orthogonality relations, this equality can be rewritten in the form

L mi(PI(j)/ki- p2(j)/k2) = 0 jEF

(3.4.1) implies that p 1 (j) > 0 for all j -::; n - 1, and p1 (j) < k1 for j > 0. Therefore, due to Lemma 3.4.2, we have

P1(j)/k1- pz(j)/k2 > 0 for all j = 1, · · ·, n -1.

Then the equality

L mi(PI(j)/kl- P2(j)/kz) = 0 jEF

can be correct only if n E F, or F = {0}. Since F was chosen tobe an arbitrary basis set, we get that the rank of the second partition of W is at most two. This contradicts the fact that W is a proper subcell ~

Remark Though Theorem 3.4.3 was proven under the assumption n-::; m/2, it is correct also if this restriction is removed. This can be easily shown if one notices that the cells Uq(n,m) and Uq(m- n,m) are isomorphic. An isomorphism can be defined on the space Vm(Fq), where for any subspace S of Vm(Fq) we map it to its annihilator in the dual space with respect to a fixed nondegenerate bilinear form.

3.4.2. q-Analoques of Hamming hinary association schemes

Let us consider a finite dimensional vector space Vm(Fq) over the finite field Fq of odd characteristic p. Let f(x, y) be a nondegenerate reflexive bilinear (or semilinear) form on Vm(Fq), and let Gm(!)= {g E GLm(q): f(x9,y9) = f(x,y)} be the classicallinear group associated with f. Let PGm(f) be the quotient group of Gm(!) by its center. Let n be the dimension of a maximal isotropic subspace of the space Vm(Fq) with respect to the form f. It is well known that PGm(f) acts


transitively on the set Xn of maximal isotropic subspaces, and that the action is faithful.

For subspaces A,B E Xn, let p(A,B) = n- dim(A n B). Then the 2-orbits of the permutation group (PGm(f),Xn) have the following form: W; = {(A, B) E Xn X Xn : p(A, B) = i}, i = 0, 1, · · ·, n. Therefore the relations {lli;}, i = 0, 1, · · ·, n, form an association scheme on the set Xn which is (P and Q)-polynomial. This association scheme is called the q-analogue of the Hamming binary scheme [St 1]. By Ln we denote the BM-algebra of this association scheme. It is clear that Ln= V(PGm(f),Xn).

The cells Ln and their automorphism groups were studied in [*Zd 1]-[*Zd 8], [*U s 9], [*U s 12]. In partiewar, an asymptotic description of the lattice of subcells of the cell Ln is obtained in [*Us 12]. Herewe present a precise statement of the complete description of the lattice of subcells of the cell Ln obtained by M.E. Muzichuk. The proof of the result is quite involved, and it can be found in [*Mu 7].

The structure of the lattice of subcells of Ln is completely determined by its structure constants, which depend (see [St 1]) on the type of the group SGm(f) = {g E Gm(!) : det g = 1} considered as a Chevalley group. Therefore, in what follows, we speak of the type of Ln, understood as the type of the group SGm(f). The types Bn, Cn, Dn, 2 Dn+l, 2 A2n, 2 A2n+l are considered.

Theorem 3.4.4 Let U be a proper subcell of the cell Ln. Then one of the following holds:

• Let U1 = (Ao, A1 + A3 + · · ·, A2 + A4 + · · ·), U2 = (Ao, A2, A4, · · ·, A1 +

A3 + · · ·), U3 = (Ao, A1 + A2, A3 + A4, · · ·)

Theorem 3.4.5 Let U be a proper subcell of the cell Ln of type Dn- Then U = U1 or U == U2. •

Theorem 3.4.6 Let U be a proper subcell of the cell Ln of type Bn or Cn. Then U=U3. •

By splittling the result of M.E. Muzichuk into several theorems we are able to give a rough sketch of the proof: First the types (Bn, Cn, Dn) of the cells which can be non-simple are determined and then, for each of these three types, all subcells are enumerated. We want to note that, contrary to [*Mu 4] where the computations were earried through exclusively in the first basis, here conditions on the existence

112 I. A. FARADZEV EI AL

of subcells in terms of the matrices P and Q described in section 2.2, were used substantially.

In concluding this section, we wish to point out that the cells Ln of type Bn and Cn have the same parameters, but are not isomorphic for odd q. It turns out that the subcell U3 of the cell of type Bn is of Schur type and it is isomorphic to the BM~algebra of the halved scheme of the scheme of type Dn+l· At the same time, the subcell U3 for the cell of type Cn is of non~Schur type for n ~ 3, and it defines a new d.r.g. (which is not in the list from [Ba 5]). The existence of this new family of d.r.g. follows easily from the Galois correspondence between permutation groups and cellular rings. It can be obtained (without any computation) from the group theoretic results given in [*El 1] and [*Zd 7] on the existence of nontrivial overgroups of the group 02d+l and the maximality of the groups Sp2d( q) in their action on the sets of maximal isotropic subspaces in spaces with the corresponding bilinear forms. The arguments are presented in more detail in [*Us 12]. In [Iv 1], one can find a self~contained combinatorial/geometric description of subcells U3 in the cells Ln of types Bn and Cn.

3.5. Primitive representations of nonabelian simple groups of order less than 106

U sing the package of programs described in section 2. 7, we have investigated ([*Iv 2],[*Iv 5]) primitive representations of nonabelian simple groups of order less than 106 , excluding groups from the PSL(2,q) fo,mily. The subdegrees and ranks of some primitive representations of P SL(2, q) grow so quickly as q increases that, even for groups of comparatively small orders, they are beyond the capabilities of the programs of the package. On the other hand, all subgroups of PSL(2,q) were described by Dickson [Di 1]; using this information and the method of Burnside marks, L. Tchuda [*Tchu 1], [*Tchu 2] obtained formulae for the computation of ranks and subdegrees of primitive representations of the groups of this family.

For groups not in the PSL(2, q) family, all primitive permutation representations were constructed, V~rings were computed, and lattices of cellular subrings ancl the automorphism groups of all elements of these lattices were determined. This was clone using permutation representations of minimal degree, and inducing them oll orbits of appropriate relations. There are 9 representations for which we could not complete this work: the group Sz(8) on 1456 alld 2080 points, the group PSU(3,4) on 1600 points, the group J1 on 2926 and 4180 points, the group PSL(3, 5) on 3100, 3875 and 4000 points and the group J2 Oll 10080 points. The ranks of these representations are too large for our algorithms for computing lattices of subrings of the corresponcling V ~rings.

One of the followillg three methods for obtaining gellerating permutations of


minimal degree was used:

( a) For the alternating groups An the description of the natural representation on n points is easy;

(b) the dassical groups and the group Sz(8) are described as groups of matrices over a finite field, and a permutation represent.ation of minimal degree in this case is a transitive component of their action on the lines of the corresponding space;

( c) generating permutations of the representations of minimal degree ( or algorithms for their constructions) for sporadic groups of order less than 106 are well described in the literature.

A one-to--one correspondence between the primitive representations of a group and its maximal subgroups allows us to use the catalog from [Fi 1], which contains, for every nonabelian simple group of order less than 106 , its maximal subgroups and their embeddings in the group. More complete and precise information is in [Co 1], but we did not have access to it while this work was being clone. As we noted in section 2.7, in order to obtain the permutation representation of a group G on the cosets of its maximal subgroup H by means of inducing, it is sufficient to find an H -invariant relation U in the original representation ( G, n) which is not G-invariant. If the action of the group H in the original representation is intransitive, then one can use a unary relation consisting of all elements of any orbit of the action ( a symmetrized Je-point). In case H acts imprimitively Oll n, one can use the binary relation of equivalency, generating the partition of n .into blocks of imprimitivity. Finally, when H is a uniprimitive Frohenins group with cyclic kernel and complement, the induced action can be obtained on the cyclic graph. In the remaining cases, when U = { u;} is an arbitrary k-orbit of the group (H, f!), the desired representation can be obtained by double inducing. First we construct a representation (G, U') on the k-orbit, containing some point u; EU. In this representation U C U' is a k'-point, k' = [U[, and the desired representation is obtained by inducing on this point. Complete information about generating permutations of the representations of minimal degree, as well as the inducing of other primitive representations of nonabelian simple groups of order less than 106 ,

is given in [*Iv 2]. The program for verifying celi isomorphisms was used in the context of this work, not only to search for isomorphic subrings in the V -rings of the groups under investigation, but also to establish the similarity of representations of the same group having the same degrees.

In order to identify automorphism groups of subrings by using their orders (which were already computed), the following lemrna was used.

Lemma 3.5.1 Let (G, D) be a primitive group of degree n, n = mp and m > 1, where p is prime and p2 doesn't divide JGJ. Then H <1 G:::; Aut(H), where His a nonabelian simple group. •

In all of the cases we considered, knowledge of the order of the automorphism group of a primitive cell allowed us to find its normal subgroup, i.e., a nonabelian simple subgroup H. Since we did not use the results of the classification of finite simple groups, the following sources were employed: the description of nonabelian simple groups of order less than 106 [Ha 7], the description of simple groups whose 2-Sylow subgroups have order less than 210 [Be 2], and the list of orders of known simple groups [Cr 1]. To finish the identification, we used the classical results of Steinberg [St 2] on automorphisms of finite groups of Lie type, and facts on automorphisms of sporadic simple groups (see the survey [*Sy 1]).

The main results of our computations for uniprimitive representations are collected in Table 3.5.1. The subrings of each representation are numbered in decreasing order of their ranks ( the V -ring of the representation has number 0 ). For a subring W we give its rank r, its subdegrees, the number associated to each of the subrings W' into which W is embedded, and its automorphism group. The notation zk+2m in the subdegrees columns is used to show the existence of k symmetric and m pairs of antisymmetric 2-orbits, each having subdegree l. Coincidence of the automorphism groups of the subrings W and W', when W is embedded in W', means that the subring W is of non-Schur type.

This work generated a number of both group theoretic and combinatorial results. Let us describe several of the most interesting of these.

(1) The maximality of some primitive representations of nonabelian almost simple groups in the corresponding symmetric and alternating groups was established.

(2) Embeddings G C G' of some nonabelian simple groups into automorphism groups of other nonabelian simple groups were realized via primitive permutation groups. The cases we discovered are collected in Table 3.5.2. All cases of such embeddings were described later in [Li 1].


n w r Subdegrees W' Aut(W)

A5 = P5L(2,4) = P5L(2 5)

10 0 3 1,3,6 5s

A6 = P5L(2, 9)

15 0 3 1,6,8 56

A1

21 0 3 1, 102 57

35 0 4 1, 4, 12, 18 s7 1 3 1,16,18 0 s8

PSL(3, 3)

144 0 8 1, 131+2 2 , 392 PSL(3,3) 1 6 1,13,262 ,392 0 PSL(3,3) 2 3 1,39,104 1 PSL(3,3)

234 0 18 1, 3, 42.1 ' 6, 124+2·2 ' 241+''· 2 PSL(3,3) 1 12 1,3,6,8,122,242+2·1 ,482 0 PSL(3,3)

--PSU(3, 3)

36 0 4 1, 72.1 ' 21 PSU(3, 3) 1 3 1,14,21 0 PfU(3,3)

63 0 4 1, 6, 24,32 PfU(3,3) 1 3 1,30,32 0 PfU(3,3)

63 0 5 1,6,1621 ,24 PSU(3, 3) 1 4 1,6,24,32 0 PfU(3,3) 2 3 1,30,32 1 PSp(6, 2)

j\111

55 0 3 1,18,36 Su

66 0 4 1,15,20,30 Mu

1 3 1,20,45 0 S12

165 0 8 1, 8, 12, 242+2·1 , 48 Mu

1 4 1,24,56,84 0 Su

A8

28 0 3 1,12,15 s8

35 0 3 1,16,18 s8

56 0 4 1,10,15,30 58

Table 3.5.1



PSL(3,4)

56 0 3 1,10,45 PL.L(3, 4)

120 0 4 1,21,42,56 PL.L(3,4) 1 3 1,42,77 0 PL.L(3, 4)

280 0 8 1, 9, 183 , 721+ 21 PSL(3,4) 1 7 I, 9, 183 ,72,144 0 PSL(3,4) 2 7 1,9,18,36,721+2·1 0 PL.L(3, 4) 3 6 1,9,18,36,72,144 1,2 PL.L(3, 4) 4 6 I, 9, 54, 721+21 2 Pf L(3, 4)

5 5 1,9,54,72,144 3,4 PrL(3,4) 6 3 I ,36,243 3 PfU(4,3)

There are three subrings which are isomorphic to each of subrings 2, 3 and 6.

PSU(4,2)

27 0 3 1,10,16 PfU(4,2)

36 0 3 1,15,20 PfU(4,2)

40 0 3 1,12,27 PfU(4,2)

45 0 3 1,12,32 i PfU(4,2)

Sz(8)

560 0 17 I, 133 , 262 3, 527 Sz(8) 1 7 1,39,52,78~ 1 ,1562 0 Aut(Sz(8)) 2 3 I ,208,351 1 Aut(Sz(S))

1456 0 79 1' 53' 102·3' 2021+2·24 Subrings are not determined.

2080 0 165 1, 77+2·12' 1429+2·52

Subrings are not determined.

Table 3.5.1 continued


n I

w I

r I

Subdegrees W' Aut(W)

PSU(3,4)

208 0 5 1,12,602 ,75 PSU(3,4) 1 4 1,12,75,120 0 PfU(3,4) 2 3 1,75,132 1 PfU(3,4)

416 0 9 1,15,25~ 2 ,752 ,150 PSU(3,4)

1 7 1,15,502,752,150 0 PSU(3,4) 2 5 1, 15,100, 1502 1 PfU(3,4) 3 4 1,100,150,165 2 PfU(3,4) 4 3 1,100,315 3 Aut(G2(4))

1600 0 48 1' 131+2-4' 394+2-1 7


M12

66 0 3 1,20,45 s12

144 0 5 1, 112'1, 55,66 M12 1 4 1,22,55,66 0 Aut(M12 )

2 3 1,66,77 1 Aut(M12)

220 0 5 1,12,27,72,108 M12 1 4 1,27,84,108 0 s12

396 0 10 1,10~ 1 ,15,30 2 ,601 + 2 · 1 ,120 M12 1 7 1,15,20,602,1202 0 Aut(M12)

495 0 11 1,6,16,24,322'1,482'1,963 M12 1 8 1,6,16,24,64,962,192 0 Aut(M12)

495 0 11 1, 6, 16, 24, 322' 482' 96 1+2·! M12 1 8 1,6,16,24,64,962 ,192 0 Aut{M12) 2 5 1,32,70,168,224 0 512 3 3 1,238,256 1,2 po-{lo,2)

There are two subrings isomorphic to subring 2.

1320 0 29 1, 8, 9, 12, 181+21 ' 242, M12 363+2·2' 725+2·4

1 22 1,8,9,12,18,242,364 , 0 Aut(M12) 725+2·1' 1.444

Table 3.5.1 contimied



PSU(3,5)

50 0 3 1, 7,42 PI;U(3, 5)

175 0 4 1,12,72,90 PI;U(3, 5) 1 3 1, 72, 102 0 PI;U(3, 5)

525 0 8 1,20,483 ,1203 PSU(3, 5) 1 7 1,20,48,96,1203 0 PI;U(3, 5) 2 6 1, 20,1203 , 144 1 PfU(3,5) 3 5 1,20,120,144,240 2 PfU(3,5) 4 4 1,20,144,360 3 PfU(3,5) 5 3 1, 144,380 4 PfU(3,5)

There are two subrings isomorphic to subring 2.

h

266 0 5 1, 11, 12, 110, 132 h

1045 0 11 1,8,28,563 ,1685 J1

1463 0 22 1, 12, 152"1 , 202, 603+2·3 , h 1207

1540 0 21 1, 19,384, 574+2·1 , 1445+2·2 h

1596 0 19 1, 11,222,552, 1109+2 2 J1

2926 0 67 1, 151+2·2, 307+2 10, 6016+2·9


4180 0 107 1, 7, 14\ 214+2 1, 4227+2"34


Ag

36 0 3 1,14,21 Sg

84 0 4 1, 18,20,45 Sg

120 0 3 1,56,63 pfl-(8, 2)

126 0 5 1,5,20,40,60 Sg 1 3 1, 25,100 0 S1o

280 0 5 1, 27, 36, 54, 162 Sg 1 4 1, 54, 63, 162 0 Sg 2 3 1,117,162 1 Sg

840 0 12 1,8,242"1 ,27,36,722+2·1,2162 Ag 1 9 1,8,27,36,48,1442,2162 0 Sg




PSL(3,5)

3100 0 50 1, 101+2'1' 122' 151+2-1' 302+2·1, 606+2·1o, 1201+2·5


3875 0 57 1,6,12,161+2·1,242,322'1, 485+2'6' 964+2-13


4000 0 54 1, 313+2-6' 936+2·16 Subrings arenot determined.

M22

77 0 3 1,16,60 Aut(M22)

176 0 3 1, 70, 105 M22

231 0 4 1,30,40,160 Aut(M22) 1 3 1,30,200 0 Aut(M22) 2 3 1,40,190 0 s22

330 0 5 1, 7, 42, 112, 168 Aut(M22)

616 0 5 1,30,45,180,360 Aut(M22)

672 0 6 1,552,66,165,330 Aut(M22) 1 4 1,66,110,495 0 Aut(M22) 2 3 1,176,495 1 PfU(6,2)

h

100 0 3 1,36,63 Aut(J2)

280 0 4 1,36,108,135 Aut(J 2) 1 3 1, 135, 144 0 Aut(J2) 2 3 1,36,243 0 Aut(J2)

315 0 6 1,10,322,80,160 h 1 5 1,10,64,80,160 0 Aut(J 2)

525 0 6 1,12,32,96,1922 Aut(J 2)

840 0 7 1,15,32,24,180,240,360 Aut(J2)




h continued

1008 0 11 1,12,25,50,602,1002, h 1502 1' 300

1 8 1,12,25,50,120,200,3002 0 Aut(J2)

1800 0 18 1, 142"1' 21, 28,423, 841+2 1' ]2 1684+2'1' 336

1 14 1, 21,282, 42,842' 1683+21 ' 0 Aut(J2) 3362

2016 0 18 1, 15, 25, 502"1' 733+2"1' h 1504+2'1' 3002

1 12 1,15,25,75,100, 1504,3002, 0 Aut(J2) 600

2 3 1, 975, 1040 1 PfSp(6,4)

10080 0 191 1,6,103,124,15,202"2, 308+2·8' 6033+2·60 Subrings are not deterrnined.

PSp(4,4)

85 0 3 1,20,64 PfSp(4,4)

120 0 3 1,51,68 PfSp(4,4)

136 0 3 1,60, 75 PfSp(4,4)

1360 0 11 1,152,45,902"1,1202,144, PfSp(4,4) 3602

1 7 1,30,45,144,180,240,270 0 PfSp(4,4)

Table 3.5.1 concluded


G n GI

PSU(3,3) 63 PSp(6,2) Ag 120 pn-(8, 2) PSL(3,4) 280 PrU(4,3) PSU(3,4) 416 Aut(G2(4)) M12 495 pn-(10,2) M22 672 PrU(6,2) J2 2016 PrSp(6,4)

Table 3.5.2

(3) For some primitive representations, we dete1mined their ranks, subdegrees and the decomposition of their permutation characters into irreducibles. Some of them had been omitted or computed incorrectly in [Fi 1]. (Some of them were also omitted in [Co 1].)

( 4) Five new strongly regular graphs of rank different from 3, but with vertexprimitive automorphism groups were constructed. The parameters (v, k, .>.) of these graphs are given in Table 3.5.3.

(v,k,.>.) Aut(r)

(144,39,6) PSL(3, 3) (280,36,8) Aut(J2) (280, 117,44) Sg (280,135,70) Aut(h) (560,208,72) Aut(Sz(8))

Table 3.5.3

While writing [*Iv 5], we also assumed as unknown the graph with parameters (231,30,9) and automorphism group Aut(M22 ), but we found later that the graph had been constructed in [Ca 3].

Herewe give abrief description of the structure of one of these graphs.

The group S9 acts transitively on all 280 partitions of a set of 9 points into 3 classes of equal size. The subdegrees of this action are 1, 27, 36, 54 and 162. A pair of partitions belongs to a basis graph of valency 162, provided that the matrix of


the cardinalities of intersections of these classes can be brought to the form

via row and columnn permutations. The complement of this basis graph gives the s.r.g. with parameters (280,117,44). This construction was found independently by Mathon and Rosa [Ma 4].

Recently, B. Bagchi [Ba 1] gave an independent construction of two of the new strongly regular graphs with automorphism group Aut(]z). The vertices of these graphs are the 10-vertex cliques of the s.r.g. of Hall-Janko on 100 vertices. Let us note that Bagchi did not describe the whole group of automorphisms of the graphs he constructed. We also note that the new s.r.g. with parameters (280,36,8) demonstrates that the assumption of rank 3 is essential in W.M. Kantor's characterization [Ka 4] of the graphs of classical geometries. (A known graph of rank 3 with automorphism group PfU( 4,3) has the same parameters.)

The vertices of the new strongly regular graphs on 144 and 560 points are the cyclic subgroups of order 13 in the groups PSL(3,3) and Sz(8). Since every description ( of which we are aware) of the adjacency relations in these graphs seems quite artificial, we do not present any here.

(5) We considered 3 pairs of non-similar primitive representations of the same degree of the same group: PSU( 4,2) of degree 40, PSU(3,3) of degree 63, and M12

of degree 495. Note that the first two of these groups are members of infinite families. In each of these cases, by considering an intransitive representation of the corresponding group with two orbits, one easily sees that every symmetric basis graph of the V -ring of this representation in which endpoints of edges belong to different orbits is a nonadmissible graph ([Fo 1], [*Vi 1]), i.e., a bipartite graph, the automorphism group of which acts transitively on the edges but intransitively on the vertices. Moreover, in this case we obtain the first examples of biprimitive nonadmissible graphs, the automorphism groups of which act primitively on each bipartition. These examples allow one to reformulate the conditions for nonadmissibility entirely in group theoretic language ( the first such attempt was made in [Kl 4]) and, in particular, to give a complete description of biprimitive nonadmissible graphs of valency 3 [*Io 1 J.

We conclude this section with the remark that in this research, LV. Chuvaeva, Ya.Yu. Gol'fand, M.E. lofinova, E.A. Komissarchik, L. Tchuda and S.V. Tsaranov participated with the authors in the consideration of certain concrete groups.


3.6. Enumeration of primitive graphs

The task of constructing catalogues of graphs from sufficiently general dasses is obstructed by the phenomenon of "combinatorial explosion", the catastrophic growth in the number of graphs in these dasses when the number of vertices increases. For example, there are 27 4 668 connected simple graphs on 9 vertices [Ba 2] (see also [He 1]), and the constructive enumeration of such graphs with more than 10 vertices would probably be meaningless. The attempt to narrow the dass of graphs under consideration by imposing some conditions of combinatorial symmetry (such as regularity, strong regularity, etc.) does not remove these difficulties. Such works as [*Fa 1], [*Ba 1] on regular graphs and [*Ro 1], [*Ar 1] on strongly regular graphs illustrate this fact fairly well.

The problern of constructive enumeration of graphs whose properties are described in terms of their automorphism groups is of special interest. (Usually we speak of the transitivity or primitivity of the action of these groups on sets of certain subgraphs.) But the number of graphs with vertex-transitive automorphism groups also grows too fast in relation to the number of vertices. For example, the list computed by B.D. McKay [Mc 1] oftransitive graphs with no more than twenty vertices contains 1021 graphs. On the other hand, a further narrowing of the dass of graphs allows one to remove this restriction completely. F'or example, it is reasonable to speak about lists of distance--transitive graphs of order 103-104 . The problern of listing primitive graphs falls precisely between these extremes. ( A primitive graph is a simple connected graph, the automorphism group of which acts uniprimitively on the set of vertices.)

It is a well-known fact ( e.g., it is a consequence of the theorem of O'Nan--Scott, see [Ca 5]) that the sode of a primitive permutation group is either:

(a) a regular elementary abelian group (the affine ease), or

(b) a direct product of isomorphic nonabelian simple groups ( the nonabelian case ).

In ease ( a), wi th n = p", p a prime, the siruplest case is n = p. It is well known [Bu 1 J that any transitive permutation group of prime degree is primitive and contains a regular cydic subgroup Zp- This imples that every primitive graph with a prime number of vertices is isomorphic to the cydic graph Cp( J) = 2::: Cp(j),

jEJ

where J C {1, 2, ... , ~} and Cp(j) is the simple graph with vertex set {0, 1, ... ,p-1} and edge set { { i, i + j mod p} : 0 ::::; i ::::; p - 1}. Isomorphisms of cydic graphs can be established by Adam's criterion [Ad 1] (for n = p this is proved in [Dj 1], see also [Fa 2] and subsection 3.7.1):

Cp(J) "'Cp(J') {==? :IJ.l E {1, ... ,p--1}: J.l] = 1',

124 LA. FARADZEV ET AL.

where] = J U {p- j Jj E J}. Therefore a constructive enumeration of primitive p

vertex graphs is reduced to an enumeration of the orbits of the multiplicative group z; = {1, 2, ... , p- 1} acting via multiplication Oll the SUbsets of the elements of z; which are closed with respect to additive inverses. It is clear that the solution to this problern does not involve any major difficulties. We remark that, following this approach, it was possible to get the generating function for the number of k-regular primitive p-vertex graphs [Tu 1]. Table 3.6.1 includes the number of k--regular p

vertex graphs, which we found by constructive enumeration, where p is a prime no greater than 4 7 and k ::; ~.

p\k 2 4 6 8 10 12 14 16 5 1 7 1

11 1 2 13 1 3 3 17 1 4 7 7 19 1 4 10 14 23 1 5 15 30 42 29 1 7 26 73 143 217 128 31 1 7 31 91 201 335 429 37 1 9 46 172 476 1308 1768 2348 41 1 10 57 245 776 1944 3876 6310 43 1 10 64 285 969 2586 5538 9690 47 1 11 77 385 1463 4389 10659 21318

p\k 18 20 22 37 1367 41 8398 4654 43 14000 16796 47 35530 49752 63308

Table 3.6.1

When p and k increase, the number of primitive graphs grows so rapidly that enumeration becomes meaningless. In what follows we consider primitive graphs with a composite number of vertices.

Suppose we have a complete list 9n ( up to similarity) of uniprimitive permutation groups of degree n. Then the complete list of primitive graphs on n vertices ( together with their automorphism groups and a description of their action on both edges and non-edges) can be obtained from the scheme below.

(1) First we form Hn, which is a base of the sublattice of uniprimitive groups


of degree n, i.e., a set of transitive groups of degree n such that for every group G E 9n there exists a subgroup H ~ Gwhich is similar to a group belanging to Hn.

(2) Foreach group H E Hn, we construct its V-ring and compute the lattice W(H) of its primitive subrings of rank at least 3. Taking the union of W(H) over all H E Hn, we obtain a lattice Wn containing V-rings of all (up to similarity) 2-closed groups from 9n· Factoring this lattice by the classes of mutually isomorphic subrings, we obtain the lattice W n which contains the V -rings of all 2-closed groups from 9n·

(3) We compute automorphism groups of all cells W E W n and determine whether they are of Schur type.

( 4) We enumerate ( up to isomorphism) all simple graphs of valency k :::; n~ 1

which belang to cells from W n (for k = n~l one also checks for isomorphism of the complements). We obtain the set Tn: the complete list ( up to isomorphism and complementation) of primitive graphs on n vertices. Foreach r E Tn we determine the minimal Schurian ring W which contains the graph r. Then Aut(r) = Aut(W) and Aut(r) acts transitively on the edges ( or non-edges) of r if and only if r ( or its complement) is a basis graph of the ring W.

In order to carry out items (2)-(4) of the scheme, one can use the package of programs described in section 2. 7. At the same time, due to restrictions on the rank and number of subrings of a cell, one has tobe careful while doing part (1) in the program for computing lattices of subrings; in this case, the groups in Hn have to be sufficiently "large".

It is well known ( e.g., see [*Su 1)) that for n = pr, the permutation groups of degree n with abelian socle (the affine case) are eontained in AGL(r,p) and are exhausted by groups of the form (ZpY : H, where His an irreducible subgroup of the group GL(r,p). Nevertheless, when nissmalL one can use more elementary arguments to find a suitable basis of uniprimitive groups of affine type of degree n. It is easy to see that every simple graph invariant with respect to the group (ZpY is also invariant with respect to the group (ZpY : (r), where r is a central involution. This was sufficient for all n :::; 27. For n = 49, the rank of the group (Z7 )2 : (r) and the number of subrings in its V-ring are too large, but some special considerations allowed us to establish that the groups (D7 ) 2 and (Z7 J2: Q8 , where Q8 is the quaternion group, form a basis of uniprimitive subgroups of the group AGL(r,p) containing (Z7 J2 : (r). Table 3.6.2 contains the number of primitive graphs of affine type which we constructed for n :::; 49.

n\k 4 5 6 8 12 16 18 20 24 9 1

16 1 1 25 1 1 1 4 27 1 1 1 49 1 1 4 8 12 5 14 21

Table 3.6.2

A list of uniprimitive permutation groups with nonabelian socle can be found in [Si 3] for n :::; 20, in [*Po 1] for n ::=; 50, andin [Di 2] for n < 1000. Table 3.6.3 contains the basis of uniprimitive groups of nonabelian type used by us for n ::=; 100. Table 3.6.4 gives the computed numbers of primitive graphs.


n 'Hn 10 A5 15 AB 21 PSL(2, 7) 25 Az

5 27 PSU(4,2) 28 PSL(2, 7),PSL(2,8) 35 A1 36 A6, PSL(2, 8), PSU(3, 3), PSU(4, 2) 40 P SU( 4, 2) - two representations 45 A6,PSU(4,2) 49 PSL(2, 7) 2

50 PSU(3,5) 52 PSL(3, 3) 55 PSL(2, 11)- two representations 56 As,PSL(3,4) 57 PSL(2, 19) 60 Az

5 63 PSU(3,3)- two representations 64 PSL(2, 7) 2

65 PSL(2, 25) 66 PSL(2, 11) 68 PSL(2, 16) 77 Mzz 78 A13 ,PSL(2, 13) 81 PSL(2, 8) 2

84 Ag 85 PSp(4,4) 91 PSL(2, 13)- two representations 100 h,HS,A~

Table 3.6.3


n\k 3 4 5 6 7 8 9 10 12 14 15 16 18 10 1 15 1 21 1 2 1 25 1 27 1 28 1 2 3 2 35 1 1 1 36 1 1 2 3 3 40 2 45 1 2 3 3 49 1 50 1 52 1 1 55 1 3 1 1 7 1 1 9 56 2 1 57 1 60 1 1 63 2 64 1 65 1 66 1 4 4 68 1 1 77 1 78 2 6 81 1 84

I 1

91 2 3 2 2 11 2 7 16 100 1 1 1 1 1

n\k 20 21 22 24 25 26 27 28 30 32 34 45 3 52 1 55 1 1 11 1 56 1 57 1 1 60 1 1 1 63 2 2 65 1 1 66 7 7 8 68 1 1 1 78 9 1 14 84 1 85 1 91 7 7 30 7 12 39 12 12

100 1 1 1 n\k 35 36 38 40 42 44 45 48

78 16 84 1 91 51 12 14 56 14

100 3 2 2 2

Tab1e 3.6.4


3. 7. Miscellaneous results

This section includes brief comments on miscellaneous results which have not yet recieved much attention in this survey. Unlike earlier sections, here we usually restriet ourselves to just a mention of the main content of the result, and do not try to formulate it precisely. Some additional details can be found in our previous surveys [*Kl 10] and [Fa 2].

3.7.1. Cyclic graphs

We call a graph cyclic ( or circulant) if it is invariant with respect to the regular representation of the cyclic group Zn of order n ( this representation is considered to be fixed throughout). The isomorphism problern for these graphs, namely that of finding necessary and sufficient conditions for two cyclic graphs on n vertices to be isomorphic, goes back to A. Adam [Ad 1]. The history of this problem, tagether with some generalizations, is well described in [Pa 1].

\Ve have solved this problern for some values of n, based on a modification of the standard scheme (see section 1.3) applied to S~rings. Usually the modification was the following: After all steps had been completed, the normalizers in the syrnmetric group Sn of the automorphism groups of all rings were determined. Also determined were the representatives of the cosets of the normalizers with respect to the corresponding automorphism group of the S~ring. Then the conditions for isomorphism were formulated in terms of these representatives. The whole scheme was used for n = p2 and n = pq, where p and q are distinct primes (see [Kl 2], [Kl 5], [Po 3]). The case n = p was considered as an example in [Fa 2]. The same results were also obtained, using a different technique, by several other authors (see references in [Pa 1]).

The case n = pm, p an odd prime, was considered in [Po 2] and [Kl3]. In [Po 2] all S-rings were enumerated, and in [Kl 3] conditions for isomorphism were given which did not use knowledge of the automorphism groups and their normalizers. We note that [Kl 3] was preceeded by a machine computation for the cases n = p3 , p = 3,5, the results ofwhich can be found in [*Vy 1].

The case n = 2m turned out to be more difficult. In [Kl 6] the S-rings were first enumerated for m :::; 6, and then a conjecture describing all S-rings for all m

was made (see [Kl 6] and [*Go 2]). This conjecture was proved in [Go 1].

For n :::; 32, computer-generated catalogues of all S-rings, and their automorphism groups, were obtained in [*Za 2] and [*Za •!]. This rich body of experimental data is still awaiting theoretical analysis. The case where n is "square-free" (n = PI ···Pb the p;'s distinct primes) is considered by Ya.Yu. Gol'fand in the article [*Go 4] in this book.


3.7.2. S-Rings over nonabelian groups

Most of the recent results concerning S-rings deal with S-rings over abelian groups. Only a few results concern nonabelian groups. In [Wi 1 J it was proved that the dihedral group Dn is a B-group. The problern of finding primitive S-rings over the smallest nonabelian simple group A5 was posed in [Wi 2]. The technique developed by M.E. Muzichuk (part of which is presented in section 2.4) allows one to approach this problern without the use of a computer.

The technique was first applied to the group A4 in [*Mu 1], where all S-rings over A4 were described. lt turned out that there are exactly 12 nontrivial S-rings, and all of them are imprimitive. The same result is presented in [*Mu 7] in great detail.

In the case of A 5 , only primitive S-rings were described ([*Mu 2], [*Mu 7]). There are exactly 2 nontrivial primitiveS-rings over A5 :

U2 = (Co,C2,Ca,C~ + c~'), where the basis sets of the S-ring U1 are the conjugacy classes of elements in A 5

(the index i of the dass C; coincides with the order of the elements). There is no complete description of allS-rings over A 5 • We would like to note that the interest in this particular case is related to the problern of L. Babai about G-CI-groups (see [*Kl 10]).

3.7.3. Automorphism groups of Paley graphs

Let Fq be the finite field of q = pn elements, with odd characteristic p, and F;2 = { x2 1 x E F;} be the subgroup of squares of the multiplicative group of the field Fq. The Paley graph P(q) is the graph with vertex set Fq and edge set {(x,y)IY- x E F;2 }. It is weil known that for q = 1(mod 4), the graph P(q) is an s.r.g., and that , for q = 3(mod 4), it is a tournament. Let (H(q), Fq) be the

permutation group consisting of all permutations of the form x ~ axPk + b, where a E F;2 , b E Fq and k E {0, 1, ... ,n-1}. It is easy to checkthat H(q):::; Aut(P(q)). It was proved in [*Mu 6] that H(q) = Aut(P(q)). The proofis based on an analysis of sets of the form

{ 1/J(x)-1/J(y) I } A.p = X- y x, y E Fq, X =f. y '

where 1/J is an arbitrary function from Fq to Fq· On the basis of some intermediate results, M.E. Muzichuk could obtain a new elementary proof of Salomaa's Lemma


(see [Sa 1] and [Fa 2], section 2.3) about the 2-transitivity of a transitive group of degree p (p prime) which contains at least one nonlinear permutation. The interest in Salomaa's Lemma is due to the fact that it can be used to prove, without invoking character theory, the classical theorem of Burnside [Bu 1] concerning the structure of a unitransitive group of degree p.

3.7.4. Representing graphs as objects of communication

Vllhen catalogues of graphs are constructed (see, e.g., section 3.6), the problern of representing graphs in a form which is convenient for the transfer of information from the authors of the catalogue to the users becomes very important. The traditional way of representing graphs by their adjacency matrices is usually too cumbersome. For some well known graphs, it is possible to find a compact specialized representation ( e.g., in the form of a nice diagram, as can be clone for the Petersen graph). Finally, for many families of graphs (in particular s.r.g.'s and d.r.g. 's, see [Hu 1]), the description uses a speciallanguage. It is not always easy to reconstruct the adjacency matrix of the graph from such a description.

We have developed an approach to the representation of graphs based on the description of the graph in terms of the elements of a V -ring of a standard permutation group. The roles of such standard groups are played by semiregular cyclic permutation groups, regular and semiregular representations of the cartesian powers of cyclic groups, and cartesian products of distinct cyclic groups. Such an approach allows us to achieve compact and visual representa.tions for many sufficiently symmetric graphs. The idea behind the approach, as well as information concerning computer implementation and numerous examples of such representations, can be found in [*Kl 11]. Some isolated particular cases of the representations we are discussing appeared earlier in [Fr 2], [Bi 1], [*Ma 1] and [Le 1].

3.7.5. Constructive enumeration of two-graphs

The definition of a two-graph ( a system of 3-element subsets of an n-element set satisfying certain conditions) can be found, for example, in [Se 1]. The paper [Bu 2] contains results of a constructive enumeration of two-graphs for n :S 10. V.A. Zaichenko developed an alternate approach to the enumeration of two~-graphs (see [*Za 2], [*Za 3]) which allowed him to obtain some additional interesting information in the case n = 10. The most interesting of his results was a constructive enumeration of twenty two-graphs on 10 vertices, with each two-graph T satisfying the following conditions:

(1) Aut(T) =/:- Aut(f) for every graph r from the switching dass corresponding to T;

132 I A. FARADZEV ET AL

(ii) Aut(T) = Aut(f') for some f' from a switching class distinct from that which corresponds to T.

Each of these graphs has a noncyclic automorphism group of order 4, the involutions of which have two fixed points. For n < 10, there are no examples of such graphs.

In [Ma 1], the problern of analytical enumeration of two-graphs was reduced to the enumeration of eulerian graphs. The eulerian graphs were enumeratecl by V.A. Liskovets [*Li 1].

3.7.6. Transitiveextensions of intransitive permutation groups

Let (H, !1) be a permutation group, and let X ~ n. The problern of transitive extension is the following: Construct all transitive permutation groups ( G, n u {X}) suchthat Gx = H. The classical case, when (H, !1) is transitive, is cliscussecl in the next subsection. Here we consicler the case when H is intransitive.

The following special case of the problern was studied in [*Iv 6]:

(i) the group (G, n u {x}) is 2-closed;

(ii) the structure constants of the V -ring of the permutation group ( G, n u {X}) are given.

An algorithm for the solution to this problern was sketchecl in [*Iv 6]. It was utilized in [*Iv 1] for the construction of an automorphic graph on 65 vertices. In some particular cases, the so-callecl "methocl of fixed points" can substantially reduce the running time of this algorithm (see the survey by A.A. Ivanov [*Iv 9] in this book ancl [*Chu 2]).

Another approach to the construction of transitive extensions is presented in V.K. Medvedev's dissertation [*Me 5], mainly devoted to the transitive extensions of rank 3. This problern was quite popular in the 60's and 70's (see, e.g., [Wa 2], [Ba 4]). In this setting, the group (H, !1) is intransitive, it has 2 orbits, ancl the action of H on one of the orbits is known. The problern of rank 3 extensions of a 3-transitive permutation group is thoroughly studiecl in [*Me 5] (without the use of results from the classification of simple groups ). Some of the results of [*Me 5] are published in [*Me 1] and were announced in [*Me 2], [*Me 3], [*Me 4] and [*Me 6].

3.7.7. Transitive extensions of unitransitive permutation groups

The problern of transitive extension of transitive permutation groups has always attracted the attention of researchers (see, e.g., [Ma 2]). T. Holyoke [Ho 1] gave


necessary and sufficient conditions for the existenee of transitive extensions, but their practical value turned out to be not very substantial, being used only, m essence, for the construction of multiply transitive :Mathieu groups (see [Ha 5]).

A combinatorial approach proved to be more fruitful: Necessary and/or sufficient conditions for the existence of a transitive extension of a group (H, fl) were formulated in terms of some combinatorial objects which areinvariant with respect to (H,fl). Many interesting results were obtained in this way. We mention, for example, [Hu 3], [Lu 1], [Sh 5], [No 1], [Ca 2].

In this same direction, M.H. Klin gave necessary conditions for the existence of a transitive extension of a unitransitive permutation group (H, fl), formulated in terms of 2- and 3-orbits of (H, n) ([*Kl 5], [*Kl 6]). In the case when all 2-orbits of (H, fl) are symmetric ((H, fl) is generously transitive, in the terminology of [Ne 1 ]), these conditions are reduced to the existence of a certain partition of the collection of 3-element SUbsets of the Set fl U {X} into disjoint block designs ( triple packings, in the langnage of [De 3]), invariant with respect to (H, fl). We call such a part:ition a compatible family of a system of triples. A compatible family which satisfies some additional conditions (formulated in terms of symmetric differences of blocks) is called coherent. The concepts introduc:ed represent in varying degrees combinatorial approximations to the property of 2-transitivity of the transitive extension for which we are looking.

In [*Kl 5] these conditions were used to answer the question of existence of a transitive extension of the exponentiation Sn j Sm of symmetric groups. By using a weak condition ( compatible families) it was shown that there is no transitive extension for n-# 3. Using permutation characters it was proved that, for n = 3, a transitive extension exists only when m = 2.

The question of whether a similar theorem could be proved combinatorially by using a stronger condition ( coherent families) was considered in [*Za 5], where, with the aid of a computer, the existence of at least one compatible family invariant with respect to S3 j Sm was established for m = 2, 3, 4, 5. Out of 8 constructed families, only one turned out to be coherent (when n = 3, m = 2; and it in fact corresponded to a transitive extension). We conjeeture that invariant compatible families exist whenever n = 3, m :::: 2, and that for m > 2 they are never coherent.

Acknowledgrnents

The authors of this survey are extremely grateful to the translators (Professors J. Hemmeter, F. Lazebnik, A. Woldar) for a high-quality translation ofthe Russian version of the text which delicately preserves our style of exposition. We are also obliged to the translators for their valuable comments and corrections.


We are wholeheartedly grateful to Professor Peter Cameron, who read a preliminary version of the English translation and made numerous suggestions which greatly improved the clarity of exposition.


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ON p-LOCAL ANALYSIS OF PERMUTATION GROUPS

V.A. Ustimenko

1. lntroduction.

In [11], [12], as a corollary to the classification theorem of finite simple groups, a description of all maximal subgroups of the symmetric (Sn) and the alternating (An)

groups was given. It is interesting to know what part of this description can be obtained by the classical methods of permutation group theory. In particular, the following questions are of interest.

(i) Is it possible to prove the famous O'Nan-Scott reduction theorem without use of the classification of finite simple groups? (Notice that at one step of the present proof the Schreierhypothesis is used explicitly.)

(ii) For which farnilies of permutation groups which "appear to be maximal", can their maximality be established without use of the classification of finite simple groups?

The second question is closely related to the investigation of the V-rings of Schur of permutation groups. This question was considered in some details in the survey [4], where a number of results obtained by the so called "the method of p-local analysis on permutation groups" are mentioned.

The purpose of the present work is to illustrate this method on two families of permutation groups. One of these families consists of exponentiations of the symmetric groups. The proof of their maximality was published in [18]. Another family corresponds to the action of PGLn+l (Fp) on the set of points of the n-dimensional projective space PGn(Fp) over the simple field Fp. We believe that the proof given here (which follows

[ 17]) shows the sphere of application of this method in the study of the actions of the Chevalley groups over the field F q of characteristic p, on the maximal parabolic sub

groups. The maximality of the permutation groups (PGLn+l (F 2' ), PGn(F 2' )) for sufficiently large n was proved in [16] (see also [10]). Notice that different proofs of the maximality of (PGLn+l (F q), PGn(Fq)) without the use of the classification of finite simple groups are presented in [3], [8], [13]. The history of this problern goes back to C. Jordan [6], [7].

Let us say a few words about the peculiarities of question (ii) for the algebraic groups over the finite field FP. The most difficult case is that of small dimensions. For instance, in the families AGLn(Fp) of affine groups, the cases n = 1,2 have not been studied completely. These cases are known as the problems of Mathieu and Noboru Ito,

respectively (see [14], [15]). For the series (PGLn+l (Fp), PGn(Fp)), the case n = 1 corresponds to the well-known Galois problem. In our opinion, the investigation of overgroups of the Chevalley groups of rank 2 acting by conjugation on their maximal parabolic subgroups is also rather interesting.

153

154 V. A. USTIMENKO

In Section 2 we establish our terminology and present some classical results from [19] which are used in this paper. In Section 3, as a corollary to these results, some wellknown results about maximal permutation groups are obtained. In this way we give, in particular, elementary proofs of some results from [1], [2]. In Section 3 some lemmas conceming the p-local analysis of permutation groups are also given, and the maximality of (PGLn+1(Fq), PGn(Fq)) for n:?. 3 is proved. In Section 4 the maximality of the exponentiation of the symmetric groups is proved.

2. Preliminary information.

By a permutation group (G, W), we shall mean a subgroup G of the symmetric

group S(W).

An orbit of G on the set W is an equivalence class defined by the following equivalence relation: a - b ~ 37t E G: a" = b. A k-orbit is an orbit of the permutation group (G, Wk) acting on the k-th cartesian power of W by the rule: (a 1, ... , ak)" = (a1, .. . , aD. Since a k-orbit is a subset of Wk, it can be considered as a k-fold relation on W. A union of k-orbits of (G, W) is called an invariant relation of the permutation group (G, W). A union of orbits of (G, W) is usually called a block.

Let 'P be an arbitrary subset of wk (a k-fold relation on W). The set

Aut'P = {7t E S(W): V (a1, ... , ak) E 'P, (a1, ... , aD E 'P} is called the automorphism group of 'P. If 'P5 , s EI, is a family of relations on W, then Aut{'Ps} = n Aut 'P5 • It is

SEI

easy to see that if 'Ps is a set of invariant relations of a permutation group (G, W), then G is contained in Aut{'P5 }.

A relation <1> is said to be antireflexive if, for each (a 1 , ... , ak) E <j>, one has

ai = aj ~ i = j. In what follows weshall consider antireflexive relations only. The maximal antireflexive k-fold relationwill be denoted by wk.

lf McW then G{MJ={1tE(G,W)IVxEM, x"EM} and G(M) = {7t E (G, W) I Vx E M, x" =x) are subgroups of G, known as the setwise and the elementwise stabilizers of the set M, respectively.

Let a<M) be the permutation group induced by the action of the group G {M) on the block M. It is clear that G (M) is normal in G IM) and that G (M) = G (M) I G (M)·

Theorem 2.1. (cf. [15]). Let (G, W) be a primitive permutation group. Suppose that forM c W the group G (M) acts transitively on W- M. Then

(a) ifiMI > IW-MI,wehaveGE {A(W),S(W)};

(b) the group G (M) is primitive. 0

Part (a) of the theoremwas proved by B. Marggraf in 1892.

Let (G, W) and (X, W) be permutation groups with G <X. Then (X, W) is said tobe an overgroup of G.

ON P-LOCAL ANALYSIS OF PERMUTATION GROUPS !55

Proposition 2.2. Let G ~X where (X, W) is a transitive pennutation group. Then the orbits of G are imprimitivity blocks of X_ 0

Corollary 2.3. A nontrivial normal subgroup of a primitive pennutation group is transitive. 0

Let {'Ps}, s e J, be the farnily of k-orbits of a pennutation group (G, W). Then a korbit of (X, W) is a union u 'Ps for some Tc J_ We shall say that 'P; and 'Pj are

SET

congruent modulo X (and write 'P; = 'Pj(modX)), if 'P; and 'Pi are contained in the same k-orbit of X.

Let (X,r1) and (Y,r2) be pennutation groups on disjoint sets r1 and r2- Let us associate with a pair (1t1,1t2) E X X y a transfonnation X---+ xlt of the set r1 u r2. where 1t = 1t; for x e r;, i = 1 ,2. In this way we obtain a faithful action of the group X x Y on the set r 1 u r 2. The pennutation group corresponding to this action is denoted by (X, r!) + (Y, r 2) and is called the direct sum of (X,r1) and (Y, r 2).

Let us associate with a pair (1t1 ,1t2) e X x Y the transfonnation (a,b)---+ (an1 ,b'ltz), a E r 1, b E r 2. This defines a faithful action of the group X X y on the Set r 1 X r 2. The perrnutation group corresponding to this action is denoted by (X,r1) x (Y,r2) and is called the product of the pennutation groups (X, r 1) and (Y, r 2 )_

Theorem 2.4. (cf. [19])_ Let (G, W) be a transitive pennutation group. Suppose that for M c W, I W I =F- 2· IM I , the group G { M} acts primitively on both M and W- M. Then the pennutation group ( G, W) is primitive. 0

Lemma 2.5. Let (G, W) be a primitive pennutation group which contains a transpo-sition. Then G = S (W). 0

Lemma 2.6. The wreath product Sn wr Sm has a unique imprimitivity system. 0

We shall not make a distinction between a partition 't of the set W and the corresponding equivalence relation. For an arbitrary partition 't of W let us consider the subgroups

G't = {1t E (G, W) X'tY =*' X'tYlt} ,

G't={1te(G,W) X't)' =*' xn'tyn}

which will be called the elementwise and the setwise stabilizers of the partition 't in G, respectively.

The number of points which are actually moved by 1t is denoted by m(1t)_ The minimal degree m(G) of a pennutation group (G, W) is, by definition, the minimum of m(1t) over all non-identity 1t e G.

If (G, W) is a doubly transitive permutation group of degree n, then by the wellknown Bochert-Manning theorem,

!56 V. A. USTIMENKO

n ,-m(G) > --2 "'~n 3 .

Analogaus bounds for k-fold transitive permutation groups for certain k ~ 3 are also

available (cf. [19]).

3. Some proofs. Elements of p-Jocal analysis on permutation groups.

Theorem 3.1. An intransitive permutation group (G, W) of degree n is maximal in S(W) if and only if it is the setwise stabilizer of M c W, 2· IM I "# I W I inS (W), isomorphic to the direct sum Sm+ Sn-m• where m = IM I.

Proof. Let B be an arbitrary block of the group (G, W). Then (G, W) is contained in the setwise stabilizer X=S(W)(BJ of Bin S(W) similar to S(B)+S(W-B). If I W I = 2· IB I then Xis contained in the wreath product S 2 wr S (B), which isaproper

subgroup of S (W).

Now let (G, W) = S (B) + S(W -B), 2· IB I "# I W I. Then a proper overgroup X of G is transitive. Since X (B l acts primitively on both B and W-B, by Theorem 2.1 the group X is primitive. Now X is a primitive group containing a transposition. Hence by Lemma 2.5 it coincides with S (W). 0

Theorem 3.2. An imprimitive permutation group (G, W) is maximalinS (W) if and only if it coincides with the wreath product Sn wr S (B ), where I W I = n · I B I.

Proof. Let (G, W) be an imprimitive group. Then with a partition 't of W into imprimitivity blocks on G, we can associate embedding of G into the automorphism group of 't coinciding with Sn wr S (B ), where B is one of the blocks. Now let X be a proper overgroup of Sn wr S (B ). Then X does not preserve 't and by Lemma 2.6, X is primitive. Since X contains a transposition (i,j) where {i,j} E B, by Lemma 2.5 we have X =S(W). o

Let us consider the induced symmetric group S ~, i.e. the group S (N) acting on the set P~ of the 2-element subsets of N.

Theorem 3.3. The group S~ is maximal in S(P~) for n = IN I > 14.

Proof. Let A E P~. Then the group S ~(Al is isomorphic to S (A) x S (N-A) and is maximal in S (N) by Theorem 3.1, since IN- A I > I A I = 2. This implies that S ~ is pnrmuve. This group has two antireflexive binary orbits: <\> 1 ={(A,B)EP~xP~ I IAnBI=1} and<\Jo={(A,B) IAnB=0}.Thetransposition v = (i,j) E S(N) stabilizes the element {i,j} as weil as all 2-element subsets which have empty intersection with {i,j}. So m(v) = 2(n- 2).

Let (X,P~) be an overgroup of S~ distinct from S(P~) and A(P~). Then the primitive group (X,P~) either has the same 2-orbits as S~, i.e. <\>o and <)J 1, or it is doubly transitive. But in the latter case, for n > 14, the relation m(X)-;;;. m (v) = 2· (n- 2) contradicts the Bochert-Manning bound. Hence (X,P~)~ Aut<\>o n Aut<\J 1. Notice that the graph of

/"-.._

the relation <1>1 is the line graph of the complete relation N xN (recall that the line graph of a graph r has the edges of r as its vertices, with two such vertices adjacent if the corresponding edges have a vertex of r in common).

_./'.... Since the automorphism group of the line graph of the relation N xN isS (N) [5],

we have S (N):::; X:::; Aut <h :::; S (N). D

These argument were first presented in [9]. As a rule, for smaller values of n, the induced group is also maximal (see [17] for the details).

Let us consider the description problern of the overgroups for the group of projective linear transformations in dimension n over the simple field F P, permuting the points of the projective geometry PGn(Fp) (i.e. the lines ofthe vector space Vn+1 (Fp)).

We shall make use of the following well known propositions concerning the group PGLn+l (D) of projective linear transformations of the geometry PGn(D), where Dis an arbitrary skew field.

The group PGLn+1 (D) is contained in the group PrLn+1 (D). The latter is induced by bijections q, of Vn+1 (D) onto itself such that <jl(x + y) = <jl(X) + <jl(Y), <jl(AX") = A.cr <jl(x) where cr E AutD.

(A) The groups PGLn+ 1 (D) and Pr Ln+ 1 (D) are both doubly transitive on the points of PGn(D) and have exactly two ternary orbits: 0= {(l 1,l2,/3)E (PGn+1(D))3 I dim <1 1 h,l 3 > = 3} ("three points in general position") and e' = (PG;(i5))3 - e ("three collinear points").

(B) (The main theorem ofprojective geometry.)

Notice that in the case of simple fields F the groups PGLn(F) and PrLn(F) coin-cide.

Theorem 3.4. The permutation group (PGLn+l (Fp), PGn(Fp)) is maximal in S(PGnCFp)) for n ~ 3.

We present the proof in the case n = 3. Suppose that for P =PGn(Fp) we have G =PGLn+1(Fp):s;X < S(P), where Xis a group distinct from S(P) andA(P). In view of (A) and (B), to prove the theorem it is sufficient to show that Xis not triply transitive. Let us consider the p-subgroups Px=Sylp(Xz1 nXz2 nXz), (l1,l2,/3)E e, and Px' = Sylp (Xz 1' n Xz 2 ' n Xz;), (11',/2',!3') E e', which contain Pc = Px n G and Pc' = Px' n G, respectively. It is clear that if the groups Px and Px' arenot isomorphic then e ;t. 0' (modX).

Let e1,e2,e3,e4 be a basis of V4(Fp) such that I;= <ei>, i = 1,2,3, and <ll',/2',!3'> = <e1>e2>. In this basis Pc and Pc' consist respectively of the matrices having form

!58

1 0 0 0 0 1 0 0 0 0 1 0

* * * 1

and

It is clear that I P c' I > I P c I.

1 0 0 0 0 1 0 0

* * 1 0

* * * 1

V. A. USTIMENKO

The group K = P G is the group of transvections stabilizing each vector and each line of the hyperplane H = <1 1 ,12 , 13 >. This group has a unique orbit T of size > 1. This orbit contains all 1ines which do not belong to H. The elementary abelian group K acts regularly on T. The set T, as the unique orbit of K having size pn, is a block of the group Nx(K). Let us show that Nx(K) ac~s faithfully on T. If this is not:_ the case, then Xy #- 1 and Xy is a n_ormal subgroup of x<Tl where T = P(V)- T. But x<T) contains a subgroup

(PGL 4 (Fp))(T) similar to (PGL 3 (Fp), PG 2 (FIZ)) and hence is primitive. Since X(T) is a

nontrivial subgroup of the primitive group x<T), it is transitive on T (cf. Corollary 2.3), a contradiction to case (b) of the Marggraf theorem.

The group Nx(K) = Nx(K)(T) is determined by the relations:

PGLn+1 (Fp)\)'>J = Nc(K) < Nx(K) = AGLn(Fp)·

Here we have made use of the fact that the normalizer in S (F;) of the group (Fp/ of affine transformations of the vector space (F P) is isomorphic to (F Pi A. GLk(Fp ). Thus the group Nx(K) is contained in G. The latter implies that Npx(K) is contained in G and sta

bilizes the points 11 ,1 2 ,! 3 of general position. So Npx (K) = K. But the group Px is nilpo

tent and so satisfies the normalizer condition (i.e., the normalizer of a proper subgroup H of a nilpotent group is different from H). Thus Px = K and the groups Px and Px' have differentorders(IPx'l > IPc'l > IPcl = IPxl). D

The argument from the proof of the theorem can be subsumed into a general scheme. In order to describe this scheme we need the following definition.

Definition 3.1. A partition 't = {'P;), i E ], of the set wk will be called pdistinguished in the permutation group (G, W) if, for i #- j, y E 'P;, z E 'Pj, the permutation groups Sy1p Gy and Sylp Gz arenot similar.

Notice that to prove the maximality of a permutation group (G, W) which has a p

distinguished partition {'P;), i E J, it is sufficient to show that

(a) n Aut 'P; = G; iEJ

(b) For an overgroup X of G distinct from S (W) and A (W), the subgroup Sylp X-y coincides with Sy!P G-y.

Indeed, by (b) the partition {'P;) is p-distinguished in the group X and hence is invariant under X. So Xis contained in n Aut 'P;.

iEJ

In some cases verification of condition (b) can be reduced to the calculation of the normalizer in X of a certain p-subgroup P.

Definition 3.2. A p-subgroup P is called k-compact in ( G, W) if

(a) the elementwise stabilizer Gy of an arbitrary k-point y e ßrk contains a subgroup conjugate to P in G;

(b) all subgroups of Gwhichare similar toP are conjugate in G.

Proposition 3.5. Let (X, W) be an overgroup of (G, W) such that for some k-compact p-subgroup K

Nx(K) < G. (3.1)

Then Sylp Ga = Sylp Xa for every a E wk. Proof. Let us show that

Nx(Sylp Ga)< G for every a e ßrk. (3.2)

Since the subgroup K is k-compact in G, without loss of generality we may assume that Sylp Ga > K. Let 1t e Nx (Sylp Ga). Then the subgroup K'lt = tt-1 K 1t is similar to K and is contained in Sylp Ga. By property (b) for k-compact subgroups, there is a permutation t e G suchthat Gm =K. In view of (3.1) 1tt e Nx(K) < G. Since 1tt and t are contained in G, 1t e G.

Now let us show that (3.2) implies

(3.3)

Indeed, it follows from (3.2) that N = Nsy~X;; (Sylp Ga) is contained in G. Since N is a psubgroup of G stabilizing a , N coincides with Sylp Ga and the normalizer condition implies (3.3). D

Corollary 3.6. If the hypotheses of Proposition 3.5 are satisfied, then any partition of ßrk which is p-distinguished in G is p-distinguished in X.

Example. Let us prove Theorem 3.1. The subgroup K of transvections which stabilize the points of a hyperplane His a 3-compact p-subgroup of the group PGLnCFp)- In fact, each maximal abelian subgroup in PGLnCFp) which moves pn-1 points, stabilizes elementwise a hyperplane H'. Since PGLn(ff,) acts transitively on the set of hyperplanes, all these subgroups are conjugate in PGLnC1<-p). On the other hand, if n > 3 then an arbitrary triple / 1 ,/z,/3 of lines is contained in some hyperplane H'. So PGLnCFp)(l1 ,12 ,13 )

contains the subgroup K(H'). Thus we have shown that K(H) is a k-compact p-subgroup of PGLnCFp)-

We have seen that the normalizer of K(H) in any overgroup of PGLnCFp) other than S(PGn+1 (Fp)) and A(PGn+1 (Fp)) is contained in PGLnCFp)- In view of Proposition 3.5 this implies that X is contained in the automorphism group of the p-distinguished

160 V. A. USTIMENKO

partition of W3 into e and 8'.

4. The maximality of the exponentiation of the symmetric groups

The table wreath product of the permutation groups (X,M) and (Y,N) is the group of all "tables" [g,h (x)] where g E X, h(x) E ym, with multiplication defined as follows:

-1

[g,h (x)] x [g 1 ,h 1 (x)] = [gg 1, h (Xg 1 ) • h 1 (x)].

Let us define an action of the table wreath product on the set M i N of all partial functions from M to N by the rule:

f(x)[g,h(x)] = f(xg-1 )h(x).

The invariant blocks of this action are the sets M iN [I], 1::; l ::; m, of the partial functions fwhich are defined on sets d(l) of size l. The sets Mi N[1] and Mi N[l] coincide

with M x N and NM, respectively. The table wreath product of the groups X and Y acts

faithfully on the set Mi N[1l. Let (Y,N) i 1 (X,M) denote the permutation group

corresponding to this action. The groups Y i 1 X and Y im X are known as the wreath product and the exponentiation of the permutation groups (X,M) and (Y,N) and are denoted

by (X,M) wr (Y,M) and (Y,N) i (X,M), respectively. The notion of wreath product of

permutation groups first appeared in the book [6] by C. Jordan; the exponentiation was

defined by F. Harary in [5]. It is interesting to mention that the permutation group

S 2 i Sn corresponds to the action of the Coxeter group Bn on the cosets of the maximal parabolic subgroup corresponding to the rightmost node of the diagram.

The Hamming distance r(fl ,f2) between the functions ! 1 and h from f =NM is the number

r(fl,h) =m -l{x I fl(x) =h(x)} I· The permutations from S (N) i S (M) preserve the distance between functions. Moreover

the binary orbits of S (N) i S (M) are just the sets

~2 'I';= {(fl,h) E (N ) I r(fl,h) =m- i}.

~ /':':Proposition 4.1. The partition R of the set (NM)2 into '!'1 u 'I'm_1, '!'0 and

(NM)2 - 'I'o u 'I'1 u 'I' m-1 is 2-distinguished in the permutation group S (N) i S (M) for IN I~ 3, IM I~ 2.

Proof. The elementwise stabilizer of a 2-point from 'I'; in the permutation group

Sn i Sm is similar to the group CSn-1 i S;) x (Sn-2 i Sm_;). Let T; denote the permutation group Syl2(SniSm) where (fl,f2)E'I';, o::;i::;m-1. It lS clear that

T; =Syl2CSn-1 i S;) x Syl2CSn-2 i Sm-i)· The group Tm-1 is not similar to the group T; for i "# m - 1. The group T 0 is not similar to T; for i "# 0. D

ON P-LOCAL ANALYSIS OF PERMUTATION GROUPS 161

The base of the table wreath produet Sn i Sm is the m-th direet power of the symmetrie group S(N). Let Sk(N) denote the subgroup of this base whieh preserves the values of all funetions from F at the points of M- {k}, k E M. It is elear that Sk(N) is isomorphietoS (N). Let v = vfj be the transposition (ij) in the group Sk(N).

Proposition 4.2. If n ~ 4 then the involution v generates a 2-eompaet 2-subgroup of

Sn j Sm.

Proof. The involutions vt are all eonjugate in Sn i Sm. Let us ealeulate the degree ofvt. This transformation stabilizes the (n -2)· nm-1 funetions from F whose values at the point k are distinet from i and j, and it permutes all other elements with orbits of length 2. Thus m (v) = 2· n m-1. Sinee any other element of the group moves more than 2· nm-1 points, eaeh subgroup whieh is sirnilar to <v> eoineides with <vt> for suitable i,j E N and k E M. It follows from the deseription of elementwise stabilizers of pairs of points that, for n ~ 4, they eontain the direet produet of the symmetrie groups S (Ni- {ai,bi }), where ai and bi are the values of the funetions !I and h at the point v, respeetively. For n ~ 4 the eardinality of eaeh set Ni- {ai,bd is at most 2. This implies that the group S (N) i S (M)(j1,t,) eontains the involution v~ß where

{ a, ß} n { ak, bd = 0 and k is arbitrary. D

For a pair of partial funetions f,g Eu Mi N[kl, write f > g (f eontains g) if k

d (f) ~ d (g) and f (x) = g (x) for all x from d (g ). Let us define the symmetrie antireflexive relation e1 = {(f,g) E (Mi N(ll)2 I f > h, g > h for some h E Mi N(l-1]}, an adjaeeney relation on the set M i N[l-ll. Let us also eonsider the relation <1>1 = {(f,g) E (Mi N(ll)2 I h > f, h > g for some h E Mi N[1+1l }. (Note that <1>1 is empty for l = m). For l = 1 weshall assume that e1 is the eomplete antireflexive relation. lt is clear that for [ < m, (f,g) E <!>1 implies (f,g) E 8z.

Theorem 4.3. The automorphism group of the relation e1 on Mi N[I] eoineides with S (N) iz S (M).

Proof. Let be a flag of the relation e1 (i.e. a maximal subset of pairwise adjaeent elementsfromM i N[ll). Apermutation n of the set Mi N[l] whieh preserves e1 maps onto some other flag. It is clear that eoineides with one of the sets !!.g = {/ E Mi N[ll I f > g} where g is a fixed element from Mi N(l-11 and M = {f E M i N[I] I f < g} where g E M i N[1+1l. If l = m then the set M is empty. Otherwise eaeh pair of elements from !!.g is in the relation <1>1, while !!.h does not possess this property. This implies that the transformations n and n:-1 induee permutations on the set L = {!!.g I g E Mi N[l-1] }. But g" = h for a permutation n whieh maps !!.g onto !!.h.

In this way we obtain a faithful aetion of the group Aut e1 on the set Mi N[I-1]. The permutation n from Aut ez aeting on the set Mi N(I-IJ will be denoted by ft:. It is clear that ft: preserves the relations e1_ 1 and <1>1_ 1. In faet, let s(g,h) = j{(x,y) I x E Ag, y E Ah,

(x,y) E ez} I Then s(g,h) > s(g',h') if (g,h) E e/-1 and (g',h') (!!: el-1· The permutation

162 V. A. USTIMENKO

it preserves the relation cj>1_ 1 as well. Indeed, if l = m then cj>1 = 0, and if I # m then .1g n .1h is nonempty if and only if (g,h) E cj>1_ 1 . By continuation of this procedure, we associate with the permutation 1t a bijective transformation it of the set M i N[ 1l preserving the re1ation cj> 1 . Then it preserves also the binary re1ation cJ>t =(Mi N[1l)2 - cj> 1 , i.e. the comp1ement of cj> 1 . But cJ>t is an equiva1ence relation: (j ,g) E cj>f ~fand g have the same domain of definition. This means that Aut cj>f is isomorphic toS (N) i S (M). 0

Recall from Proposition 4.1 that R denotes a certain decomposition of F x F into

invariant b1ocks of the group Sn i Sm. So AutR ;;;;;! Sn i Sm·

Proposition 4.4. If IN I~ 3, IM I~ 2, then the group AutR coincides with S(N) i S(M).

Proof. By Theorem 4.3 the group Aut 'P m-1 coincides with the exponentiation of symmetric groups. If m = 2 then the re1ations 'Pm _1 and 'P 1 coincide and the lemma is true. Suppose m > 2. For a fixed edge (a, b) of the graph 'P; let us consider the parameter

A; = I {f E Nm I (f,a) E 'Po, (b,f) E 'Po} I It is clear that A; does not depend on the

choice of edge (a,b) in 'P;. Suppose that the group AutR acts transitively on 'Pm-1 u 'P1. Since 'Po is invariant under R, the parameters A,1 and A,m_1 should coincide. But Am-1 = (n- 2)m-1 • (n -1) while 1..1 = (n -1)m-1 (n- 2)m-1, a contradiction. Hence AutR preserves 'P m-1 and is contained inS (N) i S (M). 0

Let r 1 be the set of points fixed by the involution V = V b and r 2 = F - r 1 where F =NM. Let 't denote the partition of F into orbits of the subgroup <v> x S(N 1 - ( 1,2}) of S(N 1), and let 1:1 and 1:2 be the restrictions of 't to r 1 and r 2, respectively. Put

G =Sn i Sm.

Proposition 4.5. (a) The group Ce (v) is isomorphic to (S 2 x Sn-2) x (Sn i Sm_1 );

(b) Cc(v) stabilizes the partition 1:;

(c) Cc(v)~ = S2 xSn-2; moreover Cc(v)<r1 ) =Sn_2 and Cc(v)<rz) = S 2;

(d) the group Cc(v) acts faithfully Oll the blocks r1 and r2 and permutes the sets r1/'t1 and r2/'t2 similar to Sn i Sm_1.

Proof. It is clear that the direct product of the groups S (N 1 - ( 1, 2}) x S ( ( 1,2}) and Sn i Sm-1 commutes with the involution v = vh and possesses properties (b)-(d). On the other hand, involutions of the form vt fom1 a conjugacy class and the number of elements conjugate to v is m · C~. This implies that the groups Cc(v) and (S 2 X Sn-2) x (Sn i Sm-1) coincide. 0

Let (X,F) be an overgroup of G different from S (F) and A (F).

Lemma 4.6. The permutation group X (r1 1 stabilizes the partition 't, and the groups

X (rJl and X (r2 ) stabilize elementwise the partitions 1:2 and 1:1, respectively.

Proof. If n > 1 then the number I r 2 1 = (n -2)· nm-1 is greater than IF 1/2. By

the Marggraf theorem (Theorem 2.1(a)) this implies that X (r1) is intransitive. Since X (r2)

is intransitive and nontrivial (it contains <v> ), the group x<rz) is imprimitive. Suppose

that the group (X (r2), r1) is transitive. Then (X (r2 J, r 1) is primitive by Theorem 2.1(a), a contradiction. The groups x<r2 ) and x<rt> contain subgroups G(r,) and G(r2) which are

similar to (Sn i Sm_1) x S 2 and (Sn i Sm_1) x Sn-2• respectively. The group Xis intransitive and contains G(r2 ) which is isomorphic to S2. Let ~2 be the partition of the set r 2

into orbits of the intransitive group X (r2 ). Since X (r,) contains G (r,), ~2 is coarser than

t2. In addition ~2 is an imprimitivity system of x<r2). Hence the family ~2 is invariant under G (r2 ) as a whole. But on r 2 there is no partition into imprimitivity blocks of G (r2 )

which is coarser than -r2 . Hence ~2 = -r2. In an analogaus way it can be shown that the partition ~1 of the set r 1 into orbits of X (r2) coincides with -r1. o

Lemma 4.7. Let H be a subgroup of the table wreath product of the symmetric group S (M) (the alternating group A (M) for IM I :2:: 3) with a finite nonabelian simple group G. Suppose that Gis invariant (as a module) under A (M) and that G contains the diagonal diag GM of the base. Then either H = diag GM or H =GM.

Proof If IM I = 2 then a proper subdirect product in G x G is isomorphic to G and has the form K .p = { (~(h),h) I h E G} where ~ is an automorphism of the group G. It is clear that only the group Ke (e is the identity automorphism) is invariant under the group S (M) = S 2 . For IM I = 3 a proper subdirect product has either the form L.p = ((h,k,~(k)) I h,k E G} or the form L.p'l' = {(h,~(h),'Jf(h)) I h E G} where ~ and 'Jf

are automorphisms of G. But only Lee is invariant under A (M). The setwise stabilizer of a subset N c M in A (M) coincides with the even product of the symmetric groups S (N)

- -and S(N) (i.e. with (S(N)xS(N)) n A(M)). Let PN denote the group of tables of the form (e,h(x)) where h(x) = e for x e: N. Let us apply induction on IM I = k > 3. The groups HN = H n PN and Hjj = H n Pjj are invaria~t under § (N) and S(N). By the induction hypothesis HN E {GN, diagGN], Hjj E {GN, diagGN}. It is easy to see that only two subdirect products of HN and H'N are invariant under A (M), namely PN x Pjj and diag GM. 0

Let us consider the partition -r' of the set F =NM into orbits of the group S (NdSince 't is the set of orbits of the subgroup Sn-2 x S 2 of S (N 1 ), the partition -r' is coarser than 't (-r' > -r).

Theorem 4.8. The permutation group Sn i Sm is maximal for n > 4.

Proof Weshall use induction on m. If m = 1 then the claim of the theorem is obvious. As above, Iet (X,F) be an overgroup of Sn i Sm distinct from S (F) and A (F). We assume that the inductive hypothesis is true. Then the following intermediate propositions (Propositions 4.9-4.12) hold.

Proposition 4.9. The permutation group (X~', F !t') is similar to Sn i Sm_1.

Proof If m = 2 then the group (G~', F 1-r') coincides with Sn and the claim is trivial. Let m :2:: 3 and suppose that the proposition does not hold. Since the group Sn i s;" permutes the equivalence classes of -r' similar to Sn i Sm_1, by the induction hypothesis (X~, F l't) contains the alternating group Am_1• The group X~ contains the direct product

164 V. A. USTIMENKO

Sn x Am_1• But in this case the elementwise stabilizer of the points fixed by the centralizer in X of a transposition S (N 2) does not preserve the 2-element orbits, a contradiction to Lemma 4.6. 0

Proposition 4.10. The group X~, coincides with S (N 1 ).

Proof. Let us consider the equivalence relation n = {(fl ,/2) I h (2) = h (2)). It is clear that n is coarser than 't1

• The stabilizer of the partition n in G is similar to

Sn x (Sn x Sm_1 ). Let B be a class of n. Then the restriction of the group G" to this block is similar to Sn I Sm_1. Let us show that the restrictions of the groups X" and G 11 on the block B coincide. In fact, form= 2 the group G 111s coincides with S(B) and the claim is true. Suppose that m > 2 and that the claim is not true. Then, by the induction assumptions in Theorem 4.8, the group X 111s contains the altemating group Anm-1. But in this case (X~, F /'t') is not contained in Sn I Sm_1, a contradiction to Proposition 4.9.

So X l!IB =Sn t Sm-1 and L = x~, IB =(X ll)t'IB =Sn. Moreover, L has exactly nm-2

orbits of length n and the action of L on any of these orbits is similar to Sn. Let L 1 denote the subgroup of L isomorphic to An. Let us consider the group S = Nx· (X,,). The action of this group on the equivalence classes of n is similar to Sn (in fact S n G itself permutes blocks in a manner similar to Sn). This implies that S is contained in the wreath product of the symmetric group Sn and L, and in addition X t; contains diag Ln. By

Lemma 4.7 the group X~' n L 1 coincides either with diagL) or with Lj. In the latter case the group X (F -B) > L 1 cF- { e) is a nontrivial normal subgroup of the primitive group xCB) and it acts transitively on the set B. This is a contradiction to the Marggraf theorem. HenceX~, n Lj '=An andX~, =Sn. 0

Proposition 4.11. The permutation group (X r12 J, r 1h) is similar to Sn I Sm_1.

Proof. Let us consider the action of the group X (11 l on the elements of the sets

r1 l't1 and r2/'t2. By Proposition 4.10, X~, is contained in G. So the group X, is also contained in G and is isomorphic to S 2 x Sn_2 . This implies that X c1 Jl and X c12) are iso

morphic to S 2 and Sn-2• respectively. Let X 1 and X 2 denote the groups (X {11 l, r21't2)

and (X {12 ), r 1/'t1). By consideration of the action of X {1d on the sets r 1 and r 2, we

obtain

(4.1)

Thus X (r1 J induces a diagonal action on r 1/'t1 and r 2/'t2. This means that X rr1 l stabil

izes 't'. Suppose that the group (Xrrd,r11't1) is not contained in Sn I Sm_1. Then the

groups (X (f'd, r1 h1) and (Xf!Jl, F /'t') contain A;;'-1, a contradiction to Proposition 4.5P

By Proposition 4.11 and relation 4.1 we have the following.

Proposition 4.12. The permutation group X rr1 l is contained in G.

Corollary 4.13. The centralizer of the involution v in the group Xis contained in G.

In fact, the centralizer of the involution v permutes those orbits of the subgroup <v> which have the same size. In particular it stabilizes the set r 1 of fixed points of this involution.

Now, to conclude the proof of Theorem 4.8, it is sufficient to notice that Corollary 4.13 enables us to apply Proposition 3.1 to the 2-compact 2-subgroup <v> (cf. Proposition 4.2) and to the 2-distinguished partition R (cf. Proposition 4.1). By Corollary 3.6 the partition R is 2-distinguished in X as well and, since AutR c G by Proposition 4.4, we conclude thatX is contained in Sn i Sm. D

Remark. The Bochert-Manning bounds which are usually used in proofs of maximality of permutation groups when such proofs do not appeal to the classification of finite simple groups, arenot used in the proofs of Theorems 3.4 and 4.8.

The author is highly indebted to Dr. I.D. Suprunenko, who has read the preliminary version of the manuscript and has made a number of useful remarks.

References

1. R.A. Bairamov, On the completeness problern in the symmetric semigroup. In "Discrete analysis", Math. Irrst. No. 8, Novosibirsk, (1966), 3-26 [In Russian].

2. R.W. Ball, Maximal subgroups of symmetric groups, Trans. Amer. Math. Soc., 121 (1966), 398-407.

3. P. Bhattacharya, On groups containing the projective special linear group, Arch. Math. 37 (1981), 295-299.

4. LA. FaradZev, M.H. Klirr, M.E. Muzichuk, Cellular rings and groups of automorphisms of graphs [In this volume].

5. F. Harary, Exponentiation of permutation groups, Amer. Math. Monthly, 66 (1959), 572-575.

6. C. Jordan, Traite des Substitutions des equations algebriques, Paris, GauthierVillars (1870).

7. W. Kantor, Jordan groups, J. of Algebra 12 (1969), 471-493.

8. W. Kantor & T. McDonouch, The maximality of PSLn(q), n ~ 3, J. London Math. Soc (2), 8 (1974), 426.

9. M.H. Klirr, On an infinite family of maximal subgroups of the symmetric groups. Trudi Nikolaevskogo Korablestroitelnogo Instituta, 44 (1970), 148-151 [In Russian].

10. M.H. Klirr, Investigation of the algebras of invariant relations for certain classes of permutation groups, Ph. D. Thesis, Nikolaev, 1974 [In Russian].

166 V. A. USTlMENKO

11. M.W. Liebeck, On the orders of maximal subgroups of the finite classical groups, Proc. London Math. Soc. (3), 50 (1985), 426-446.

12. M.W. Liebeck, C.E. Praeger & J. Saxl, A classification of the maximal subgroups of the alternating and symmetric groups, J. of Algebra, 111 (1987), 365-383.

13. R. List, On permutation groups containing PSL(n,q) as a subgroup, Geom. Dedic. 4 (1975), 373-375.

14. P.M. Neumann, Transitive permutation groups of prime degree !I. A problern of Noborulto, Bull. London Math. Soc. 4 (1972), 337-339.

15. P.M. Neumann, Transitivepermutation groups of prime degree N. A problern of Mathieu and theorem of Ito. Proc. London Math. Soc. (3),32 (1976), 52-62.

16. B.A. Pogorelov, On maximal subgroups of the symmetric groups defined on projective spaces over finitefields. Mat. Zametki 16 (1974), 91-100 [In Russian].

17. V.A. Ustimenko, Induced maximal permutation groups. Ph. D. Thesis, Kiev (1979) [In Russian].

18. V.A. Ustimenko, Exponentiations of symmetric groups are maximal permutation groups. In "Topics in Group Theory and Homological Algebra", Jaroslavl' (1983), 19-33 [In Russian].

19. H. Wielandt, Finite Permutation Groups, Acad. Press, London (1964).

AMORPHIC CELLULAR RINGS

Ja. Ju. Gol'fand, A.V. Ivanov, M.H. Klin

1. lntroduction

The enurneration problern for cellular subrings of a given cellular ring has found nurnerous applications in cornbinatorics and graph theory (see for exarnple [17], [22]). Usually one uses a cornputer to solve this problern. A general idea underlying searching algorithrns for the enurneration of subrings was proposed in [19] and consists of the following.

Let W be a given cellular ring with the set Q = {Qo,Ql, ... , Qr-1} of basis graphs. An elernent B = Q; 1 + Q; 2 + · · · + Q;, of W is said to be "good" if

s B 2 = A. L, Q;i + L, ök Qk. By testing all subsets of Q, we choose those subsets

j=l k*ii corresponding to "good" elernents. After that, we construct all partitions of the set Q with the property that each part of each partition is a "good" elernent. Finally, we check whether such a partition determines a cellular subring of W. The algorithrns for subring enurneration which realize this rnethod [9], [17] are quite suitable frorn the practical point of view. This is due to the fact that in the rnajority of the cases considered the nurnber of "good" elernents turnsouttobe less than 27 • Nevertheless, the sphere of application for this algorithrn is restricted to permutation groups of rank 20-30.

A principled role in the theoretical estirnation of the cornplexity of algorithrns for subring enurneration is played by the existence problern for those cellular rings, in which every partition of the set of basis elernents gives rise to a cellular subring. The cellular rings which possess this property will be called arnorphic. In Section 3 it will be proved that there exist infinitely rnany arnorphic cellular rings on n2 points which can be constructed frorn an affine plane of order n. In what follows, the arnorphic cellular rings which can be constructed frorn an affine plane of order n will be called affine cellular rings. Sorne exarnples of arnorphic cellular rings which are not affine will also be given.

The characterization problern for arnorphic cellular rings is, in our opinion, of significant interest. Since each subring of an arnorphic cellular ring is also arnorphic, a first step in the solution to this problern is the study of arnorphic cellular rings of srnall rank.

A cellular ring of rank 3 is surely arnorphic. So the first nontrivial case consists of the rings of rank 4. It is easy to prove that the basis graphs of an arnorphic cellular ring whose rank is at least four, are strongly regular graphs.

It is shown in Section 4 that a cellular ring of rank 4 can exist only on n 2 points. For each such ring, the pararneters of each basis graph (as a strongly regular graph) are found. The pararneters belong to two series of pararneters Lg(n) and NLg(n) (see

167

168 JA. JU. GOL'FAND ET AL

definitions in [15]). An ana1ogous result is true for amorphic rings of higher rank.

In section 5, the results of constructive enumeration for all amorphic cellular rings

ofrank 4 on 4,9,16 and 25 vertices are presented.

The contributions of the authors of this paper were not uniform. All results in Section 4 were obtained by A.V. Ivanov. He also carried out the constructive enumeration of cellular rings. For this reason the original Russian version of the paper was divided into two parts with different lists of authors. Since these two parts were considered by us to represent a single work from the outset, and since this manuscript was prepared jointly, both parts have been united in the present translation into English.

2. Preliminary information and results

The notion of cellular ring is weil known (see for example [11]). The cellular ring

W whose basis graphs are Qo, ... , Q,_1 is denoted by < Qo, ... , Q,_1 >. This ring is said to have rank r. A cellular ring is called a cell if all its basis graphs are regular, i.e. if

all vertices of the graph Qi have the same valency ki for i = 0 , ... , r- 1. One of the basis graphs of a cell is the complete reflexive graph, i.e. the graph consisting of all loops. We will assume that this reflexive graph has the smallest index. All other basis graphs of a cell will be called nontriviaL A cellular ring is called primitive if all its antireflexive basis graphs are connected. If there is a disconnected basis graph then the ring is said to be imprimitive.

Let W = < Q o, ... , Q,_1 >, W' = < Q o', ... , Qs-1 ' > be cellular rings defined on the same set of elements. The ring W' is said tobe a cellular subring of W if Q/ E W for j = 0 , ... , s - 1. This means that

r-1 s-1

Q/ = L O.ji • Qi, where a.Ji E {0, 1}, L O.Ji = 1. (2.1) i=O j=O

Every collection Qo', ... , Qs_1' which sarisfies the conditions (2.1) will be called a partition of the basis elements of W. If W is a cell we will always assume that Q0 ' = Q0 . In what follows we consider only those partitions which possess this property. It is clear that a partition of W determines a cellular subring only if the product of any two elements in the partition is a linear combination of the elements in the partition.

Let r be a simple (undirected) regular graph of valency k with v vertices. The graph r is called strongly regular (abbreviated s.r.g) if the ends of an edge have exactly /.. common neighbours and the ends of a non-edge have exactly 11 common neighbours. Set l = v- k- 1. The following necessary conditions for the existence of a s.r.g. are well known (see, for example [15]):

k· (k-/..-1)=1· 11 (2.2)

a) d = ('A-11)2 -4· (k-11) is aperfect square and

(2· k + (A.-11) (v -1)) I -{I has the same parity as v -1 , or

b) k = l, 11 = A. + 1 = k I 2.

I69

(2.3)

If [' is a graph then [' denotes its complementary graph. Notice that [' is a s.r.g if and only if [' is a s.r.g. It is easy to see that if [' is a s.r.g. then [' and [' tagether with the complete reflexive graph generate a cell of rank 3.

Let r = (V,E) be a simple graph with vertex set V and edge set E. A collection { ri = (v',Ej), i = 1 , ... , k} of graphs with vertex set V such that {E 1 ,E 2 , .•. , Ekl is a partition of the set E, is said tobe a factorization of the graph 1.

Let Kn denote the complete graph on n vertices and m o Kn be a disjoint union of m

copies of the graph Kn. It is weil known that the disconnected strongly regular graphs are just the graphs m o Kn.

A cellular ring W = < Q o ,Q 1 , ... , Q,_1 > is called amorphic if each partition Q o' ,Q 1', ... , Qs-1' of the set {Qo ,Q 1 , ... , Qr-1} is the set of basis graphs of a cellular subring W' of W. Notice that each subring of an amorphic cellular ring is also amorphic.

Recall that a simple graph is called strong if the ends of each edge have A. common neighbours and the ends of each non-edge have 11 common neighbours. Every strongly regular graph is strong. Let [' = K 1 $ m o Kn be the union (in the sense of Zykov; see also [13]) of the graphs K 1 and m o Kn· It is easy to see that [' is strong but not strongly regular. It is shown in [18] that a connected strong graph is either isomorphic to K 1 $ m o Kn or is strongly regular. A strong graph can be a basis graph of a cellular ring. Nevertheless, it will be shown below that the basis graphs of all nontrivial amorphic cellular rings are always strongly regular.

Lemma 2.1. Let W = < Qo ,Q 1 ,Q 2 > be a cellular ring of rank 3. Then W is a cell and only one of its basis graph is reflexive.

Proof. At least one of the three basic graphs is antireflexive. Suppose that the

other two, say Q0 and QI are reflexive. Then Q2 is the complete graph on v vertices

and Q0 · Q2 is a proper directed subgraph of the graph Q2. Hence Q0 · Q2 rf_ W, a

contradiction to the definition of cellular ring. So Q1 and Q2 are antireflexive graphs.

Suppose that Q1 is nonregular. Then Q1 • Q1 1 rf_ W since the loops at distinct vertio~s have distinct multiplicities. Here c-1 denotes the graph obtained from G by reversing of the orientation of the arcs. 0

Lemma 2.2. Let < Q o ,Q 1 ,Q 2 > be a cellular ring of rank 3 with Q 1, Q 2 simple graphs. Then Q 1 and Q 2 are strongly regular.

Proof. By the definition of c:ellular ring we have Qf = k Q0 + i\Q1 + JLQ2 where

k. 1\, fL are appropriate integers. Sinc:e Q1 is a simple graph and Q2 = Q1 we conclude that Q1 and Q2 are strongly regular. 0

170 JA. JU. GOL'FAND ET AL.

Theorem 2.3. Let W be an amorphic cellular ring of rank r;;::: 4. Then it is a commutative cell, all of whose basis graphs are strongly regular.

Proof. Suppose that among the basis graphs of W there is a direc:ted graph, say Q2, and Iet Q2 = Q!1. Sinc:e T ~ 4 the graph QI + Q2 is nonempty and henc:e W' = <Qo, Q1, Q2, Q1 + Q2> is an amorphic: c:ellular subring of W and W" = <Q0 , Q1 + Q1 + Q2, Q2> is a c:ellular subring of W'. Butthering W" does not satisfy the sec:ond axiom of c:ellular rings, sinc:e Q21 = Q1 is not a basis graph of W". This means that all antireflexive basis graphs of W are simple graphs. Henc:e W is commutative. Now Lemma 2.1 implies that W is a c:ell and Lemma 2.2 implies that all the basis graphs are strongly regular. 0

In what follows, the term amorphic cellular ring will stand for a nontrivial amorphic ring, i.e. one of rank r ;;::: 4.

Corollary. A nontrivial amorphic cellular ring is a commutative cell, all of whose basis graphs are strongly regular.

Thus a necessary condition for the existence of an amorphic cellular ring of rank r on v points is the existence of a factorization of the complete graph Kv into r - 1 strongly regular graphs. It is clear that this condition is sufficient for r = 3. Sufficiency can also be proved for r = 4. ./ '! /' •

,___,

• '_x.-.2 • ---;~

a,

Figure 1

-·~ ·~~s 5 •

Figure 2 For r;;::: 6 this condition is not generally sufficient. In Fig. 1 a factorization of the

complete graph K 6 into 5 s.r.g's of valency 1 is presented. This factorization does not lead to a cellular ring. For instance the graph Q 1 • Q 3 (see Fig. 2) does not belong to the linear span of the graphs Q o , Q 1 - Q 5. Notice that all factorizations of the graph K 6 are isomorphic (see [7]).

3. The affine cellular rings

Let W = < Q o , Q 1 , ... , Qr_1 > be a cell where Q o is the reflexive graph. Let us give a sufficient condition for W to be amorphic.

Lemma 3.1. Suppose that multiplication of the basis elements of the cell W = < Q o , Q 1 , ... , Qr -1 > is determined by the following formulas

AMORPHIC CELLULAR RINGS 171

(3.1)

Then W is an amorphic cellular ring.

Proof. Let

R*={1,2, ... ,r-1},l,JcR*,IIlf=0, III=s, IJI=t,X=R*\X. Set A = L, Q;, B = L, Qj. Using formula (3.1), it is easy to calculate that

iel jel

A2 =(8+(s-l)(s-2)o+2(s-l)p)A +s(s-l)o L, Qk+seQo kel

A·B =t((s-l)o+p)A +s((t-l)o+p)B +sto L, Qk. ke Iul

(3.2)

This implies immediately that every partition of the set R* determines a cellular subring ~w o

Recall the following well-known definition.

Definition. A set P = { P 1 , ••. , P n} of points and a collection L of subsets of P is called an affine plane of order n if

1) for any two points there is exactly one line passing through these points;

2) for a point which does not lie on a line l there is exactly one line m passing through this point such that m 11 l = 0;

3) on each line there are exact1y n points.

The set L of lines of an affine plane can be divided into n + 1 parallel dasses with n lines in each dass. With a parallel dass Lk we associate the graph Bk with vertex set P. The vertices P; and Pj will be adjacent in Bk if the line passing through P; and Pj

belongs to the dass Lk. It is dear that the graph Bk is isomorphic to the graph n o Kn.

Lemma 3.2. Let B 1 , ... , Bn+1 be the set of graphs which are associated with the parallel dasses of an affine plane of order n and let B 0 be the reflexive graph. Then is a cell of rank n + 2.

Proof. Each graph Bk is regular of valency n - 1. It follows direc:tly from the axiorns of an affine plane that

{(n-2)B;+(n-1)B 0 , ifi=j;

B;Bj= ""B 'f·. ..... k 1 l "# J. kE i,j

(3.3)

0

Theorem 3.3. Starting with an affine plane of order n an amorphic cellular ring of rank n + 2 can be constructed.

The graphs B 0 ,B 1 , ... , Bn+1 in Lemma 3.2 determine a cellular ring. Now it fol-lows from (3.3) and Lemma 3.1 that this ring is amorphic. 0

In what follows, the cellular ring constructed from an affine plane of order n will be called the complete affine ring of order n and each of its subring will be called an affine ring.

A direct definition of the complete affine ring can be given as a factorization of the complete graph Kn2 into n + 1 graphs of the form n o Kn. It is clear that the existence of an affine plane of order n is equivalent to the existence of a complete affine ring of order n.

An infinite class of affine rings of rank 5 is presented in the book by H. Wielandt [27] as examples of non-Schurian rings. Nevertheless, as far as we know the general definition of this construction did not appear in the Iiterature 1).

Let G be a basis graph of an affine ring, G =Bi, + Bi2 + · · · +Bi,. Then it follows

from (3.2) and (3.3) that

G 2 =(n-2+(g-1) (g-2)) G +g(g-l)G +g(n-l)Bo,

i.e. Gis a s.r.g. with the parameters

v=n 2 ,k=g(n-l),A.=n-2+(g-l)(g-2),J..l=g(g-1). (3.4)

This s.r.g. is usually called the graph ofLatin squares and is denoted by Lg(n). This name is motivated by the following. The graph Lg(n) can be constructed from a system of g - 2 pairwise orthogonal Latin squares of order n. The vertices of the graph are the items of a square. Two items are adjacent if they are in the same row, in the same column or the symbols in these items coincide. The above definition of the graph Lg (n) is more combinatorial. The existence of Lg(n) is equivalent to the existence of a system of g

graphs of the form n o Kn with the same vertex set and pairwise disjoint edge sets. Such a system of graphs is said tobe a net, i.e. a particular case of a partial geometry [4].

A s.r.g. with the parameters (3.4) is called a pseudolatin graph. R. Bruck has proved in [5] that for n > T (g -1) (g 3 - g 2 + g + 2), a pseudolatin graph is a graph of

Latin squares. If g = 2 this bound is strict since there are two nonisomorphic s.r.g .' s with the parameters v = 16 , k = 6, A. = 2 , J..l = 2. One of these two graphs is not a graph of type L4(2) (see [23]). For g = 3 the cases n = 5 [24] and n = 6 [6] were studied. Here numerous examples of pseudolatin graphs, which are not graphs of Latin squares, are

1) When the present paper had already been prepared, the authors had learned that Lemma 3.2 (in slightly different terms) is given in the paper: Hayden John L., Association algebras for finite projective planes, J. Combin. Theory, 1983, A34, 360-374. Notice that the notion of amorphic ring is not considered in this paper.

constructed.

The characterization problern for the pseudolatin graphs can be generalized.

Definition 3.4. Let W = < Q0 ,Q 1 , ..• , Q,_1 > be a cell on n2 points with the following multiplication table

Qr = <<n- 2) + (gi -1) (gi- 2)) Qi + gi(gi -1) I, Qj + gi(n -1) Q o , j~i

i=1,2, ... ,r-l,

Qi • Qj = gj(gi -1) Qi + gi(gj --1) Qj + gi gj I. Qk hi,j

(3.5)

where 0 < gi ~ n, g 1 + g 2 + · · · + g,_1 = n + 1 and Qo is the reflexive graph. Then W

will be called the pseudoaffine cellular ring of rank r and order n with the parameters

(n; g 1 ,g2, ... , g,_!).

It is clear that each pseudoaffine ring is amorphic. Each affine ring is pseudoaffine. The converse Statement is not generally true. Some examples of pseudoaffine rings of rank 3 which are not affine ones are indicated above. In [ 16] an example of a pseudoaffine ring ofrank 5 with parameters (12,1,1,5,6) which is not affine is presented. In this paper a procedure is given for the construction of an infinite family of pseudoaffine cellular rings of rank 5 from Hadamard matrices of order 4m. Some additional examples of pseudoaffine rings are given in [10].

It follows from a result due to R. Bruck that the pseudoaffine rings, the valencies of whose basis graphs are bounded by a constant, are affine if n is sufficiently large. The existence problern for pseudoaffine, nonaffine cellular rings with parameters (n ; g 1 ,g 2 , .•. , g,_1) has, in many cases, independent interest. In particular the situation when r is large, and gi ~ 3 for all 1 ~ i ~ r- 1, is of interest.

4. The parameters of amorphic cellular rings

Let W = < Qo ,Q 1 ,Q2 ,Q3 > be an amorphic cellofrank 4 and of order v. By the corollary in Section 2, the basis graphs Qi, i = 1,2,3, are s.r.g.'s. Let (ki,Ai,!li) and (ri,Si)

derrote the parameters of the s.r.g. Qi and the eigenvalues of the adjacency matrix Ai of Qi, respectively Cri > 0, Si< 0). Since W is a commutative cellular ring, the matrices A o ,A 1 ,A 2 ,A 3 are pairwise commuting. It is known (see [8], [20]) that a set of pairwise commuting mat:ices can be simultaneously diagonalized by a certain orthogonal matrix P. Let /i =PA, P, i = 1,2,3. We can assume that the first element on the diagonal of the matrix Ii coincides with ki. Here ki is the eigenvalue of Ai with multiplicity 1. Let p{ derrote the j-th diagonal element of the matrix /i,i=1,2,3, j=2,3, ... ,v. Then p{ E {r1,si }. The following relations hold:

174

3 L ki =v -1, i=l

3 . L p{=-1, 2~j~v. i=l

JA. JU. GOL'FAND ET AL

(4.1)

(4.2)

In fact, the matrix A 0 has a unique eigenva1ue 1 with multiplicity v and the all-ones

matrix A 0 + A 1 + A 2 + A 3 of order v has the eigenvalue v with multiplicity 1 and the eigenvalue 0 with multiplicity v - 1.

Lemma 4.1. The eigenvalues Ti,Si of the graph Qi are integers for i = 1,2,3.

Proof It is well-known that a s.r.g. has a noninteger spec:trum only if k =

l = 2 · {L = 2 · (>. + 1). It follows from the relations (4.2) that if some of the eigenvalues

Ti, Si arenot integers, then at least two of the three graphs have a noninteger spectrum.

Now ( 4.1) implies that the third graph has valency 0. This c:ontradic:tion proves tlw Iemma. 0

In each relation from (4.2) three variables p{ are involved. The variable p{ is equal

either to Ti or s;. So each of the relations is reduced to one of 23 possible equalities which

tie tagether the eigenvalues of the graph Qi. For the cellular ring W, some of these 8 equalities will appear more than once (if v - 1 > 8). Let us choose all distinct equalities from (4.2) and denote by S the obtained system of equalities.

Lemma 4.2. The system S sarisfies the following conditions:

a) each of the numbers Ti ,si, 1 ~ i ~ 3, is involved in at least one equality;

b) neither of the equalities contains three eigenvalues of the same sign;

c) every two equalities contain in all at least 5 numbers from the set

B = {rJ,T2,r3,SJ,S2,s3l·

Proof. Eac:h of the numbers c:ontained in B appears in the diagonalized form

of the matric:es Ai and hence in some of the relations in (4.2) andin S. The claim h)

holds sinc:e Ti, si are integers. It is c:lear that an equality from S contains in all at least

4 numhers from B. So if the c:laim c) fails, then there are two equalities whic:h diffpr

hy only one item in the left side. This implies immediately that Ti = s, for some i, a

c:ont.radic:tion to t.he inequalities Ti > 0, si < 0. 0

Lemma 4.3. Up to the ordering of the graphs in { Qi), only the following three systems satisfy the claims in Lemma 4.3:

{T 1 +s 2 +s 3 =-1

s 1 +T 2 +r3 =-1 (4.3)


{r 1 +r2+s3 =-1

r1+s2+r3=-l

s 1 +r2 +r3 =-1

{sl+s2+r3=-1

s1+r2+s3=-l

r 1 +s2+s3=-l.

175

(4.4)

(4.5)

Proof. Using the definition of S and the conditions a) - c), we associate with each system of rn equalities, an (m x 3)-matrix with entries from {1, -1}, whic:h satisfieR the following conditions. All rows of the matrix are distinct; in each row and in each column there are numbers of different signs; every two rows differ in at leaRt two c:olumnR. Let us c:all two such matrices equivalent if one c:an be obtained from the other by permutation of the rows and/or columns. Now the proof of the lemma can be reducecl to a small enumeration of such matrices up to equivalence. ancl a Rnbsequent exc:hange of all 1 's in the i-th c:olumn of eac:h matrix by T; and -1 's by s;. 0

Let fi (respectively g;) denote the multiplicity of the eigenvalue r; (respectively s;). It is easy to see that the multiplicity of an eigenvalue is equal to the number of times it appears in equations ( 4.2). This simple remark enables us to obtain some relations on the multiplicity from the system S. In particular we have the following.

Corollary 4.4. For the system ( 4.3) the equalities h = h and g 2 = g 3 hold.

In fact r 2 and r3 (and also s 2 and s 3) are involved in the same equality in (4.3) and are not involved in any other case. Hence they are involved in the same number of equalities in (4.2). o

Let us now proceed to the study of the structure constants Ylj of the ring W. First let us write down the well-known relations (see [26]) which hold for a commutative cell

Yfj = y);, i,j,l E {0, 1,2,3} ,

3 L ij = k;, i,l = o, 1,2,3, j=O

kl·Ylj=k;·y}j, i,j,IE {0,1,2,3}.

Let us express the structure constants of the ring W in terms of the s .r.g. Q;:

y{;=!!;, i:t:.j, i,jE {1,2,3}.

(4.6)

(4.7)

(4.8)

In view of (4.6) and (4.7) let us set yl.z = a., Y~. 3 = ~. YL = y. We write down the values of the structure constants in Table 1. Here the value yfj is contained in the /-th row of the (i,j)-th entry of the square. We use known relations for s.r.g.'s (see for example [14])


k + rs = ~. A. = r + s + ~. k- A.-1 = -(r + 1) · (s + 1). Applying again the relation (4.7) for (i,l) = (3,1), (1,2), (2,3), we find the expressions for a,ß,y in terrns of the eigenvalues

ri ,si of the adjacency matrices of the graphs:

2a = k 1 - A.1 - 1 + k2- ~2- (k3- ~3) =- (r1 + 1) (s 1 + 1)- r2 s2 + r3 s3 ,

2ß =- (k 1 - ~~) + k2 - A.2 - 1 + k3 - ~3 = r 1 s 1 - (r2 + 1) (s 2 + 1)- r3 s 3 , (4.9)

2y= k 1 - ~~ - (kz- ~z) + k3 - A.3 - 1 =- r 1 s 1 + rz Sz - (r3 + 1) (s3 + 1).

Table 1

\I 0 I 2 3

I 0 0 0

0 0 I 0 0 0 0 I 0 0 0 0 I

0 k, 0 0

I I "'· a; k,-11.,-1-a:

0 l1t k,-11.,-1-fl k,-11,-fl 0 l1t k,-11,-Y y

0 0 k, 0

2 0 a; t ko~11•-a; I k,-11..-l-fl fl 0 k,-11,-Y 11• k,-11.,-1-y

0 0 0 k,

3 0 k,-11.,-1-a: k•-11•-a: 11a 0 k,-11.-fl fl 11a I y k,-11.,-1-y "'·

Application ofrelation (4.8) for (i,j,l) = (1,3,2), (1,2,3) gives

(k3 -~3 -ß)kz = (kz -~z -a)k1,

(k~-~~-y)k3 =(kz-~z-a)kJ.

Finally, relation (4.8) for (i,j,l) = (2, 1, 1), (3, 1, 1) has the following form:

k 1 a=k2~1;

kl(kJ-Al-1-a)=k3 ~1·

(4.10)

( 4.11)

Now, using relations (4.9) - (4.11), we show that a realization of the system (4.3) is

impossible and we find all parameters of cellu1ar rings of rank 4 which lead to the systems (4.4) and (4.5).

Lemma 4.5. The system (4.3) cannot be realized as the system S for an amorphic cellular ring of rank 4.

AMORPHIC CELLULAR RINGS !77

Proof. Substitution in ( 4.9) of the expressions for r 1 , s 1 ohtained from ( 4.3) gives

Now let us substitute the expressions for a, ß, y in ( 4.10):

(r 2 s 3 + r 3 s 2 + 2r 3 s 3 + r 3 + s 3) k 2 = - (r 2 s 3 + r 3 s 2) • k 1 ,

(r2 s3 + r3 s2 + 2r2 s 2 + r2 +s 2) k3 =- (rr s 3 + r3 s2) · k 1.

(4.12)

(4.13)

The expressions in the parentheses on the left sides of (4.13) are nonzero. lndeed, we would otherwise have (r 2 s 3 + r 3 s 2) = 0, a contradiction to the definitions of ri and si.

It follows from (4.11) that 1-Lt = a· k 1 I k2 = (k 1 - /..1 -1-a) k 1 I k3. Substitution in this equality of the expressions for k2 and k3 obtained from (4.13) gives

or

= (2r2 s2 +r2s3 +r3 s 2) (r2 s 3 +r3 s 2 +2r3 s3) + (r2 s3 +r3 s2 +2r3 s3) (r2 +s2).

This imp1ies

An evaluation gives:

r 2 r 3 s 2 - r 2 r 3 s 3 + r~ s 2 - r~ s 3 + r 2 s 2 s 3 - r 3 s 2 s 3 + r 2 s~ - r 3 s~ = 0.

Factoring we obtain

i.e.

Since the ri arepositive and the si are negative, the first factor is nonzero. Hence

Let us multip1y the known relations (see for example [7])

k2+hr2+g2s2=0

k3+!3r3+g3s3=0

(4.14)

by s 3 and s 2 respectively. In view of (4.14) and the aforementioned equalities we obtain:

k2 S3 = k3 S2.

Now the relations (4.8) for (i,j,l) = (2,2,3),(2,2, 1) imply that

k3 J.l2 = k2 ß

kl J.l2 =k2(k2-A,-1-ß),

which gives

From this equality, in view of (4.12)- (4.15), we have the following

A transformation of the left side using ( 4.14) gives

s2 r3 s2 - (2r 2 s 3 + r 3 + s 3) = 2r 2 s 2 + -- + s 2 = 2r 2 s 2 + r 2 + s 2. S3 S3

Nowwe have

(2r 2 s2 +r2 +s 2) (2r2 s 3 +2r3 s3 +r3 +s3) =

= 2r2 s3(2r2 s2 +2r2 s3 +2r2 +r3 +2s 2 +s 3 +2),

or

Evaluation gives

4r 2 r3 s 2 s 3 + 2r 2 r3 s 2 + r2 r 3 - 3r2 s 3 + 2r3 s 2 s 3 + r 3 s 2 + s 2 s 3 -

4r~ s1-2r~ s3 -2r2s1 =0.

Replacing r 3 s 2 by r 2 s 3 , we obtain

(4.15)

This equality cannot hold since ri > 0, si < 0 for i = 1,2. This completes the proof. D

Lemma 4.6. Let W be an amorphic cell of rank 4 which realizes a system S of the form (4.4). Then

v = (r 1 +r2 +r3 + 1)2 , ki = ri(r 1 +r2 +r3 +2)

Ai= (ri + 1) (ri + 2)- r 1 - r 2 - r 3 - 3, J.li = r;(ri + 1).


Proof. Let us find the solution to the system ( 4.4), considering TJ, 1'2, T:J as independent variables:

s 1 =- (r2 + r3 + 1) ,

s 2 = - (r 1 + r 3 + 1) ,

s 3 =-(r 1 +r2+1).

After substitution of these solutions into ( 4.9) and further evaluation we obtain:

Substitution of these equalities into (4.10) and use of the relation ki- ~i = -ri si Ieads to

(4.16)

Now it is sufficient to insert the values of a, k2 and ~1 = k 1 - r 1 (r2 + r3 + 1) into the first equality of ( 4.11 ). This gives

k 1 r 2(r 1 + 1) = (k 1 - r 1 (r2 + r 3 + 1)) • r 2 k 1 I r 1 .

Hence

k 1 =r 1(rt +r2+r3+2).

In view of (4.16) this implies:

ki=ri(r1+r2+r3+2), i=2,3.

Now the computation of the remairring parameters is quite easy:

~i = ki + ri si = ri(r 1 +r2 +r3 +2) + ri Si= ri(ri+ 1),

v = k 1 + k 2 + k 3 + 1 = (r 1 + r 2 + r 3 + 1 )2.

0

Lemma 4.7. Let W be an amorphic cellofrank 4 which realizes a system S of the form (4.5). Then

v =(s 1 +s2+s3+1)2 , ki =si(s 1 +s2+s3+2);

Ai= (si + 1) (si + 2)- s 1 - s2- s3- 3, ~i = si(si + 1).

Proof. If we replac:e T; by s; then the system ( 4.4) will be transformed into the system ( 4.5). All relations which were used in the proof of the previous Iemma can be applied here. So the proof of the present Iemma follows automatically from the proof of the previous one. 0

Theorem 4.8. Let W be a nontrivial amorphic cellular ring of rank 4. Then v = n 2 ,

and the basis graphs have parameters of one of the following two forms:

180

a) ki = gi(n -1),

Ai= (g; -1) (gi- 2) + n -2,

lli = gi(gi -1). (the type Lg(n) -latin square graphs);

b) ki = gi(n + 1),

Ai= (g; + 1) (g; + 2)- n- 2,

lli = g;(g; + 1).

JA. JU. GOL'FAND ET AL.

(the type NLg(n)- negative latin square graphs). Here gi, 1:::; i:::; 3, arepositive integers such that g 1 + g 2 + g 3 = n + 1 in the first case and g 1 + g 2 + g 3 = n - 1 in the second

case. Proof. By Lemmas 4.3 and 4.5 it is suffic:ient to study the two situations considPn~d

in Lemmas 4.6, 4.7. Let us put g; = -,<;; in tlw c:ase of Lemma 4.7 and g; =I'; in tlw c:asP of Lmnma 4.6. ThPn we c:ome to thP graphs of typP L9 (n) and NLg(n), rPSJWC

tivP!y. 0 Theorem 4.9. Let W be a nontrivial amorphic cellular ring of rank r :2: 4. Then

v = n 2 , and the basis graphs Q; , i = 1, 2, 3 , ... , r - 1, have the parameters of one of the

following two forms:

a) ki = gi(n -1),

A; = (gi -1) (g;- 2) + n- 2,

Jli = gi(gi -1);

b) k; = gi(n + 1),

Ai= (g;+ 1) (gi+2)- n -2,

Jli = g;(gi + 1), where the gi, 1:::; i:::; r -1, arepositive integers suchthat g 1 + g2 + · · · + g,_1 = n + 1 in the first case and g 1 + g 2 + · · · + g,_1 = n - 1 in the second case.

The proof can be easily carried out by induction using Theorem 4.8 and the fact that each partition of the basis elements of an amorphic ring determines an amorphic ring. 0

The amorphic rings of the first type were already considered in Section 3 where they were called pseudoaffine. It is natural to call the amorphic rings of the second type type negative pseudoaffine rings.

The systematic description of the amorphic cellular rings on n 2 points for arbitrary

n is of significant interest. In the next section, as a first step in this description, the results of a constructive enumeration (with the aid of a computer) of the amorphic cellular rings of rank 4 on n 2 points for n :::; 5 are presented.

5. A constructive enumeration of the amorphic cells of rank 4 on n 2 point for n :::; 5

Prior to the existence of Theorem 4.8 a search for small amorphic cells was carried out. The results of this search served as experimental evidence for a conjecture concern-

ing the parameters of amorphic cells. This conjecture turned out to be true and constitutes

Theorem 4.8. We believe that the information concerning small cells is of independent

interest and for this reason we have included it in this article. Note that the algorithm for

constructive enumeration of cells is presented in a modified form which takes Theorem

4.8 into account.

As was already mentioned, a pair of complementary s.r.g's determines an amorphic

cellofrank 3 and vice versa. Many papers are devoted to the constructive enumeration of

s.r.g.'s; all s.r.g.'s with v ~ 28 have been enumerated (see [2]). Forthis reason we start

with consideration of the rank 4 case. Nevertheless, for the sake of completeness let us

give some information about s.r.g. 's from the families Lg(n) and NLg(n ).

A s .r.g. of type L 1 (n) is unique. It is the imprimitive graph n o Kn with automor

phism group Sn wrSn.

A s.r.g. of type L2(n) is also unique if n ;t 4 (see [23]). It is the line graph of the

complete bipartite graph Kn,n which is usually called the lattice of ordern. The automor

phism group of the lattice is the exponentiation Sn i S 2 of the symmetric groups of order

n and 2. The exponentiation of permutation groups is considered in detail in [10]. For

n = 4 there is an additional graph of this type. This graph will be described below and is

known as the pseudolattice.

A s.r.g. of type L 3(n) is the graph of latin square type. There are exactly two pair

wise nonisomorphic latin squares of order 4. They can be represented by Caley tables for

the cyclic group of order 4 and for the Klein four-group, respectively. Let G 1 and G 2 be

the graphs of type L 3(n) constructed by means of these squares and Iet G 1 and G 2 be

their complementary graphs. It is easy to see that in G2 each vertex is contained in

exactly two maximal cliques of order 4. This implies immediately that G2 is the lattice of

order 4. In general the graph G 1 does not contain cliques of order 4 and hence this graph

is the pseudolattice.

The ring < Q o , G 1 , G 1 > is the siruplest example of an amorphic pseudoaffine

nonaffine ring. The automorphism group of the pseudolattice has order 192 and is iso

morphic to the semidirect product Z~ "AD 6 of the cartesian product of the cyclic group

Z 4 and the dihedral group of order 12. The rank of this group is equal to 4.

Significantly Iess is known about the graphs of type NLg(n). Examples of such

graphs are presented in [21]. The smallest graph in this family, i.e. the graph of type

NL 1 (4), is the well-known Clebsch graph. From differing points of view this graph is

considered in the papers [10], [12].

Let us proceed to the consideration of amorphic cells of rank 4. It follows from

Theorem 4.8 that the parameters of all the basis graphs of such a cell

W = < Qo ,Q 1 ,Q2 ,Q3 > are uniquely determined by the partition of n2 into the sum

1 + k 1 + k2 + k3 . The partition 1 + (n -1) + (n -1) + (n -1)2 corresponds to the cell

generated by two graphs of the type n o Kn (the "verticals" and "horizontals") whose

union is the lattice, and by the complement of this lattice. All cells with such a partitio'1

are isomorphic and their automorphism group is the direct product Sn x Sn. For n = 2,3 these cells exhaust the amorphic cells of rank 4.

Let us describe a class of cells corresponding to the partltlon 1 + 2(n- 1) + n - 1 + (n -1) (n- 2). Let Ln be a latin square of ordern. By [1] Ln can be considered as a set {(a,ß,y)} of n 2 ordered triples with values from the set {0, 1,2,, , , , n -1} which possesses the following property. If we omit any one of the three coordinates in all of the triples, then there will be n 2 distinct ordered pairs.

Starting with Ln, we can construct a cellular ring W 5(Ln) of rank 5 with the basis graphs Q0 ,Q 1 ,,,,, Q4 whose vertices are the triples of Ln; Qo is the reflexive graph, Qi, i = 1,2,3, are the graphs isomorphic to n o Kn: two triples are adjacent in Qi if their i-th coordinates are equal, Q4 = Q 4 + Q2 + Q3 is the complementary graph. An easy check shows that W 5 (Ln) is an amorphic ring with the subdegrees 1,n-1,n-1,n-1,(n-1)(n-2). Set W4(Ln)=<Qo,Q1+Q2,Q3,Q4>. Then W 4 (Ln) is an amorphic ring of rank 4 corresponding to the partition n 2 = 1 + 2(n -1) + n- 1 + (n -1) (n- 2).

Notice that this construction can be generalized to the case of a family of orthogonal latin squares.

Let us now present, without proof, a description of the automorphism groups Aut(Ln), Aut(W 5(Ln)), Aut(W 4 (Ln)) in the case when the latin square Ln comes from the multiplication table of a group K of order n.

We have the following: Ln={(a,ß,y),a,ß,yEK, aßy=1}, Aut(Ln) = K 3 A.(S 3 x Aut(K)) IN The group K 3 A.(S3 x Aut(K)) acts on the elements of Ln by the following rule. For (x,y,z) E K 3 we have (a,ß,y)(x,y,z) = (x-1ay, y-1 ßz, z-1y.t);

for cr E Aut(K) (a, ß, y)" = (a", ß", f); the elements of S 3 are generated by the transformations (a,ß,y) ~ (ß-1 ,a-1 ,y-1), (a,ß,y) ~ (y,a,ß). The normal subgroup N is the kerne] of this action. For instance if K is an abelian group of order n ~ 4, then N = K, Aut(Ln) =. K 2 A.(S 3 x Aut(K)),

Let us denote the corresponding amorphic rings of rank 5 and 4 by W 5 (K) and W 4 (K), respectively. Then Aut(W 5(K)) = K 3 A.Aut(K) IN,

Aut(W 4 (K)) = K 3 A. (S 2 x Aut(K)) IN For a commutative group K we have the following: Aut(W5 (K))=K2 A.Aut(K), Aut(W4 (K))=K2 A.(S 2 xAut(K)) (n~3) where the group S 2 permutes the two coordinates of K 2 ,

For n = 4 it turns out to be possible to describe an additional cell corresponding to the partition 1 + 6 + 3 + 6. Let us consider the regular representation of Z~ acting on the set {0,1,2,3}2. Let us define the graphs Q0 ,Q 1 ,Q 2 ,Q 3 as the Cayley graphs of the group Z~ corresponding to the subsets, {(0,0)}, {(0,1), (0,3), (1,0), (3,0), (1,3), (3,1)}, {(0,2), (2,0), (2,2)}, {(1,1), (1,2), (2,1), (2,3), (3,2), (3,3)}, respectively. It can be shown that the ring < Q 1 , Q 1 , Q 2 , Q 3 > is an amorphic cell of rank 4 which is isomorphic to the V-ring of the automorphism group of the pseudolattice. Notice that the interpretation

of the pseudolattice (i.e. the graph Q 3 ) in these terms is contained in the book [3] by N.

Biggs.

For n :2: 4 the described cells do not exhaust the amorphic cells of rank 4. For these

values of n, a search for amorphic cells was carried out by use of a computer for each

partition n2 = 1 + k1 + k2 + k3.

Let us describe the searching procedure. From the known complete list of s.r.g.'s with v = n 2 (v ~ 25) vertices we choose a pair A and B of graphs having valency k 1 and

k2, respectively. After that, we construct all nonequivalent embeddings of the graph B

into the graph A such that C = A + B is a strongly regular graph of valency k 3 . As was

mentioned above, every such factorization of the graph Kv deterniines a cellular ring. For

each constructed ring we determine its automorphism group.

The main results of these computations are contained in the Table 2 where, for the

sake of completeness, all cells of rank 4 with v ~ 25 are presented. Notice that for some

partitions the computations were also carried out for n = 6 (10 pairwise nonisomorphic

factorizations were constructed).

The table contains the following information: the value of n; the partition of n 2 into

4 parts; the number of the cell with the given partition; the type (I means pseudoaffine, II

Table2.

'" " Nurober • !E rl Partition Type '" «$ I Aut I Au! Rank Remarks " ' of ring !E " 0 < " 2 I+I+I+1 I l + 4 s.xs. 4 3 I+2+2+4 I l + 36 s.xs, 4 4 1+3+3+9 I l + 576 s.xs. 4 4 I+3+6+6 I I + 192 E10 >-.D• 4 W.(E.) R+R 4 1+3+6+6 2 l - 64 z.'>-.E. 7 w.(Z.) R+R 4 I+3+6+6 3 I - I92 z.•>-.D. 4 R+R 4 1+3+6+6 4* I - 32 - R+R 4 1+5+5+5 I* ll - 160 E,.XD, 4 [28], I602 4 1+5+5+5 2* ll - 32 IO 5 I+4+4+16 I I + 14 400 s.xs. 4 5 1+4+8+12 I l + 200 z.•x 5 w.(Z,)

5 I+4+8+12 2* I 24 >-.(Z.XS,)

- -[1 0], Table 6 5 1+8+8+8 I l + 400 4

~4 5 1+8+8+8 2• l + 200 4

means negative pseudoaffine); for cells of type I the sign "+" means that the cell is affine;

the order of the automorphism group of the cell; the notation for this group (its

identification up to isomorphism); remarks. The following notations are used in the table:

E 16 is the elementary abelian group of order 16, R is the lattice L 2 (4); R is the pseudolattice.

For the rings corresponding to the partition 1 + 3 + 6 + 6, the form of two basis


graphs is given. The first ring with partition 1 + 5 + 5 + 5 is Schurian; generators of its automorphism group are given in [25].

In Table 3 we give the adjacency matrices of the coloured graphs for all rings whose structure does not fol.low immediately from the above exposition or from the referenced literatÜre. In Table 2 under the column "The rings number", these rings are marked by " n *·

Table3. Partition 1 +3+6+6. Number of ring 4.

• 3 3 3 3 3 3 2 2 2 1 1 1 1 1 1 3•33211311332211 3 3 • 2 1 3 1 1 3 1 3 1 3 1 2 2 3 3 2 • 1 I 3 I 1 3 1 3 1 3 2 2 3 2 1 1 • 3 3 3 1 1 1 1 2 2 3 3 3 1 3 I 3 • 2 1 1 3 2 2 3 1 3 I 3 1 I 3 3 2 * I 3 1 2 2 1 3 1 3 2311311•22331133 2 I 3 1 1 I 3 2 • 2 3 I 3 3 I 3 2 1 1 3 I 3 I 2 2 • 1 3 3 3 3 1 I 3 3 I 1 2 2 3 3 1 • 2 1 3 3 1 1 3 I 3 1 2 2 3 1 3 2 • 3 1 1 3 I 2 3 I 2 3 I I 3 3 I 3 • 2 I 3 I 2 I 3 2 I 3 I 3 3 3 I 2 • 3 1 I I 2 2 3 3 I 3 I 3 3 I 1 3 • 2 1 I 2 2 3 1 3 3 3 1 1 3 3 1 2 •

Partition 1 +5+5+5. Number of ring 1.

• 3 3 3 3 3 2 2 2 2 2 I 1 1 1 I 3 • 2 2 I I 3 3 2 2 I 3 3 2 1 I 3 2 • I 2 1 3 2 3 I 2 2 I 3 3 1 3 2 I • I 2 2 3 I 3 2 I 2 3 I 3 3 I 2 I • 2 2 I 3 2 3 3 I I 2 3 3 I I 2 2 • I 2 2 3 3 I 3 I 3 2 2 3 3 2 2 I * I I 3 3 2 I 2 I 3 2 3 2 3 I 2 1 • 3 I 3 I 2 2 3 1 2 2 3 1 3 2 1 3 * 3 1 2 3 I 2 I 2 2 I 3 2 3 3 I 3 * 1 3 2 1 I 2 2 I 2 2 3 3 3 3 I I • I I 3 2 2 1 3 2 1 3 1 2 I 2 3 1 • 2 3 3 2 I 3 I 2 I 3 I 2 3 2 1 2 * 3 2 3 I 2 3 3 1 1 2 2 I I 3 3 3 * 2 2 1 1 3 1 2 3 I 3 2 I 2 3 2 2 * 3 1 I I 3 3 2 3 I I 2 2 2 3 2 3 •

Partition J-1.4+8+12. Number of riug 2.

•333333332222111111111111 3•33321113111333222111111 33•3312111311211311332211 333•311211131121131213132 3333•11121113112113121323 32111•3333111111222311331 312113•331311213131112213 3112133o31131321113231112 31112333•1113132311123121 231113111•222333111311331 2131113112o22113331331113 21131113122•2311133133131 211131113222o131313113313 1321112313131•33113132211 13121112331133•3311213112 131121312331133•131121123 1231121131313131•22333111 12131231113311132o2113133 121132131113331122·131313 1132131213311121311•31332 11312113213313123133•1121 112311213113323133111•231 1121332113113211113312o33 11132311231311121313233ol 111231321131312313321131.

Number of ring 2.

.333332222211111 3•22113322133211 3 2 * I 2 1 3 2 3 1 2 2 I 3 3 1 3 2 1 * I 2 2 3 I 3 2 1 2 3 1 3 3121•22123331132 3 1 1 2 2 * I 2 3 2 3 1 3 I 2 3 2 3 3 2 2 1 • 1 I 3 3 2 I 2 I 3 2 3 2 3 I 2 I • 3 I 3 1 2 2 3 I 2 2 3 I 2 3 I 3 • 3 I 3 2 I 2 I 221332313ol23112 2122333311•11322 I 3 2 I 3 I 2 I 3 2 I • 2 3 2 3 1 3 I 2 I 3 1 2 2 3 I 2 * 3 3 2 1233112211333•22 I I 3 I 3 2 I 3 2 I 2 2 3 2 * 3 111323311223223•

Partition 1+8+8+8. Number of ring 2.

•111111112222222233333333 1•11122331122333311222233 11•1123232333112323112322 111•132323213231222233112 1111o33223231323132321221 12233ol112332331131213222 123231o113321213312322312 1323211•13113322322132231 13322111•1233133223221123 212332331ol23123211331122 2133233121•12232311123321 2231332132lo3321233122111 223312133323o212113231211 2312332311232ol2332111132 23132312323211o3212113213 232131323321223ol21312113 2332113322321321•21211331 31223312211313122ol232313 313221223113322111o313232 3212323123112113223ol3231 32132123232231111311•2323 322313221132113212332ol31 3231223211312121332231ol3 33212213222113113133231•2 332212213211123313213132·


References

1. V.L. Arlazarov, A.M. Baraev, Ja. Ju. Gol'fand, LA. Farac!Zev, Construction with the use of a computer of al/ latin squares of order 8, In "Algorithmic Investigations in

Combinatorics", pp. 129-141, Moscow, Nauka, 1978 [In Russian].

2. V.L. Arlazarov, A.A. Leman, M.Z. Rosenfeld, Construction and investigation of graphs on 25, 26 and 29 vertices by use of Computer, Preprint, Institute of Manage

ment Problems, Moscow, 1975 [In Russian].

3. N.L. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.

4. R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math., 13 (1963), 389-419.

5. R.H. Bruck, Finite nets II, Uniqueness and embedding, Pacific, J. Math., 13 (1963), 421-457.

6. F.C. Bussemaker, J.J. Seidel, Symmetrie Hadamarti matrices oforder 36, Ann. N.Y.

Acad. Sei. 175 (1970), 66-79.

7. P.J. Cameron, J.H. van Lint, Graph Theory, Coding Theory and Block Designs, London Math. Soc. Lect. Note Ser., 19, 1980.

8. P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research report suppl. 10, Eindhoven, 1973.

9. LA. Farad.Zev, A program complex for computation in V-ring of permutation groups, In "Methods and Programs for Solution of Optimization Problems", Part I,

Novosibirsk, 1982, pp. 218-222 [In Russian].

10. LA. Farad.Zev, Cellular subrings of thc symmetric square of a cel/u/ar ring of rank 3 [In this volume].

11. LA. Farad.Zev, M.H. Klin, M.E. Muzichuk, Ce/lu/ar rings and groups of automorphisms of graphs [In this volume].

12. Ja. Ju. Gol'fand, M.H. Klin, On k-homogeneous graphs, In "Algorithmic Investiga

tions in Combinatorics", pp. 76-85, Nauka, Moscow, 1978 [In Russian].

13. F. Harary, Graph Theory, Addison-Wesley, 1969.

14. M.D. Hestenes, D.G. Higman, Rank 3 groups and strongly regular graphs, SIAM

AMS Proceedings, Providence, 4 (1974), 141-160.

15. X.L. Hubant, Strongly regular graphs, Discrete Math., 13 (1975), 357-381.

16. LV. Chuvaeva, A.A. Ivanov, Action of the group M 12 on Hadamard matrices [In this volume].

17. A.A. Ivanov, M.H. Klin, LA. Farad.Zev, Primitive representations ofthe nonabelian simple groups of order /ess than 106 , Preprint, Part II, Moscow, VNIISI, 1984 [In Russian].

186 JA. JU. GOL'FANO ET AL

18. A.K. Kelmans, Graphs with equal number of paths of length 2 between adjacent and nonadjacent vertices, In "Problems of cybernetics", pp. 70-75, Mosc:ow, 1973

[In Russian].

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20. M. Markus, H. Mink, A survey of matrix theory and matrix inequalities, Boston, 1964.

21. D.M. Mesner, A new family of partially balanced incomplete block designs with some Latin square designs properties, The Annals of Math. Stat., 38 (1967), 571-581.

22. R. Pöschel, L.A. Kaluznin, Funktionen- und Relationenalgebren, Berlin, 1979.

23. S.S. Shrikhande, The uniqueness of the L 2 association scheme, Ann. Math. Statist. 30 (1959), 781-798.

24. S.S. Shrikhande, N. Bhat Vasante, Non-isomorphie solutions ofpseudo-(3,5,2) and pseudo-(3,6,3) graphs, Annals ofthe New York Acad. Sei., 175 (1970), 331-350.

25. C.C. Sims, Computing methods in the study of permutation groups, In "Computing Problems in Abstract Algebra", pp. 169-184, Pergarnon Press, 1970.

26. B. Weisfeiler (editor), On construction and identification of graphs, Lect. Notes Math., No. 558, 1976.

27. H. Wielandt, Finitepermutation groups, Acad. Press, 1964.

28. V.A. Zaichenko, An algorithmic approach to the synthesis of combinatorial objects and to computation in permutation groups based on the method of invariant relations, Ph. D. Thesis, 1981 [In Russian].

This paperwas originally published in IATC0-85 as two separated papers: "Amorphie: cellular rings. I" by .Ja . .Ju. Gol'fand and M.H. Klin, pp. 32-38 and "Amorphie: c:dlular rings. Il" by A.V. Ivanov, pp. 39-49.

THE SUBSCHEMES OF THE HAMMING SCHEME

M.E. Muzichuk

The present paper contains the complete characterization of all the subschemes of the Harnming scheme H (n, 2). The characterization problern for these subschemes is closely related to the study of the lattice of overgroups of the exponentiation S 2 i Sn in the symmetric group S 2". For this reason the results of the paper can be used in the study of symmetry in algebraic codes, and in the classification of Boolean functions. Some examples of subschemes having two classes for even n were indicated in [4]. All subschemes of H(n, 2) for n ~ 16 have been classified by computer. A synopsis of these results on enumeration can be found in [5]. An analysis of them led us to a conjecture that the number of subschemes becomes stable when n is sufficiently large. The validity of this conjecture follows from the list of the subschemes of H(n, 2) presented below.

1. Preliminary information

Be1ow we present some known properties of an association scheme (X, 'P) on the set X, having n classes. Details can be found in [1] for instance. In what follows only symmetric association schemes will be considered.

The intersection numbers of the association scheme, i.e. the numbers I {z e X I (x,z) e 'Pi, (z,y) e 'Pj} I where (x,y) e 'Pt. will be denoted by PL·

An association scheme (X, 'P'), 'P' = {'P0', 'P1', · · · , 'P m'l is called a subscheme of the association scheme (X, 'P), 'P={'P0 , 'Pt. ... , 'Pn} if for each relation 'Pj e 'P, there exists a relation 'P{ e 'P' suchthat 'P{ :2 'Pj. It is. clear that the subscheme (X, 'P') determines a partition 't = { 'to , t 1 , • · · , 'tm) of the set { 0, 1, ... , n} possessing the following properties:

1) to = {0}.

2) Foreach triple i,j,k e {0,1, ... ,n} and each pair l,r e 'tk the following equality holds

"" PI -L. t,s-t E 'ti, SE 'tj t E 't;, s E 'tj

3) 'P{ = u 'Pj. jE~~

The converse is also true. Namely each partition 't = { to , t 1 , • · · , 'tm} of the set (0, 1, ... ,n} satisfying the properties 1)- 2) determines a subscheme (X, 'P') of the associa-

tion scheme (X, 'P) with 'P{ = u 'Pj. jE~~

187

188 M. E. MUZICHUK

The Hamming scherne H(n, 2) consists of the family of relations qsi on the set of the n-dirnensional vector space Vn(F 2) defined as follows:

qsi = {(a,b) e Vn(F 2) x Vn(F 2) I# {J I aj'#: bj} = i }. In other words two vectors (J andb are in the relation qsi if the nurnber of coordinates in which they differ is equal to i. It is clear that this scherne is syrnrnetric. The intersection nurnbers can be cornputed by the following formula:

Pkij-

0 otherwise.

if i + j + k = 0 (rnod 2) and lj-i l~k~rnin(i+j,2n-i-j);

The problern of characterizing the subschernes of the Hamrning scherne H(n, 2) can be formulated in other terms. All the notation and definitions used below are contained in [2].

First of all it should be noticed that the relations qsi are the 2-orbits of the permutation group (S 2 i Sn ; V 2 (F 2) ), where the exponentiation S 2 i Sn = V n (F 2) : Sn acts on Vn(F 2) by the following rule:

b(1t,~) = P 1t b + a.

Here P 1t is the permutation rnatrix corresponding to the permutation 1t in the basis e1, e2, · · · , en where ej = (0,0, ... 0, 1,0, ... ,0). So the Bose-Mesner algebra of the association scherne H(n, 2) is the V-ring of the permutation group (S2 i Sn, Vn(F 2)) and the problern of characterizing the subschernes of H(n, 2) is equivalent tothat of characterizing the cellular subrings of the V-ring.

An additional formulation of the problern results frorn the fact that the permutation group (S 2 i Sn; Vn(F 2)) contains a regularnormal subgroup isornorphic to Vn(F 2). This subgroup acts on itself by right translations. By a result of I. Schur [3] (see also [2]) this irnplies that each cellular subalgebra of the V-ring of the group (S 2 i Sn; Vn(F 2 ))

corresponds to an S-ring over Vn(F 2). Hence the classification problern for subschernes of the Harnrning scherne is equivalent to that for S-subrings of the transitivity rnodule of the group S 2 i Sn with respect to Vn(F 2). This transitivity rnodule is anS-ring itself and will be denoted below by H n.

Now let us define this transitivity rnodule precisely. In order to avoid confusion between addition in the group ring :tZ [Vn(F 2)] and addition in the group Vn(F 2), the latter will be written in rnultiplicative form. This rneans that the coordinates aj of a vec-

-f' . a· ... _,. . tor a = (a 1 , ••• , an) wlll be replaced by (-1) ' and the surn of two vectors a and b wlll be indicated by their coordinatewise product ci · b = (a 1 • b 1 , •.. , an • bn). The group

THE SUBSCHEMES OF THE HAMMING SCHEME 189

Vn(F 2), with this new notation for group multiplication, will be denoted by E2·· After this notational transformation, the relations lf'; are determined in the obvious way. If we use the notation .fi = ( 1, ... , 1,-1, 1, ... , 1) then it is easy to see that the basis values T k of the transitivity module Hn have the following form:

T 0 = 1 ' Tk = ~ /;1 • /;2 •.... A ' 1Si1 <i2 < ... <i"Sn

In other words T k is the elementary symmetric function of degree k in the variables ft , h , ... , fn· Since the structure constants of the S-ring Hn are equal to the corresponding constants of the V-ring of the group <Vn<F 2): Sn, Vn(F 2)) and coincide with the intersection numbers Pt of the Hamming scheme H (n, 2), we conclude that

T· • T· = ~ -lr- · Tk where -Jr. · =p~ ·. I J ~ 1i,J "fi,J I,J k::O

2. The main result. A scheme of the proof

The main result of the present paper consists of proving the completeness of the !ist of S-subrings of the S-ring Hn for n ~ 9 and is presented below. Let us first introduce notation which is necessary for the formulation of this result.

Let us consider the partition of the set { 0, 1, ... , n } into the subsets Ao ,A 1 ,A2 ,A3, {0}, {n} where

Ak={i I O<i<n,i = k(mod4)}forke {0,1,2,3}.

In addition let Bk= {i I OS i Sn, i = k (mod4)} and let Sk denote those elements of the group ring JZ [E 2"] defined by the formula:

Let W be the free Z -submodule of the group algebra JZ [E 2"] with basis To,Tn,Rk= ~ T;,k=0,1,2,3.

ieA"

Theorem 2.1. For n ~ 9 the Iist below exhausts all S-subrings of the S-ring Hn (the Iist depends on the parity of n).

Forneven:

1. W;

2. <To ,Ro +R 2 , R 1 +R3, Tn >;

190

3. <To,Ro,R2,Rl +R3, Tn>;

4. <To,R 1 ,R3,Ro+R2,Tn>;

5. < T 0 ,R 0 +R 1 +R2 +R3, Tn >;

6. <T0 ,Ro +R2 +Tn, R1 +R3 >;

7. <To,So+S1,S2+S3>;

8. <To,So+S3,S2+S1>;

9. <To,Ro+R1+R2+R3+Tn>;

10. <To ,T2, ... , T21c, ... , Tn, T1 +Tn-1, ... , T2k+l +Tn-2k-l , ... >;

11.<To,T2•···•T2k•····Tn,Tl+T3+ ··· +Tn-1>;

12. < To ,T 1 + T2, T3 + T 4, ... , Tn-1 + Tn >;

13. < T o , T 1 + Tn , ... , Tk + Tn-k+l , ... >;

M. E. MUZICHUK

14. < To ,Tl+ T2 + Tn + Tn-1, ... , T21c-1 + T21c + Tn-2k+l + Tn-2k+2 , ... >;

15. < T o ,T 1 + Tn-1 , ... , Tk + Tn-k, . .. , Tn >;

16. <To ,T2 +Tn-2, ... , T2k +Tn-2k• T1 +T3 + · · · +Tn-1 ,Tn >;

17. <To ,T2 + Tn-2, ... , T21c +Tn-2k• Tn ,R1 ,R3 > for n = 0 (mod4).

Fornodd:

1. W;

2. <To,Ro+R2,R1+R3,Tn>;

3. <To.Ro+R 1 +R2+R3,Tn>;

4. <T0 ,R0 +R 1 ,R2 +R3 ,Tn>;

5. <To,Ro+R3,R 1+R2,Tn>;

6. <To,Ro+R2,R1+R3+Tn>;

7. < T o , R o + R 2 , S 1 , S 3 >;

8. < T o, R o, R 2, S 1 + S 3 >;

9. <To,Ro+R2+R3+R1+Tn>;

lO.<To,T2, ... ,T2t.····Tn-l•T1+T3+ ··· +Tn>;

11. < To ,Tl+ T2, ... , T21c-1 + T21c, ... , Tn >;

12. < To ,T 1 + Tn-1 , T 2 + Tn-2, ... , Tk + Tn-1 , ... , Tn >;

13. < To ,T 1 + T2 + Tn-2 + Tn-1, ... , T21c-1 + T21c + Tn-2k + Tn-2k+l, ... , Tn >;

14. < To ,T 1 + Tn, ... , Tk + Tn+l-k , ... >;

15. < To ,T 2 + Tn-1 , · · · , T 2k + Tn+l-2k, T 1 + Tn-2, ... , T 2k-l + Tn-2k, ... , Tn >;

16. <To ,T2 +Tn-1, ... , T2k+Tn+l-2k• T1 +T3 + · · · +Tn >.


Remark. For small n (n :5: 12) the list is degenernte in the sense that some subrings

with different numbers can coincide.

The proof of Theorem 2.1 is divided into steps. First it is proved that the free Z

submodule W is itself an S-ring, and its subrings are enumerated. For this purpose it is

convenient for us to choose a basis for W such that the product of any two basis elements

can be represented as a simple linear combination of basis elements. Section 5 is devoted

to the consideration of the S-ring W.

After that all other S-rings are enumerated. A direct search shows that any such ring

contains one of the 6 basis elements which are determined by the following subsets of

numbers of the basis elements of Hn:{2}, {1,2}, {2,n-2}, {2,n-1},

{ 1,2, n- 2, n -1}, { 1,2, n- 2, n}. The subrings whose bases contain either {2} or { 1,2}

are described in Section 6. The final step in the description of S-rings is carried out

separately for even and odd values of n. It is contained in Sections 7-8.

Before starring the proof we will need two preliminary sections, Sections 3 and 4,

where a number of necessary properties of the S-ring Hn are proved. In particular, these

sections contain an essential description of certain basis elements of the S-rings to be

classified.

3. Some properties of the S-ring Hn

Let us start with the simplest properties of the S-ring Hn. It follows directly from the

formula for structure constants that Yt,1 = 0 if i + j + k is odd. Hence Yfj+l = 0 if i + j is

even and YfJ = 0 if i + j is odd. In other words Hn is a Z 2-graded algebra. A subset of

the set {0, 1, ... ,n} will be called even if it consists of even numbers only. Odd subsets are

defined analogously.

For an arbitrary I!:: {0,1, ... ,n} let us set T1 = !: Ti. Elementsofthis form will be je/

called the simple values of the S-ring Hn. If is easy to check that Tn • T1 = Tr where I'={n-j I je!}.

It follows from the definition of S-ring that anS-subring of the S-ring Hn determines

a partition 't = { 'to , 'tt , .•• , 'tm} of the set { 0, 1, ... , n} such that

1. -ro = {0},

2. For any i,j,k = 0, l, ... ,m and any pair u, v e 'tk the equality

!: Y~.r SE 'tj, t E 'tj SE 'tj,tE 'tj

holds. The converse is also true. So each partition 't = { -ro , -r1 , •.. , 'tm} of the set

{0, 1, ... ,n} satisfying properties 1-2 determines an S-subring H with the basis values

T t; , i = 0, 1, ... ,m. The sets 'ti will be called basis sets of the S-subring H.

192 M. E. MUZICHUK

As was mentioned above, before classifying the S-rings we indicate some of their basis sets. Let A be a subset of the set { 0, 1, ... , n} and let Ai be the coefficient of Ti in the decomposition of the square of the simple value TA in terms of the basis values. A number of interesting properties of the S-ring Hn can be formulated in terms of these

coefficients Ai·

Let A,B be any two subsets of the set of nonnegative integers IN u { 0}. For any k

and n let us define the number A,k(A,B; n) by the equality

Ak (A,B; n) = L Yf.j ieAn[O,n],jeBn[O,n]

where Yt,j are the structure constants of the S-ring Hn. If one of the sets A n [O,n], B n [O,n] is empty, then it is assumed that A,k(A,B; n) = 0.

The following proposition contains some simple properties of the function

Ak(A,B; n).

Proposition 3.1. For any subsets A,B,C of the set IN u {0}, the following relations

hold:

1. A,k(A,C; n) + A,k(B,C; n) = A,k(A u B, C; n) + A,k(A n B, C; n);

2.Ä,o(A,B;k)=Ä,o(AnB,AnB;n)=. L [;]. 1eAnB

3. Ä,o(A,A; n);;:; A,k(A,B; n) with equality attained if and only if for each i E A and

j E {0, 1, ... ,n}, yf.j';t: 0 implies j E B.

Proof. The former two claims follow directly from the definition of the function

A,k(A,B; n) and from the relation rb,i = Bi,j· (Here Öi,j is the Kroneckerdelta symbol.)

The third claim follows from the formula A,k (A,B ; n) =

and the relations Yt,j ;;:: 0 , L Yt,j = [;] .

L Yt.j i E An [O,n], jE B n [O,n]

0

In what follows the numbers A.k({i} , {j} ; n) and A.k(A,A ; n) will be denoted by Yt,j(n) and Ak(A ; n ), respectively.

For arbitrary k E IN u { 0} and I s;;;; IN u { 0} let us define the set I k by the following equality: I k = {j E IN u { 0} I j + k E I} . In other words I k can be obtained from I by means of translation along the number line on k followed by exclusion of negative elements.

Proposition 3.2. For any two subsets A and B of the set IN u { 0}, the following equalities hold.

1. ')...k (A,B; n) = A.k(A,B; n -1) + A.k(A 1 ,B 1 ; n -1) for all k, 0 :-:;; k :-:;; n -1;

2. A.k(A,B; n) = A.k_1 (A 1 ,B; n -1) + Ak-1 (A ,B 1 ; n -1).


Proof. Since

i E An [O,n], jE B n [O,n]

it is sufficient to prove the equalities for the case A = { i} , B = {j}, A.k(A,B ; n) = yf.j(n ).

It follows from the definition of the function A.k(A,B; n) that yf.j(n) is equal to the coefficient of Tk in the product of the basis values Ti and Tj of the S-ring Hn. Here Ti and Tj are the elementary symmetric functions in the variables ft ,fz , ... , fn having degrees

i and j, respectively. It is obvious that To = Qo, Tn = fn Qn-1,

Tk = Qk-1 fn + Qk, 1:5: k :5: n -1 where Qk is the elementary symmetric function of degree k in the variables ft, ... , fn-1· Since ft ,fz, ... , fn-1 are generators of the subgroup E 2·-' of the group E 2., and the Qk are the elementary symmetric functions in these variables, we conclude that the Qk are the basis values of the S-ring Hn_1• Now it follows

n-1 from the definition of the function A.k(A,B; n) that Q8 Qt = I; -fs.r(n -1) Qk. If

k=O 1 :5: i,j :5: n - 1 then

n-1 n-1 = I; yf.j(n -1) Qk + I; 0-1,j-1 (n -1) Qk +

k=O k=O

n-1 n-1 + fn( L 0-1,j (n -1) Qk + I; 'Yf.j-1 (n -1) Qk).

k=O k=O

On the other hand

n n-1 TiTj= L yf.j(n)Tk= L if.j(n)lfnQk-1 +Qk)+y?,j(n)Qo+'Yf.j(n)fnQn-1-

k=O k=1

Since fn Qi-1 and Qi are linearly independent, we havt: that

yf.j(n)=0-1.j-1(n-1)+yL(n-1) for 0:5:k:5: n-1, and

Yt,j(n) = 0-1,j(n -1) + if.j-1 (n -1) for 1 :5: k :5: n.

Thus the claim is proved for the case 1 :5: i,j :5: n -1 and it is sufficient to check it in the case {i,j} n { O,n} '# 0. This check will be carried out by direct computation of the constants Yt,j(n) for each of the possibilities: 0 E {i,j} and n E {i,j}. Weshall check only the first relation in the proposition. The second relation can be checked in an analogous way.

Now suppose that one of the elements i,j is equal to zero. Since the function if.j(n)

is symmetric under permutation of the indices i and j, we can assume that i = 0. Then -/o.j = ök,j and A.k( { 0} 1 , {j} 1 ; n - 1) = 0. In addition k :5: n - 1, so it follows from the definition that the number -/o.j(n -1) is also equal to Öj,k· Now substitution of the obtained expressions in the left and right parts of the desired equality leads to an identity.

194 M. E. MUZICHUK

Now Iet us turn to the case i = n. It follows from the formulas for the structure constants of the S-ring Hn that

'fn.j(n) = Ök,n-j; 'Yf-1,j-1 (n -I)= Ök,n-1-(j-1) = Ok,n-j·

Moreover, A.k({n}, {j}; n -I)= A.k(0, {j}; n -1) = 0. Hence

'fn.j(n) = 'fn-1,j-1 (n -1) + Ak ({n} , {j} 1 ; n -1)

and the claim is proved. 0

The following Iemma plays a crucial role in the enumeration of the basis sets of the subrings of the S-ring Hn.

Lemma 3.3. Let A be an arbitrary subset of the set { 0, 1, ... , n } , n ~ 2. Then for each k , 1 ~ k ~ n - 1, the inequality "-2 (A ; n) ~ A.k(A ; n) holds. Moreover, if A.2 (A ; n) = A.k(A ; n ), then the set A satisfies the following conditions:

1. For arbitrary i E A and je [O,n- 2] the inequality 'Yf~L<n- 2) :#= 0 implies An {j,j +2} :#= 0.

2. For arbitrary i e I n 12 and j e [O,n- 2] the inequality yf.j1 (n- 2) :#= 0 implies j + 1 E /.

Proof. It follows from the second claim of Proposition 3.2 that

"-1(A ; n) = 2 A.1 (A 1 ,A ; n -1) = 2 [A.o(A 1 , n- 2) + A.o(A ,A 2 ; n- 2)].

Moreover, A.o(A,A 2 ; n- 2) = A.o(A n A 2 ; n- 2) by Proposition 3.1. Analogously,

A.k(A ; n) = 2 "-k-1 (A 1 ,A ; n - 1) = 2 [A.k-1 (A 1 ,A ; n - 2) + "-k-1 (A 2 ,A 1 ; n - 2)] (here we have also used the first claim of Proposition 3.2). Notice that since n ~ 2 and 1 ~ k ~ n - 1, all functions on the left and right sides of the above equality are uniquely determined, so all the evaluations are correct.

Thus we come to the following two relations:

A.2(A; n) = 2 [A.o(A 1 ; n- 2) + A.o(A n A 2 ; n -2)] }

A.k(A; n) = 2 [A.k-1(A 1 ,A ;n-2) +A.k-1(A 2 ,A 1 ; n -2)] (3.1)

Using the first claim of Proposition 3.1 we can rewrite the lauer of the above equalities in the following form: A.k(A; n) = 2 [A.k-1 (A 1 ,Au A 2 ; n- 2) + A.k_1 (A 1 ,An A 2 ; n- 2)]. Since A.o (A 1 ; n - 2) ~ "-k-1 (A 1 ,A u A 2 ; n - 2) and A.o(A n A 2 ; n - 2) ~ A.k_1 (A 1 ,A n A 2 ; n - 2) (see claim 3 of Proposition 3.1 ), we have A.2 (A ; n) ~ Ak(A ; n ). So the first claim of the Iemma is proved. Now Iet A2 (A ; n) = Ak(A ; n ). Then it follows from the above inequalities and the relations (3.1) thatA.o(A 1 ;n -2) = "-k-! (A 1 ,A uA 2 ; n-2) and

A.o(A n A 2 ; n- 2) = A.k_1 (An A 2 ,A 1 ; n- 2). Now to complete the proof we use the

third claim of Proposition 3.1 and the definition of the sets A 1 and A 2 . 0

Corollary 3.4. Let I be an arbitrary subset of the set {0, 1, ... ,n}, n '2!: 2, such that ').-2(/ ; n) = "-n-1 (I ; n ). Then I sarisfies the following condition: for each j e I\ { 0, n } there is i e I such that Ii + j - n I = 1.

Proof If I ~ { 0, n} then the claim is trivial, so we assume that I q { 0, n}. Let j e I- { 0, n}. Then the numbers j- 1 , n - j- 1 are contained in the segment [0, n - 2] and hence ij~f.n-j-1 (n -2) #: 0. Now, by Lemma 3.3, In {n- j -1, n- j + 1} #: 0, so the claim follows. o

Proposition 3.5. Let I be an arbitrary subset of the set {0, 1, ... ,n}, n '2!: 9. Then the equality A.2 (/ ; n) = At; (I ; n) implies that I is a union of subsets from the following list: Ao ,A 1 ,A2 ,A3, {0}, {n}. (Here A 1 ,A2 ,A3 ,Ao are the subsets defined in Section 2 of the paper.)

Proof. A set I can be presented as a union of subsets as in the lemma statement only if it sarisfies the following condition: for each i , 0 i + 4 e /. Thus we should prove that the equality "-2(/; n) =At;(!; n) implies the above equivalence for /. In fact it is sufficient to show for all i e I , 0 i+4e/. Let us consider the set I'={n-j I je!}. Then

'A2 (I' ; n) = At; (I' ; n) since Tl' = (T n • T1 )2 = T'f. Hence, in view of our assumption, if j e [1,n- 5] is contained in I' then j + 4 is also contained in I'. But the map j ~ j' = n - j is a permutation of order 2 which reverses the ordering of the elements from [O,n ]. Hence we have the following implications. which prove the claim:

jE / ::::> n- jE J' ::::> n- j + 4 E J' ::::> j -4 E /.

Thus we are to prove that if I contains an element i e [1,n -5] then it also contains the element i + 4. Suppose to the contrary that there exists i e In [1,n- 5] with

i + 4 ~ I. Let us[ n ~~nlsider the case i '2!: 2. Since 2 ~ i ~ n - 5, the number

YT-1,i+2 (n- 2) = 5 i _2 is nonzero. Hence by the first claim of Lemma 3.3,

In {i +2, i +4} #: 0. By the choice of i, we have i + 2 e I and hence i eIn 12. If

[n-7] i < n- 6, the inequality yl,i +3 (n- 2) = 5 i _1 #: 0 and claim 2 of Lemma 3.3 together

imply that i + 4 e /. This contradicts the choice of i.

[n-7] Now suppose that i = n -5. Then the value Yn-s,n-8 (n- 2) = 5 n _9 is nonzero

and by claim 2 of Lemma 3.3, n -7 e /. Hence I;;;;;;! {n-3, n-5, n-7}. In this case n - 7 e I n / 2 and Yn-7,n-2 (n- 2) = 1. This implies that n - 1 e /. The obtained contradiction proves the proposition for i '2!: 2. To prove the proposition in the case i = 1 one should use Lemma 3.3 and the relations YÖ.s(n-2)= 1, 11.s(n-2)=5(n-8),

196 M. E. MUZICHUK

[n-7] ~.3(n-2)=5 2 ,y1,4(n-2)=5. 0

4. The basis sets ofsubrings of Hn

As was shown in Section 3, each subring of the S-ring Hn is determined by the partition of the set {0, l, ... ,n} into the basis sets of this subring. Let us establish the essential form of some of these sets.

Proposition 4.1. Let H be an S-subring of the S-ring Hn , n ~ 7, and let I be the basis set of this S-subring containing the element 2. If I contains an element j such that 3 ~ j ~ n- 3, then 6 E I.

Proof. Since I is a basis set of the S-ring H , A.2 (I; n) = A.j(l; n ). In addition

j-lei1 and yj=l,4(n-2)=[j;l] [n-;-l] ;t:O. Hence by Lemma 3.3 the set

I 11 { 4, 6} is nonempty. If 6 E I then the claim is proved, so suppose that 6 ~ I. W e obtain a contradiction as follows. Since I 11 { 4, 6} ;t: 0, we have 4 E I. Since I is a basis

~~:; :e/ ~:;:,:·2:e :~:ty ~~ ~·:~ ~:: ·:.::: ~: :~ :: r~r:::::: and hence 5 E I 1 (see Lemma 3.3, claim 2). It now follows from the definition of I 1 that 6 EI. 0

The above proposition and Proposition 3.5 enable us to write down all subsets which contain 2 and which can be basis sets of some subring of the S-ring Hn.

Corollary 4.2. Let H be anS-subring of the S-ring Hn, n ~ 9, and let I be the basis set of H containing 2. Then one of the following two conditions holds:

1. {2} r;;;;,I r;;;;, {1,2,n -l,n -2,n};

2.Hr;;;;,W. Here W is the submodule defined in Section 2 of this paper.

Proof. Suppose that I does not satisfy 1. Then there exists j E I such that 3 ~ j ~ n- 3. By the previous proposition, 6 E I. Now application of Proposition 3.5 gives H r;;;;, W, and the claim follows. 0

Before proceeding to the enumeration of all S-subrings of the S-ring Hn, Iet us introduce some additional notation.

For any two elements I;= L Cg g and 11 = L dg g of the group algebra geE~ geE~

q; [E 2"] and any complex number a, Iet us define the elements I cx(s) and I; o 11 by the following equalities:


Ia(~)= L g; {g e E2• I c, =a)

~ o Tl= L Cg dg go g E E,•

(Here q; denotes the complex number fieldo) It is clear that an arbitrary S-ring over the group E 2• is closed under the binary operation o and the family of the operations I a•

where a e q;o

5. The S-ring Wand its subrings

By Corollary 402 the classification problern for all subrings of the S-ring Hn for

n ~ 9 is reduced to the following two subproblems: to describe all subrings which are contained in W and to describe all subrings having a basis set I such that

{ 2} r;;;, I r;;;, { 1 ,2, n - 2, n -1, n}. Here a solution to the first subproblern is presented. First let us prove that W is itself an S-ring.

Proposition 5.1. The IZ -module W is an S-ring, i.e., W is a subalgebra of the group

algebra IZ [ E 2"].

Proofo If is convenient for us to embed the IZ -module W into the vector space W over the field q;o It is clear that by this embedding each submodule which is an S-ring

becomes a subalgebra of the group algebra q; [E 2·]. Hence W is closed under multiplica

tion if and only if W is a q;-subalgebra of the group algebra q; [E 2• ]. To prove the latter

let us choose in W a new basis possessing the property that the product of any two basis vectors can easily be expressed as a linear combination of basis vectorso

n Let us consider the elements a(t) from q; [ E 2.] of the form a(t) = TI ( 1 + t .ti ),

i=l t e q;, where .ti is a generator of the group E 2• 0 It is easy to see that

n n 0

TI (1+t/i)= I: t'h i=l i=O

The following equality is a direct consequence of the definition of a(t):

n 1iL

a(t1)· a(t2) =TI o +tlfi) • (1 + t2.ti) = rr o + (t1 +t2)fi + t1 t2) = i=l i=l

{(tt+t2)nTn,

[ tl+t2] (1+tt•t2)na 1 + t 1 t2

(5.1)

In addition we have the following

198 M. E. MUZICHUK

n n { Tn, t =0 a(t)·Tn=fl(l+fit).ti=p(fi+t)= [ 1 ]

•=1 •=1 tn a - , t # 0. t

(5.2)

Let i be a primitive fourth root of unity and set Sk = :E Tj, k = 1,2,3,0. It is easy j = k(mod4)

to see that the vectors S 1 , S 2 , S 3 , So , T n , T o also form a basis onV. On the other band, it

follows from the formu1as presented below that the vectors T 0 ,Tn, a(1),

a( -1) , a(i) , a( -i) also form a basis of W.

So= 1.. (a(1) + a(-1) + a(i) + a(-i)) , 4

S 1 = t (a(1)- a(-1)- i a(i) + i(a(-i))),

S2 = t (a(1) + a(-1)- a(i)- a(-i)) ,

S 3 = t (a(1)- a(-1) + i a(i)- i a(-i)) .

Now using the formulas (5.1)- (5.2) it is easy to prove that W is closed under multiplica

tion. The multiplicative table of the vectors of the new basis is given below.

{0, t=-1,

a(1) · a(t) = a(t) · a(l) = (1 + t)n a(l), t *" _1;

{0, t=1,

a(t) • a(-1) = a(-1) · a(t) = (1- t)n a(-1), t *" 1;

a(i) · a(i) = (2it Tn; a(-i) • a(-i) = (-2i)n Tn ;

a(i)· a(-i)=a(-i)· a(i)=2n To,

Tn • a(t) · Tn = tn • a [ + l· It follows directly from this tab1e that W is closed under multiplication. Thus W is a rrsubalgebra ofthe group algebra q; [E 2·] and hence Wis anS-ring. o

Proposition 5.2. The S-rings numbered 1-9 in the Iist from Theorem 2.1 exhaust the

subrings of the S-ring W.

Proof Let H be anS-subring of W. There are four possibilities:

Let us consider these possibilities separately.

1. So + S 2 , Tn E H , H c W. Then H coincides with one of the following submo

dules (see definition in Section 2):

< T o, R 1 , R 3 , R o + R 2, Tn >.

We check which of the above submodules are closed under multiplication. Forthis purpose we use the second basis of W. If follows from the formulas transforming the first basis into the second one that

< To, Ro +R2, R 1 + R3, Tn > = < To, a(1), a(-1), Tn >,

< T o , R o , R 2 , R 1 + R 3 ,T n > = < T o , a(ll) , a( -1) , a( i) + a( -i) , T n > ,

< To, R 1, R3, Ro +R2, Tn > = < To, a(l), a(i)- a(-i), Tn >.

Now it is easy to see that the first submodule is closed under multiplication for all n while the remaining two are closed only if n is even. Hence there are exactly three subrings satisfying 1:

(the latter two for even n only).

2. So+ S2 Ii'= H, Tn EH. In this case H cannot contain Ri , 0 S i S 3. Indeed, if Ri E H for some i , 0 S i :5: 3, then

[~] 2

Rr = L 'A2i R 2i is also contained in H. Since Ri is th1~ sum of the basis values of an Si=O

ring having fixed parity, we conclude that

~ = T 2 + L T 2i is also contained in H. {i I ).;,oO}

The basis values T 2i are polynornials in T 2 having degree i. This implies that all the coefficients Jli in the decomposition

[~] !fl ~ 2 = L J1J2i

i=O

[~] 2

are nonzero. Hence So + S 2 = L T 2i E H, a contradiction. i=O

Thus Ri Ii'= H for all i , 0 S i S 3. In view of condition 2 we conclude that H coincides with one of the submodules below:

200 M. E. MUZICHUK

< T o , R o + R 1 , R 2 + R 3 , T n > = < T o , a(1) , a(i) + i a( -i) , Tn > ,

<To,Ro+R3,R1 +R2,Tn>=<To. a(1), a(i)-ia(-i), Tn>·

Since Tn(a(i) ± i a(-i)) = in(a(-i) ±i(-1)n a(i)), the latter two submodules areS-rings only if n is odd. The first submodule is closed under multiplication in any case. So there

are exactly three S-subrings W satisfying the condition 2:

<To ,R 0 +R 1 +R 2 +R3 +R4, Tn >;

< T o , R o + R 1 , R 2 + R 3 , Tn > and

<To.Ro+R3,R1 +R2,Tn>

(the latter two for odd n only).

3. So+ s2 EH' Tn eH.

Since So + S 2 E H , H contains the following subring of rank 3: < T 0 , So + S 2 - T o , S 1 + S 3 >. The fact that < T o , So + S 2 - T o , S 1 + S 3 > is a subring follows directly from the equality < T o , So + S 2 - T o , S 1 + S 3 > = < T o, a(1), a(-1) >. So we have 3 ~ rank (H) ~ 5. If rank(H) = 3 then H=<T0 ,So+S2-To,S 1 +S3>.

Suppose that rank(H) = 5. Then H has nonempty intersection with the subspace generated by the vectors a(i), Tn. This means that there are complex numbers A.1 , "-2 such that A.1 a(i) + "-2 Tn E H, and at least one "-i· i = 1,2, is nonzero. Since Tn ~ H we have A.1 ~ 0 and we can assume that H contains an element of the form ~ = a(i) + A. Tn =So+ i S 1 -S2 -i S3 + A.Tn. In addition, since H is an S-ring we have ft...(~) = Tn + L sk EH. Since Tn ~ H, Ä. E {i, -i, 1, -1}. Hence ~= a(i) + A.Tn EH

1\:=i"

where A. e { i, -i, 1, -1}. Thus ~2 is also contained in H. We have:

~2 = [a(i) + Ä. Tnf = (2it Tn + 2 Ä. in a(-i) + A.2 T o =

= (2i)n Tn + 2 Ä. in(S o + i S 1 + S 2 + i S 3) + A.2 T O·

Since n > 1 and I A.l = 1, I (2i) In= 2n > I 2"-in I = 2 and hence Tn E H, a contradiction. So rank(H) ~ 5.

Now let us consider the case rank(H) = 4. In this case H should contain R j for some j. Since T 0 , a(l), a(-1) EH, and the vectors T 0 , a(-1), a(l), Rj are linearly independent over (;, the vectors T 0 , a(-1), a(1) and Rj form a basis of H. Moreover, it follows from the equality Rj =Sj-'6o,j To -'6j,n Tn* = [(-i)j a(i) + (-1)j a(-1) + ij a(-i) + a(l)]·l. -'6o ·T0 -'6· • T that the vectors

4 ,J J,n n

(-1)-i a(i) + ij a(-i) To, a(-1), a(1), ~= 4 -'6j,n* Tn also form a basis of H (here

n* = n (mod 4), 0~ n* ~ 3). From formulas (5.3) it follows that the subspace with basis T 0 , a(1), a(-1), ~ is a subalgebra if and only if

THE SUBSCHEMES OF THE HAMMING SCHEME

~2 = ~ T 0 + A.1 a(1) + Lt a(-1) + A.~ ~

Let us calculate ~2 .

2n+l -t Öj,n* [(-i)iina(-i)+ij(-ita(i)]+[Öj,n* +16]To.

201

Hence the element 11 = (~it (-1Y [1 + (-1)n] Tn -t Öj,n* ij+n X

x [(-1Y a(-i) + (-l)n a(i)] is also contained in H. Since 11 is a linear combination of the

vectors T 0 , a( 1) , a( -1) , ~. we conclude that 11 = A. ~ for suitable A. e C. This implies that

the rank of the matrix

(-1)n ij+n

2 (-10+1 _ij+n Ö· * (2i)n (-1'; [1+(-1)]n

Öj,n* ,- 2 J,n 16 ,-

HY 4

-Ö· * J,n

is equal to one. By calculating the determinant of the corresponding submatrix, we come

to the following equality:

· ij+n 2 (2it ij · · (-1)' - 2- Ö j,n* - 4. 16 • (-1)' (1 + (-1t) = 0.

2n Hence Öj,n* = öJ.n• = 25 (1 + (-l)n). But since n ~ 9, n must be odd and Öj,n* = 0. So

the vector space with basis T 0 , a( 1) , a( -1) , R j is a C-algebra if and only if n is odd and

j =1. n (mod 4). This implies that each S-subring H c W which contains S 0 + S 2 and

does not contain Tn coincides with one of the following subrings (this list does not

depend on J):

<To,Ro+R2,R 1 ,R3+Tn>, n = 3(mod4);

<To,Ro+R2,R3,R1 +Tn>, n = 1(mod4);

4. S 0 + S 2 e= H , T n E H.

It is clear that in this case H coincides with one of the following submodules:

<T0 ,Ro+R 1 ,R2 +R 3 +Tn>; <To,Ro+R3 ,R 1 +R 2 +Tn>;

<To,Ro+R,+Tn,R2+R3>; <To,Ro+R3+Tn,Rt+R2>· So we should only

202

check which of these submodules are subalgebras. It follows from transforming the first basis into the second H = < T 0 , Ri1 + Ri2 , Ri, + Ri4 + Tn > = < T 0 , o.(1), ~ > where

M E. MUZICHUK

the formulas one that

o.(i) + i o.(-i) + (1 +i) (Öo,n* + Öt,n*- Ö2,n* -Ö3,n* -l)Tn, if {i l•i2} = {0, 1};

o.(i) + i o.(-i) + (1 + i) (8o,n* + Öt,n* - Öz,n* - ÖJ,n* + 1) Tn, if {i 1 .iz} = {3,2} ; ~= o.(i)- i o.(-i) + (1- i) (Öo,n* - Öt,n* - Ö2,n* + ÖJ,n* -1) Tn, if {i t.iz} = {0,3} ;

o.(i)- i o.(-i) + (1- i) (Öo,n* - Öt,n* - Öz,n* + ÖJ,n* + 1) Tn, if {i 1 .iz} = { 1,2}.

The subspace < T 0 , o.(1), ~ > is a subalgebra if and only if ~2 e < T o, o.(1), ~ >. Since ~ = o.(i) + (-1l i o.(-i) + A.Tn, k = 0, 1, we have ~2 = [(2i)n- (-2i)n] Tn + 2iA(-1)k (-i)n o.(i) + 2 A in o.(-i) + ['A2 + 2i(-1)k 2n] T 0· Hence the vectors [(2it- (-2it] Tn + 2i 'A(-1)k (-it o.(i) + 2 A. in o.(-i) and o.(i) + ( -1 )k i o.( -i) + A. T n are linearly independent. So we have the following equalities: 2iA(-1l (-i)n A = [(2i)n- (-2i)n], 2 A.i(-1)k (-i)n • i(-1)k = 2 in A.; or equivalently 2A.2 i(-1)k+n=2n-(-2t,2A.(-1)n+l=2A.. Since IA.I < 11±i 1·2=4 and n ~ 9, I A. 12 ~ 16 < 29 + 29 . Hence n is even and A. = 0. Thus the submodule H is a subalgebra (i.e. an S-ring) if and only if n is even and A. = 0. This implies that si,,n* + si •. n• = 1.

Finally we conclude that in the present case W contains the following subrings: <To,Ro+R 1 ,R2+R3+Tn> for n = 2(mod4); <To.Ro+R3,R 1 +R2+Tn> for n = 2(mod4);<To,Ro+R3+Tn,R 1 +R2>forn = O(mod4). D

6. The subrings with basis set {2}, {1,2}

Thus far we have enumerated all S-subrings of the S-ring W for n ~ 9. It follows from Corollary 4.2 that all remaining subrings have a basis set I such that { 2} c;;;. I c;;;. {1, 2, n - 2, n - 1, n } . The enumeration of these subrings will be carried out in the following way. First we will find all the subsets I , { 2} c;;;. I c;;;. { 1, 2,n, n - 1, n -2}, such that the equality A.2 (I ; n) = A.j(l , n) holds for all j e I. This can be done by a direct search. It turns out that there are exactly six such subsets: ( 2}, ( 1,2}, ( 2, n- 1), {2,n- 2), ( 1,2,n -1,n- 2}, { 1,2,n -1,n }. Hence a subring H which is not contained in W has a basis set from this list. So to describe the remaining subrings one should determine, for each of the above subsets, all subrings which contain this subset as a basis set.

Proposition 6.1. The S-rings numbered 10,11 if n is even, or 10 if n is odd, exhaust the S-rings with basis set {2}.

Proof Let H be a subring with basis set I = { 2}.

It follows from the equality


(6.1)

that T 2i is contained in H for all i , 0 ~ i ~ [ n /2]. Hence { 2i } is a basis set of the subring

H for all 0 ~ i ~ [n /2]. Now suppose that 11 ,12 , ... , ls are the remaining basis sets of

the ring H. It is clear that the ft are odd sets. Wehave the following three possibilities:

1. n is even and each Ij is of the form Ij = { k, n - k} for suitable k. (Note that when

k = n/2, we get/j = {n /2}.)

2. There exists a set Ij consisting of one element k , k "* n /2.

3. There is a set Ij containing two numbers k and m such that k + m "* n.

In the first case the ~ -module with basis T{ = T 2i , 0 ~ i ~ [n /2],

H'!!,.1+j=T2j+1 +Tn-(2j+1), O~j~ n- 2 , is anS-ring. This fact can be checked by 2 4

direct computation.

In the second case H=Hn·[:-~~l~,if Tk[~+~]' k = 1 (mod 2) and k *n/2 then

equality (6.1) and the inequality 2 _ "* 2 together imply that Tk+2 and Tk-2

are also contained in H. By means of inductive arguments, we obtain T 1 E H. Now, since

T 1 generates Hn, we have H = Hn.

Let us consider the third case. Let I be a basis set containing 1. If I = { 1} then we

are in the second situation and hence H = Hn. If I= { l,n -1} then, starting with k = 1,

we[~a~;~~Jute the products T2(Tk +Tn-k) = [k+ 2]

= 2 • (Tn-k+2 + Tk-2) + k(n -k) (Tk + Tn-k) + 2 (Tk+2 + Tn-k-2)

(k odd). If we compare the coefficients of Tn-k+2 + Tk-2 and Tk+2 + Tn-k-2· we obtain

Tk + Tn-k E H for any odd k. In this case n should be even since, otherwise, n- 1 is even

and hence T 1 E Hand H = Hn. Thus each odd basis set of the subring His contained in

the set { k, n - k} for suitable k. So we arrive at situations which were treated above.

Now suppose that I contains a value m distinct from 1 and n - 1. Since I is a basis

set, the equality A.1({2},I;n)=A.m({2},/;n) holds. If t is not contained in the set

{i- 2, i, i + 2}, then Y:z.i = 0. Hence

and

Am({2},I;n)= L 'YT.i= L yT,i;::rT,m=m(n-m) i EI i e {m-2,m,m+2) n I

1...1({2},/;n)= L YL;~Yl2=n-1. ie{l,2}ni

As m "* 1, n-1, we havem(n-m) > n -1., and so 1...1 ({2},I ;n) =Yl.2 +yb = [ ;] .

204 M. E. MUZICHUK

But then I: 'YT.i is also equal to [~]· Since 'YT.m-2 + 'YT.m + 'YT.m+2 = [~] , i e {m-2,m,m+2} ()I

we have m - 2, m + 2 e I. By repetition of this argument we obtain I= {1,3, ... ,2[n/2]-l}. o

Proposition 6.2. The subrings with basis set { 1,2} are the S-rings numbered 11 for n even, and 12 for n odd.

Proof. Let H be a subring with basis set {1,2}. By induction on i we showsuch a subring contains the elements T2i + T 2i-1 for all i, 1 ~ i ~ [n/2]. By a direct computation of the product (T 2; + T 2i _1) (T 2k + T 2k_1) it can be verified that the values T 2i + T 2i_1 for 1 ~ i ~ [n/2] and n even, and T 2i + T 2i-1 , Tn for n odd, form a subring Ho of the S-ring Hn. Hence H;;;). HO· If H-::::> Ho, then T2;, T2i-1 are contained in H for some i. Hence T 2i • T 2i-1 E H. But 'Yii. 2i-1 ;e 0 while Y1i, 2i-1 = 0. This is impossible since T 2 + T 1 is a basis value. Hence H =HO· 0

7. Completion of the proof: n even

To complete the proof of Theorem 2.1 we must consider four possibilities, namely when I is one of the sets {2,n-1}, {2,n-2}, {1,2,n-1,n-2}, {1,2,n-1,n}. In this section the case when n is even will be considered.

Proposition 7.1. If I= {2,n -1} or I= { 1,2,n,n -1}, then the corresponding S-ring isthat numbered 13 or 14 in the first list.

Proof. Let I be a basis set of the S-ring H which contains 2. Let us first consider the case I= {2,n-1}. By induction on k it can be proved that H contains the values Tk + Tn-k+1 for all k. Now by direct computation it can be checked that the Z -module with basis Tk + Tn-k+1 , T 0 is an S-ring. Hence H;;;). H 0 , and it follows from Corollary 3.4 that H = H 0 • In fact let I be a basis set of the ring H. Then I~ {k,n + 1-k} for suitable k. Since l·di; n) = An-1 (I; n), I satsisfies the hypothesis of the Corollary 3.4 and hence I = { k, n + 1- k } .

Now suppose that I= { 1,2,n,n -1 }. By arguments analogous tothat of the previous case, it can be shown that H contains the S-ring Ho= < T o. T 1 + T 2 + Tn + Tn-1, T 3 + T 4 + Tn-2 + Tn-3, ... >. Suppose that H -::::>HO· Then there is a basis set I which is a proper subset of one of the following sets: Sk = { 2k, 2k- 1, n + 1 - 2k, n + 2- 2k},

S (n+2)t4 = {; + 1, ; }. Since A.2(I; n) = An-1 (I; n), the set I satisfies the hypothesis of

Corollary 3.4 and hence is one of the sets: {2k, n + 1- 2k}, {2k -1, n +2- 2k} (if

k ;e n : 2 ). But for any of these sets, 0 = /..1 (I ; n) ;e /..2 (I; n) > 0, a contradiction. Hence

n+2 H = H 0 . The case k = - 4- can be treated analogously. 0

Proposition 7.2. There are noS-rings with basis set { 1,2,n -1,n- 2}.

Proof. Let H be such a subring. Since

[(To+Tn)(Tl +T2)f=2(To+Tn) ([ n; 1] To+2(n-1) (Tl +T2)+6(T3+T4)), we

conclude that (T o + Tn) (T 3 + T 4) = T 3 + T 4 + Tn-3 + Tn-4 E H. By induction on k it can

be shown that (T o + Tn) (T 2k + T 2k-l) = T 2k + T 2k-l + Tn-2k + Tn-2k+l is contained in H

for all k, k:::; ~. Consider k = ~. Then Tn + To + T 1 + Tn-l E H. On the other hand,

since I is a basis set, In{0,1,n,n--1} is either empty or equal to I. But In {0, 1,n -1,n} = { 1,n -1}, a contradiction. Hence S-rings with basis set

{1,2,n -1,n- 2} do not exist if n is even. D

Proposition 7 .3. If I = { 2, n- 2}, then the corresponding S-ring coincides with one

of the subrings numbered 15-17.

Proof. Let I be a basis set of the subring H. In this case H contains the simple

values T 2k + Tn-2k for all k, 0:::; k:::; [n /4], and Tn. These values are basis values, i.e.,

they cannot be presented as sums of two simple values of H. Indeed, H would otherwise

~~=n< ili; ::::: :.::.::·: <fo; ::~kk<* ;; ~ :• ::.:: :~ e~[;k ~ f .:d m

'f2k:1k ={ 0[~ ~~4lk < n- 2; 2k , if4k=n-2,

we have -?zk. 2k > 'fz.t:1k- B ut I = { 2, n - 2} is a basis set, so the lauer inequality is impossible.

Let I be the basis set which contains 1. Then I is an odd set, i.e.

I !;;;; { 1, 3, 5, ... , n - 1}. Since T n E H, Tn • T1 = Tr where I' is also a basis set of the subring H. Hence either I' n I = 0 or I' n I =I. Set

- {I, ifi=I', I = I u I' , if I "*I'.

It is clear that Tn Tj = Tj. Moreover the product

(T 2 + T n-2) Tj = T 2 (T o + T n) Tj = 2 T 2 Tj is also contained in H. In addition, it is easy to show that the equality T 2 Tj = A. Tj + ~ holds where ~ E H and ~ o Tj = 0. This means that

- - -for all i E I the equality A.1 ( {2}, I)= A.;({2}, I) holds. It follows from Proposition 6.1 that

there are exactly three possibilities for the set I; namely { 1}, { 1,n -1}, and

{ 1, 3, ... ,n- 1}. The first case is impossible since I'=/. In the second case if, I' n I = 0

then T 1 E Hand H = Hn, while if I'= I then His the S-ring with basis values Tk + Tn-k

206 M. E. MUZICHUK

for k!! {O,n}. To and Tn.

Let us consider the latter case: I= { 1, 3, ... , n- 1}. If I'= I then I = { 1, 3, ... , n- 1},

and His the S-ring with basis sets {2k,n-2k}, 0 < k < ~, {0}, {n} ({n/2} if n = 0

(mod 4)) and { 1,3, ... ,n -1 }.

Now suppose that I' n I = 0. Then I I' u I I = 21 I I = ~ . Hence n is divisible by 4

and II'I=III=E... Let us show that I={1,5, ... ,n-3}. Since I'ni=0 and 4

Iul'={1,3, ... ,n-1}, it is sufficient for this purpose to prove that kE I implies {k+2, k-2} <;;;;.I'. The sets {2,n-2} and I are basis sets of the S-ring H, hence /..,z (I; n) = 'An-Z (I; n ). Let k be an arbitrary element from I n [3, n- 3]. Then the numbers -/n::J_z,n-3 (n- 2) and -/n=ln-3 (n- 2) are nonzero. Hence, by Lemma 3.3, the intersection of I with each of the sets { n - k - 2, n - k }. { n - k + 2, n - k } is nonempty. Since n-k!! I, {n-k-2, n-k+2} <;;;;.I. Hence {k-2,k+2} <;;;;.I' and the claim is proved.

Finally, if n is even and I= {2,n -2}, then H is one of the following subrings:

< To, Tl+ Tn-1• Tz +Tn-Z• ... , Tn >, < To, Tz +Tn_z, T4 + Tn-4• · · ·, T1 +T3 + · · · +Tn-1 >; <To, Tz+Tn_z, T4+Tn-4, ... , T1 +Ts+ · · · +Tn-3• T3+T1+ · · · +Tn-1 > for n = 0 (mod 4). D

8. Completion of the proof: n odd

Proposition 8.1. There are exactly two S-rings (numbered 12 and 13 in the second list) with basis sets {2,n- 2} and { 1,2,n-2,n -1}, respectively.

Proof. Let H be an S-ring and I its basis set containing 2. Suppose first that I= {2,n- 2}. In this case H should contain all simple values of the formT Zk + Tn-Zk for

1:::; k:::; n ~ 1 , and Tn. It is easy to check that the ~ -module Ho with basis

To,Tz+Tn_z, ... ,Tn_1+T1 is closed under multiplication, i.e. that Ho=<To,Tz+Tn_z, ... ,Tn-l+Tl> is anS-ring. It is clear that H;;;;J.Ho. Let us show that H coincides with H 0 . Indeed, if H ::J H 0 then H contains the value T Zk for

some k , 1::;; k :::; n ~ 1 . Since 0 < Y~k, Zk #- rlk:~k = 0, this implies that { 2, n - 2} cannot

be a basis set, a contradiction. Hence if n is odd there is a unique subring with basis set {2,n-2}. namely < To, Tz +Tn_z, T4 +Tn-4• ... , Tn-! +T1 >.

Now suppose that I= {1,2,n -1, n -2}. By induction on k, it can be shown that H

contains the simple values of the form T 2k-l + T 2k- Tn-Zk+l + Tn-2k for all k,

1:::; 2k::;; n ~ 1 , and Tn. In addition, a direct check shows that the ~ -module H 0 with

basis Hk'=T2k-1 +T2k+Tn-2k+Tn-2k+l is anS-ring. Hence H;;JHo. Suppose that H ::::> H 0 . Then there exists a basis set I of the S-ring H such that Ic{2k,2k-1,n-2k+1,n-2k}. The set {1,2,n-1,n-2} is a basis set, hence I sarisfies the hypothesis of Corollary 3.4. So I coincides with either I 1 = {2k -1, n- 2k} or I 2 = {2k,n- 2k + 1 }. But since ldlt; n) > 0 and 1..1 (/t; n) = 0, I cannot be a basis set. This contradicrion proves that H = H 0 and we are done. 0

Proposition 8.2. There are noS-rings with basis seti = { 1,2,n -1,n }.

Proof. Let H be anS-ring with basis set I= { 1,2,n -1,n }. Since H contains the element T 1 + T 2 + Tn-l + Tno it can be shown by inducrion on k that H contains the ele

n-1 ments T 2k-l + T 2k + Tn-2k+l + Tn-2k+2 for all k, 0 ;S; k ;S; - 2-. But then H contains the

simple value T n-2 + T n-1 + T 2 + T o (norice that n- 1 is even). { 1,2,n -1,n} n I is neither I nor 0, this implies that I is not a basis set.

Now since 0

Proposition 8.3. There are exactly three subrings (numbered 14-16 in the second list) with basis set I= {2,n -1 }.

Proof. Let H be an S-ring with basis set I = { 2, n - 1}. lnducrion on k implies that

T2k+Tn-2k+l eH for all k;S; n; 1 • Let us consider the cases Tn eH and Tn EH

separately.

If Tn e H, then it is easy to check that H = < To, T2 +Tn-1• ... , Tn-2 +T1, ... >. Now suppose that Tn E H and that J is the basis set of the S-ring H, containing n. If J = { 1,n }, then H = < To, T 1 +Tn, T3 + Tn-2•· .. >. Let J contain an element i, disrinct from 1 and n. We show in this case that 1={1,3, ... ,n}. Since J is a basis set, A.i(I, J; n) = /..n(I, J; n), and it follows from the relarions

A.i(I, J; n);::: YL = i(n -i),

i/,j = [;] Öi+j,n ,

thatJ n {l,n-2} ;t:0.

If J () {l,n- 2 I = {1 I, then A,. (I,J ~) = [;] < A.(I,J ; n ), ~~ the !atteds impos~ble. If J n {l,n -2} = {n -2}, then 1 e J = { 1,3, ... ,n} \J. But J is a union of basis sets - - -and hence ~(J; n) = An-1 (J; n), i.e., J sarisfies Corollary 3.4. So the intersecrion of the sets J and { n, n - 2} is nonempty. This is impossible since { n, n - 2} is contained in J.

Let us consider the final possibility: Jn {l,n-2}={1,n-2}. In this case

A.i(I,J; n) = /..n(I,J; n) = [;] + [ ~ l· On the other hand,

208 M. E. MUZICHUK

Ai(/,1; n) = L ~.j + L -ln-1.} = jeJ jeJ

= L rL + L in-z.j· je (i-2,i,i+2)nJ je (n-i+l,n-i-l)nJ

These equalities are valid, since ~.J '# 0 ~ I i - j I :::;; 2 and

Y~-l,J '# 0 ~ I i + j- n I = 1. But the right side of the latter equality is equal to

[;] + [ ~] if and only ifl contains the subset {i- 2,i,i + 2,n -i + 1, n- i -1}. By repeti

tion of these arguments for the elements i-2,i,i+2,n-i+1,n-i-1 (and so on) we obtain I= { 1,2, ... ,n }. This proves the proposition. D

The proof of Theorem 2.1 is now complete.

References

1. P. Delsarte, An a/gebraic approach to the association schemes of coding theory,

Philips Research Reports Supplement, No. 10, 1973.

2. LA. Faradzev, M.H. Klin, M.E. Muzichuk, Ce/lular rings and groups of automor

phisms of graphs. [In this volume].

3. I. Schur, Zur Theorie der einfach transitiven Permutationsgruppen, S.-B. Preuss Akad. Wiss., 1933, S. 598-623.

4. V.A. Ustimenko-Bakumovskii, Strongly regular graphs which are invariant under

the group Ym i Ynfor m:?: 3. In: Computation in Algebra and Combinatorics, Kiev, Institute of Cybemetics, 1978, p. 101-113 [In Russian].

5. V.A. Zaichenko, An algorithmic approach to the synthesis of combinatorial objects

and computation in permutation groups based on the method of invariant relations,

Ph. D. Thesis, 1981, Moscow, MFTI [In Russian].

This paper was originally published in IATC0-85, pp. 49-65.

A DESCRIPTION OF SUBRINGS IN V(SPI X SP2 X ••• X Sp,)

JaJu. Gol'fand

1. Preface

The construction of V-rings of permutation groups and a description of their subrings are very important in the development of permutation group theory as well as combinatorics and graph theory. One can see about these connections in [1,3,6].

V-rings of primitive permutation groups usually have only a few subrings (see, for example, [3,4]). But in the case of imprimitive - and especially intransitive - groups the lattice of subrings can be very rich. In our opinion a description of this lattice of subrings is a very interesting problem.

In this paper we consider subrings of the V-ring of the group G = SPI X SP2 X ••• sp .. of degree n = Pl P2 ... Pm, where Pi's are distinct prime

numbers and sp, is the Symmetrie group of degree Pi· This problern is very interesting

because of the fact that the subrings of V(G) are in one to one correspondence with the traces of S-rings of Zn (see definitions in [1]). Having an exhaustive description of Srings of Zn, one can solve the isomorphism problern for cyclic graphs on n vertices in the case when n is square-free. More detailed information on this theme can be found in [5].

The description of cellular subrings of V ( G) is obtained in terms of finite topologies (in other words - relations of quasiorder) on the set M = { 1 , 2, · · · , m}. It tums out that the lattice of subrings of V(G) is isomorphic to the lattice of all topologies on M. Elements of the basis of the subring which corresponds under this isomorphism to the topology T are in a bijection with the closed sets from T. These results are contained in three theorems stated in section 7. The theorems are proved in sections 8-12. The algebraic structure of the ring V(G) is described in sections 2, 3. These sections also contain a formulation of the problern of subrings enumeration. Some important properties of subrings of V(G) are established in sectiom: 4, 5. Section 6 contains general facts about finite topologies which are necessary for our purposes. Finally section 13 contains results conceming the automorphism groups of the subrings being constructed. It is proved that these groups are wreath products of symme1ric groups over the quasiorders related to the topologies; the subrings are V-rings of their automorphism groups.

All elementary facts conceming cellular rings which are used in the paper can be found in [1].

209

210 JA. JU. GOL'FAND

2. The algebraic structure of the ring V(Sp, x Sp 2 x · · · Sp,.)

Let Sq be the symmetric group of degree q. The ring V(Sq) has a basis which consists of two elements x 0 , x 1 where x 0 is the reflexive relation and x 1 is the complete antireflexive relation (the empty and the complete graphs on q vertices, respectively). The ring V(Sq) is a commutative ring, x0 is its identity element, and xy = (q- 2)x1 + (q -1)xo (equivalently, (xo +x d = q(xo +xt)).

Let (Gi, Xi), i = 1, 2, be permutation groups and let (G, X)= (G 1 x G2, X 1 xX 2) be the direct product of G 1 and G 2 acting on the Cartesian product of the sets X 1 , X 2. Then it follows from [6] that V(G) = V(G 1) ® V(G2), where ® denotes the tensor multiplication of cellular rings. Recall that if V ( G 1) = < 'I' 1 , 'I' 2 , · · · , 'I' k > and V(G2) = < 'l'1, 'l'2, · · · , 'l'1 > then V(G1)®V(G2)=<'l'i®'I'i I i=O, 1, ··· ,k,j=O, 1, ... ,l>. Here 'I'i®'I'i is the graph with vertex set X 1 x X 2 and edge set Ei x Fj where Ei, Fj are the edge sets of 'Pi, 'I'j respectively. Multiplication in V(G) is defined by the following rule: ('I'i ®'I'j)· ('l's ®'I't) = 'I'i 'I's ® 'I'j 'I't.

Later we will denote by BR a set of basis elements of a cellular ring R.

Lemma 2.1. LetRo = V(Sp, xSp 2 x · · · xSPm). Then

a) R 0 = V(Sp,) ® V(Sp) ® · · · ® V(SpJ;

b) R o is a commutative ring with identity element 1 and generators x 1 , · · · , Xm satisfying the relations

(2.1)

c) The set B R 0 consists of the elements of the form XA = rr Xj where A is any Subset of ieA

M = { 1 , 2, · · · , m}; for A = 0 we put x 0 = 1. In particular dim R o = 2m and

L != I, XA=(l+x1)(1+x2) ··· (1+xm). (2.2) feBR, A~M

Proof. a) follows immediately from the above description of V ( G 1 x G 2 ). b) and c) follow from a) and the description of V(Sp) since x 0 ® x 1 is denoted by Xi Xj and the reflexive graph of the ring R 0 is denoted by 1. o

3. The problern

Let ft , h be polynomials which are the sums of distinct monomials XA. We denote by f 1 A h the sum of monomials which are members of both f 1 and f 2·

Having in mind the algebraic structure of the V-ring R 0 described above, we can formulate the problern of describing all cellular rings R < R 0 in the following way:

A DESCRIPTION OF SUBRINGS IN V(Sp 1 x Sp2 x ... x SPm) 211

Find all possible systems B R = {.fi } of polynomials in the variables x 1 , x 2 , • • · , Xm satisfying the relations (2.1 ), such that:

(3.1) Each polynomial fi e BR, i :2: 1., is a sum of distinct monomials XA, A !::: M; fo = 1 E BR.

(3.2) If.fi ,/je BR, i ::l=j, thenfi A!; = Oand

L /=(1+x1)(1+x2) .. · (1+Xm)· feBRo

(3.3) The elements of BR form a basis of an algebra (over the ring Z of integers): fi /j = L ct fk , cfj e Z , ct :2: 0. In this case we will write R = < 1 , f 1 , · · · , fk >

k

where fi are the elements of the basis.

Example 3.4. Let us consider the case m = 2. We have R o = < 1 , x 1 , x 2 , x 1 x 2 >. There are four possibilities for subrings R < R o:

It is not difficult to prove that R 1 , R 1 , R 3 are really subrings of R o while in the case of R 4 we have

Since the coefficients of x 1 and x 2 are distinct, x 1 + x 2 does not belong to the basis. So R o has only three proper subrings: R 1 , R 2 , R 3.

Before stating the solution of the problern in the general case, we will state some important properties of the subrings R < R 0 .

4. The operation f ~ /P•l

Let us define an endomorphism f ~ f[p.J of the ring R o by the equalities: x~•1 = Xi

if i ::!=k, and xr•1 =-1. Let f, g e Ro. We will write f = g (mod p) if f- g = ph ' h E R 0·

Lemma4.1.Letfe Ro. Then/'• = t[p•l (modpk).

Proof. It is sufficient to check the statement for the generators Xi. It follows from the equality (1 + x;)2 = Pi(l + xi) that (1 + xiY' = pfP-1> (1 + xi)· On the other hand (1+xi)P = 1 +xf (modp). Hence

xf = 1 +pf-1 (1+xi) (modp).

Letp =pk ~Pi· Using the fact that for all a ;;f: 0 (modp), aP-l - 1 (modp) (Srnall Fermat theorem from elementary number theory), we obtain

xf" = Xi =xY'"1 (modpk).

For i = k we have ~"-I = 0 (mod Pk). So~" = -1 = xr"1.

Let R < R o be a cellular subring of R o and B R the basis of R.

Lemma 4.2. The operationf ~ :/" is an endomorphism onR.

0

Proof. We show that f[p"l e R for each f e BR. Let f e BR. Then :/" e R, :/" = 'L ci fi, fi e BR, ci e Z. ByLemma 4.1,:/" =/[p"l + h where h e Ro, h = 0 (mod Pk). Since the coefficients of the polynomial f[p"J are equal to ± 1 and Pk divides the coefficients of h, we have :/" e R. 0

5. The properties (C 1) and (C 2)

Let f e R 0 • For a fixed k, 1 ~ k ~ m, any monomial of feither does not contain Xk or contains xk to the first power. Hence f can be uniquely represented as f = fi + xk fz where the polynomials fi, fz do not depend on Xk. We will use the expression at 1 i'Jxk =fz.

Let R() derrote the set of all polynomials with nonnegative coefficients. Let us construct a quasiorder onRo by f~ g ~ g- f eR().

Lemma 5.1. a) Let fi,gieRÖ.fi~gi for i=1,2. Then ft+fz~gl+g2, fi fz~ gl g2. b) LetA c.Mandi e A. Then 1~xi and, ifpi ~2. thenxi~xi,

Proof Statement a) can be checked directly. For b) we have that xi is the product of (pi- 2) Xi + (pi -1) for all i e A. So the Statement holds. o

Let RÖ c. R() derrote the set of all polynomials with coefficients 0, 1. Below we will often use the following obvious property of cellular rings.

Lemma 5.2. Let R < R o be a cellular ring, f e R. Let f = c 1 !I + c 2 fz + · · · + c k !k where .fi e Rb for 1 ~ i ~ k and, for 1 ~ i < j ~ k , Ci and CJ are distinct integers and fi 1\ t1 = 0. Then fi e R for all 1 ~ i ~ k. o

As a corollary of Lemma 5.2 we have the following lemma. Let/= 'L cA xA, cA e Z. Putr = L d xA where d =max(cA, 0).

A~M A~M

Lemma 5.3. If f E R then r E R. 0

The following important property is proved using the operation f ~ f[p 1.

Lemma 5.4. (The property (C 1)). Let je BR, R <Ro. Let f=!I +xkf2 where fz = at I ()xk ~ 0. Thenft ~ fz.

213

Proof. Let cp = /[p1l = (fi + xkh)[p1l = h - f2. Then cp e R and by Lemma 5.3 we have q,+ e R. On the other hand, q,+::;; h whence q,+ < f = fi + xkh· Now f e BR forces q,+ = 0. Henceh ::=;f2, 0

Lemma 5.5. Let f E B R , cp < f, d f I d Xk '# 0, d cp I d Xk = 0. Then Xk cp < f.

Proof. Wehave f=h +xdz,cp<f,dcpldxk=O. So <P<fi. By (C1)f1::;;h holds, hence cp < h and xk cp < Xk fz < f 0

As a corollary we have

Lemma 5.6. Let f E B R and suppose there exists A <;;;; M such that a f I a Xj '# 0 for ieAanddjldxi=Oforie;A. ThenxA<f 0

We need another one property of the elements of B R.

Lemma 5.7. (The property (Cz)). Let m ~ 3 and let x, y, z e {x1, Xz, · · · , Xm} be distinct variables. Let f e B R and cp a monornial in the variables Xi :t- x , y , z such that (x +yz) cp::;; fholds. Then either y <1J < f or z cp < f

Proof. Suppose to the contrary that neither y cp::;; f nor z cp::;; f. Then there are g,heBR suchthat ycp::;;g, zcp::;;h and f=t-g,h. Let us show that dg I dx =dh I dX =0. lndeed, if ()g I dX :t-0 or dh I dX =t-0 then, by Lemma 5.5, xy cp < g or xz cp < h. On the other hand x 4> < (x +yz) cp::;; f, so by Lemma 5.5, xy cp < f and xz <IJ < f. Then we have either f A g :t- 0 or f A h :t- 0. This contradicts the conditions f :t- g , h and f, g , h e B R.

Therefore a g I a X = d h I d X = 0. Let g 1 ' h 1 E B R such that yz ::;; g 1 and <P::;; h 1. By Lemma 5.l(b) yz::;; yz cp2 ::;; gh. This implies that d g 1 I dX = 0. If d h1 I dX :t- 0 then h 1 ~ x cp and hence h 1 = f. In this case /~ y<!J, z<IJ. So we can suppose () h 1 I () x = 0. Then yz cp < g 1 h 1> d(g 1 h 1) I d X = 0 and f A g 1 h 1 = 0 since f E B R and d f I d X '# 0. This is a contradiction to yz cp::;; f. 0

6. Finite topologies

The structure of a topological space on a finite set M can be defined in several equivalent ways. As an original one we will use the following definition.

Definition 6.1. A topology T on a finite set M is a collection of subsets T = {V I V c M} such that 0 e T and

1) u V=M; VeT

The elements ofT will be called closed and the Operation U ~ U = n V will be Ur;;;,VeT

called the closure of U <;;;; M. It follows from the definition that a closed set coincides

with its closure and that U 1 u U 2 = U 1 u U 2· U 1 n U 2 = U 1 n U 2·

Another definition of topology is formulated in terms of quasiorder relations on M.

Recall that a quasiorder is a reflexive and transitive binary relation. For a topology T let

us define a relation $ by the rule u $ v ~ v e Ii. It is not difficult to check that $ is a

quasiorder induced by the topology and conversely the quasiorder $ determines the topology: V e T ~ (u e V, u $ v :::> v e V). Let u - v ~ u $ v , v $ u. It is easy to see that - is an equivalence relation and hence M is the union of equivalence classes

u 1 , u 2, ... , uk, ui n uj = 0. These classes will be called components and the set of all components will be denoted by C(T).

The quasiorder $ induces a partial order on C(T): U 1:::;; U 2 ~ U 2!:;; U 1,

U 1 , U 2 e C (T). So we can graphically express a topology as a directed acyclic graph. Vertices of the graph are components from C (T) and arcs are pairs ( U 1 , U 2 ), U 1 < U 2. Closed sets correspond to sets of vertices with no arc coming out of it.

Example 6.2. Let us consider the topology Ton 6 points in Figure 1. Here

C (T) = {{ 1 , 2} , { 3 }, { 4} , { 5 , 6} } ,

T={{1,2,4,5,6}, {3,5,6}, {4}, {5,6}, {1,2,3,4,5,6}}.

We will need the collection N(T) of all subsets of M which are unions of pairwise

incomparable components.

Example 6.3. For the topology T of Figure 1

N (T) = { {1 , 2} , { 1 , 2 , 3} { 3} { 3 , 4}, { 4} { 4 , 5 , 6}, { 5 , 6} } .

Fig. 1 Fig. 2

Example 6.4. There are only four topologies on the set consisting of two elements. These topologies T o, T 1 , T 2, T 3 are represented in Figure 2. Hen~

T o = {{ 1 } , { 2} , { 1 , 2} } , (l} = { 1 }, (2} = { 2}, C (T o) = T o , N (T o) = T o ;

T 1 = {{ 1 , 2} , { 2} } , (I} = { 1 , 2}, { 2} = { 2} , C (T 1) = { { 1 } , { 2} }, N (T 1) = C (T 1) ;

T2={{1,2}, {1}}, (1}={1}, (2}, {1,2},C(T2)={{1}, {2}},N(T2)=C(T2);

A DESCRIPTION OF SUBRINGS IN V(SP1 X SP2 x ... X SPm)

Let us Iist some properties of finite topologies.

Lemma 6.5. a) 'itV e T, V= u Ui where Ui e C(T);

b) 'itV e T, V= Wwhere W e N(T);

c) Let V= W, W e N(T), U ~ W, U e C(T). Then V\ U e T.

215

d) Let V1,V2eT,V1(;;;;;V2,'V1=W1,W1eN(T). Then 3U(;;;;W1,

U e C(T), suchthat U ~ V2•

Proof follows from the graphical representation of a topology. 0

Definition 6.6. A topology T 1 is weak:er than a topology T 2· written T 1 ~ T 2· if V e T 2 for each V e T 1·

Remark 6. 7. Evidently T ~ T o for each T where T o is the discrete topology

({i} = {i} for alli e M).

7. Main results

W e will express the connection between the subrings of the ring R o and the topologies T ~ T 0 in the following three theorems.

ForA ~M setPA =TI (l+xi), P 0 = 1. ieA

Theorem 7.1. Let T ={V} be a topology on M. Let us associate with T the subspace R = r(T) over Z generated by the polynomials of the form

fv =Pv,w TI (Pu1 -1), fo = 1 (7.1)

where V= W, W e N(T), W = u Ui, Ui e C(T). Then R = r(T) is the cellular subring ofthe ringRo having basisBR = Uv I V e T}.

Theorem 7 .2. For each cellular subring R ~ R o there is a topology T = t(R) on the set M suchthat 1) r(t(R)) = R; 2) t(r(T)) = T.

Theorem 7.3. The lattice of topologies on M is isomorphic to the lattice of cellular subrings of R 0 , i.e.

Example 7.4. The rings Ro, R 1 , R2, R3 from Example 3.4 correspond to the topologies T o , T 1 , T 2 , T 3 from Example 6.4.

Example 7.5. The discrete topology corresponds to the original ring R 0 •

8. The proof of Theorem 7.1.

Let T be a topology. Let us checkthat the polynomials fv, V E T, satisfy conditions (3.1)-(3.3).

Lemma8.1.LetV1 '#V2, v1, v2 E T. Thenfv, 1\ fv2 =0.

Proof. Without loss of generality we can suppose V 1 q; V 2· By Lemma 6.5(d) 3Ue C(T), U~V1, uq; V2. By Lemma 6.5(c) V1 \Ue T. Then fv, =(Pu-l)f

where f is a polynomial in the variables Xi, i E V 1 \ U. So if XA :S: fv, for A ~ M, then

3iE u with OXAIOXi-#0. Since uq; v2 we have ofv21oxi=0Vie U, hence

fv 1 " fv 2 = 0 and the proof is complete. o

Lemma 8.2. Let V 1 be a closed set. Then

Py1 = L XA = L fv. A ~V1 V~V~>Ve T

Proof. It is sufficient to show that VA ~ V 1 , 3 V E T, V ~ V 1 such that XA :S: fv. Since V 1 = u ui, ui E c (T) (Lemma 6.5(a)) and A ~ V 1, we can find components

U 1 , U 2 , · · · , Uk E C (T) such that A 11 Ui '# 0 and A ~ W = u Ui. Then fW is the 1~ r$k

polynomial we are seeking. lndeed fW has the form fW = Pw\W ll /i, where each /i is 1~ i~ k

equal either to (Pu, -1) or to Pu,. Let Ai= A 11 Ui. Then XA = XA, XA 2 • • • xA., XA, < Ii, XA ~nfi <fW. 0

Let R+ (respectively, R+(Q)) be the set of allsums of polynomials fv, V E T, with nonnegative integer (respectively, nonnegative rational) coefficients.

Lemma 8.3. Let V 1 , V E T andf=fv, Pv. Thenf E R+(Q).

Proof. We proceed by induction on the number of components contained in V. If V=0 then Pv= 1 and f=fv, ER+. By Lemma 6.5(c) V can be represented as

V= U u (V\ U) where U ~V, V\ U E T, U E C(T). Hence by (2.1)

f = fv, Pv = c 1 (Pu -1) Pu\U fv, Pv\U + fv, Pv\U =

=c2fv,uuPv\U+fv, Pv\U

for some positive rationals c 1 and c 2 . The set V\ U E T contains fewer components than V does, hence f E R + (Q) by induction. 0

Proof ofTheorem 7.1. Property (3.1) follows immediately from the definition of fv

given by formula (7.1).

Property (3.2) follows from Lemmas 8.1, 8.2 for V1 =M. In order to verify (3.3) we have to show that V V 1 , V 2 E T, fv 1 fv 2 E R +. For A ~ M let nA = I1 Pi ; n 0 = 1.

i E A

Equalities (2.1) imply

A DESCRIPTION OF SUBRINGS IN V(Sp 1 x Sp2 x .. x SPm)

PÄ =nA PA;

(PA -1)2 = (nA -2) (PA -1)+ (nA -1);

p A (PA -1) = (nA -1) p A·

Let W = u Ui, Ui E C(T). Set

Qw = I1 (Pu, -1).

It is not difficult to conclude from relations (8.1) that

217

(8.1)

(8.2)

where cw E Z, cw;;:: 0, and the sum ranges over all subsets W ~ M which are unions of components and suchthat W 1 ~ W 2 ~ W ~ W 1 u W 2·

Now let V 1 , V 2 E T. Put Vi = Wi = Wi u Xi where Wi E N (T), Xi E T,

wi n xj = 0, i = 1 ' 2. Then

fv 1 fv2 = Qw1 Px1 Qw2 Px 2 = c Qw1 Qw2 Px.

Here W 1 = W 1 \X 2 , W 2 = W 2 \X 1 , X =X 1 u X 2 and c is a positive integer. Notice that W1 u W2 e N(T).

Now (8.2) and Lemma 8.3 imply that f = fv 1 fv 2 E R + (Q). Since f is a polynomial

with integer coefficients and since (3.1) and (3.2) hold for {fy} as was already checked, we havef ER+. o

9. The main proposition

Let R be a cellular subring of R o and let B R be its basis. Set LR ={jE BR I 3i, Xi~f}.

The following proposition plays the key role in the proof of Theorem 7 .2.

Proposition 9.1. Let/ E LR. Then

forsomeA, B ~M, An B =0.

(9.1)

The statement is based on properties (C 1), (C 2 ) of elements of BR (see section 5). We will write f E (C) to indicate that f satisfies a property (C). A property (C) will be called hereditary if f E (C) implies 11 , h E (C), where as above, f = f 1 + xkh and h = a 1 1 a xk '# o.

Lemma 9.2. a) The property (C 1) is hereditary.

b) For fe(CI), if !=h+xkh,f1,h'i'O then v 1 , a 11 1 a xr '# o ~ a h 1 a xr '# o.

Proof, Letl1 '# 0, f2 '# 0, a f I axz '# 0. Let us representl1 as 11 = fu + Xz f12 and h asfz =h1 +xzh2· Then !=11 +h=fu +xzf12+xkif12+xzh2)=fu +xd21 +xzif12+xkh2). By (Cl) we have fu +xd21 ~ !12 +xkf22• so fu ~ !12. h1 ~ h2· Hence 11, h e (C 1), !1 '# 0 andattaxz'#O ~ al1taxz'#O,ahtaxz'#O. o

Lemma 9.3. The property (C 2) is hereditary.

Proof. Letusverifythatl1 e (C2). Let(x+yz)c1><11.Sincefl <f, (C2)implies that either y < f or z < f. Furthermore a<l> I axk = 0 implies either y < /1 or z < fl·

Now let (x +yz) <1> < f2, so (x +yz) Xk < f implies either y xk <1> < f or z Xk <1> < f, It follows that either y <1> < h or z <1> < f2, i.e. h e (C 2). 0

Now let us prove a general statement on polynomials satisfying properties (C1), (C2).

Lemma 9.4. Let f e RÖ be a sum of distinct monomials, f e (C 1 , C 2) and suppose f has linear part, i.e. 3xi, Xi~f, Then f=PA,B=(PA-1)PB where A,Br;;;,M, An B = 0. In the case A = 0 we suppose PA,B tobe PB.

Proof. By induction on m = IM I. The statement is trivialform = 1. Suppose it is true for all k < m.

Let us consider a polynomial f in the variables x 1 , x 2 , • · · , Xm, f e (C 1 , C 2), and suppose 3 Xi ~ f, Without loss of generality we can assume i '# m. Let f = f 1 + Xm f 2. Then Xi~l1 and Xi~h since f1 ~f2(C 1 ). By Lemmas 9.2, 9.3 we have 11 ,f2 e (C 1 , C2). Hence by induction we have 11 =(PA, -1)PB, and h=(PA 2 -1)PB 2 where A 1 ,B 1 ,A 2 ,B 2 cM\{m}. Let us study the connection between these two pairs of sets.

a) If A2 '# 0 then A 1 r;;;, A2. lndeed, assuming A 2 '# 0, we have / 1 = I, Xi +high

degree monomials, h = I, Xi + high degree monomials, and f 1 ~ h. ieA1

b) A 1 u B 1 = A 2 u B 2. The result follows from Lemma 9 .2.(b ).

c)l1 <h ~ A2=0. Let11 <h andA2'#0. ThenA1 r;;;,A2 andA 1 '#A2. Let j e A 2 \ A 1· Then Xi + Xj Xm < f, By our choice of j, Xj <t 11, so we have Xj <t f, Furthermore Xm <t f since 1 <t h (A 2 '# 0). This is a contradiction to f e (C 2).

Thus we have the following possibilities:

1) 11 = fz. Then /= 11 +xmh = 11 (1 +xm) =(PA, -1)PB, (1 +xm) = PA,B• where A = A 1 , B = B 1 u {m}, A n B = 0.

2) 11<fz,A2=0,B2=A 1 uB 1 ,PB2 =PA,PB,· In this case

f= 11 +Xmh =(PA, -l)PB, +xmPA, PB,= (PA -l)PB, where A =A 1 U {m}, B = B 1, A n B = 0.

A DESCRIPTION OF SUBRINGS IN V(SP1 x Sp2 x ... x SPm) 219

So the induction hypothesis is true for k = m as weiL 0

Proof of Proposition 9.1 follows immediately from the fact that f e LR c B R

sarisfies (C 1 , C2) by Lemmas 5.4, 5.7 and 9.4. o

10. The construction of a topology from a subring

Set C(LR) = {(A, B) I PA,B e LR} and C1 (LR) = {A I 3B, PA,B e LR }. We want to show that the collection C(LR) uniquely defines a topology on M. First, let us list some important properties of C(LR).

Lemma 10.1. a) A 1 , A 2 e C 1 (LR) :::> A 1 n A 2 = 0;

b) u Ai =M, Ai e C1(LR);

c) Let (Ai, Bi) e C(LR), i = 1, 2. Then B 1 n A2 * 0 :::> A2, B2 ~B 1·

Proof. Properries a) and b) follow from the definition of LR, Propositon 9.1 and the fact that

L f = x 1 + x 2 + · · · + Xm + high degree monomials. feLR

In order to verify c) we will use the equalities (8.1). They imply PÄ,B =(PA -1)2 PtJ = c1 PA,B + c2PB, where c1 = (nA -2) nß, c2 = (nA -1) nß. Since c 1 * c 2' f = p A,B E R :::> p B E R. Let/; = p A;B;' i = 1 ' 2, = p BI E R. lf B 1 (") A 2 * 0 then <1> 1\ f2 * 0. As f2 belongs to the basis we have f2 ~ <IJ. Hence we have the inclusion A2,B2 ~B1. 0

Definition 10.2. Let us define the closure of a point x e M in the following way:

x=A uB where (A 'B) E C(LR), XE A.

The definition is correct because ofLemma 10.1(a), (b). - -

For U ~ M define U = u .X. Set T = { U I U ~ M}. XE U

Lemma 10.3. The collection of sets T = t(R) is a topology on M.

Proof. We should verify that T sarisfies the properties 1 ), 2) from Definition 6.1.

Property 1) follows from Lemma lO.l(b). By Lemma 10.1(c) if z e .X then z ~x. Now it is easily seen that 2) holds as well. 0

11. Proof ofTheorem 7.2.

Previously, for a topology T and a subring R, we introduced a subring r(T) and a topology t(R) respectively. Hereweshow that the mappings r and t areinverses of each other, i.e. r(t(R)) =Rand t(r(T)) = T.

220

Lemma 11.1. a) C (t(R)) = C 1 (LR );

b) C(T) = C 1 (Lr(T)).

JA. JU. GOL'FAND

Proof a) follows immediately from Lemma 10.1 and Definition 10.2 of the topology t(R). b) The formula (7.1) implies that

Lr(T) = {flj =(Pu-l) Pu\U I U e C(T)}. (11.1)

Hence

C(Lr(T)) = { (U, U \ U), U e C(T)} and C 1 (Lr(T)) = C(T). (11.2)

The proof is complete. 0

Lemma 11.2. Let R = r(T), T' = t(R). Then T' = T.

Proof Lemma 11.1 implies that C(T') = C 1 (L,(T)) = C(T). By the definition of T' = t(R) we have A u B e T' for A e C (T'), (A , B) e C (LR ). It follows from equality (11.2) for C(LR) = C(Lr(T)) that closures of components in T, T' are the same. Hence T=T'. o

Lemma 11.3. Let T = t(R), R' = r(T). Then R' = R.

The proof will consist of several parts.

Part 1. LR' = LR. This Statement follows from the expression (11.1) for LR' = Lr(T),

as well as from Proposition 9.1 and equality C (LR) = C (LR') (Lemma 11.1 ).

Part 2. B R' !::;;; R. Let f e B R', f = fW with fW defined by formula (7 .1 ), W e N (T), w = u Ui, ui E C(T). Then by equalities (8.1)

nru, =const· Jw.

Therefore fW E R since ru, E LR' = LR.

Letf eR and V(f) = {i I dj I dXi # 0}.

Part 3. V(f) e T. It is easy to see that we may suppose f eR+. Let i e V(f) and

vi E T suchthat Xi ~ !v, E LR. We prove that vi ~ V(f). By Lemma 5.6 X =Xv(f) ~ f.

Letx =xi y. Suppose Pi*- 2. By Lemma 5.l(b) Xi ~ x 2 ~ / 2 . Hence Vi!::;;; V f ~ Vt in this case. If Pi = 2 then y ~ x 2 ::; ! 2• Let g e B R such that y ::; g. Then y ::; ! 2 implies that V(g) ~ V(f). Now xi ~ xy ~ fg by Lemma 5.l(b). Hence Vi!::;;; V(fg)!::;;; V(f). So V(f) is closed.

Part4.fe BR' <:;=>je BR.

We will proceed by induction on m(f) = I V(f) I. The Statement is evident for m(f) = 1. Suppose it is true if m(f) < l. Let now m(f) = l. Suppose firstthat f e BR'• i.e. f = fv for some V e T. By Lemma 5.6 we have xv ~ f Suppose fv e B R. Then there are

4>1, 4>2 E BR suchthat 4>1 + 4>2::; f Since 4>1 A 4>2 = 0, by Lemma 5.6 either a 4>1 I a Xj = 0 or a <1>2 I a Xj = 0 for some i E V. Let a <1>1 I a Xj = 0. By the induction hypothesis there is

V 1 e T such that <1>1 = /v1 • Furthermore V*- V 1 since fv 1 < f; this contradicts Lemma

221

8.1. Sofe BR.

Let now f e BR. By part 3, V= V(/) e T. Wehave already proved that fv e BR. Since xv S: f " fv we have f = fv. This completes the proof of Lemma 11.3 and that of Theorem 7.2 as well. D

12. ProofofTheorem 7.3.

Let T1, T 2 be two topologies on M and let R1 =r(Tt), R2=r(T2) be the corresponding cellular rings. Theorem 7. 3 states that T 1 < T 2 <=> R 1 < R 2. The proof is based on the following lemmas.

Lemma 12.1. If LR 1 !:: R 2 then R 1 S: R 2·

Proof. By Lemma 11.4, elements of LR generate R, so the lemma holds. D

Lemma 12.2. Let T 1 S: T 2• Ri = r(Ti), i = 1, 2. Then LR 1 !:: R 2·

Proof. Letf=JA =(PA -1)Pn e LR whereA e C(T2), A =AU B, Be T2. Since T 1 S: T 2 we have A , B e T 2·

ByLemma8.2

PA.= l: fv, Vs;;A,VeT1

Pn = l: fv. Vs;;;B,Ve T1

Hence.fA. =(PA -1)Pn =PA. -Pn e R2.

Lemma 12.3. R 1 S: R 2 => T 1 S: T 2.

D

Proof. Let V e T1, fv e R1. Then fv =l:fw where Iw e R2, W e T2. So V= u WeT1.

fw<fv

The proof of Theorem 7.3 is completed.

13. Automorphism groups

D

Here we give an interpretation of cellular subrings R < R 0 as V-rings of their respective automorphism groups. In particular we get a description of 2-closed overgroups of the group Sp 1 x Sp 2 x · · · x Sp,. in the symmetric group Sn,

n=PtP2'''Pm·

We need the construction that was considered in [2] in the more general setting of infinite permutation groups.

Definition 13.1. Let A = (A, <) be a finite partially ordered set and let (G cx, X cx), a e A be a family _ of permutation groups marked by elements of A. Set - ~X- - ~ -

Xcx = TI X~, Gcx = Gcx" = {Xcx ~Ga} if Xa :~:- 0 and Gcx = Gcx if Xcx = 0. Let ~<cx

X= n X <X, G = n Ga. The action of G Oll X is defined in the following way: aeA aeA

xg = (x~·(X)) for g = Cia) E G, ia E Ga.

So we have a permutation group (G, X) which is called the wreath product of permutation groups (Ga, X a) over a partially ordered set A. It is denoted by W r <XE/\ Ga·

Example 13.2. Let A = (a1 , a2), a 1 < a2. Then the wreath product W r aeA Ga coincides with the ordinary wreath product of permutation groups G a 1 , G <Xz.

Example 13.3. In the case A = (a1 , a2) with a1 and a2 incomparable, we have WraeAGa=Ga1 xGaz.

Let R < R 0 be a cellular subring, T = t(R) the corresponding topology, and ~ and -the quasiorder and the equivalence defined in section 6. Let 1t : M ~ M be the projection of M onto M' =MI- and < the partial order which is induced naturally on M. If a e M then1t-1(a)=UaE C(T).Setna= TI Pi·

ie u. Theorem 13.4. R = V(G), G = Aut(R) where G = W r aeM Sn •.

Proof. We first study the basic graphs fv, V e T of the ring R. Let X a be a finite set of cardinality n a• X= TI X a• I X I = n = p 1 P2 · · · Pm· Using the formula (7. 1) for fv and the interpretation of generators Xi (section 6) we come to the following description of the graph fv where V= W, W e N(T): the vertex set of fv coincides with the set X; the edge set E (/v) consists of pairs of vertices x , y e X, x = (x a), y = (y a), a e M, such that (1) x a = y a for all a ~ 1t(V) and (2) x a * y a for all a E 7t(W).

Let us show that G = W r aeM Sn. ~ Aut(fv ). Let a ~ 1t(V \ W) = 7t(V) \ 7t(W). Then

it follows from the definition of the quasiorder (section 6) that 'dß < a, ß ~ 1t(V). So, if (x, y) E E(fv) then \iß < a, Xa =Yß· Then, by Definition 13.1, \ig e G ga(x)=ga(y)e Sn •. Hence\ig E G, (x,y)e E(fv) =:> (xg,yg)eE(Jv).

The transitivity of Gon E(Jv), as well as the inverse inclusion Aut(Jv) k G, V e T, can be verified analogously. o

Remark 13.5. The ring R = r(T) can be viewed as the wreath product of trivial subrings V(Sn) over the partially ordered set A = M. Using a more complicated construction we can define a cellular ring R = W r <XEA Ra where the Ra are arbitrary cellular rings. Then Aut(R) = W r <XE/\ Aut(R a)· Moreover, if Ra= V(G a) then R = V(G) where G = W r aeA Ga·

Remark 13.6. For the ring V(Sdl X sd2 X .•. X sd..) where d 1 ' d2' . , . ' dm are distinct but not prime, the main result of the paper does not hold. For example Iet R o = V (S 2 x S 3 x S 6 ). The generators x 1 , x 2 , x 3 of R o satisfy the relations: xy = 1, x~ = x 2 + 2, xj = 4x2 + 5. It is not difficult to verify that R 0 contains the subring R'=<l,x1+x2+x1x2+x3, x3(x1+x2+x1x2)>=V(S6IS2) in addition to the subrings described in sections 7, 13.

A DESCRIPTION OF SUBRINGS IN V(Sp 1 x SP2 x ... x SP) 223

It seems that in the general case the subrings correspond to V-rings of permutation groups which can be constructed from symmetric groups by means of wreath products and exponentiations.

References

1. LA. FaradZev, M.H. Klin, M.E. Muzichuk, Cellular rings and groups of automorphisms of graphs. [In this volume].

2. V.Z. Feinberg, The wreath product of permutation groups over partially ordered sets andfiltres. Izv. AN BSSR (1971), 28-38 [In Russian].

3. A.A. Ivanov, M.H. Klin, LA. Faraclzev,Primitive representations ofthe nonabelian simple groups of order less than 106 . Part 2. Preprint, Moscow, VNIISI, 1984 [In Russian].

4. L.A. Kaluznin, M.H. Klin, On certain maximal subgroups of symmetric and alternating groups, Mat. USSR Sb. 16 (1972), 95- 123.

5. M.H. Klin, R. Pöschel, The König problem, the isomorphism problern for cyclic graphs and the method of Schur rings. - Colloq. Math. J. Bolyai, 25, Algebraic

methods in graph theory, Szeged, 1978; North Holland Amsterdam 1981, pp. 405-434.

6. B. Weisfeiler (edited by), On construction and identi.fication of graphs. Lect. Notes Math., 1976, v.558.

7. H. Wielandt, Finitepermutation groups. Academic Press, 1964.


CELLULAR SUBRINGS OF THE SYMMETRIC SQUARE OF A CELLULAR RING OF RANK 3

I A. F aradiev

1. Introduction

The present paper contains a treatment of the V-rings of the symmetric powers G i Sm of a group G and of the symmetric: powers of cellular rings. All necessary information about V-rings ofpermutation group and cellular rings can be found in [7].

Subring lattices of V-rings of permutation groups from different classes were studied intensively during recent years in connection with the characterization problern of the 2-closed overgroups of a given permutation group (G, N) in the symmetric group S(N). All investigations previously carried out belong to one of the following two principle schemes.

1. For a finite set of permutation groups their V-rings, and the corresponding lattices of cellular subrings, are determined by means of computer calculations (see for example [12], [21]). A failing of this approach isthat it does not give a proof of the results in the usual sense. The sphere of application for this approach is limited to searching for counterexamples to certain conjectures, creating matherial for further consideration, and studying exceptional cases.

2. An infinite series of V-rings, whose parameters can be described analytically, is considered. In this case the complete or asymptotic description of the lattice of subrings is obtained without use of a computer (see for example [13], [16], [18]). Up to now investigations along this line were carried out for V-rings having more or less simple structure. These are so called metric V-rings, one of whose basis graphs is distancetransitive. In this paper, as well as in [8], one of the first attempts to describe analytically the lattices of subrings of non-metric cellular rings is made.

The choice of V-rings of symmetric powers of permutation groups and their combinatorial generalization (symmetric powers of cellular rings) as the object of investigation is not accidental. These objects play a significant role, both in the theory of permutation groups and in applied combinatorics.

1. By the O'Nan-Scott theorem (see [4]), the structure of the socle of a primitive permutation group is the most complicated when its degree is apower of some integer. If the V-rings of the groups G i Sm could be investigated completely, then the graphs arising in the case 2a) of the theorem could be classified.

225

226 LA. FARADZEV

2. Many series of V-rings which are extremely interesting in coding theory [6] (the

Harnming schemes, the Lie schemes, the spectral schemes) are just the V-rings of the

groups G i Sm.

3. In connection with the isomorphism problern for graphs, interest in the classical problern conceming bounding the order of a primitive permutation group has increased.

L Babai [2] has shown that the solution to this problern heavily depends on the precision of the estimation of certain invariants of primitive cellular rings. It follows from a result

due toP. Cameron ([4], Theorem 6.1) that this estimation is the most difficult for the case of the V-ring of a group G i Sm, where G belongs to certain series of primitive permutation groups. L. Babai believes that the investigation of subrings of the V-ring of the group S~ i Sm, where S~ is the induced action of the symmetric group Sn of the set of k

element subsets, is of particular interest. For instance, the following question is believed

to be of interest. For what numbers k and m are there infinitely many n such that

V(S~ i Sm) has a nontrivial subring?

In this paper the lattice of subrings of the symmetric square of a cellular ring of rank 3 (in particular of the V-rings of the symmetric square G i S 2 of a group G of rank 3) is determined. In particular, this result gives the answer to the question of L. Babai for k = m = 2. The determination of subrings of the symmetric power of a cellular ring is

reduced to the formulation of certain polynomial equations in certain variables (the parameters of the original ring) and searching for all integer solutions. When the rank of

the original ring and the exponent of the investigated power contribute to the growth of

exponents in the equations, the number of variables and the number of equations to be considered grows very rapidly. But the methods described in Section 3 can be easily prograrnmed. So we hope that in the future, computer calculations can be used for the inves

tigation of higher powers of cellular rings of higher rank.

2. The V-ring of the symmetric power of a permutation group and the symmetric power of a cellular ring

Let G be a transitive permutation group of a set N , I N I = n, having rank r and subdegrees Si, 0 ~ i ~ r- 1. Let V(G) = < <1>0 , <1>1 , · · · , <1>,_1 > be the V-ring of the group G, where <l>o , <l>1 , • • · , <1>,_1 are the basis graphs and let A-t , 0 ~ i , j , k ~ r - 1 be the structure constants of V(G).

Let G i Sm be the exponentiation (see [10]) of the group G and the symmetric group Sm. This is a permutation group on the set Nm and it is called the m-th symmetric

power of the group G. An element w e G i Sm can be represented by a sequence

(!11 , · · · , !Lm; v), !Li e G, 1 ~ i ~ n, v e Sm. If such an element acts on the vector X = (x 1 , · · · , Xm) e Nm, then !Li acts on Xi and v permutes the coordinates. In this section we give a description of the V-ring of the group G.

CELLULAR SUBRINGS OF THE SYMMETRIC SQUARE OF A CELLULAR RING OF RANK 3 227

Let X = (x 1 , · · · , Xm), Y = (y 1 , · · · , Ym) be a pair of vectors from Nm. Define the vector t(X, Y) = (t 1 , · · · , tm), 0$ ti $ r- 1, as follows.

If the pair (xi, Yi) is contained in the basis graph <Pj of the ring V(G) then ti = j, 1 $ i $ m. Let /(X, Y) be the multiset {0~, · · · , (r -1)a,._1 } consisting of numbers of basis graphs of V(G). Here ai is the number of coordinates in t(X, Y) equal to i. It is clear that I /(X, Y) I =I, ai = m. Fora given multiset I of numbers of basis graphs with I I I = m, let us consider all pairs X , Y e Nm such that I (X , Y) = I and let T (!) be the set of vectors t(X , Y) for all these pairs. It is dear that T(/) is the set of all orderings of the multiset/.

Proposition 1. The ordered pairs (X, Y) and (X', Y') of vectors from Nm are contained in the same 2-orbit of the group G i Sm if and only if I (X , Y) =I (X', Y').

Proof. Let (X, Y) and (X', Y') contained in the same 2-orbit of the group G i Sm. Then there is an element ro = (~1 ' ... ' ~m; v) E G i Sm suchthat xro =X' and yro = Y'. Rewriting these equations coordinatewise, we have x~ = x/ and yrJ = y{ for 1 $ i $ m. This means that (xp , Yi·) and (x/, y{) are contained in the same 2-orbit of the group G. So t(X', Y') can be obtained from t(X, Y) by permutation of coordinates in accordance with v, i.e. /(X, Y) =/(X', Y').

Let /(X, Y) =/(X', Y'). Then the vector t(X', Y') can be obtained from t(X, Y) by means of a certain permutation v of Coordinates: ti• = t{ for 1 $ i $ m. This means that (Xi•, Yi•) and (x{, y{) are contained in the same 2-orbit of the group G. So there are ele

ments ~i e G , 1 $ i $ m, such that x~ = x{ and y~ = y{. Then ro = (~1' ... ' Jlm; v) E G i Sm and xro =X', yro = Y'. Hence (X' Y) and (X'' Y') are contained in the same 2-orbit of the group G i Sm. 0

Notice that for each vector t = (t 1 , · · · , tm), 0$ ti $ r -1, there isapair of vectors X , Y from Nm such that T(X , Y) = t. This fact and Proposition 1 imply the following.

Corollary 1. The 2-orbits of the group G i Sm are in one-to-one correspondence with the multisets of cardinality m containing numbers of 2-orbits of the group G.

Corollary 2. The rank of the group G i Sm is determined by the formula

_ [r +m -1] R- 1 . r- (1)

It should be mentioned that Corollaries 1 and 2 appear without proofin [14, 15].

If I = { 0~ , · · · , (r -1) llr-1 } is a multiset of numbers of basis graphs of V ( G i Sm)

and I I I = m, then the corresponding basis graph of V(G i Sm) will be denoted by 'PI= {<l.>Ö", ... , Cl>~{}.

Let A and B be binary relations on the sets N and M, respectively. The tensor product A ® B of the relations A and B is the relation on N x M defined as follows:

228 I. A. FARADZEV

((x, y), (x', y')) e A ® B <=> (x, x') e A and (y, y') e B. It is easy to see that the tensor product is an associative operation and that I A ® B I = I A I • I B I .

Proposition 2. The 2-orbit of the group G i Sm corresponding to the roultiset I of cardinality m of nurobers of the 2-orbits of the group G is determined by the formula:

m

'1'1 = U ® r. · reT(/) i=1 •

(2)

Proof. It follows froro the definition of the tensor product that the set of ordered pairs (X, Y) of eleroents X and Y froro Nm such that t(X, Y) = t = (t1 , · · · , tm) coin-

m cides with _® <1>1;. The union of all these sets over all orderings t of the roultiset

1=r

I, I I I = m, gives the set of all pairs (X, Y) such that I (X, Y) =I, i.e. the 2-orbit '1'1 of the group G i Sm. 0

Corollary 3. The valency S1 of the basis graph 'P1 of the group G i Sm corresponding to the roultiset I = { oao , · · · , (r - 1) a,._, } , I I I = m, can be calculated by the forroula:

r-1 S1 =q(l)· Tis(, (3)

i=O

where q(/) = r-1 m! is the nurober of different orderings ofthe roultiset /.

n (a; !) i=O

Proposition 3. Let I, J, K be roultisets of nurobers of the 2-orbits of the group G, each roultiset having cardinality m. Then the structure constant A5 is determined by the formula:

K m r. A/J = L n Ar:•rt '

r'eT(/) i=l r"eT(J)

where t = (t 1 , • • • , tm) is an arbitrary ordering of the roultiset K.

(4)

Proof. Let us fix a pair (X, Y) of vectors froro Nm such that /(X, Y) = K. Then t(X, Y) = t = (t 1 , · · • , tm) isanordering ofthe roultiset K. The constant A~ is equal to the nurobers of vectors Z e Nm such that I (X , Z) =I and I (Z , Y) = J. For each pair of orderings t' e T(/) and t" e T(J), the nurober of vectors Z e Nm such that

m t(X, Z) = t' = (tt', · · · , tm') and t(Z, Y) = t" = (tt", · · · , tm") is TI/..~;.,._.. By suro-•.

t=r

rning over all orderings of the roultisets I and J we obtain the clairoed formula. 0

Proposition 2 was proved in other terms in [14]. In this same paper it was suggested that a formula of type (4), in terms of tensor product of roatrices consisting of structure constants, rnight be attainable.

The construction described above enables one to obtain the V-ring of the symmetric power G i Sm frorn the V-ring of a transitive permutation group G. This construction can be generalized on arbitrary cells, i.e. cellular rings with regular basis graphs.

Let W = < <l>o, <1>1 , · · · , <1>,_1 > be a cell of rank r on n points. Let us define cellular ring wm on n m points as follows. The basis graphs of wm correspond to the rnultisets of cardinality m consisting of nurnbers of basis graphs of the ring W and are determined by formula (2). In this case the rank R, the valencies S1 and the structure constants A~ of the ring wm can be calculated by fotmulas (1), (3) and (4), respectively. The ring wm will be called the m-th symmetric power of the ring w.

An analogous construction of association schernes, called extension, was introduced in [6] and was used in the study of Hamming schernes, i.e. V-rings ofthe groups Sn i Sm.

In [6) the formula (1) is given, but the formula for the structure constants is not presented. The formula for the structure constants of V(Sn i Sm) was rediscovered rnany tirnes. In connection with the enurneration problern of subrings, this formula was first obtained in [14].

The lattice of subrings of the ring wm, for m > 1, r > 2, always contains at least three elernents: the ring wm itself, the ring W(Snm) of rank 2 and the ring V(Sn i Sm) of rank m + 1 whose basis graphs are characterized by the rnultiplicity <XQ of the reflexive basis graph <1>0 in the corresponding rnultiset. These rings will be called trivial.

The enurneration problern for cellular rings w for which wm contains nontrivial subrings is of interest. In the next section this problern is solved in the ease of symmetrie ringsform = 2 and r = 3. Narnely, the list of parameters of symmetrie eellular rings of rank 3 whose symmetric square has nontrivial subrings is obtained. In Seetion 4 the subring lattices of the symmetrie squares of these rings are presented.

3. The parameters of cellular rings of rank 3 whose symmetric square has nontrivial subrings

For the deseription of the pararneters of symmetrie eellular rings of rank 3 the following parameters are usually used: v = n , k = s 1 , I = s 2 , A = A }1 and Jl = A.h. These pararneters satisfy the relations v = k + l + 1 and

k(k-J..-1) =I Jl.. (5)

Up to reordering of the basis graphs <1>1 and <1>2 , we ean assurne that k ~I, and if k = I then A ~ Jl - 1.

Theorem. Let W be a symmetric eellular ring of rank 3. The cellular ring W2 has nontrivial subrings if and only if the parameters (v , k, I, A., Jl) of W belong to one of the following five series for integers x , y:

Series I: v = xy , k = y - 1 , I = (x - 1) y , A == y , Jl = 0 (x ~ 2, y ~ 2);

230 I. A. FARADZEV

Seriesii: v=4x+1,k=l=2x,l.=x-1,Jl=X (x2:: 1); Series Ill: v =x2 , k = 2(x -1), I= (x -1)2 , I.= x- 2, Jl = 2 (x 2:: 2); Series IV: v =4x2 , k =x(2x-1), I= (x+ 1) (2x-1), 1.= Jl=X(x-1) (x2=: 1);

Series V: v =4x 2 , k = (x-1) (2x+ 1), I =x(2x+ 1), A.= (x-2) (x+ 1), Jl=x(x-1) (x2:: 2).

Proof Let us express the structure constants t.t, O:s; i, j, k:::; 2, of the ring Was polynomials in the parameters k, I, I., Jl. Since A.fj = t.ji for a symmetric ring we present

t.t for i:::; }:

t.8o = 1 A.Öo = 0

t.81 = 0 t.ln = 1

t.82 = 0 "-Ö2 =0

t.?1 = k t.l1 =I.

"-?2 =0 t.b =k-A.-1

"-~2=1 A.b.=l-k+/.+1

A.Öo =0

"-Öl =0

A.Ö2 = 1

AI! = ll AI2 =k-ll

A.b = 1- k + jl- 1

(6)

For the antireflexive basis graphs of the ring W2 we use the following numeration. 'P1 = {<l>o, <l>d, 'P2 = {<l>o, <1>2}, '!'3 = {<l>y}, '!'4 = {<1>1, <l>2}, 'l's = {<1>~}. Now formula (4) gives us expressions for the structure constants At, 1:::; i, j, k:::; 5, of the ring W2 as polynomials in the parameters k , I , A., ll of the ring W. It is easy to see that At = A ji, so it is sufficient to do the calculations only for i :::; }.

Ah =A. Ab=k-A.-1 Ab=k A}4 =0 A}s =0

Ab.=l-k+A.+1

Ab=O A!4 =I Ab=O Ah =kA.

A14=k(k-A.-1) A1s =0 AL =k(/-k+A.+l)+l'A.

Aas = I (k - A.- 1)

A ~5 = I (I - k + A. + 1)

At1 = ll At2 = k-ll At3 =0 At4 =k Ats =0 Ab=l-k+ +jl-1 Ab=O

A14 =0 Ab =l Aj3 = kll Aj4 =k(k-ll) A~s =0 A4 =k(l-k+ +jl-1)+/jl A~s = l (k -ll)

A~s =l(l-k+ +jl-1)

AI1 =2 Ah=O

AI3 = 2A. AI4 = 2(k-A.-1)

Ais =0 Ab=O

Ab= 2(k-A.-1) A~4 = 2(1-k+A.+ 1) Ais =0 A~3 = A.2

A~4 = 2A.(k- A.-1) A~s = (k- A.-1)2

AL= 2(k -A.-1) X

X (k-21..-1)+2/A. Ais = 2(k - A.- 1) x x(/-k+A.+1) A~s =(l-k+A.+I)2

Af1 =0 Af2 = 1 Af3 =~ Af4 =k-~+A. Ats = k-A.-1 Ah=O Ah =k-~ A~=1-A.+~-2 At =1-k+A.+1 A~3 = A.~ A14 = kA.+ ~(k- 2A.-1) A~s = (k-A.-1) (k-~) A~ = A.(1-k+~-1)+~(l-k +Ä.+ + 1)+2(k- Ä.-1) (k -~) Ais= (k -A.-1) (1-k+~-d)+ +(k -~) (1-k+Ä.+ 1)

A~s = (1-k +A.+ 1) (1-k +~-1)

AI1 =0 Ar2=o AI3 =0 AI4 = 2~ Ars= 2(k-~> A~2=2 A~=O A~=2(k-~) A~ =2(1-k+~-1) A~3 = ~2 A~4 = 2~(k-~) A~s = (k-~)2 A4 = 2~(1-k+~-1)+ +2(k-~)2

Aas= 2(k-~> <1-k+~-1>

A~5 = (1-k+~-1)2

231

(7)

Each cellu1ar subring of the ring W2 corresponds to a partition of the set { 1 • 2 • 3 • 4. 5} of numbers of the antireflexive basis graphs of the ring w2 • Let T = { t 1 , • • • , tr _ d be a partition of the set { 1 , 2 , 3 , 4 , 5}. This partition deterrnines a subring of w2 having rank r if and on1y if the following relations hold:

V tx , ty , tz E T , I tz I > 1 , V k , k' E tz

k k' l: l: Aij = l: l: Aij. (8) ietxjet, ietxjet,

Because of the symmetry At = A ~i of the structure constants and of the relations s :I: At= Si- Öik• the relations (8) are not independent. It can be shown that if relations j=l

(8) hold for all unordered pairs tx and ty of elements of the partition T which do not contain an arbitrary fixed element t* of this partition, then these relations hold for all ordered pairs of elements of this partition.

U sing this fact Iet us write down the relations (8) for different types of partitions T.

1. T={{id. fi2l. {i3}, {i4,isll Ai4 _Ais Ai• _Ais Ai• Ais i,i, - i,i,• i,i2- i,i2• i,i3 = i,i3. Ai4 Ais Ai4 _Ais Ai4 _Ais i2i2 = i2i2. i2i3 - i2i3. i3i3 - i3i3 (81)

232 I. A. FARADZEV

(8s)

Each of the conditions (81)- (85) with the relation (5) gives a system of algebraic equations in the variables k , 1 , A., Jl. An integer solution to such a system, with the natural restrictions k~ 1, 1~ 1, k > A.~ 0, k~ Jl~ 0, gives parameters of a cellular ring whose symmetric square has a subring corresponding to the partition T. It should be mentioned that in each of these systems there are at least 4 equation in 4 variables. So the existence of an integer solution should be considered as somewhat probable.

The permutation 't = (1, 2) on the set of numbers of the basis graphs of the ring W 1eads tO the ring W't with parameters V 1 =V, k' = 1, [' = k, A1 = 1- k + J.L- 1, Jl' = 1- k + A.+ 1. Hence 't induces on the set of numbers of the basis graphs of the ring W2 , the permutation 1t = (1,2) (3,4). The latter permutation leads to the ring (lV't)2 . In this case a partition T determines a subring of W2 if and only if the partition Tft determines a subring of (~)2 •

There are exactly 52 partitions of a 5-element set. Three of the partitions correspond to trivial rings: { { 1}, { 2}, { 3}, { 4}, { 5} } , {{ 1, 2}, { 3, 4, 5} } and { { 1, 2, 3, 4, 5} } . In view of the correspondence between the subrings determined by the partitions T and Tft, we have 29 nontrivial partitions: 6 partition into 4 classes, 15 partitions into 3 classes, and 8 partitions into 2 classes (see Table 1). To prove the theorem it is sufficient to investigate these 29 partitions. The results of this investigation are formulated in a series of lemmas. (For each partition, the number of the corresponding lemma is indicated in Table 1.)


Table 1.

i Ti T't type Lemma

1 ((1}, (2}, (3}, (4, 5}} {{1}, (2}, {3,4}, (5}} 1 1

2 {{1}, (2}, (3, 5}, (4}} 1 1

3 {{1}, (2,5}, (3}, (4}} ({1,3}, (2}, (4}, (5}} 1 2

4 ((1}, {2,4}, (3}, (5}} ( ( 1, 4}, (2}, (3}, (5}} 1 3

5 {(1}, (2,3}, (4}, (5}} {{1,5}, (2}, (3}, (4}} 1 4

6 ( {1, 2}, (3}, (4}, (5}} 1 2

7 ( (1}, (2}, {3,4, 5}} 2 1

8 ({1}, (2,4,5}, (3}} {{1,3,4}, (2}, (5}} 2 3 9 {{1}, (2, 3, 5}, (4}} {{1,3,5}, (2}, (4}} 2 1

10 {{1}, (2,3,4}, (5}} ({1,4,5}, (2}, (3}} 2 1

11 ( {1, 2, 5}, (3}, (4}} {{1,2,3}, (4}, (5}} 2 2

12 ({1,2,4}, (3}, (5}} 2 2

13 ((1}, {2,5}, {3,4}} ( {1, 3}, (2}, {4, 5}} 3 1

14 {{1}, {2,4}, {3,5}} {{1, 4}, (2}, (3, 5}} 3 1 15 {{1}, (2,3}, (4,5}} (I 1, 5}, (2}, (3, 4}} 3 2 16 ((1,5}, {2,4}, {3}} {{1, 4}, (2, 3}, (5}} 3 5 17 ( {1, 5}, {2, 3}, (4}} 3 6 18 ((1,3}, (2,4}, {5}} {{1, 4}, {2, 5}, {3}} 3 3 19 ((1,3}, (2,5}, (4}} 3 2

20 ( {1, 2}, (3}, {4, 5}} ({1,2}, {3,4}, (5}} 3 2 21 ({1,2}, (3,5}, (4}} 3 7

22 ( (1}, (2, 3, 4, 5}} ({1,3,4,5}, (2}} 4 1 23 ((1,2,4,5}, (3}} {{1,2,3,4}, (5}} 4 8 24 ({1,2,3,5}, (4}} 4 9

25 ((1,4,5}, (2,3}} {{1,5}, {2,3,4}} 5 10

26 ((1,4}, (2,3,5}} {{1, 3, 5}, {2, 4}} 5 11 27 ({1,3}, (2,4,5}} {(1, 3, 4}, (2, 5}} 5 3 28 ((1,2, 5}, (3,4}} {{1, 2, 3}, (4, 5}} 5 12

29 ((1,2,4}, (3,5}} 5 13

Lemma 1. The partitions Ti and Tf, for i = 1 , 2 , 7 , 9 , 10, 13 , 14 , 22, do not lead to a cellular subring of the ring w2•

Proof For each of the listed partitions Ti, there is a relation in (8) of the form At = A~ where At and Af} are distinct constants by (7). See Table 2. D

234 I. A. FARADZEV

Table 2.

i T;,T'/ condition relation substitution

of values from (7)

1 ( (1), (2), (3), (4, 5)) (8,) A1z =Ah 1=0 ( ( 1), (2), (3, 4), (5))

2 ((1), (2), (3,5), (4)) (8!) M,=Ai, 2=0

7 ((1), (2), (3,4, 5)) (82) Ar, =Ar, 2=0

9 ( (1), (2, 3, 5), (4)) (82) Ar, =Ar, 2=0 (( 1, 3, 5), (2), {4))

10 ((1), (2, 3,4), (5)) (82) A?,=A1, 2=0 (( 1, 4, 5), (2), (3))

13 ((1), (2,5), (3,4)) (83) Ar, =A1, 2=0 ((1, 3), (2), (4, 5))

14 (( 1), (2, 4), (3, 5)) (83) Ar, =Ar, 2=0 ({1,4), {2), (3,5))

22 ( { 1), {2, 3, 4, 5)) (84) A?,=A1, 2=0 { (1, 3,4, 5), (2))

Lemma 2. The partitions Ti and Tf, for i = 3, 6, 11, 12, 15, 19, 20, do not deterrnine cellular subrings of the ring W2•

Proof For each of the partitions in the Iemma, there are re1ations from (8) whose common solutions contradict restrictions on the parameters of a rank 3 ring. See Table 3.

0


Table3.

i Ti,T'/ condition relation Substitution restriction of values from (7)

3 { {1}, {2, 5], {3}, {4}} (8,) Ar, =Ay, !J-=0 k>O {{1,3}, (2}, {4}, {5}} Ar4 =AI4 k =2!J.

6 {{1,2}, (3}, {4} {5}} (8!) Ah =A~3 k(!L- i..) = 0 k>O Ak=AL k(!J.-i..-1)=0

11 {{1,2, 5}, {3}, {4}} (8z) Ah =A~3 k(!L-i..)=O k>O {{1,2,3}, {4}, {5}} Ak=AL k(!L- i..-1) = 0

12 {{1,2,4}, {3}, {5}} (8z) A~3 =A~3 k(!L-i..) = 0 k>O A!s =A~s Züt-i..-2)=0 1>0

15 { {1}, {2, 3}, {4, 5}} (83) A1z +A13 =Arz + !!+1=0 !J.;?! 0 {{1,5}, {2}, {3,4}} +AI3

19 {{1,3}, {2,5}, {4}} (83) AI4 +AL =AI4 + (k- 2!J.)(k- !!) + k;?!!!

+A~ +1)= 0 k>O Af, + 2Ar3 +A~3 = !L(k-!L+ 1) = 0 =Ar, +2AI3 +A~3

20 {{1, 2}, (3}, {4, 5}} (83) A~13 +Ab =AI3 + k=O k>O {{1,2}, (3,4}, {5}} +A~3

Lemma 3. The partitions Ti, for i = 4 , 8 , 18 , 27, detennine cellular subrings of W2 if and only if the ring W has parameters v =.xy, k = y- 1, I= (x -1)y, A. = y- 2, ll = 0 for X~ 2, y ~ 3. In this case the partitions Tf detennine subrings of the ring (Wt)2 .

Proof Among the relations in (8), for each of the partitions in question, there is an equation, which after Substitution of At from (7), implies that either ll = 0 or k = A. + 1. Using (5) we obtain ll = 0 in both cases (se:e Table 4). It is known [11] that for ll = 0 the parameters of a symmetric ring of rank 3 are those presented in the Iemma. The remaining relations from (8) are satisfied for the partitions in question after substitution of At from (7). 0

236 I. A. FARADZEV

Table 4.

i T;, T'[ condition relation substitution

of values in (7)

4 ( (1), (2, 4), (3}, (5)) (8t) Aft =A1t j.J.=O ( ( 1, 4), (2), (3), (5))

8 ((1}, (2,4,5), (3)) (8z) AT!= A11 j.J.=O ( ( 1, 3, 4), (2), (5))

18 { ( 1' 3}, { 2, 4), { 5)) (83) Als +A~s =A?s + (k-1..-1)2 = 0 ((1,4), (2,5), (3)) +A§s

27 ( {1, 3), {2,4, 5)) (8s) Af! + 2Af3 + A~3 = j.J.(k-1..-1)=0 { { 1' 3, 4), {2, 5)) = A1t + 2A13 + Aj3

Lemma 4. The partition T 5 = { { 1} , { 2, 3} , { 4} , { 5}} determines a cellular subring of the ring W2 if and only if the ring w has the following parameters: v=x 2 ,k=2(x-1),1=(x-1)2 ,A.=x-2,Jl=2 forx~2. In this case the partition n = { { 1 , 5 l , { 2 l , { 3}, { 4 l} ctetermines a subring of the ring (W~)2 •

Proof. The relations Ay1 =AI1 and At4 =AI4 from (81), after Substitution of At from (7) give J.1 = 2 and k = 2(A.+ 1). By Substitution of these equalities in (5) we obtain l = (A.+r)2 . The remaining relations from (81) are satisfied after substitution of Afj from (7) and substitution of the previously obtained values for the parameters. 0

Lemma 5. The partition T 16 = {{ 1 , 5} , { 2 , 4} , { 3} } determines a subring of the ring w2 if and only if the ring w has parameters V= 4, k = 1, I= 2, A = J.l = 0. In this case the partition T'l6 = { { 1 ' 4} ' 2' 3} ' { 5} } determines a subring of the ring w2 •

Proof. The relations A!J = A~3, A~3 = A~3 and A l1 + 2 Als + A!s =AI 1 + 2 AIs + A3s from (83 ), after substitution of At from (7), imply the following:

kA.-Jl2 =0

Jl(k- A.) = 0

A. + l(l-k+A.+ 1)- 4(k-J.1)- (l-k+Jl-r)2 = 0.

(9)

(10)

(11)

It follows from (9) and (10) with the restrictions k > A.~ 0 that A. = J.l = 0. This equality and (5) imply that k = 1. By substitution of these values of k, A., J.l in (11) we have I= 2. These values of parameters satisfy the remaining three equations from (83 ) after Substitution of At from (7). 0

Lemma 6. The partition T 11 = { { 1, 5} , { 2, 3} , { 4} } determines a subring W 2 if and only if the ring W has parameters v = 9, k =I= 4, A. = 1, J.l = 2.

Proof. After the substitution of At from (7), the relations (83 ) take the following form:


k(l-k +A.+ 1) + l A.- 2 J.L(l-k +J,l-1)- 2(k -J,l)2 = 0

k(l-k+J.L-1) + IJ.L- 2(k-A.-1) (k-2A.-1)- 2/A.=O

l(k -A.-1)- 2 J,l- 2(k-J,l) (1-k +J,l-1)= 0

k + l(k-J,l)- 2(k-A.-1)- 2(k-A.-1) (1-k+A.+ 1) = 0

A.+l(l-k+A.+ 1)- 4(k-J,l) + (1-k+J,l-1)2 =0

J.l+ 1(1-k+J,l-1)- 2- (1-k+A.+ 1)2 = 0.

237

(12)

(13)

(14)

(15)

(16)

(17)

Let us consider two linear combinations of the equations (12) - (17) with the coefficients -1, 1, -2, 2, -1, 1 and 1, 1, -2, -2, 0, 0:

(J,l- A.) (J.l+ A.+k + 1) = 4(k- A.-1)

(1-k) (2k-J,l-A.-2) = 6(A.-J.t+ 1).

(18)

(19)

It follows from (18) that A.:S: J.l for k > A.. The value 2k- J.L- A.- 2 for k > A. and k ~ J.l can be negative only if J.l = k , A. = k - 1. But in this case (5) is not satisfied for 1 > 0. Hence 2k- J.L- A.- 2~ 0 and it follows from (19) that J.l:S: A.+ 1 for 1 ~ k.

1. Let J.l = A.. Then it follows from (18) that J.l = A. = k - 1, but in this case equality (19) does not hold.

2. Let J.l = A.+ 1. Then (19) gives (1-k) (2k-2A.-3) = 0 and hence k = 1. The Substitution of this equality in (5) gives k = 2 J.l. Now if we substitute the latter equality in (16) we have J.l2 - J.L- 2 = 0. So J.l = 2 and k = 1 = 4, A. = 1. These parameters satisfy the equalities (12)- (15) and (17). D

Lemma 7. The partition T 2l = { { 1 , 2} , { 3 , 5} , { 4} } determines a subring of the ring W2 if and only if the ring W has parameters V = 4x + 1, k = 1 = 2x, A =X - r, J.l =X

for x~ 1.

Proof. The relations (83), after substitution of At from (7), assume the following form:

k(J.L-A.-2) + l(J.L-A.) = 0,

(J.L-A.) (6k-21-4J.L-4A.)- 4k + 6A.+ 2J.l + 2 = 0,

k=1.

(20)

(21)

(22)

The Substitution of (22) in (20) gives J.l = A. + 1. Now by Substitution of this value in (5) we have k = 2(A.+ 1). In this case equality (21) also holds. D

Lemma 8. The partition T 23 = { { 1, 2, 4, 5}, {3}} determines a subring of the ring W2 if and only if the ring W has the following parameters: V= 2X, k = 1, I = 2x - 2, A. = J.l = 0 for x ~ 2. In this case the partition T'lt = { { 1 , 2 , 3 , 4}, { 5} } determines a subring of the ring (W't)2 .

238 I. A. FARADZEV

Proof. Relations (84 ), after substitution of At from (7), have the following form:

A.(k -J.l.) = 0'

J.l.(k-A.) = 0'

J.l.(k-J.l.) = 0.

(23)

(24)

(25)

To satisfy (24) for k > A., we should have J.l. = 0. But now it follows from (23) that A. = 0. By substitution of these parameters in (5) we have k = 1. In this case equality (25) is also satisfied. 0

Lemma 9. The partition T 24 = { { 1 , 2, 3, 5} , { 4}} determines a subring of the

ring W2 if and only if the ring W has parameters V = 4x + 1 , k = [ = 2x , A =X - 1, J.l. =X

for x ~ 1.

Proof. The relation A,L = A4 from (84), after substitution of At from (7), has the

form k (J.l.- A.- 2) + l (J.l.- A.) = 0. For k > 0, l > 0, this relation can be satisfied only if J.l. - A. is positive and J.l.- A.- 2 is negative. This implies that J.l. = A. + 1, k = /. Substitution in (5)

gives k = 2(A.+ 1). Theseparameters satisfy the rest of the equalities in (84 ) after Substitu

tion of At from (7). 0

Lemma 10. The partition T 25 = { { 1 , 4, 5} , { 2 , 3} } determines a subring of the ring W2 if and only if the ring w has one of the following sets of parameters.

a) v = 4 , k = 1 , I = 2 , A. = J.l. = 0;

b) V= 9, k = [ = 4, A = 1, J.l. = 2;

c) V = 16 ' k = 5 ' I = 10 ' A. = 0 ' J.l. = 2.

In this case the partition 715 = { {1 , 5} , { 2, 3 , 4} } determines a subring of the ring (W't)z.

Proof. The equations (8s), after Substitution of At from (7), have the following form:

A. 2 - k J.l. + 5k - I - 4 A.- J.l.- 3 = 0,

J.l.2 - k A. + k -[- A. + 1 = 0'

J.l.(J.l.- A.+ 2)- 2(k-l) =0.

It follows from (28) that

2k = J.l.(J.l. - A. + 2) + 2.

Substitution of (29) in (5) gives

J.l.[(J.l.- A. + 2) (J.l.2 - J.l.A + 2J.l.- 2A. + 2)- 41] = 4A..

(26)

(27)

(28)

(29)

(30)

1. If J.l. = 0, then it follows from (30) and (29) that A. = 0 and k = 1. Substitution of these values in (27) gives l = 2. In this case, relation (26) is also satisfied. These


parameters correspond to the case a) of the Lemma.

2. If J.L > 0, then for the validity of (30) it is necessary that J.L divides 4 A.. Put 4A.=J.LS. lf we rewrite (28) in the form 2(k-A.-1)=J.L(J.L-A+2)-2A. and substitute

A.= 1145 in the right side, we obtain 8(k-A.-1)=J.L(J.L-2)(4-s). Since A.;:::O and

k- A.- 1 > 0 for J.L > 0, we have 0 s; s s; 3. Now by Substitution of A. = J.L; in (29), we

obtain 8k = J.L2(4-s) + 8 J.L + 8 while Substitution in (27) yields 321=J.L\s 2 -4s)+J.L2 (48-12s)+J.L(32-16s)+64. Finally, by substitution in (5) of the expressions for A., k, l in terms of J.L and s, we obtain the equation (J.L- 2) [(16-s 2 ) 112 + 8s + 32] = 0. For OS: s s; 3 this equation has the unique root J.L = 2.

In this case A. = ~ . Since A. is an integer it is necessary for s to be even. For s = 2 and

s = 0 we have the sets of parameters presented in cases b) and c) of the lemma. In both cases relation (26) is also satisfied. D

Lemma 11. The partition T 26 = { { 1 ,. 4} , { 2 , 3 , 5}} determines a subring of the ring W2 if and only if at least one of the following condition holds:

a) w~ has parameters V = 2 y , k = y - 1 , l = y, A = y - 2 , J.L = 0 for y ;;::: 2;

b) Whas parameters v = 4x2 , k =x(2x-1), l = (x+ 1) (2x-1), A.= J.L=x(x-1) for x;::: 1;

c) w~ has parameters v = 4x 2 , k = (x -1) (2x + 1), l = x(2x + 1), A. = (x- 2) (x + 1), J.L = x(x -1) for x;;::: 2. In this case the partition 716 = { { 1 , 3 , 5} , { 2 , 4} } determines a subring of the ring (W~)2.

Proof. The equations (8s), after Substitution of At from (7), have the form

(k - J.L) (3k - l - 4 A.- 1) = 0,

(k - J.L) (3k - l - 4 J.L - 1) = 0,

(J.L- A.) (2J.L+2A.- 3k + l + 1) = 0.

(31)

(32)

(33)

1. If J.L '# A. then the equality k = J.L is necessary for the validity of (31) and (32), while the equality

2 J.L + 2 A.- 3k + I + 1 = 0 (34)

is necessary for validity of (33). Substitution of k in (34) gives l = J.L- 2 A.- 1. Now Substitution of these parameters in (5) gives A. = 0, I = J.L - 1. If we rewrite these equalities in terms of the ring w~ we obtain case a) in the lemma.

2. Let J.L = A.. Then the equality (33) is satisfied and for validity of (31) and (32) for k;;::: A., one should have I = 3k - 4 A.- 1. Substitution of the latter in (5) gives

240

k 2 - k(4 A.+ 1) + 4A.2 + 4 = 0. Hence

k=.!.. (4A.+1±.V4A.+1). 2

I. A. FARADZEV

(35)

Since k is an integer we have 4 A. + 1 = u 2 , i.e. A. = .!.. (u 2 - 1 ). The condition on A. to be 4

an integer implies that u = 2x - 1 or A. = x (x - 1 ). Substitution in (35) gives two series of solutions corresponding to the cases b) and c) ofthe lemma. 0

Lemma 12. The partition T 28 = { { 1 , 2 , 5} , { 3 , 4} } determines a subring of the ring W2 if and only if the ring W has parameters v = 4, k = 1 , [ = 2, A. = J.l. = 0. In this case the partition 11s = { { 1 , 2, 3} , { 4, 5}} determines a subring of the ring (W~)2 .

Proof The relations (8s). after the Substitutions of At from (7), have the following form:

(A. -J.l.) (/- k + A. + 3)- 2k + 2 + 4/J.l. = 0'

(A. -J.l.) l = 0'

(k -I+ 1) (k - 2J.l.) -J.l.(/-J.l.) = 0.

(36)

(37)

(38)

For the validity of (37) one should have A. = J.l.. Substitution of the latter equality in (36) gives 2/A. = k- 1. Now Substitution in (5) gives 2k 2 - k(2A.+ 3) + 1 = 0. Hence

k =t (2A.+ 3 + ,j(2A.+3)2 - 8). (39)

For k to be integer it is necessary for (2 A.+ 3)2 - 8 to be a square of an integer. This is possible only if A. = 0. In this case it follows from (39) that k = 1 and from (5) that l = 2. For these values of parameters relation (38) is satisfied. 0

Lemma 13. The partition T = { { 1 , 2 , 4} , { 3 , 5} } determines a subring of the ring W2 if and only if the ring W has the following parameters: V = 4x + 1 , k = [ = 2x, A =X - 1 , J.l. =X for X ~ 1.

Proof. The relation Ah + 2Ah +Ass= A~3 + 2A35 from (85), after substitution of At from (7), has the form k(J.l.-A) + /(J.l.-A.-2) = 0. It was shown in the proof ofLemma 9 that this condition holds only if J.l. = A. + 1 and k = l. These parameters satisfy the remaining equations from (85) after substitution of Afj from (7). 0

Now to complete the proof of theorem it is sufficient to check that the parameters of the series of cellular rings from the theorem satisfy the rationality conditions from [ 11].

The author is grateful to M.Z. Shapiro, who has checked the proof of the theorem very carefully, and to G.M. Adelson-Velskii, who has simplified the proof of some of the lemmas.

CELLULAR SUBRINGS OF THE SYMMETRIC SQUARE OF A CELLULAR RING OF RANK 3 24I

4. The lattices of subrings of the symmetric squares of cellular rings of rank 3 and the 2-closed overgroups of the symmetric squares of groups of rank 3

In this section we summarize the information contained in Lemmas 1-13 and give the lattices of subrings of the symmetric square of the cellu1ar rings of rank 3 when these

lattices are nontriviaL Notice that the lattict::s are nontrivial if and only if the parameters belong to the series presented in the theorem. In some cases, when the initial rings are Schurian, we give an interpretation of the 2-closed overgroups of the symmetric square

of their automorphism groups.

The cellular rings belonging to the series I - V will be denoted by W 1 (x , y) and Wi(x), i = 2, 3, 4, 5. Notice that the series I- IV have nonempty intersections:

W1(2, 2)=W3(2)=W4(1),

W2(2) = W3(3),

W4(2)=W3(4).

The series I. v = xy , k = y - 1 , l = (x- 1) y , A. = y - 2, J.l. = 0 (x ~ 2, y ~ 2).

The cellular rings W 1 (x , y) of this series are uniquely determined by their parameters. Their basis graphs are isomorphic to x o Ky (x copies of the complete graph with y

vertices) and x o Ky = Ky, ... ,y (the complete m-partite graph with y vertices in each part). Since Aut(W 1 (x , y)) = Sx wr Sy acts transitively on the edges of the graphs x o Ky and

x o Ky, we have W 1 (x, y) = V(Sx wr Sy) and W2(x, y) = V((Sx wr Sy) i S2). The lattice of subrings of the ring W I<x , y) is given in Fig. 1. The symbols A , D and 0 in the figure mark the subrings which arise only for x = 2, y = 2 and x = y = 2, respectively. The appearance of the additional subrings in the x = 2 and/or y = 2 case is due to the iso

morphism of the groups S 2 wr S 2 and S 2 i S 2, to the existence of nontrivial subrings in the V-ring of the group S 2 i S 4 [21] and also to the containment of the ring W 1 (2, 2) in the series III and IV mentioned above. In Table 5, for each subring of the lattice, we present a partition which generates this subring, the rank of the subring, the condition for its existence, its automorphism group, the number of the lemma which asserts its existence (for the nontrivial subrings) and the parameters (V, K, A) for the rank 3 subrings.

242

subring partition rank

0 ((1), (2), (3), (4), (5)) 6 1 ((1), (2,4), (3), (5)} 5 2 ((1,5), (2), (3), (4}} 5

3 ((1}, (2,4,5), (3}} 4 4 ((1, 3), (2, 4}, (5}} 4 5 ( (1, 5), ( (2,4), (3} J 4

6 ( ( 1, 3), (2, 4, 5} J 3

7 ((1,2,4,5), (3}) 3 8 ((1,3,5), (2,4}} 3 9 ((1,4), (2,3,5}} 3

10 (( 1, 2), (3, 4, 5}) 3

11 ((1,4,5), (2,3}} 3

12 ((1,2,5), (3,4)) 3

13 ((1,2,3,4,5}} 2

Fig. 1. 0

TableS.

condition

12

Aut

es. wr S,) i S 2

es. i Sz) wr es, i Sz)

x=2, S2 i S4

y=2 s; wr es, i S2)

es. i S2) wr s; x=2, S2 wr S4 wr S2

y=2 s; wr s;

y=2 s~ wr s2 x=2 S 2 wr S~1 x=2, S4 i Sz

y=2

S., i Sz

x=2, ~ :Ss y=2 x=2, ~ :Ss y=2

Sx.''l

I. A. FARADZEV

Lemma (V,K,A)

3 4

3

3 5

3 exzyz,yz-l, y2-2)

8 e4x2,1,0) 11 (4y2, y2-1, y - 2),

11 (16, 6, 2)

exzyz, 2xy- 2, xy-2)

10 (16, 5, 0)

12 (16, 5, 0)

The series n. V= 4x + 1, k = l = 2x, A. =X- 1, Jl =X (x~ 1). Cellular rings with the parameters from this series exist on1y for certain values of x. For example it is proved in [1] that the cellular ring W 2(5) does not exist. In the cases when these ringsexist (for instance if v = 4x + 1 is a prime power), they are not generally characterized by their parameters. As examples, many nonisomorphic basis graphs with the parameters W 2(6) and W 2(7) are contained in [1].

The lattice of subrings of the ring W~(x) is presented on Fig. 2. The symbol 0 in the figure indicates the subrings which arise only for x = 2. The appearance of the additional subrings in the case x = 2 is due to existence of nontrivial subrings of the V-ring of the group s 3 i s4 [14] (see below) and also to the Containment of w z(2) in the series III. In Table 6, for each subring, we give a partition which generates it, the rank of the subring, the condition for its existence, the number of the lemma which asserts its existence (for the nontrivial subrings) and the parameters (V, K, A) for the subrings of rank 3.

It is easy to see that the parameters of the basis graphs <1>1 and <1>2 of the ring W 2 (x)

coincide. In the case when the graphs <1>1 and <l>z are isomorphic, certain additional information about subrings of the ring W~(x) can be obtained.

Fig. 2.

Subring partition rank condition Iemma (V,K,A) number number

0 { {1}, {2}, {3}, {4}, {5)) 6 1 { (1, 5}, (2}, (3}, (4)) 5 x=2 4 2 ( (1}, (2, 3}, {4}, (5}} 5 x=2 4 3 {{1,5}, {2,3}, {4}} 4 x=2 6 4 {(1,2}, (3,5}, (4}) 4 7

5 ( (1,4, 5}, {2, 3}} 3 x=2 10 (81, 24, 9) 6 ( (1, 5}, (2, 3,4)} 3 x=2 10 (81, 24, 9) 7 ((1,2,3,5}, (4)) 3 9 ((4x + 1)2 , 8x2,

4x2 -2x+1) 8 {{1,2,4}, {3,5}} 3 13 ((4x + 1)2 , 8x2,

4x2 -2x+1) 9 {{1,2}, {3,4,5}} 3 ((4x+ 1)2 ,

10 ({ 1, 2, 3, 4, 5}} 2 Sx, 4x-1)

244 I. A. FARADZEV

Proposition 1. If the basis graphs <1> 1 and <1>2 of the ring W 2 (x) are isomorphic, then the subrings of W~ (x) determined by the partitions T 24 = {{ 1 , 2 , 3 , 5}, { 4} } and T 29 = { { 1, 2, 4} , {3, 5}} are isomorphic.

Proof Let J.l. be a permutation on the points of the ring W 2 (x) which maps <1>1 onto <1>2 . Let us Iabel the points of the ring W~(x) by pairs of points of the ring W 2(x). Then the permutation v : (i 1 , i 2) ~ (i~ , i 2) determines the isomorphism of 'P 4 = { <1>1 , <1>2 }

and 'P3 u 'P5 = {<l>y} u {<1>~}. o

Proposition 2. If the basis graphs <1>1 and <1>2 of a Schurian ring W 2(x) are isomorphic, then the subrings of the ring W~(x) determined by the partitions T 24 = { { 1 , 2 , 3 , 5} , { 4} } and T 29 = { { 1 , 2, 4} , { 3 , 5} } are non-Schurian if x '* 2.

Proof By Proposition 1 it is sufficient to show that the subring of W~(x) determined by the partition T 24 is non-Schurian. Let us choose a pair of points in W~(x) which are joined by an edge from 'Pi. Let Ni,j denote the nurober of complete four-vertex subgraphs containing the choosen pair of vertices, all of whose other edges are from 'Pj. Since the ring W~(x) is Schurian, these numbers do not depend on the choosen pair of vertices. If the subring determined by the partition T 24 is Schurian, then N 1,4 = N 2,4 = N 3,4 = N 5,4· We will show that N 1,4 '* N 3,4 for x > 2.

The numbers N 1,4 and N 3,4 can be calculated from the numbers mi for i = 1, 2, 3 and ni for i = 1, 2, ... , 14. Here mi and ni are the numbers of three- and four-vertex configurations with one or two labeled vertices in the basis graph of the ring W 2 (x) (Fig. 3).

Fig. 3.

vvr

A simple combinatorial calculation gives the following.

N 1,4 = m 1 (n 1 + n 14) + m2(n2 + n 13) + m3(n6 + n9);

N 3,4 = n 1 n 14 + n 2 n 13 + n 3 n 12 + n 4 n 11 + n 5 n 10 + n 6 n 9 + n 7 n s .

The numbers mi can be easily calculated from the parameters of the ring W 2 (x ):

m1 =x(x-1), m 2 =x 2 , m 3 =2x 2 .

(40)

(41)

The numbers ni for a selfcomplementary basis graph of a Schurian ring W 2 (x) can be expressed in terms of the parameters of the ring and the number n 1 = a (see [11]):

n 1 = an 14 = .!.. (x- 1) (x - 2) - a. 2

n2 =.!.. (x -1) (x- 2)- an 13 = x- 1 + a 2

n3 = 2(x-1) (x-2) -4an 12 = 4(x-1) + 4a

n4 = 4(x -1) + 4a n 11 = 2x + 2(x -1) (x- 2)- 4a

n 5 = 2(x- 1) + 2a n 10 = x + (x - 1) (x - 2) - 2a

n6 = 2(x-1) + 2a n9 = (x-1) (x-2)- 2a

n7 = 2(x-1) + 2a n8 = (x-1) (x-2)- 2a

(42)

Aftersubstitution of (41) and (42) into (40), the equality N 1,4 =N3,4• which gives a necessary condition for the subring determined by the partition T 24 to be Schurian, takes the following form:

46a2 - (23x2 -104x + 91) a + ·f (x -1) (x -2) (2x 2 - 12x + 15) = 0.

Since the coefficient of the highest power of the determinant of the equation is negative, the equation has at most finite number of integer solutions. A detailed study shows that these solutions are exhausted by the values x = 1 , 2 and a = 0. 0

For v = 4x + 1 = p s where p is a prime, we get Schurian rings W 2 (x) with isomorphic basis graphs <1>1 and <1>2 • These graphs are the Paley graphs P(ps) [5]. The vertices of the graph P(ps) are the elements of the Galois field GF(ps) with edges joining those pairs of elements whose difference is a nonzero square of the field. The automorphism group GP(ps) of the Paley graph P(ps) contains the group Zp': (Z(p'-1)12 • Zs)

where Z(p'-1)12 is the multiplicave group of nonzero squares of the field and Zs is the group of field automorphisms. M.E. Muzichuk has announced that P(ps) has no other automorphisms.1) In the case when the basis graphs of the ring W 2 (x) are the Paley graphs P(4x + 1), the properties of subrings of the ring W~(x) can be described in detail. Notice that, for x ~ 4, the rings W 2(x) are characterized by their parameters and have the Paley graphs as basis graphs.

Let f.l be a permutation of the vertices of the Paley graph which maps the graph onto its complement. Then the permutation v : (i 1 , i 2 ) --t (i~ , i~) is an automorphism of the subring numbered 4, which preserves each of 'I' 1 u 'I' 2 = { o , <1>1 } u { o , <1>2 } , 'I'3 u 'I's = ( y} u { <D~} and '!'4 = { <1>1 , 1l>2}. So the automorphism group of the sub-

1) This result is now published: M.E. Muzichuk, The automorphism group of the Paley graph. In problems of Group Theory and Homological Algebra., pp. 64-69. Jaroslavl', 1987 [In Russian)

246 I. A. FARADZEV

ring numbered 4 contains the group H = (G P(ps) I S2) · < v >. To prove that His the full automorphism group of this subring it suffices to show that His 2-closed. In the general case wehavenot proved this fact, but using the computer package presented in [12] thiswasdoneforps=5,9,13,17,25.

The ring W 2(1) is unique and its automorphism group is GP(5) = D 10· In this case the isomorphic subrings numbered 7 and 8 are Schurian with automorphism group S 5 I S 2. It is interesting to notice that the antireflexive basis graphs of the subring numbered 4 (they are basis graphs of the subrings numbered 7, 8, 9) form a partition of the complete 25-vertex graph into 3 graphs, each isomorphic to the lattice graph L(K s,s). Apparently, this partition was first discovered in 1981 by T.N. Mitina, a student at the Kaluga Pedagogical Institute, in her diploma worked out under the guidance of M.H. Klin.

The ring W 2(2) is also unique and its automorphism group is GP(9) = S 3 I S 2· The subrings numbered 1 and 2 in the ring Wh2) are isomorphic with automorphism group isomorphic to the group S3 I S4. The subrings numbered 5 and 6 are also isomorphic and are non-Schurian. As was proved in [22], the automorphism group of the subring numbered 3 is zj : PS04 (3). The isomorphic subrings numbered 7 and 8 are, in this case, Schurian with the automorphism group isomorphic to zj : PGO! (3).

Theseries m. v =x2 , k = 2(x-1), I= (x-1)2, A-=x- 2, Jl = 2 (x:::?: 2). The cellular rings W 3(2) and W 3(3) are members of the series I and II, respectively, and their lattices of subrings were discussed above.

For x > 4 the ring W 3 (x) is characterized by its parameters [19]. It is the V-ring of the group Sx I S 2 and has the lattice graph L(Kx,x) as a basis graph. In this case the ring W~ (x) has a unique nontrivial subring of rank 5, which is determined by the partition { { 1} , { 2 , 3} , { 4} , { 5} } (Lemma 4 ). This subring has a simple interpretation as the four-dimensional Hamming scheme H(4, x) over an x-element subset, and its automorphism group isomorphic to Sx t S4.

In the lattice of subring of the ring W~(4), an additional rank 3 subring arises. This subring has parameters (v , k , A.) = (256, 120, 56) and is determined by the partition { {1 , 4} , { 2, 3 , 5} } (Lemma 11 ). The appearance of this subring is due to the fact that W3(4) belongs to the series IV. In the case when W3(4) = V(S 4 I S2), the automorphism group of this subring isomorphic to Z~ : PSOt (2). The existence of this subring in the V-ring ofthe group S4 I S4 was mentioned in [14].

For x = 4 there is an additional cellular ring with the parameters of W 3(4). This is a non-Schurian ring, one ofwhose basis graphs is the pseudo-lattice on 16 vertices [19]. Its automorphism group is za : D 12· A computation carried out using the Computer package from [12] enabled us to show that the ring W~(4) and its rank 5 subring are nonSchurian, while the rank 3 subring has automorphism group of order 218 · 32 · 5.


The series IV. v = 4x2 , k = x(2x -1), l = (x + 1) (2x -1), A. = J.1 = x(x -1) (x~ 1).

The cellular rings W 4 (1) and W 4 (2) are members of the series I and m, respectively, and

the lattices of subrings of W~(l) and W~(2) were discussed above.

For x > 2 the question of existence and uniqueness of cellular rings with the param

eters of W 4 (x) has gone almost completely unstudied. It is known only that a ring with

these parameters can be constructed from a system of (x- 2) pairwise orthogonal Latin

squares of order 2x [3].

The lattice of subrings of the ring W~(x) for x > 2 contains a unique nontrivial rank

3 subring determined by the partition { { 1 , 4} , { 2, 3 , 5} } (Lemma 11 ). This subring has

parameters v = 16x4 , k = 2x2 (4x2 -1), A. == 2x2(2x 2 -1), i.e., it belongs to the same

series. Hence the existence of a ring W 4 (x) implies the existence of an infinite family of

rings W 4 (xi ), where Xi = 22'-1 x 2' , i ~ 0, with the parameters vi = 22'+1 x 2', ki = 22'-1 x 2' (22' x 2' -1), Ai== 22'-1 xi (22'-1 xi -1).

For x = 2s--1 , s ~ 1, an infinite series of Schurian cellular rings with the parameters

of W 4 (x) is known. These rings are V-ring of the groups Z~ : PS01,(2) [9]. The non

trivial subrings in the symmetric squares of these rings belong to the same series and

have automorphism groups Zf : PSO;t(2).

The series V. v =4x2 , k = (x-1)(2x+ 1), l =x(2x+ 1), A.= (x-2) (x+ 1),

J.1 = x(x - 1) (x ~ 2). The question of existence and uniqueness of the cellular rings W 5 (x)

for x ~ 4 is unstudied. The basis graphs of these rings are known to be negative Latin square graphs [17].

The lattice of subrings of the ring W3(x) for x > 2 contains a unique nontrivial rank

3 subring determined by the partition { { 1 , 3 , 5} , { 2 , 4}} (Lemma 11 ). This subring has

parameters v = 16x2 , k = 2x2(4x 2 -1), A. == 2x2(2x 2 -1) and belongs to the series IV.

For x = 2s-1 , s ~ 2, a series of Schurian cellular rings with the parameters ofW 5(x)

is known. These rings are the V-rings of the groups Z~ : PS02..(2) [9]. The only non

trivial subrings (s > 2) in the symmetric squares of these rings have the automorphism

groups isomorphic to Zf : PSO!s(2).

The ring W s(2) is unique. It is the V-ring of the group Z~ : S 5 and has the Clebsch

graph as a basis graph. In the lattice of subrings of the ring W~(2), one additional subring

of rank 3 arises. This subring has the parameters (v , k , A.) = (256, 45, 6) and is deter

rnined by the partition { { 1 , 4, 5} , { 2, 3} } (Lemma 1 0). It is shown in [20] that the

automorphism group of this subring contains ZS : 510 _ Using the computer package from

[12] it is proved that this subring has no additional automorphisms.

References

1. V.L. Arlazarov, A.A. Leman, M.Z. Rozenfeld, Construction and investigation of

graphs on 25, 26 and 29 vertices by use of a computer, Preprint, Institute of

248 I. A. FARADZEV

Management Problems, Moscow, 1975. [In Russian].

2. L. Babai, On the order of uniprimitive permutation groups, Ann. Math., 113 (1981), 553-568.

3. R.H. Bruck, Finite nets. /I Uniqueness and embeddings, Pacific J. Math., 13 (1963), 421-457.

4. P.J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc:., 13 (1981), 1-22.

5. P.J. Cameron, J. van Lint, Graphs, codes and designs, London Math. Soc. Lecture Notes 43, Cambridge Univ. Press, Camhridge, 1980.

6. P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplement, No. 10, 1973.

7. LA. Farad.Zev, M.H. Klin, M.E. Muzichuk, Cellular rings and groups of automorphisms of graphs [In this volume].

8. Ja. Ju. Gol'fand, A description of subrings in V(Sp, x Sp 2 x · · · x SpJ [In this volume].

9. Ja. Ju. Gol'fand, M.H. Klin, On k-homogeneous graphs, In Algorithmic Investigations in Combinatorics, pp. 76-85. Nauka, Moscow, 1978 [In Russian].

10. F. Harary, Exponentiation of permutation groups, Amer. Math. Monthly, 66 (1959), 572-575.

11. M.D. Hestenes, D.G. Higman, Rank 3 graphs and strongly regular graphs, SIAM AMS Proc., 4 (1971), 141-160.

12. A.A. Ivanov, M.H. Klin, I.A. Farad.Zev, Primitive representations of the nonabelian simple groups of order less then 106 , Part /I, Preprint, Intstitute for System Studies, Moscow, 1984. [In Russian].

13. L.A. Kaluznin, M.H. Klin, On certain maximal subgroups of symmetric and alternaring groups, Mat. USSR Sh., 16 (1972), 95-123.

14. M.H. Klin, On the method of construction of primitive graphs, Trudi NKI, No. 87 (1974), 3-8. [In Russian].

15. M.H. Klin, Investigation of algebras of invariant relations of certain classes of permutation groups, Ph. D. Thesis, Nikolaev, 1974 [In Russian].

16. M.Ch. Klin, R. Pöschel, The isomorphism problemfor circulant graphs with pk vertices, Preprint, Zentralinst. Math. und Mech. Berlin, 1980.

17. D.M. Mesner, Negative Latin square designs, V.N.C. Inst. Stat., Mirneo Ser., 1964, No. 410.

18. M.E. ~1uzic:huk, The subschemes of the Hamrning scherne, [In this volume].

19. S.S. Shrikhande, The uniqueness ofthe L 2 association scheme, Ann. Math. Stat., 30 (1959), 781-798.


20. V.A. Ustimenko-Bakumovskii, On the automorphism groups of strongly-regular graphs which are invariant under exponentiation of symmetric groups, In Computation in algebra, number theory and combinatorics, pp. 59-72, Kiev, 1978. [In Russian].

21. V.A. Zaichenko, An algorithmic approach to the synthesis of combinatorial objects and to computation in permutation groups based on the method of invariant relations, Ph. D. Thesis, 1981. [In Russian].

22. V.A. Zaichenk:o, M.H. Klin, Construction and investigation with the use of a computer of some block-designs and strongly regular graphs invariant under exponentiation of symmetric groups, In Permutation groups and combinatorial objects, 18-37, Preprint, Kiev Institute ofMathematics, 1982. [In Russian].


lntroduction

THE INTERSECTION NUMBERS OF THE HECKE ALGEBRAS H(PGLn(q), BWiB)

VA. U stimenko

Let G be a finite group of Lie type, r be the set of flags of its geometry and E>c be the coherent configuration corresponding to the action of G on r. The orbits of the permutation group (G, r), i.e. the minimal reflexive relations from E>c, are the sets of flags having fixed type. With an arbitrary triple a, ß, y e E>c of relations on r, a constant p ~ is associated. This constant is known as the intersection number of the configuration. The action of the pt:rrnutation group (G,r) on an invariant block r' (i.e. on a union of orbits of (G, r)) gives rise to a coherent configuration E>~ consisting of the relations from E>c lying in r' X r'. The intersection numbers of the configurations E>c and e~ corresponding tO a triple a, ß, y E E)~ COincide.

The computation problern for the intersection numbers of E>c and, in particular, of e~ is very interesting. This interest Sterns from coding theory (here e~ is a metric scheme, G is a classical group and r' is the set of flags having a given type [8], [9]) and from the representation theory of finite simple groups of Lie type. In particular the intersection numbers play a significant role in the computation of the characters corresponding to the action of G on the flags of a fixed type [3], [9], [12] and in the description of the subconfigurations of E>c and e~ [13]. This problern is very close to the description problern for the structure constants of the Tits algebra and its standard subalgebras [2].

The intersection numbers of the coherent configurations of the groups PSLn(q) and PGLn(q) acting on the Grassman manifold were computed in [14] (see also [16]). For the case of the unitary groups of different types (in the sense of [4]) acting on the maximal totally isotropic subspaces, the problern was solved in [15], [16]. An algorithm for the computation of the intersection numbers of E>c is presented in [1], in the case when Gis a Chevalley group corresponding to a diagram without multiple edges (i.e. An, Cn, Dn, Ei, i = 6, 7,8) and r' is its set ofmaximal parabolic subgroups.

Let G be a group over the field Fq and suppose that its Weyl group is not the dihedral group of order 16. Then it is known (see for example [13]) that the intersection numbers of E>c are po1ynomial expressions in q. In this case the computation problern for these numbers can be reduced to a deterrnination of certain numerical parameters of the Weyl group of G [1]. Usually the numbers p~ are presented in terms of some special functions related to G.

In the present article the considered problern is solved for the case G = PSLn(q). Notice that Substitution of q = 1 in the expressions for the intersection numbers of E>psL.(q) enable:s one to obtain the intersection numbers of the configuration related to the

251

252 V. A. USTIMENKO

symmetric group Sn acting on the ordered partitions. Recall that Sn is the Weyl group of PSLn(q).

The intersection numbers of a coherent configuration corresponding to a transitive permutation group can be obtained from a description of the orbits of this group on the triples of elements. Just in this way were results in [16] obtained. For a solution to the problern for 8psL.(q) this method is not effective, since it is very hard to describe the

orbits of P SLn (q) on the triples of flags. We will use a suitably coarser partition. R 2 of the set r3 , than the partition into orbits of (PSLn(q), r3 ) (see Section 5). The partition R 2 does not depend on the particular field, and the "contribution" of a relation from R 2

into the expression for the intersection nurober is a product of certain functions from the following family:

In Section 1 a combinatorial interpretation of this problern is given. In Sections 2 and 3 some Connections between the configuration e and the study of the Hecke algebras H(PSLn(q), BW,B) and the generaring algebras AA. are considered.

In Section 6 the problern is reduced to the determination of the constants r[ ;J. In

Section 7 a polynornial F~ (x) with the property F~ (q) = r[ ;J is constructed (Theorem

1). Some consequences of Theorem 1 are also contained in Section 7. The main assertians and relations are given in Section 7 without proof. In Section 8 a proof is given of Proposition 1 (which is a consequence of Theorem 1), and then Theorem 1 itself is proved.

1. A combinatorial interpretation of the problern

Let n be an integer and q be a prime power. Let V denote the vector space F~ and P(V) be the totality of subspaces of V (i.e. the projective geometry). A flag is a subset F

of P (V) such that for all subspaces W 1 , W 2 in F either W 1 :::;; W 2 or W 2 :::;; W 1 holds. It will be convenient for us to assume that each flag contain V and does not contain the null subspace. The elements W; of a ftag F are assumed to be indexed by elements from the set J = { 1, ... ,m}, where m = I F I is the length of the ftag. Moreover we assume that i > j implies W; > Wj. Let F 1 = { U;} , 1 :::;; i :::;; s, F 2 = { Wj} , 1 :::;; j :::;; t, be flags. The s x t matrix D (F 1 ,F 2) = (dim(U; n Wj)) will be called the type of the flags F 1 and F 2 .

THE INTERSECTION NUMBERS OF THE HECKE ALGEBRAS H (PGL.(q), BWß) 253

The binary relation "(F 1 ,F 2 ) and (F 1 ',F 2') have the same type" is an equivalence relation on the set of pairs of flags. For an integer matrix M = (mij) of size s x t, put [M] = {(F t.F2} e r x r I D(F t.F2) = M}. H [M] ;to 0 then M will be interpreted as a binary relation on r. It will be shown in Section 2 that these relations generate a coherent

configuration e and the value P!~hLJ =#{FE r I (F t.F) E [K]' (F,F 2) E [L]} for a fixed pair (F 1 ,F 2) of flags from [M] does not depend on the particular choice of (F 1 ,F 2)

in [M]. In the present paper the computation problern for these constants p[~}[L] is considered.

2. The configuration E> and the Heckealgebra H(PGLn(q), BWJ B)

Let (G,X) be a permutation group on the set X. Let us consider the action of Gon

the set xk (k-th cartesian power of X) defined by the following rule: (XJ, ... ,xk)'t = (xlf, ... • x~). where (XJ, ... ,xk) e xk and 1t e G. The orbits of the action of Gon xk will be considered as k-fold relations on X and are called the k-orbits of the per

mutation group (G,X). The collection of binary orbits of an arbitrary permutation group form a coherent configuration. More precisely, these orbits form a basis of the configuration in the sense ofD. Higman [6]. In what followsf e E> means thatfis an element of the basis of the configuration e.

A step basis of a chain V 1 < V 2 < · · · < Vk of subspaces is a basis which is

obtained by extension of a basis of Vi-1 to a basis of Vi where Vo = {0} and i = 1 , ... , k. ut (F 1 ,F 2> e r x r, F 1 = { Wd , i e I, F 2 = ( W/} , j e J, let < be the lexicographical order relation on the set I x J, and let Lij = (Wi n W/) u Wi_1. It is clear that ((i,j) > (s,t)) ~ (Lij ~ Ls~). A step basis of the chain of subspaces Lij will be called the step basis of the pair of flags (F 1 ,F 2). The group PGLn (q) of linear transformations of the space V can be considered as a permutation group on the set r, since a

nonsingular linear transformation maps a flag onto a flag. If 1t e PGLn(q), {Wi} = F 1,

(W/} = F 2· then dim Wi n W/ = dim Wf n Wj't. Let (4) and <4') be step bases of (F 1 ,F 2) and (F 1 ',F 2'), where (F t.F 2), (F 1 ',F 2') e [M] for some [M] from E>. Then

for a linear transformation ID which maps <4> onto <4'), it follows that Ff =F 1',

F'f = F 2'· Hence the coherent configuration E> is generared by the set of 2-orbits of the permutation group (PGLn(q), r). Every group G satisfying PSLn(q) $, G $, PrLn(q) has the same set of binary orbits on r. This means that (G, r) and (PSLn(q), r) are 2-equivalent in the sense of [7].

Let F = {W 1, ... , Wz} be a flag of length /. The integer vector (ai), 1$, i $,I, where ai = dim Wi, is called the type of the flag and is denoted by t(F). Let (F 1 ,F 2 ) be contained in [M] where M = (mij) is a matrix of size s x t. Then the vectors t 1 (M) = (mi1)

and t2(M) = (msj) of length s and t respectively, determine the types of the flags F 1 and F 2. The flags having a fixed type form an orbit of the permutation group (PGLn (q) , r).

Let a = (ai), i = 1 , ... , s, be an integer vector such that s $, n, a5 = n and

254 V. A. USTIMENKO

[(i <j) :::> (ai < aj)]. Let r~ = rä denote the set of flags in r having type a. Let us consider the permutation group (PrLn(q)' rä) which is the Testrietion of the group (PrLn(q)' n to the set rä. The binary orbits of the group (PrLn(q)' r ä) are the Testrietions of the binary orbits of the group (PrLn(q), r) (i.e. the relations [M] over r) to the set rä. These relations form a homogeneaus coherent configuration Sa which is a subconfiguration of E> in the sense of [6]. Notice that the restriction of the relation [M]

over r to the set r ä is nonempty if and only if t 1 (M) = t 2 (M) = a. The latter condition implies [M] c rä x ra-. Thus if [M], [K], [L] are all contained in ra- thenpf~[LJ is the intersection number ofthe configuration E>a-.

Let F = {W 1 , ... , Wn} be a flag from I/ where I= (1, ... , n) (i.e. Fis a maximal ftag). The stabilizer B of a point in the permutation group (PGLn(q), I/) is a Borel sub-

- -group of the group PGLn(q). In the basis h of the space V, where /i e Wi, the group B

consists of the upper triangular matrices. The stabilizer Pa- of an arbitrary point in the permutation group (PGLn(q), rä) contains a subgroup which is conjugate toB, and this stabilizer is a parabolic subgroup of PGLn(q).

The subgroup Pa- is conjugate to a subgroup B W1 B, where W1 is the subgroup generated by some subset J of transpositions from the set S = {(12) , (23) , ... , (n - 1, n)} of generators of the Weyl group W =Sn (the group of permutation matrices). The adjacency a1gebra K e.- [6] is the Schurian V-ring of the permutation group (PGLn(q), r~). In other

words, it is the Heckealgebra H(PGLn(q), B W1 B).

3. e and the generating algebra AA.

The intersection numbers of the configuration E> are polynomial expressions in q. This is also the case for the numbers of the configuration of binary orbits of an arbitrary finite Chevalley group whose Weyl group is not the dihedral group of order 16, acting by left transformations on the set of left cosets of its parabolic subgroups (see for example [13]). By Substitution of an indeterminate x (instead of q) in the expressions for the intersection numbers of the Heckealgebra H(PGLn(q), B), we obtain the structure constants of the generating algebra AA._1 of the Coxeter group An_1 (i.e. of the symmetric group

Sn). The notion of a generaring algebra was introduced by J. Tits. The Substitution of x instead of q in the expressions for the numbers for E>a- enables one to obtain the structure constants p[if~[LJ (x) of the standard block subalgebras in AA._1 , as well as of the univer

sal block algebras. Thesenotions were introduced in [13].

4. The intersection numbers and feasible partitions

Let (E>,X) be a coherent configuration. A ternary relation cp over X will be called feasible with respect to e if there is a triple f,g,h of elements from e such that

THE INTERSECTION NUMBERS OF THE HECKE ALGEBRAS H (PGLn(q), BWß) 255

(x,y,z)e <1> ~ [(x,z)e.f] & [(z,y)eg] & [(x,y)eh] and the value p(cp,h) = # {z I (x,y,z) e <1>, (x,y) e h} does not depend on the particular choice of (x,y). The relations f,g,h will be called the sections of <1> and will be denoted by si(<l>) for i = 1 ,2, 3. A prutition 't of the set X 3 into feasible relations will be called "feasible for 8".

Let (8,X) = Orb 2(G,X). It is easy to see that the family R' of 3-orbits of the group (G,X) is feasible for 8. If <1> is a 3-orbit of (G,X), then the nurnber p(cp,g) is the projection coefficient of <1> onto g in the sense of [10]. It follows frorn the definition of the intersection nurnber pjg of the configuration 8, that if 't is feasible partition for 8, then

pjg = L P<<l>.h), cpe R},,

whereRj,g = {~e 't I s3(4>)=h, s1(<1>)=/, s2(<1>)=g}.

Let R" be the partition of X 3 into the temary relations

(1)

Dj,g = { (x,y,z) E x 3 I (x,z) E f' (z,y) E g' (x,y) E h }. Now it is clear that a partition R which satisfies the condition

R':~ R ~ R" (2)

is feasible. Here 't ~ 't1 rneans that 't' is either coarser than or identical to 't.

The known formulas for the intersection nurnbers of the Hecke algebras of a Chevalley group (or of its Weyl group) acting on the cosets of a parabolic subgroup were obtained by rneans of relation (1) where 't is the collection of 3-orbits of the permutation group. In the considered case ofthe action of PGLn(q) on ftags, the description ofthe ternary orbits is a very hard problern. The nurnber of these orbits depends on the field F q

and, under cert~tin restrictions, on the dirnensions of subspaces (i.e. on the types of ftags). This problern is equivalent to a well-known problern conceming a pair of rnatrices. The latter problern is to find necessary and sufficient conditions for a pair of n x n rnatrices to be equivalent over Fq.

Let us give another example of a partition which is feasible with respect to 8. For A,B e P(V), letA-B denote the affine space of vectors frorn A which does not lie in

A 11 B. Let us consider the set ~ of syrnbols -, 1., 11 (here l. and 11 rnean surn and intersection, respectively), of individual variables xi, j = 1,2,3, i ~ n, and of the brackets (, ). Let H be the set of correctly constructed formulas over ~- Let us define the equivalence relation R 1 on r 3 by the following rule. Suppose that Fi = { W~ , W~ , ... , wt } and

F{= {U~, U~, ... , UtJ. Then [(Ft>F2,F3), (F1',F2',F3')] e R1 if l(Fi)=l(F{)

and, for any formula P(xl, ... ,xt ,xf, ... ,xt ,xf, ... ,X~3 ) frorn H, the dirnensions of the

quasiprojective manifolds (see [11]) P(Wl, ... , wt.wf. ... , WI2 ,Wf_ ... ,Wt) and

P<Ul, ... , ut ,ur, ... , ut.ur, ... , ut) coincide.

It is easy to see that R 1 is a feasible partition. In fact, since a nonsingular linear transforrnation preserv1~s dimensions and H contains the formula x[ 11 xJ, we see that R 1 satisfies (2) and hence R 1 is a feasible partition. Notice that the nurnber of classes of R 1

256 V. A. USTIMENKO

does not depend on Fq. Nevertheless, the number of parameters which is necessary to define a class in the partition grows exponentially in n, and calculation of the coefficient p(<\>,g) for arbitrary <1> from R 1 is a hard problem. We present below a coarser feasible partition which is free from the above encumbrances.

5. The partition R 2

Let < and $ be the relations of strict and nonstrict lexicographical order on I x J,

wherel= {1, ... ,kd,J= {1, ... ,k2}.

Let us consider the following formulas from H:

and finally

Tij = ..l (xi n xr) , (s,t)~ (i,j)

Ti/= ..l (x! n xh, (s,t) < (i,j)

Aij = Tij - Ti/ ,

Wso = ..l (x} n xy), (i,j) I j <s

Wst=Wso..ix?, t=1,2, ... ,k2,

rf} = Aij n W st , j < s

rst ( 1 3 1 3 2 3 2 3) ij X1 (")XI , ... , Xk1 (")XI , X1 (")XI , ... , Xk 2 (")XI .

Let us define the equivalence relation R 2 on r 3 by the following rule. Suppose that Fi = {W{, ... , W~.} and F/= {Ui, ... , U~.}, i = 1,2,3. Then (F1,F2,F3) and

' ' (F 1',F 2',F 3') are equivalent if ki = ti for i = 1,2,3, and dim Gf}I<Wl , ... , wt) = dim

Gf}I<Ul , ... , ut) for each choice i,j,s,t,l of parameters such that j < s. With an

integer function h = h(i,j,s,t,/) defined on all sets (i,j,s,t,l) E Z 5 such that j < s, let us associate a temary relation

{h} = {(F1,F2,F3) E r 3 I Fj = {W{, ... , W{j}, j = 1,2,3 &

dimGtjl(Wl, ... , Wk 1 , Wy, ... , wt) = h(i,j,s,t,/)}.

We will consider only such functions h with {h} ;t:0. It is clear that the relations {h} form the set of equivalence classes of R 2.

Let F = { W 1 , ... , Ws} be a flag from r and C E P (V). Let F n C denote the flag { W 1 n C , ... , Ws n C} of the space C. The following lemma is a direct consequence of

THE INTERSECTION NUMBERS OF THE HECKE ALGEBRAS H (PGLn(q), BWß)

the definition of { h } .

Lemma 1. Let (F1oF2,F3) E {h} andF3 = {W1, ... , Ws}. Then (F 1 !1 Wt, F 2 !1 Wk) E [(at)l where

at = l: h(i,j,jo + 1,io,k).

Corollary 1. s3( {h}) = [h(afj)], s = l(F 3), s 1 ({h}) = [(bik)l where bik = a~, t = l(F 2), s2({h}) = [(cjk)l where Cjk = ai, I= l(F 2).

Corollary 2. The partition R 2 is feasible. In fact, by Corollary 1 R 1 ~ R 2 ~ R", and hence R 2 satisfies (2).

The constantp({h}, s 3({h})) will bedenoted by r(h).

6. Reduction to r X r x P (V)

257

The set P (V) will be identified with the set of flags of length 2 (recall that if F = { W 1 , W 2 } then W 2 = V). We will show that the computation problern for r(h) can be reduced to the case when {h} is contained in r X r X P(V).

Let {h}crxrxP(V) and (Fl>F 2,F3)E {h}. Then F 3 =(Wi,V} and h(i,J,s,t, 2) = dim r~<Wl , ... , Wr). h(i,J,s,t, 1) = dim <rf]<wl , ... , w!;. Wy , ... , Wf2 ) n WI) where Ii = l(Fi), i = 1,2.

Notice that in the considered case the function h(i,j,s,t, 2) is completely determined by the matrix (a~) = s 3 ( { h }). This means that the function h is determined by the pair of functions M = (a~) and g(i,j,s,t) = h(i,j,s,t, 1). The ternary relation {h} which lies in

r x r x P (V) will be denoted by {!}. where n = dim V, whilu { h } will be denoted by

rn [~] orr [~]· For a flag F = {W1 , .•• , Wd let us define a set F(1) = {W1, W2},

F<2> = {W 2 , W 3}, ... , F(l-1) = {W1-1, Wd of flags of length 2 in the spaces P(W 2), ... , P(Wz), respectively. Now let (F 1oF 2,F 3) E {h}, F 1 = {W} , ... , wt }, F 2 = {Wt , ... , WT2 }, F 3 = {WI , ... , Wf3 }, where /3 (the length of F 3 ) is an arbitrary

number. Then

{ ga.(i,j,s,t)}

(F1 n w& . F 2 n w&, F~-1 ) E M a. ,

a = 2, ... ,1, where g a.U.},s,t) = h(i,j,s,t, a), M a. = (aij) (see Lemma 1). Since the func

tions g a. and M a. can be computed from h(i,j,s,t,l), the set ofrelations { !: } does not

258 V. A. USTIMENKO

depend on the particular choice of (F 1oF 2 ,F 3) from { h}, and the following lemma holds.

Lemma 2. Let h(i,j,s,t,l) be an integer function such that {h} * 0. Then

1 [Mal ((F1,F2,F3)E {h}) =:> t(F3)=(m1····•mz)andr(h)= !J/m· ga . In fact, t(F 3) is determined by the function h and, for a fixed pair offtags (F l>F 2),

the expression rm [Mal gives the number of ways of choosing W a-1 in W a suchthat • 8a

for F 3 = {W 1 , ... , Wz}, the inclusion (F 1 ,F 2.F 3) E {h} holds.

7. The constant r [ ;J The purpose of this section is to define a polynomial F~ (x) whose value at the point

x = q is equal to r [ ;J. The polynomial F~ (x) will be produced as a product of certain

polynomials from the following families:

a)

b)

Y =xt

Hj(x)=(x 1-l) (x 1-x) · · · (x 1-x1- 1),

c) [ nl Hn(X) (.x) = < l , where H1(x) = Hj(x) and n,m,l,t are nonne-m Hm(X)· Hn-m(X)· xm n-m

gative integers with n 2:: m , l 2:: t.

.. [ nl l:i,-t(m-1)m

(x) = I: x'=' m 0<i 1 <···<i .. ~n .

(3)

Relation (3) shows that [;] (x) is in fact a polynomial.

Let a,bbe vectors fromNk suchthat a;;;:: b; for i = l, ... ,k. Put

[ ä'l k [ a;l b; L (ai-bi) - (x) = TI (x) x i<' . b i=1 b;

A matrix M and a function h are said to be feasible if the relations { M } and { h } are nonempty. A function g(i,j,s,t) is feasible if g(i,j,s,t) = h(i,j,s,t,/0 ) for a fixed 1° and a feasible function h. For a feasible relation {h} put l;(h) = l(F;), i = 1,2,3, where (F1,F2,F3)E {h}. For (i,j) EI xJ put Q;j= {(s,t) I j<s, t<i}. Let (l+,m+) be the element from Qij which follows (l,m) in the lexicographical ordering. Let a;;;; (a 1 , ... ,a1)

be an element from N 1 such that (i > j) =:> (a; > aj)· Put

a* = (aba2 -al ' ... ' al-al-1). With a matrix A = (aij), i EI' jE J, let us associate the vector t(A) = (a 11 , ..• , am 1m2 ) oflength m1 • m2, where m1 = I I I, m2 = I J I and

the indices of the components are ordered lexicographically. We will also consider the

vector Ä = t(A)* where t(A)* e Nm,xmz.

Definition. Let { !} be a feasible relation where M = (mij), M 1 = (mfj). Let us

consider the function

ms,r =. I1 n [ [g~<{:;.:::;)] xH~ • x-d'·r] l>t+1j<S

where r = mt+ls- m}+1s and

d = I: I: (g(i,J,s+ ,r+)- g(i,J,s,t)), d' = I: I: g(i,J,s,t). i>t+1j<s

The function IDst is not in general a polynomial.

Theorem 1. If {!} is a feasible relation then

F~ (x) = [:1] ll ij IDst t=O s-2

isapolynomialandF~(q)=r [;].

i>t+1j<s

Corollary. The polynomial F~(x) is a product of a finite number of polynomials from the families a), b) and c).

As a consequence of Theorem 1 the following proposition can be obtained.

Proposition 1. Let F = {Wd, i = l, ... ,k, be a flag and (t(F))* = (1 1, ... ,/t). Let

s = (s 1> ... ,sk) e Nk where li >Si for all i. Then the number of subspaces X in P(V) such

that (t(F 11 X))* = s coincides with [~ (q).

Example. For m 1 = m 2 = 3 Iet us present all functions IDsr corresponding to {!} . Herewe put (p'ij) = M- M 1 and write gijst instead of g(i,j,s,t).

260 V. A. USTIMENKO

ffi30 = H:~3 [ ~:~:: ]· [ ~:::: ]· [ ~:~:: ]· [ ~::::] x -pndz',

wheredz=g2131-g213o+g2231-g223o+g3131-g3130, dz' = gznl + gzz31 + g3131 + g3231 ;

whered4 =g3131 -g3121, d4'=g3131·

Remark. Let the Hecke algebra H(PGLn(q), B W1 B) correspond to the action of the group on ftags of type (I I> ... ,h). Then its basis elements correspond to the relations [M] from e suchthat the matrix M 1 = (aij) has size k x k and the (ij)-th component aiJ

of the vector t(M) satisfies the following relations:

k k I, asj = I, ais = ls , s = 1 , ... , k.

}=1 i=l

By means of the formulas (1) and Lemma 2, the intersection numbers pfiftrLJ can

be expressed in terms of the coefficients r [ ;J. In particular, the generators of

H(PGLn(q), B) correspond to the relations [M] from e such that M 1 is a permutation matrix of size n x n.

8. Proof of Theorem 1 - -

Let us fix a basis (ll ,lz, ... , ln) of the space V.

Lemma 3. For each element W of P (V), there is a unique basis of W having the

form CXi 1 , ••• , Xim) where Xi, = /;, + L ai,J Ii and ai,J = 0 if j E { i 1 , ... , im}. j < i~

Proof Starting with an arbitrary basis we can obtain a basis of the desired form using the method of Gauss. On the other hand, different sets of vectors satisfying the conditions of the lemma generate different subspaces. o

Definition. The basis :Xi, , ... , :X;m and the set {i 1 , ... , im} which satisfy the con

ditions of Lemma 3 will be called the Gaussian basis and the Gaussian sets of the subspace W.

Remark. The set S = { i 1 , , . , , im} is an element of the Coxeter geometry r of the

group Sn (the Weyl group of the group PGLn(q)). The basis Xi 1 , , •• , Xi,. can be

identified with a function fs defined on the set { (i,j) I i E S , j ~ S} of transpositions,

and which assumes values in Fq. The pair (S,fs) determines a combinatorial covering of

the geometry r in the sense of [13).

Lemma 4. The cardinality of the Grassman manifold r~(q) is equal to [:] (q).

Proof. The nurober of different Gaussian bases corresponding to a given Gaussian { . . } . I (i 1-t)+ · · · +(i,.-m) B · all { · · } set l 1 , ... , lm 1s equa to q . y summmg over sets l 1 , ... , lm ,

we obtain the nurober of different rn-dimensional subspaces. 0

Lemma S. The nurober of n x m matrices over F q having rank r is equal to

R(m,n,r)=[~.l [~]·Hr(q). Proof. Let us consider the action of the group G = GLn(q) x GLm(q) on the set of

all n x m matrices by the rule: (A,B) E GLn(q) x GLm(q) maps X into A X B. By means

of G each matrix of rank r can be carried to the canonical matrix J = (aij) such that

a 11 = a 22 = · · · = a77 = 1 and all other entries are equal to zero. Thus G acts transitively on the set of matrices having rank r. The subgroup of G which stabilizes J consists of the pairs (A,B) having the following form:

[ Al A2] [A11 0] A == 0 A 3 , B = B 1 B 2 , where

A1 E GL7 (q),A3 E GLn_7 (q),B2e GLm-r(q) and A2,B1 are arbitrary matrices. Hence the nurober R(n,m,r) is equal to the indexoftbis subgroup in G:

0

The proof of Proposition 1. Let F = {V1, ... , Vd} be a flag from r, (t(F))* = (t 1 , ... , td), and let X be a subspace from P(V) such that

(t ( {V 1 n X , . . . , V d n X }))* = ( 11 , ... , ld) = T. Let us consider the basis T 1 , ... , Tn in

F~. Without loss of generality we may assume that Vi = < T1 , ... , lc, >. Leti'i 1 , ••• , Xi,.

be a Gaussian basis of X. Then for a Gaussian set M of the space X, we have

IM n {ti-1 , ... , ti} I = li where to = 1. The nurober of the Gaussian bases satisfying

this condition is equal to [ ~ (q ). 0

The proof ofTheorem 1. Let (F 1,F2,F3) E { !} and T1 , ... , Tn be a basis of F~. Without loss of generality we may assume that F 1 = {B 1 , ... , Bm} where

262 V. A. USTIMENKO

Bi=< l 1 , ... , /,, >, F 2 = {D 1, ... , Dm2 }, where Di are generated by the basis vectors and Bill Dj ::::> < /,1-1+1 , ..• , /,i-l+aij > = Aij· The cardinality of the set

A= {X I (F 1 nX, Fz roX) e [M ,]);, "'!Ual to [ ~], M = t((aij)).

Let :Xi, , ... , Xit be a Gaussian basis of X. Since g(i,j,s,t) = dim(X ll (Aij +Ds +B1)), then, for the rank r' of the matrix V1s = llxpqll where p E S(Ai+1,s)- S(X ll At+1,s), q E S(Aij + Ds + B1), s > j, t < i, we have

' 1 r = mt+1,s- mt+1,s· (4)

In addition

(5)

It can be shown using Lemma 5 that the nurober of matrices satisfying (4) and (5) is equal to

[ (. . + +)] g l,J,S ,t d E = TI TI (' . ) • Hr'· . 1 . g l,J,S,t l>t+ j<S

The nurober of matrices whose size is that of (Xpq) is equal to Z = qd'r where d = L L g(i,j,s+ ,t+). It is easy to see that the nurober of X from A such that

i>t+1j<s (Xpq) = u1s satisfies (4) and (5), can be obtained by multiplying I A I by EI Z. Since con-ditions (4) and (5) should be satisfied by u18 for all t and s, the nurober of X such that

{g (i,j,s, t)}

(F 1 ,F 2,X) E M for a fixed pair (F 1 ,F 2), can be computed from the formula

given in Theorem 1. 0

References

1. A.E. Brouwer, AM. Cohen, Computation of some parameters of Lie geometries, Stichting math. centrum, Amsterdam ZW, 198/83 September, 32p.

2. C.W. Curtis, Representation of finite groups of Lie type, Bull. Amer. Math. Soc. (New Series) 1 (1979), 721-757.

3. P. Delsarte, Association schemes and t-designs in regular semilattices, J. Comb. Th., 20 (1976), 230-243.

4. J. Dieudonne, La Geometrie des Groupes Classiques, Springer-Verlag, Berlin, 1971.

5. LA. Farad.Zev, M.H. Klin, M.E. Muzichuk, Cellular rings and groups of automorphisms of graphs [In this volume ].


6. D.G. Higman, Coherent configuration, Geom. Dedic., 4 (1975), 1-22.

7. L.A. Kalumin, M.H. Klin, On certain maximal subgroups of the symmetric and

alternating groups, Mat. USSR Sb., 16 (1972), 95-123.

8. L.A. Kalumin, V.I. Suschanskii, V.A. Ustimenko, Use of the computers in the

theory of permutation groups and its applications, Kibemetika 6 (1982) 83-94 [In

Russian].

9. W. Kantor, Furtherproblems concerning finite geometries and finite groups, Proc.

of Symp. ofPure Math., AMS, Providence, Rhode Island, 37 (1980), 479-486.

10. M.H. Klin, Investigations of algebras of invariant relations for certain classes of permutation groups, Ph. D. Thesis, Nikolaev, 1974 [In Russian].

11. I.R. Shafarevich, Basic algebraic geometry, Springer Verlag, Berlin, 1977.

12. D. Stanton, Some q-Krawtchouk polynomials on Chevalley groups, Amer. J. Math.

102 (1980), 625-662.

13. V.A. Ustimenko, On some properlies of geometries of Chevalley groups and their

gen~ralizations [In this volume].

14. V.A. Ustimenko-Bakumovskii, Algorithmsfor the construction ofblock-designs and

strongly regular graphs with given automorphism group, In Computation in Alge

bra and Combinatorics, pp. 137-148, Kiev, 1978 [In Russian].

15. V. V. Zdan-Pushkin, V.A. Ustimenko-Bakumovskii, V-rings of Schur of finite unitary groups acting on maximal isotropic subspaces. In VIII All Union. Symp. on

Group Theory, Kiev, 1982, pp. 36 [In Russian].

16. V.V. Zdan-Pushkin, V.A. Ustimenko, Classical groups and metric association

schemes, Kibemetica 5 (1985), [In Russian].

This paperwas originally published in IATC0-85, pp. 95-104.

RANKS AND SUBDEGREES OF THE SYMMETRIC GROUPS ACTINGON PARTITIONS

I A. F aradiev & A. V. Ivarwv

1. The rank and subdegrees (i.e. the number of orbits of the point stabilizer and their respective lengths) of a transitive permutation group are quite simple and very significant characteristics of the group. In certain cases the knowledge of the rank and subdegrees of a group is sufficient for the resolution of the following problems:

(a) identification ofrank: 3 graphs [6];

(b) proof of the existence and the nonexistence of distance-transitive graphs [7];

( c) computation of the decomposition of the permutation character of the group into irreducibles by means of the character table.

There are a few different approaches to the computation of the subdegrees of a transitive permutation group. One of these approaches relies on construction of the orbits of the point stabilizer of the group under consideration in an essential way. To realize this approach one should know a system of generators for the point stabilizer or should construct such a system using the generators of the whole group. An algorithm realizing this method is described in [10]. By means of this algorithm, the rank:s and the subdegrees of the primitive representations of the nonabelian simple groups of order less then 106 were computed [9]. Since this method requires a large reserve of computer resources, its sphere of its application is restricted to permutation groups having degree at most 104-105.

The second approach (see [8], [11]) relies on the solution of a system of linear equations arising from the computation, in two ways, of the numbers of points being fixed under the action of subgroups of the point stabilizer. This method is quite effective for representations of groups on the cosets of their "small" subgroups. It can be applied not only to concrete groups, but also to infinite series of groups. In [8] some examples are presented which give a computer-free computation of the rank:s and the subdegrees of two representations of the group J 1 and of the representations of the groups PSL 2(q) on the cosets of A4. The sphere of application of this method is restricted by the amount of detailed structural information one has concerning the group under investigation.

Finally, the third approach relies on consideration of the relevant group (G, N) in its action on the orbit of some relation in another representation (G , M) of G (an induced action). In this case the 2-orbits of the group (G, N) are certain equivalence classes of pairs of relations under the action of (G, M). By finding representatives of these classes

265

266 I. A. FARADZEV AND A. V. IVANOV

and computing the order of the subgroup in G which stabilizes both of the relations, one can determine the subdegrees of the group (G, N) using Lagrange's theorem. If the rela

tion has a simple structure then it turns out to be possible to obtain not only the rank and

subdegrees, but also a formula for the structure constants of the corresponding V-ring. In [13] this method was realized for the group sr;: (which is the action of the symmetric

group Sn on the set of all subsets of size m, i.e. the representation of Sn on the cosets of

the direct sum Sm$ Sn-m) and for the group Sn i Sm (the action of the wreath product Sm wr Sn on the set of rn-dimensional vectors). A less trivial example ofthistype is the action of the group PGLn(q) on the Grassman manifold r'f:(q) [15]. It should be men

tioned that each of the above three examples corresponds to a metric association scheme. These schemes are the Johnson scheme, the Hamming scheme and the q-analog of the

Johnson scheme, respectively. In more complicated cases the classical enumeration tech

nique can be applied. The procedure is as follows. Using the cyclic index of the stabilizer of a relation U in the original representation of the group, the rank of the induced

representations can be calculated by means of the Bumside lemma. After that, the pairs

of relations in the orbit containing U can be classified by determination of additional more detailed invariants. In [3] the ranks and the subdegrees of certain representations of

the group M 12 were calculated using this method.

For the classification of pairs of relations or their full invariants, the constructive enumeration technique [5] can be applied. As far as we know this technique was not used

for computation of ranks and subdegrees before.

2. The present paper is devoted to computation of the rank and subdegrees of the group slio 1). which is the representation of the symmetric group s/d on the cosets of the

wreath product Sk wr S1. It is known [2] that the subgroup Sk wr S1 is maximal in Ski, so Slio 1> is primitive. This fact stimulates interest in the investigation of these groups. The

degree of the group S~o 1> grows very rapidly with the growth of k and l. Forthis reason, application of the first method is not effective in this situation. The second method is also not effective since the lattice of subgroups of the group Sk wr S1 is quite rich. On the

other hand, application of the third method is well motivated since, in the natural

representation of the group Ski on m = k • I points, the subgroup Sk wr S1 preserves a simple combinatorial object. This object is the partition of the set of m points into k classes, each of size /. From this point of view, it is convenient to study the group S~~o 1> as the induced action of the symmetric group on the set of all such partitions.

In [ 16] a description of t-orbits of the group S~~o 1> for arbitrary t is given. For the 2-orbits a more elementary description can be obtained.

3. Let us recall some facts conceming directed multigraphs and their automorphism

groups. Fora directed multigraph r let V(r) denote its set ofvertices andA(r) denote its

set of arcs. Two multigraphs r and L1 are isomorphic if there exists a pair (Jl1 , Jl2), where Jl1 is a bijection of V(r) onto V(L1) and Jl2 is a bijection of A (r) onto A (L1), such that for

a e A(r) and x, y e V(r), the vertices x~1 and y~1 are incoming and outgoing vertices

RANKSAND SUBDEGREES OF THE SYMMETRIC GROUPS ACTINGON PARTITIONS 267

of the arc a112 if and only if x and y are incoming and outgoing vertices of a, respectively.

The set of all isomorphisms of a multigraph r with itself constitutes the automorphism

group (in a broad sense) of the multigraph and is denoted by Aut(r). The group Aut(r)

induces two permutation groups on the sets V(r) and A(r), respectively. These groups

are called the vertex and the arc groups of the multigraph and are denoted by V Aut(r)

and AAut(r), respectivelyl). For a multigraph r and x, y e V(r), let '}(x, y) be the

number of arcs from x to y. We shall say that r has no isolated vertices if, for each

x e V(r), there is at least one y e V(r) suchthat either '}(x, y) > 0 or y(y, x) > 0. The

following lemma is obvious

Lemma 1. If a multigraph r has no isolated vertices then its automorphism group

acts faithfully on the set of arcs and Aut(r) = AAut(r). o

In what follows we shall consider only multigraphs without isolated vertices. For

this reason we can more simply denote the isomorphism (~1, ~2) by ~2· Indeed by

Lemma 1, the action of this isomorphism on the vertex-set is determined by its action on

the arc-set. The group V Aut(r) contains all permutations A. of the vertex-set such that the

number of arcs from x to y coincides with the number of arcs from x). to y). for all

x,y e V(r). These permutations are said to be automorphisms of r in the narrow sense.

The kemel of the action of Aut(r) on V(r) consists of all isomorphism which aretrivial

on V(r). It is easy to see that this kerne! is isomorphic to the direct sum

l: S~x,y) x,yeV(I}

of symmetric groups. Here fory(x, y) = 0 it is assumed that Sy(x,y) is the identity group.

So the following proposition holds.

Lemma2.

I Aut(r) I = I V Aut(r) I • IJ '}(x , y) ! :x,yeV(I}

0

Now we are ready to formulate a criterion for isomorphism of directed multigraphs.

Notice that this criterion, as weH as Lemma 2, is valid for arbitrary directed multigraphs.

Lemma 3. Two directed multigraphs r and A are isomorphic if and only if there

exists a bijection A. of the set V(r) onto the set V(A) such that for all x, y e V(r) the

number of arcs from x to y coincides with the number of arcs from x). to y)..

Proof. The necessarity of the condition is clear. To show that the condition is

suflicient one should fix, for any pair x, y E V{f), a bijection between the arcs from

x to y and the arcs from x>. to y>-. In this way the pair of mappings which determines

the isomorphism of multigraphs will be well-defined. 0

1) It should be pointed out that in other papers of this book the automorphism group of a multigraph r is understood to be the group V Aut(r).

Let us return to consideration of the group s~· I). As was mentioned above, this group can be realized via the action of the symmetric group Skl of a set M of m = kl points on the set of all partitions of M into k subsets, each of size I. Let

R = {r 1 , • · · , rd and T = {t 1 , • · · , tk} be two such partitions. With the ordered pair (R , T) we associate a directed multigraph r(R , T) by the following rule. The set of vertices of r(R , T) is R u T and the set of arcs is M where the arcs from the set r; n tj join r; with tj. The multigraph r(R , T) will be called the intersecnon graph of (R , T).

Lemma 4. Two pairs (R , T) and (R' , T') of partitions are contained in the same 2-orbit of the group s~· I) if and only if their intersection graphs are isomorphic.

Proof. Suppose that the pairs (R 1 T) and (R' 1 T') are eontained in the sanl<' 2-orhit of the group si7ol)_ This means that there is a permutation IL E S~.;:ol) such

that R' = R~" and T' = T~". Tlwn it is clear that Jl induc:es an isomorphism from

f(R 1 T) to f(R'1 T').

Let r = r(R, T) and r' = r(R', T') be isomorphic and 1.1. be corresponding isomorphism. Since the sets A (r) and A (r') are marked by elements of the set M, the restriction of 1.1. on A (r) can be considered as a permutation from s~· I) which maps the pair of partitions (R, T) onto the pair (R', T'). o

Corollary 1. A pair of partitions (R , T) is contained in a symmetric 2-orbit of the group s~· I) if and only if the multigraphs r(R , T) and r(T, R) are isomorphic.

4. Let us consider the set T(k, I) of all directed bipartite multigraphs with k vertices in each part such that exactly I arcs come out of an arbitrary vertex in the first part and exactly I arcs come into an arbitrary vertex in the second part.

Lemma 5. Every graph r from the set T(k , I) can be realized as the intersection graph of some pair of partitions.

Proof. Let us mark the arcs of a graph r from the set T( k, I) by tlw distinct

elements of the set l\.1. There are two equivalenc:e relations on the set of arcs. Thf'

first relation is equivalenc:e at each incoming vertex and the second one is equivalt>nc<' at each outcoming vertex.

This pair of partitions of the set M into equivalence classes possesses the claimed properties. o

For a pair (R , T) of partitions, the graph r(R , T) is isomorphic to a graph from the set T(k , 1). Hence Lemma 4 and Lemma 5 imply the following proposition.

Theorem 1. There is a one-to-one correspondence between the 2-orbits of the group s~· I) and the isomorphism classes of graphs from T(k , 1).

Corollary 2. The rank of the group s~· /) is equal to the nurober of pairwise nonisomorphic graphs from T(k, 1).

Corollary 3. The 2-orbit of the group s~· /) corresponding to a graph r E T(k ' I) is symmetric if and only if r is isomorphic to the graph r' which is obtained from r by reversing orientation of all arcs.


Theorem 2. Let r E T(k , l) and Yij be the number of arcs joining the i-th vertex from the first part with the j-th vertex from the second part of the graph r, 1 ~ i,j ~ k. Then the valency of the 2-orbit corresponding to r is deterrnined by the formula:

S(r) = (I !)k. k ! = --,------'-(1=-!"--)k_. _k=! =-------:-1 Aut(r) I I VAut(r) I· I1 Yij!

(*)

15.i,~k

Proof. For an arbitrary transitive permutation group (G, N), the valency of the 2-orbit of the group which contains the ordered pair ( a, b) of elements from N is equal to the index [Ga : Gab]· In the considered case, Ga ~ Sk wr Sc and Gab is the stabilizer in Skc of the pair of partitions Rand T corresponding to r, i.e. Gab ~ A Aut(f(R, T)). Now Lemmas 1 and 2 imply the claimed formula. D

Thus the deterrnination of the rank and the subdegrees of the group s}j· /) can be reduced to the: enumeration of pairwise nonisomorphic graphs from T(k , I) and to the computation of the orders of the corresponding automorphism groups, respectively.

The graphs from T(2, I) and T(k, 2) can be easily described and a formula for their ranks and subdegrees can be obtained in these cases.

5. A graph from the set T(2, l) is uniquely deterrnined by each constant Yij· If y11 = y then Y12 = Y21 = l - y, Y22 = y. By isomorphic transformations of the graph we can force Yn ~ Yl2· Hence the rank of the group S~]· I) is equal to [112] + 1. The order of the automorphism group of a graph from T(2, l) is equal to 2 if I "# 2 y and to 4 if I = 2 y. A substitution in (*) gives the following formula for the subdegree corresponding to a graph r E T(2, I) forO~ y~ [//2]: [ ~r for z "* 2y;

t [~r for /=2y.

Each graph from T(k , 2) splits into connected components which are cycles of even length where arcs and antiarcs altemate. A graph is uniquely determined by the sizes of these connected components. So the rank of the group s~· 2l is equal to p(k), i.e. the number of ways to decompose k into a sum of positive integers. For information conceming the computation of p (k) see, for example, [ 1]. The vertex group of a connected component of size 2i is the dihedral group of order 2i if i > 1 and the identity group if i = 1. The arc group of the multigraph is the direct sum of wreath products of the automorphism groups of connected components via symmetric groups on components of the same size. By Substitution of the order of the automorphism group in (*) we obtain the following fonnula for the subdegrees corresponding to a graph r E T(k , 2) having ai

270

connected components of size 2i:

2k· k! S(r)=-----

IJ (2i)""' · (ai !) i > 1

I. A. FARADZEV AND A. V. IVANOV

6. For k ;:: 3 , l ;:: 3 we have not obtained a formula for the rank and subdegrees of the group sl1o I). Apparently this is due to the fact that the groups S~7o I) and s!}fo 2) are exceptional actions of the symmetric groups on partitions since their permutational characters are multiplicity free [14].

For k ;:: 3 , l ;:: 3 we have applied constructive enumeration of the graphs from the set T(k , l) using a computer. An algorithm for this enumeration can be obtained by a slight generalization of the algorithm for constructive enumeration of the incidence systems presented in [12], [17].

The results of the computation of the ranks and subdegrees of the groups sl1o I) for small k and l (k;:: 3, l;:: 3) are given in Table 1. As in [9] here sp+2q stands for p symmetric and q pairs of antisymmetric orbits of length s. The ranks of the groups sl1o /) for greater values of k and l are given in Table 2.

Table 1.

k l n degree rank subdegrees

3 3 9 280 5 1,27,36,54, 162

4 12 5775 9 1,48,54, 128,216,576,1296,1728

5 15 126 126 13 1, 75, 250, 300, 1 500, 2 000, 6 000, 8 000, 9 oooz, 27 ()0()2, 36 000

6 18 2 858 856 22 1, 108, 432, 600, 675, 3240, 6750, 8000, 16 200, 27 000, 32 400,43 200, 60 750, 108 000, 121 500, 162 ooo1+:/, 1, 216 ooo, 324 ooo, 432 ooo. 972 ooo

7 21 665 512 160 30 1, 147,686, 1323,3675,6174, 18 522,37 044, 74 088, 85 750, 92 610, 154 350, 205 800, 277 830, 463 050, 514 500, 555 660, 926 100, 1 234 800, 1 389 150, 1 852 200:1, 1,2 315 250,2 744 000, 3 087 000, 4 630 500, 6 945 750, 9 261 ()0()2' 18 522 000

8 24 1 577 585 295 45 1, 192, 1024,2352, 7350,9408, 10 752,43 904, 75 264, 175 616, 225 792, 343 000, 351 232, 451 584,752 640,940 800,987 840, 1 580 5442 '

4 741 632,4 939 200,5 268 4802 ,6 585 6001+2· 1 '

7 902 720'1, 1 ' 9 483 264, 13 171 200, 15 805 440, 16 464 000, 18 522 000,31 610 880, 35 562 240, 39 513 6002' 1' 43 904 000, 74 088 000, 118 540 800, 131 712 000, 158 054 400, 175 616 000, 197 568 000, 395 136000

RANKS AND SUBDEGREES OF THE SYMMETRIC GROUPS ACTING ON PARTITIONS 271

Table 1. (continuation)

k I n degree rank subdegrees

4 3 12 15 400 12 1, 54, 144, 216, 243,486, 6482• 1296, 3888:3

4 16 2627 625 43 1,96, 108,512,768,864,972,1536,1728, 2304, 3888, 5184, 6144,69122• 13 8242• 18 432, 27 6483+:1,l' 31104:1,2,41472,62 2082,

82 9442+:1,l, 124 4163 • 165 8882 ,248 832l+:l,l,

331 776

5 20 488 864 376 106 1,150,600,1000,1875,3750,6000,8000,

15 000, 24 000, 30 OOQ2' 32 000, 36 OOQ2. 60 0003'

108 OOQ2' 120 0003 • 144 000, 180 0003 ' 240 000, 270 000, 360 0001+:1, I, 480 0003 , 720 0003+:1,2 ,

960 0003 , 1 080 ()()(f+:~, 1 , I 440 ()()(f+:~, 3 , 1 620 OOQ2,

2160~1 • 2430 000,2 88000Q2, 3 240 000, 3 840 000, 4 320 ()()(f+n, 4 860 000, 5 760 OOOI+:I, I, 6 480 (}()(f+:l, I, 8 640 0001+:1,2,

12 960 oool+n, 17 280 000,25 920 000,

38 880 0003 • 51 840 000

5 3 15 1401400 31 1,90,360,540, 1215,1620,2430,32402,4860,

5832, 6480, 14 580, 19 4403 • 29 1602. 38 880, 58 3203+:1,l, 93 312, 116 64<?' 233 280

4 20 2 546168 625 264 1,160, 180,1280,2160,3840,4860,5760,7680,

8640, 12 960, 17 2802' 19 440, 20 480, 23 040, 24 576, 30 7202,34 560,38 880,69 1202,921602,93 312, 103 680, 138 2403+:1, 1 , 155 520:1,2, 184 320,

207 3602. 233 280, 276 4802. 311 040". 368 6402. 414 7203+:1,l, 552 9603+:1,l, 622 0803,829 44()5+B,

933 1201+:1, I, 1 105 92<ft:1, I' I 244 1602+:1, I.

1 658 88<f+H, 1 866 2401+:1,2 , 2 211 840,

2 488 32Q3+:1,4 ' 2 985 984, 3 317 7604+:1,3 •

3 732 48cf'":1,2. 4 423 680, 4 976 6407+:1,8. 5 971 968, 6 635 52<ft:1,1 , 7 464 96<f+:1,5 ' 7 962 624, 9 953 2808+:1,5 • 14 929 9209+:1,5 • 19 906 56Q4+H. 29 859 8403+:1,6 • 39 813 120,59 719 6807+:1,l,

119 439 3603

6 3 18 190590400 103 1,135, 720,1080,3240,3645,7290,9720, 10 935, 12 960, 19 4402. 29 1602' 34 992, 38 880, 58 3202 ' 65 610, 87 4803 • 116 640, 174 96(f'

233 280, 262 4402' 349 9203+:1,2. 524 880S' 559 872, 699 84Q6, 1 049 7608+:/, I, 1 399 68~, 2 099 5208+:1.4 , 2 799 3601+:1. I,

4 199 0407+:~. 4 • 8 398 o80S. 16 796 160

k l n degree rank subdegrees

7 3 21 36 212 176 ()()() 383 1, 189, 1260, 1890,5670,8505, 17 010,22 680, 45 630, 68 040, 76 545,90 720, 102 060, 122 472, 136 0803• 204 120, 272 160, 306 180Z. 459 270Z. 612 36<f. 816 480Z, 918 540, 1102 248, 1224 72!1*" 1, 1 377 810, 1 574 640, 1632 960, 1 837 08<JZ, 1 959 552, 2 449 4406+'), I , 2 755 620, 3 6741607 ,4 898 880S, 5 511240Z, 7 348 32013+'), I ,

8 398 080, 9 797 76<JZ, 11 022 4807+'), I, 14 696 6408+"4 • 17 635 968, 19 595 52QI+'),I, 22 044 96018+"5 • 29 393 2808+2'6 •

44 089 92<J24+" 15 ,

58 786 5606,88179 84~+"20• 100 776960, 117 573 120, 176 359 68025+" 16, 235 146 240, 352 719 36013+"7• 470 292 480,705 438 72!1*'21,

Tablel.

k l rank k l rank k l rank k l rank

3 9 61 3 23 1354 3 37 8 560 4 6 321 10 85 24 1585 38 9462 7 787 11 111 25 1833 39 10 418 8 1960 12 149 26 2193 40 11461 9 4 354 13 189 27 2570 41 12 563 10 9386 14 244 28 2986 42 13 761 11 18790 15 304 29 3424 43 15 025 12 36362

16 381 30 3909 44 16 392 5 5 1856 17 465 31 4423 45 17 832 6 12 703 18 571 32 4991 46 19 384 7 71457

19 685 33 5593 47 21016 6 4 2804 20 825 34 6257 48 22 769 5 71053

21 977 35 6962 49 24 609 7 4 44524

22 1158 36 7738 50 26 579 8 3 1 731

References

1. G. Andrews, The Theory of Partitions, Addison-Wesley, London, 1976.

2. R. Ball, Maximal subgroups of the sym.metric groups, Trans. Amer. Math. Soc., 121 ( 1966), 398-407.


3. LV. Chuvaeva, On some combinatorial objects which are invariant under the Mathieu group M 12 , In Methods for Complex Systems Studies, pp. 47-52, Moscow,

VNIISI, 1983 [In Russian].

4. A.M. Cohen, A synopsis of known distance-regu/ar graphs with /arge diameter,

Math. Cent. Afd. Zuevere wisk, 1981, N 168.

5. LA. FaradZev, Constructive Enumeration of Combinatorial Objects, In Algorithmic

Investigations in Combinatorics, pp. 3-11, Moscow, Nauka, 1978 [In Russian].

6. X.L. Hubaut, Strongly regular graphs, Discrete Math. 13 (1975), 357-381.

7. A.A. Ivanov, Combinatoric-algebraic methods for the investigation of distance

regular graphs, Ph.D. Thesis, Moscow, MFTI, 1984 [In Russian].

8. A.A. Ivanov, Computation of lengths of orbits of a subgroup in a transitive permutation group, In Methods for Complex Systems Studies, pp. 3-8, Moscow, VNIISI,

1983 [See English translation in this volume].

9. A.A. Ivanov, M.H. Klin, LA. FaradZev, Primitive representations of the nonabelian

simple groups of order less than 106 , Part/, Preprint, Moscow, VNIISI, 1982 [In Russian].

10. A.A. Ivanov, M.H. Klin, I.A. FaradZev, Primitive representations of the nonabelian

simple groups of order less than 106, Part I/, Preprint, Moscow, VNIISI, 1984 [In Russian].

11. A.A. Ivanov, M.Kh. Klin, S.V. Tsaranov, S.V. Shpektorov, On the problern of computing the subdegrees of transitive permutation groups, Russian Math. Surveys, 38 (1983), 123-124.

12. A.V. Ivanov, LA. FaradZev, Constructive enumeration of incidence systems, Rostock Math. Kolloq., 24 (1983), 4-22 [In Russian].

13. M.H. Klin, lnvestigations of algebras of invariant relations for certain classes of permutation groups, Ph. D. Thesis, Nikolaev, 1974 [In Russian].

14. J. Saxl, On multiplicity-free permutation representations, London Math. Soc. Lect, Note Ser., 49 (1981), 337-353.

15. V.A. Ustimenko-BakumovskÜ,Algorithmsfor the construction ofblock-designs and symmetric graphs with given automorphism group, In Computation in Algebra and Combinatorics, pp. 137-148, Kiev, 1978 [In Russian].

16. V.A. Vishenskil, Action of symmetric groups on partitions. Preprint 84.3. Mathematical Institute, Kiev, 1984, pp. 45-53 [In Russian].

17. V .A. Zaichenko, LA. FaradZev, An algorithm for testing the canonicity of incidence

systems, In Algorithmic Investigations in Combinatorics, pp. 126-129, Moscow, Nauka, 1978 [In Russian].


COMPUTATION OF LENGTHS OF ORBITS OF A SUBGROUP IN A TRANSITIVE PERMUTATION GROUP

A.A.lvanov

Let (G, f.t) be a transitive permutation group and F be a subgroup of G. We denote by n 1 , n 2 , · · · , n, the lengths of the orbits of F in its action on Q. In the present paper a method for the description of (F , Q) as a permutation group and, in particular, for the computation of the numbers ni, 1 :5: i :5: r, is given1>. To realize this method, certain information about the structure of the group Gis necessary. Knowledge of the length of orbits of certain subgroups in a transitive permutation group enables us to effectively investigate this group as well as certain combinatorial objects which admit this group as an automorphism group. For example, in Section 3 this method is used to construct a new cubic graph on 110 vertices which is edge- but not vertex-transitive and which admits PGL2(11) as automorphism group.

The important particular situation is when (G , Q) is the representation of G on the cosets of the subgroup F. In this case, the numbers ni , 1 :5: i :5: r, are the subdegrees and r is the rank of the group (G, Q). In Section 4, as an illustration of the method, the ranks and the subdegrees of two primitive representations of the sporadic simple group J 1 and of the representations of the groups from the farnily PSL2(q) on the cosets of subgroups isomorphic to A 4 are computed.

1. Main Notions and Formuration of the Problem

Let (G, n) be the representation of a group Gon the cosets of its subgroup Hand let ~1 , ~2 , · · · , ~r be the orbits of a subgroup F :5: G in its action on Q. Then the action of F on each such orbit is similar to the action of F on the cosets of some subgroup K depending on the orbit, and K :5: g-1 Hg for some element g E G. Let K = (K 1, K 2, · · · , Ks) be a maximal set of subgroups of F such that Ki :5: g-1 Hg, 1 :5: i :5: s, and suchthat Ki and Kj arenot conjugate in F for i '# j. It will be assumed that the set H is ordered in such a way that if Ki :5: r 1 Kj f for f E F then i ~ j. In this case Ks = E is the identity subgroup. Let Qi denote the number of orbits of F in which the action of F on each such orbit is similar to its action on the cosets of its subgroup Ki, 1:5: i :5: s.

The problern of computation of the lengths of orbits of the subgroup F on the set Q

consists of the determination of the numbers Qi, i.e. the characterization of (F, Q) as a

l) A special case of this method is described in A.A. Ivanov, M.H. Klin, S.V. Tsaranov, S.V. Shpectorov, On the problern of computing the subdegrees of transitive permutation groups, Uspehi Mat Nauk 38 (1983), No. 6, 115-116 [In Russian, translated in Russian Math. Surveys, 38 (1983), 123-124].

275

276 A. A. IVANOV

permutation group.

Detailed information concerning permutation groups is contained in the book [10]. Weshall use this information freely.

2. Method of Computation

The proposed method relies on the solution of a certain system of linear equations which arises from calculating, in two ways, the numbers of elements of 0 which are fixed by subgroups of F.

Following [3] we introduce the following definition.

Definition 1. The mark of a subgroup F in the representation (G, 0) of G on the cosets of His the number m(F, H, G) of elements of 0 that are fixed by every permutation in the action of F on 0. The following lemma is simple but useful.

Lemma 1. The mark m (K, H, G) is nonzero if and only if K is conjugate in G to some subgroup of H.

Proof. This follows from the fact that in the representation (G, 0) the stabilizer of each element is a subgroup conjugate to HinG. We introduce the following abbreviations:

m} (F) = m(Ki, Ki, F), mi(G) = m(Ki, H, G), 1 ~ i, j ~ s. 0

Theorem 1. The numbers Qi , 1 ~ i ~ s, satisfy the following system of linear equations:

i . l: Qj m{ (F) = mi (G), 1 ~ i ~ s. j=l

(1)

Proof. By definition all permutations from Ki fix m{ (F) elements of an orbit ~ on which F acts in the manner similar to which F acts on the cosets of the subgroup K1.

Since there are Q1 such orbits ~. there are Q1 m{ (F) fixed elements in these orbits. In addition it follows from Lemma 1 and the ordering of K that m{ (F) is nonzero only if j~ i. 0

The system (1) is already in triangular form. Moreover, the number of equations in the system coincides with the number of variables and is equal tos.

Lemma 2. For some i' j, let the inclusion Ki < ] 1 KJfhold for f E F, and suppose mi (G) = mi (G). Then Qi = 0.

Proof. Since K is an arbitrary set of pairwise nonconjugate subgroups, we can assume thatf= e, i.e. Ki < K1. Then all elements of 0 which are fixed by K1 are certainly fixed by Ki. If, in this case, Qi :t: 0 then there is an orbit ~ such that Fa = Ki for some

COMPUTATION OF LENGTHS OF ORBITS OF A SUBGROUP IN A TRANSITIVE PERMUTATION GROUP 277

element a e A. So Ki has a fixed element in the set !!. while Kj has no such elements. Hence mi (G) > mj (G), a contradiction. 0

For computing marks of a subgroup we will use the following assertion.

Theorem 2. Let (G, 0) be a transitive permutation group, G u = H be the stabilizer of an element a e 0 and L be a subgroup of H. Let {L 1 , L2 , · · · , Lm) be a maximal set of subgroups of H which are conjugate to L in G and such that Li and Lj are not conjugate in H for i '#- j. Then

m m (L, H, G) = :I:, [NG (Lj) : NH (Lj)].

j=1 (2)

Proof. Put M = {g-1 Hg I g e G) and = {g-1 Lg I g e G ). Let E (H') denote the set of elements of 0 which are fixed by all permutations from H'. Then the set {E (H') I H' e M) forms a system of imprimitivity of the group (G, 0). So, by Theorem 3.5 in [10], we have I E (H) I = [NG (H): H]. Moreover, a subgroup L' e G fixes the element ae E(H') if and only if L' S: H', and if L' S: H' then L' fixes all elements in E (H' ).

So if k is the number of subgroups from the set M which contain L then m(L , H, G) = I E (H) I • k. To find k let us consider the set S = {(H', L') I H' e M, L' e N, L' S: H'). The cardinality of the set S can be calculated in two ways. On the one band, the number of subgroups H' such that (H' , L) e S is equal to k, so

I S I = IN I • k = [G :NG (L)] • k. (3)

On the other band, the number of subgroups L' such that (H, L') e S is equal to m :I:, [H :NH (Lj)]. Hence j=1

m I S I = [G :NG (H)] • (:I:, [H :NH (Lj)]).

j=1 (4)

After evaluating k from (3) and (4) we obtain the expression for m(L, H, G) stated in the theorem. 0

In the situation when L is conjugate in H to each subgroup L' of H to which it is conjugate in G, equality (2) takes the moresimple form

m(L, H, G) = [NG (L) :NH (L)]. (5)

Theorem 2 was first proved by Alperin [1]. Our proof differs slightly from the original one.

278 A. A.IVANOV

3. The Construction ofan Edge- But Not Vertex-TransitiveGraph

It is well known that if a graph is edge- but not vertex-transitive then it is bipartite and the automorphism group of the graph acts transitively on each of its parts. J. Folkman was the first who studied edge- but not vertex-transitive regular graphs. In [6] he posed the problern (Problem 4.7) concerning the existence of an edge but not vertex-transitive regular graph whose valency is a prime number which does not divide the number of ver

tices of the graph.

In [2] an edge- but not vertex-transitive cubic graph on 112 vertices is mentioned,

but a description of such a graph is not presented. Below, an edge- but not vertextransitive graph on 110 vertices is described.

It is known (see [4]) that the group PSL 2(11) of order 660 contains two conjugacy classes of subgroups of index 55. If H 1 and H 2 are representatives of these classes then

H 1 = A4 and H 2 = D 12· Let W 1 and W 2 be the set of cosets of H 1 and H 2 in G, respectively. Let us compute the lengths of the orbits of H 1 on W 2 . It is easy to see that the lengths of orbits of H 2 on W 1 are the same. In this case the set K is the following:

(Z2 X z2' z3' z2' E). Using some information about the structure of G, we obtain the system of equations

3Q1 = 1

Q2 = 1

2Q3 +3Q1 =7

12Q4 +6Q3 +4Q2 +3Q1 =55.

So Q 1 = 1, Q2 = 1, Q3 = 2, Q4 = 3, and H 1 has an orbit of length 3 on W 2. This means that we can construct a bipartite cubic graph r on the vertex set W 1 u W 2 on which G acts edge- but not vertex-transitively. By means of a computer package worked out by LA. FaradZev, it was checked that the automorphism group of r has order 1320 and that it acts on r edge- but not vertex-transitively.

The graph r has a simple combinatorial description. Let us consider G as the automorphism group oftheblock design with parameters v = b = 11, k = r = 5, Ä. = 2. Such a design is unique and is presented in Fig. 1.

COMPUTA Tl ON OF LENGTHS OF ORBITS OF A SUBGROUP IN A TRANSITIVE PERMUTA Tl ON GROUP 279

"1 "2 "3 v4 vs 116 V? Vg "9 V1Q vn

b1 1 1 1 1 0 0 0 0 0 0

b2 1 0 0 0 1 1 0 0 0

b3 0 1 0 0 1 0 0 1 1 0 b4 1 0 0 1 0 0 1 0 1 0

bs 1 0 0 0 1 0 0 1 0 1 1

b6 0 1 1 0 0 0 1 0 0 1 b? 0 1 0 1 0 0 0 1 1 1 0

bs 0 1 0 0 1 1 0 0 1 0 1

b9 () 0 1 0 1 0 1 0 0 1

blO () 0 1 0 1 0 1 1 0 0

bn () 0 0 1 1 1 0 0 1 0

Fig. 1.

Then W1 = {(Vi,bj) I Vj E bj}, W2 = {({Vn, Vm}, {br, bk}) I Vn, Vm E br, bk}, and the vertices w1=(vi,bj)E W1 and w2=({vn,Vm}, {br,bk})e W2 are adjacent if and only ifvi e br (') bk, Vn, vm e bj. Forexample the vertex ({v1, v2}, {b1, b2}) is adja-cent to each of (v9, b 10). (v 10, b 11 ), (v 11 , b9). It is easy to see that if 't is a polarity of the design, then 't e Aut(r). Since < 't, G> = PGL 2(11) and I PGL2(ll) I = 1320, we conclude that 1\ur(r) = PGL 2(11). Finally, it should be mentioned that PGL2(ll) acts primitively on both W 1 and W 2. So r is a so-called bi-primitive edge- but not vertex-transitive graph. Such graphs are of particular interest in applied combinatorics.

4. Computing the Ranks and Subdegrees.

In this section the situation when (G , .Q) is the representation of G on the cosets of F is considered. Then the number of orbits of F on .Q and their lengths are the rank and subdegrees ofthe group (G, .Q), respectively.

In this Situation K 1 =Fand m 1 (G) ~ 1, so by Lemma 2 and equality (5) we obtain the following important assertion.

Lemma 3.. Suppose for some i, 1 s; i s; s, that NG(Kj} = Np(Ki) and that Ki is conjugate in F to every subgroup of F to which it is conjugate in G. Then Qi = 0. Let us illustrate the proposed method on three examples which are of independent interest.

Example 1. Let G = J 1 and F be the normalizer of a Sylow 7-subgroup in G, isomorphic to the Frobenius group F~ (with Frobenius kerne! of order 7 and complement of order 6). The lattice of subgroups of the group F~ is given by Fig. 2.

280 A. A. IVANOV

Fig. 2.

Let us put K = (F~ , F~ , F1 , Z7, Z6 , Z3 , Z2 , E). Foreach subgroup Kj, 1 ~ j ~ 8, the corresponding marks can be calculated by equation (5).

Since each of the subgroups F~, F,, Z 1 contains a characteristic subgroup of order 7, its normalizer coincides with F. Hence by Lemma 3, Q 2 = Q 3 = Q 4 = 0. The orders of normalizers of other subgroups of F in J 1 can easily be calculated from the character table of the subgroup J 1 (see for example [9]). For direct calculations in the group F~ we find the marks m{ (F), 1 ~ j ~ i ~ 8. Finally, system (1) is the following

Q1=1

Qs+ Q1=2

2Q6+ Qs+ Q1=10

3 Q 7 + Q 5 + Q 1 = 20

42 Qs + 21Q7 + 14 Q 6 + 7Q 5 + Q 1 = 4180.

Hence Q 1 =1,Qs=1,Q 6 =4,Q 7 =6,Qs=95. So the rank of the primitive representation of the group J 1 of degree 4180 has rank 107.

Example 2. G = J 1, F is the normalizer of a Sylow 5-subgroup (which coincides with the normalizer of a Sylow 3-subgroup) and is isomorphic to D 6 x D 10 , the direct product of dihedral groups of order 6 and 10.

The group F contains one subgroup of order 5 and one subgroup of order 3. So if (I K I, 15) "# 1 for some subgroup K of F then Nc(K) ~ F. In addition, such a subgroup K

is conjugate in F to these subgroups of F to which it is conjugate in G. So we can use Lemma 3 and exclude such subgroups K from consideration. Let us order the remaining subgroups as follows: D 6 xD10, Zz xZz, <'t1>, <'tz>, <1:3>, E. Here 1:1 , 'tz, 't3 are representatives of the three conjugacy classes of involutions of D 6 x D 10 such that CF('t1) = <1:1 > x D 10, CF('tz) = <'tz> x D6, CF('t3) = Zz x Zz. Since J 1 has a unique

COMPUTATION OF LENGTHS OF ORBITSOFA SUBGROUP IN A TRANSITIVE PERMUTATION GROUP 281

conjugacy class of involutions, all these involutions are conjugate in G. By equation (2) we obtain m3(G) = m4 (G) = m5(G) = 46. Since the centralizer of an involution in J 1 is isomorphic to Z2 xA 5 , we have NG(Z2xZ2F=Z2 xA4 and hence m2(G)=6. After calculation ofthe marks m;(F), we obtain to the following system

Q1=1

Q2+ Q1=6

10Q3 + 5Q2+ Ql =46

6Q4+ 3Q2+ Q1 =46

2Qs + Q2+ Q1 =46

So Q 1 = 1 , Q 2 = 5 , Q 3 = 2, Q 4 =5 , Q 5 = 20, Q 6 = 34 and the rank of the primitive representation of J 1 of degree 2926 is 67.

It is weil known (see for example [10]) that the rank of a permutation representation is equal to the sum of the squares of the coefficients in the decomposition of the permutation character into a linear combination of irreducible characters. In [5], decompositions of the permutation characters of the primitive representations of the nonabelian simple groups of order less then 106 are given. In the case of the representations of the group J 1 of degree 4180 and 2926, these decompositions are incorrect, since it follows from these decompositions that the ranks are 133 and 51, respectively. The correct decompositions can be obtained from the subdegrees using the character table of J 1 (see pp. 22-23 in [8]).

Example 3. Gis isomorphic to a group of the family PSL2(q), q ~ 5. All subgroups of such a group were described by L. Dickson in [4]. In particular, if q :#. 22n+1 then G contains a subgroup which is isomorphic to the altemating group A4. We will consider the situation when A4 isamaximal subgroup of G. This is the case if and only if either q is a prime such that q = ±3 (mod 8) and q ~± 1 (mod 10), or q = 3r where r is an odd prime number.

Let H = (A 4 , Z 2 x Z2 , Z3 , Z 2 , E). For the value of q under consideration, NG(Z2 xZ2) =A4 and hence Q2 = 0. The system (1) has the following form.

Q1=1

Q3 + Ql =m3(G)

2 Q4 + Q1 =m4(G)

12Qs+ 6Q4+ 4Q3+ Ql=ms(G).

282 A. A.IVANOV

The order of NG(Z3) is equal tothat number from the set {q -1, q, q + 1) which is divisible by 3, while the order of NG(Z2) is equal tothat number from the set {q- 1, q + 1) which is divisible by 4. So the marks mi(G) for i = 3, 4, 5 can be easily calculated. The subdegrees are presented in the table below.

q

8k ± 3; 3 I (q- 1)

8k ± 3; 3 I (q + 1)

3r = 4k ± 1

Acknowledgement

subdegrees (_q_.::i_) ( q-4±1 ) ( q3-51q+176+ 18)

1 • 4 3 ' 6 8 ' 12 288 (~) ( q-4±1 ) ( q3-51q+112+ 18)

1 ' 4 3 ' 6 8 ' 12 288 (rl) ( q-4+ 1 ) ( q3-51q+144±18 )

1 • 4 3 • 6 8 ' 12 288

The author is highly indebted to M.H. Klin, S.V. Tsaranov and S.V. Shpectorov for fruitful discussions.

References

1. J.L. Alperin, On a theorem of Manning, Math. Z, 88 (1965), 434-435.

2. I.E. Bouwer, On edge but not vertex transitive regular graphs, J. Combin. Theory (B), 12 (1972), 32-40.

3. W. Burnside, Theory of Groups of Finite Order, Camb. Univ. Press, Cambridge, 1911.

4. L. Dickson, Linear Groups with an exposition of the Galois Field Theory, Dover New York, 1958.

5. J. Fischer, J. McKay, The nonabelian simple groups G, I G I < 106 -maximal sub-groups, Math. Comput., 32 (1978), 1293-1302.

6. F. Folkman, Regular Une-symmetrie graphs, J. Combin. Theory 3 (1967), 215-232.

7. M. Hall, Jr., Combinatorial Theory, Blaisdell, 1967.

8. A.A. Ivanov, M.H. Klin, LA. FaradZev, Primitive representations ofthe nonabelian simple groups of order less then 106 , Part I. Preprint Institute for System Studies, Moscow, 1982.

9. J. McKay, The simple non-abelian groups G, I G I < 106 -character tables, Comm. Algebra, 7 (1979), 1107-1445.

10. H. Wielandt, Finite Permutation Groups, Acad. Press, New York, 1964.

This paperwas originally published in "Methods of Complex Systems Study", Moscow, Institute for System Studies, 1983, pp. 3-8.

Part 2. Distance-Transitive Graphs

DISTANCE-TRANSITIVE GRAPHSAND THEIR CLASSIFICATION

A.A. Ivanov

1.1. The main notions 1.2. A historical essay

2. Some generat properties of distance-transitive graphs 2.1. Intersection array and feasibility conditions 2.2. Reduction to primitive graphs 2.3. Relation on subdegrees 2.4. Multiplicity freeness 2.5. DTG's and s-transitive graphs 2.6. Moore graphs and generalized n-gons 2.7. The line graphs 2.8. Multiple metric structures 2.9. Distance-transitive graphs and involutions

3. Bounding the diameter 3.1. Prelirninary remarks 3.2. The valencies 3 and 4 3.3. The bipartite case 3.4. The Terwilliger bound 3.5. The Sims conjecture 3.6. The case of infinite diameter 3.7. The st-theorem 3.8. A theorem ofWeiss 3.9. On diameter of a DRG

4. Distance-transitive graphs of small valencies 4.1. Trivalent and tetravalent graphs 4.2. A problern of N. Biggs

283

285 288

292 292

295 298 300 301 303 305 306 307

308 308 310 311 311 313 314 314 316 317

319 319 321

284

4.3. Valendes from 5 to 7 4.4. Valendes from 8 to 13 4.5. Approach of A. Gardiner 4.6. A fixed points method

5. A local approach to the classification of DTG's 5.1. The subconstituents Ak and Sk 5.2. s-transitive DTG's for s :2:: 4 5.3. DTG's admitting elations 5.4. A characterization of the odd graphs 5.5. Perspectives in the classification of s-transtive DTG's

6. A global approach to the classification 6.1. A characterization of the Hamrning graphs 6.2. The symmetric and altemating groups 6.3. The linear groups 6.4. The classical and exceptional groups of Lie type 6.5. The sporadic groups 6.6. The affine case 6.7. Construction of DTG's of a group from its character table 6.8. Imprimitive graphs

References

A. A. IVANOV

324 325 326 327

330 330 332 333 336 339

343 343 345 348 352 355 358 360 366

370

DIST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA TION 285

1.1. The main notions

In the present survey the classification problern of distance-transitive graphs is considered.

Let r be an undirected, connected, finite graph with vertex-set V(I), edge-set E(I) and autornorphisrn group Aut(I). The nurnber of edges in a shortest path joining u, v E V(I) is called the distance between u and v and is denoted by d(u, v). The diameter d of r is the rnaxirnurn of the distances between pairs of its vertices. Let G be a subgroup of Aut(I). The graph r is said tobe G-distance-transitive if, for each quadruple x,y,u, v of vertices such that d (x,y) = d(u, v ), there is an autornorphisrn g E G which rnaps x onto u and y onto v. In other words r is a G-distance-transitive graph if G acts transitively on the set { (x,y) I x,y E V(r), d(x,y) = i} for each Os; i s; d. A distance-transitive graph (DTG for short) is a graph which is G-distance-transitive for sorne subgroup G of its autornorphisrn group. If r is a G-distance-transitive graph, then the permutation group induced by the action of G on V(I) is said to be a distance-transitive representation (DTR for short) of G.

Arnong the classes of graphs which are characterized in terms of certain properties of the autornorphisrn group, the class of distance-transitive graphs has a special position. For this class the classification problern is tenable, yet nontrivial. If for instance we assurne rnere transitivity of Aut(I) on V(I) then the classification problern is intractable. On the other hand if Aut(I) acts doubly transitively on V(I) then the classification problern becornes trivial.

Distance-transitive graphs have rich cornbinatorial structure. This structure alone, with no additional assurnptions on the autornorphisrn group, enables one to develop an interesting theory and to carry out a classification.

Before formulating of the cornbinatorial properties of DTG's we recall sorne notions frorn algebraic cornbinatorics (for details see Bannai & Ito (1984), Brouwer et al. (1989), Farad.Zev et al. (1991)).

Let V be a set of n elernents and R o. R 1, ... , Rd be relations on V satisfying the following conditions:

(i) R o = { (x,x) I x E V}, the diagonal relation;

(ii) VxV=RouR 1 u ··· uRd,RinRj=0fori:#j;

(iii) tRi= Ri' for sorne i' E {0, 1, ... ,d}, where tRi= ( (x,y) I (y,x) E R;};

(iv) for each triple Os; h,i,j s; d and for each pair (x,y) E Rh, the nurnber of elernents z E V suchthat (x,z) E Ri, (z,y) E Rj depends only on h,i,j and is denoted by pt.

286 A. A.IVANOV

Then Y =(V, {Ri }Q;;i:s;d) is called an association scheme (or simply scheme) with d classes on the set V. If d = 1 then the scheme Y is said tobe trivial.

The scheme Y is called symmetric if it satisfies the additional condition

(v) i' = i. The collection of 2-orbits of a transitive permutation group forms an association

scheme. Namely, let (G,V) be a transitive permutation group on a set V and R 0 , R 1, ... , Rd be the 2-orbits of this group, i.e. the orbits of G on V x V. Then Y =(V, {Ri }Q;;i:s;d) is an association scheme, known as the scheme of2-orbits of the group (G, V), and is denoted by Y(G). The scheme Y(G) is symmetric if and only if (G, V) is generously transitive. The latter means that for each pair x,y e V there is an element g e G which maps x onto y and y onto x. The scheme Y ( G) is trivial if and only if ( G, V)

is a doubly transitive group. An association scheme which is the scheme of 2-orbits of some group will be called Schurian.

The adjaceney matrices of the scheme Y are the matrices A o =I, A 1, ... , Ad whose rows and columns are marked by the elements of V and whose entries satisfy

{ 1 if (x,y) E Ri ;

(Ai)xy = 0 otherwise.

Here x,y e V , 0:::;; i :::;; d.

The following equality is a direct consequence of property (iii):

d Ai·Aj= L PtAh (O:s;i,j:s;d).

h=O

The adjacency matrices generate an algebra A(Y) of dimension d + 1 over IR which is known as the Rose-Mesner algebra. Fora scheme of 2-orbits, the Bose-Mesner algebra is called the V-ring of the corresponding permutation group.

Suppose that r is a DTG. Let us define relations Ro. R 1> ••• , Rd on V(r) by the rule:

(u,v) E Ri ~ d(u,v) = i for 0:::;; i:::;; d. (1.1)

Then Y =(V, {Ri }Q;;i:s;d) is an association scheme.

An arbitrary graph with the property that (1.1) deterrnines an association scheme is called a distance-regular graph (DRG for short). The association scheme constructed from a distance-regular graph is called metric or P-polynomial. A DRG is distancetransitive if and only if the corresponding scheme is Schurian.

The complete set of structure constants Pt, o:::;; i,j,k:::;; d, of a P-polynomial scheme can be calculated from a certain set of 2d structure constants known as the intersection array of the DRG. As a first step in the classification of DRG's it is natural to enumerate all possible intersection arrays which satisfy certain feasibility conditions. In practice it is

DISTANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA TION 287

usually the case that an intersection array which sarisfies these feasibility conditions in fact corresponds to a DRG.

Recently many strong and significant results concerning DRG's have been obtained. Among these are the classification of the DRG's of valency 3 (Biggs et al. (1986)) and the results of Bannai & Ito (1987 a,b,c,d) who have made significant progress in bounding the diameter of a DRG in terms of its valency.

Nevertheless, in complete generality the classification problern for DRG's is too hard. For instance, if d = 2 then the problern is equivalent to the classification of all strongly regular graphs. The classification of all finite projective planes is a subproblern for the case d = 3. So it is necessary to restriet the class of graphs under classification. The study of DTG's is one such possibility. A few words should be said about another possibility. This is the study of (P and Q)-polynomial schemes.

The Bose-Mesner algebra of a symmetric scheme Y is an algebra of linear operators in the space W of all functions from V into IR. Under the action of this algebra, W splits into a direct sum of eigenspaces W 0 , W 1o ••• , Wd. Let PiU) be the eigenvalue of the matrix Ai on the space Wi. The scheme Y = (V, {Ri }os;isd) is P-polynomial if and only if

PiU> = vj(ej) (i,j = o, ... , d),

for some polynomials Vj(X) of degree i, where ej = p 1 U). This definition explains in particular the terminology introduced above. In addition it enables one to introduce a notion which is dual to the notion of P-polynomiality.

Let Qj(i) = (mj I ki) PiU> where mj = dim(Wj) and ki = # {y I y e V, d(x,y) = i} for x e V. The scheme Y is called Q-polynomial if

qj(i) = v1 (ef) (i,j = o, ... , d)

for some polynomials v1 (x) of degree j where er = Ql (i).

Notice that P-polynomiality depends on the ordering of the relations while Qpolynomiality depends on the ordering of the eigenspaces.

A systematic introduction to the theory of (P and Q)-polynomial schemes can be found in Bannai & Ito (1984). In this book a classification program for these schemes is proposed as well. A significant step in the realization of this program is contained in a theorem by D. Leonard (see Theorem 5.1 in Bannai & Ito (1984)). This theorem describes the class of polynomials Vi(X) and vt (x) which can arise in (P and Q)polynomial schemes. It turns out that this class coincides with the class of so-called Askey-Wilson polynomials. This implies in particular that the polynomials vi(x) and vt (x) (and hence all numbers in the intersection array of the corresponding DRG) are determined by six parameters only. Significant progress in the classification of (P and Q)-polynomials schemes was achieved by P. Terwilliger. The formulation of some of his results are presented in the survey paper Bannai & Ito (1986).

288 A. A. IVANOV

It should be noted that many of the known (P and Q)-polynomials schemes correspond to distance-transitive graphs.

The classification of distance-transitive graphs is extremely interesting from the group-theoretical point of view. Hopefully, such a classification can give a new understanding of the nature of nonabelian simple groups.

In addition, pursuant to the classification of DTG's, some intermediate problems of independent interest arise. Two examples are the description problern of multiplicity-free permutation representations of groups and the description problern of s-transitive graphs with small girth.

1.2. A historical essay

Investigation of certain special classes of DTG's has a rather long history. These classes include the Tutte's cubic cages, the incidence graphs of the Desarguesian projective planes, the rank three graphs, etc.

The first general result in the classification of DTG's was obtained in Biggs & Smith (1971) where the DTG's of valency k = 3 were classified. During the subsequent ten to fifteen years the classification of DTG's having small valencies was the central problern in the study of DTG's. The DTG's of valency k = 4 were completely classified in Smith (1973, 1974, a,b).

As a first step in the classification of DTG's of valency 3 and 4 it was proved that there are at most finitely many such graphs. It was conjectured that for all k :2: 3 there are finitely many DTG's of valency k. This conjecture is equivalent to the existence of a function d(k) with the property that the diameter of a DTG of valency k is at most d(k). In Smith (1974 c) the existence of such a function d(k) was proved for the bipartite DTG's. In Cameron (1982) it was proved that the diameter of a DTG is bounded in terms of the order K of vertex stabilizer in its automorphism group. As follows from the main result of Cameron et al. (1982), proved modulo the classification of finite simple groups, the value of K is bounded by a function of the valency k. At the same time the infinite, locally finite DTG's were classified in Macpherson (1982).

In Ivanov (1983 a) certain relations on numbers in the intersection array of a DRG were established. These relations imply, in particular, that the diameter of a DRG (and hence of a DTG) is bounded by a certain function depending on the valency k and on the geometric girth g. It gives an independent proof of the main result of Cameron (1982), as well as the extension of the result in Macpherson (1982) to infinite, locally finite DRG's.

Finally, in Weiss (1985 a), without appealing to the classification of finite simple groups, it was proved that the geometric girth g, as well as the order K of the vertex stabilizer in the automorphism group of a DTG, is bounded in terms of the valency k. By this result the diameter bounding of DTG's in terms of the valency was solved

completely on the theoreticallevel. But many related problems are yet to be solved. One of these is to improve the existing bounds, which are far from ideal. Another problern is to extend diameter bounding to the class of DRG's. Very interesting results on diameter bounding of DRG's are contained in the papersTerwilliger (1982, 1983, 1985) andin a series ofpapers Bannai & Ito (1987 a,b,c, 1988).

The success in diameter bounding of DTG's has stimulated activity in the classification of DTG's with small valencies. In A.A. lvanov, A.V. Ivanov & LA. FaradZev (1984) the DTG's of valency 5, 6 and 7 were classified. In A.A. lvanov & A.V. Ivanov (1988) the classification was done for 8 ~ k ~ 13. In these papers, as well as in the previously mentioned papers on valencies 3 and 4, some computer calculations were involved. An Independent computer-free classification of DTG's of valencies 3, 4, 5 and 6 was obtained in Gardiner (1975), Gardiner (1985), Gardiner & Praeger (1987) and Gardiner & Praeger (1986), respectively.

To carry out a classification of DTG's having valency k one should obtain a reasonable bound on diameter. This problern is related to an estimation of the order of the stabilizer G(x) of a vertex x in the automorphism group Gof the graph in question. The estimations which are available nowadays improve significantly if one excludes the following two Situations from consideration:

(1) the permutation group, induced by G(x) on r(x), contains the altemating group ofdegree k;

(2) the group G contains an elation that is a nontrivial automorphism which stabilizes elementwise the set r(x) u r(y) for an edge {x,y} of the graph r.

This observation stimulated an interest in the classification of DTG's corresponding to situations (1) and (2). For situation (1) the classification follows from Cameron (1974), Praeger (1980), Armanios (1981), Ivanov (1984) and Yokoyama (1987). In situation (2) the result was obtained in Weiss (1985 b) and lvanov (1989).

These results, and others which will be discussed below, have opened a new era in the classification of DTG's wherein more general classification problems have come under consideration. At approximately the same time, lists of known DTG's became available: Cohen (1981 b), Bannai & Ito (1984), Brouwer et al. (1989). A natural conjecture is that each DTG is isomorphic to a DTG occurring in one of these lists. This conjecture seems to be close to accurate since, after the appearence of the aforementioned lists, only a few new examples have been found.

All of the previously mentioned results can be considered as part of a "local approach" to the classification of DTG's. Here the starting point is the structure of the stabilizer G(x) of a vertex x in a group G acting distance-transitively on r. Sometimes the available information is just the permutation group G(xl<x) induced by G(x) on the set r(x). The results on diameter bounding form a theoretical foundation for this approach. Ne:vertheless, it is doubtful that the complete classification of DTG's can be

290 A. A.IVANOV

obtained in this way since there are no ideas on how to restriet the class of transitive groups which can arise as subconstituents G(xl<x) in distance-transitive actions. A more

specialized problern within this framework is to consider the situation when G(x)r(x) is a

doubly transitive permutation group. This is the case if and only if G acts 2-transitively on r. that is transitively on the set of paths of length 2 in r. lf r has girth at least five and

G acts distance-transitively on r, then G(xl<xl is doubly transitive. Nowadays, rather effective methods for the consideration of this situation are being developed. These methods rely on applications of group amalgams and we believe that in the nearest future

the complete answer will be obtained.

A more promising approach to the classification of DTG's is the so-called "global approach". Within this approach we first study the abstract structure of the group G act

ing distance-transitively on r, then determine the permutation action of G on V(r) and finally construct ras a graph which is invariant under the permutation group (G, V(r)).

In a rudimentary form, the global approach was used in the resolution of a problern on the existence of 11 DTG's with given intersection arrays, proposed in Biggs (1976).

Here, using the presentation of the number of vertices as a product of primes, it was

shown that T ~ G :s; Aut(T) for some nonabelian simple group T. From the intersection array of the graph, the order ofT was estimated. This enabled one to identify T with a certain known simple group. After that, the transitive permutation representations of T

with degrees equal to the number of vertices were studied, and the DTG's, invariant under these actions were enumerated.

The general foundation for the global approach to the classification of DTG's was

formulated in the paper Praeger et al. (1987). It was proved in that paper that, if G is a

group acting primitively and distance-transitively on a graph r. then either (1) r is a Hamming graph; (2) T ~ G :s; Aut(T) for some nonabelian simple group T; or (3) G con

tains an elementary abelian normal subgroup N acting regularly on V(r). By this result the classification of primitive DTG's was reduced to a consideration of almost simple groups (case (2) above) and affine groups (case (3)).

In the almost simple case, one can use very effectively the classification of finite

simple groups. Indeed, for T one considers the alternating groups, the Lie type groups and the 26 sporadic simple groups. A very significant role here is played by the fact that each distance-transitive representation (DTR) of a group is a multiplicity-free representation. This means that each irreducible character of G arises in the character of a DTR of

G at most once.

For the alternating groups An• the multiplicity-free permutation representations for n > 18 were listed in Saxl (1981). This result formed a starting point for the description

of DTR's of those groups G satisfying An ~ G :s; Aut(An). This description was obtained in Liebeck et al. (1987) and Ivanov (1986 c).

In the case of linear groups, i.e. PSLn(q) ~ G :s; Aut(PSLn(q)), the primitive multiplicity-free representations for n ~ 8 were classified independently in Inglis et al.

DISTANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICATION 291

(1986) andin van Bon & Cohen (1988 a). In van Bon & Cohen (1989) the case 2~ n ~ 7 was done completely. The case n = 2, including also the imprimitive representations, was considered independently in FaradZev & Ivanov (1990).

The situaltion when T is a classical Lie type group was studied in Inglis (1986). Here the multiplicity-free representations of such groups of sufficiently large dimension (greater than or equal to 13) are described. For determination of all DTR's in this case there is still a lot of work tobe done. The DTR's of the exceptional Lie type groups are studied in Cohen et al. (1989). Here the classification is reduced to a consideration of a few concrete representations. A very significant result concerning DTR's of Lie type groups is contained in the book Brouwer et al. (1989) where all DTR's corresponding to parabolic subgroups are determined.

The methods for the description of DTG's which have been developed in the aforementioned papers are more effective when the groups under consideration are sufficiently large. For the study of small groups certain ad hoc arguments are required. On the other hand, small groups have a small number of irreducible characters. By the property of multiplicity-freeness, the character of each DTR is a {0, 1 }-linear combination of the irre.ducible characters. In FaradZev & lvanov (1988) a computerprogram for the determination of DTR's of a group from its character tableis presented. By means of this program the DTR's of all nonabelian simple groups of order up to 109 were obtained. This result gives, in particular, a description of the DTR's of 12 sporadic simple groups. For instance, if T ~ G ~ Aut(T) where T is Janko's group fJ, Held's group He or the O'Nan-Sims group O'N, then G does not possess a DTR. For Held's group this result was proved independently in van Bon et al. (1987) without the use of a computer.

In the affine case the methods are completely different. The multiplicity-freeness condition does not impose any restriction in this case, since a permutation group with an elementary abelian regular normal subgroup is automatically multiplicity-free. A principal role here is played by the following observation. Let N be the elementary abelian normal subgroup of G. Since N acts regularly on V(r), the vertices from V(r) can be identified with the elements of N. Let x e V(r) correspond to the identity element of N.

Then since r is primitive, the elements corresponding to the vertices in r(x) generate N. This implies that the dimension of N as a vector space over GF (p) does not exceed the valency of r. On the other hand it is easy to show that the diameter of r does not exceed the dimension of N. By distance-transitivity the number of orbits of G(x) on the nontrivial elements of N is just equal to the diameter of r. In this way we come to description problern for groups which act on rn-dimensional vector spaces over GF(p) in such a way that the number of orbits on the set of nontrivial elements does not exceed m. This condition is rather restrictive and it enables one to start the classification. The complete answer is now available for the case when the diameter of r is 2 (see Liebeck (1987)).

The global approach to the classification of DTG's is effective for the classification of primitive graphs. It follows from Smith (1971) that each imprimitive DTG is either

292 A.A.IVANOV

bipartite or antipodal or both, and that starting with an imprimitive DTG, a primitive one

can be constructed. So after classifying the primitive DTG's one should determine which

imprimitive graphs correspond to each of the primitive ones. The construction of bipar

tite graphs is related to the study of certain classes of cliques in primitive DTG's and this

problern was treated in Remmeter (1984, 1986, 1988) where allinfinite series of known

DRG's were studied from this point of view. The construction of antipodal graphs was

studied in van Bon & Brouwer (1988). The methods developed in the above papers fail if

the primitive graph is just a complete graph. In this case there are a lot of interesting

open problems. One of these problems is the classification of the k-fold antipodal covers

of the complete bipartite graph Kk.k·

The classification of distance transitive k-fold antipodal coverings of Ku has just

recently been completed (cf. Gardiner (1974), Chuvaeva & Pasechnik: (1990), Liebler

(1991)).

Our view on the history of the classification of DTG's determines the content of the

present survey. In Chapter 2 some general results conceming the structure and

classification of DTG's are presented. In Chapter 3 the results on diameter bounding of

DTG's are considered in an historical retrospective. Chapter 4 contains a survey of

results on the classification of DTG's with small valencies. In Chapter 5 the local

approach to the classification of DTG's is considered. The results in this chapter concem

the situation when G(xl<x> is a doubly transitive permutation group. Finally in Chapter 6

the most promising global approach is discussed.

2. Some generat properlies of distance-transitive graphs

2.1. Intersection array and feasibility conditions

A significant nurober of the combinatorial properties of a DTG follow directly from

its distance regularity. Forthis reason it is natural to study these properties in the class of

DRG's.

Thus let r be a DRG of diameterd and Y =(V, {Ri losisd) be the corresponding

association scheme with structure constants Pt, 0 ~ i,j,k ~ d. These constants have a

simple combinatorial interpretation. Namely

Pt= lri(u) n rj(v) I for u e rk(v),

where P?.i = I ri(u) I is denoted by ki, while the nurober ofvertices ofr is denoted by n.

It follows from the triangle inequality that

Pfi = 0, if I k- j I ;;::: 2.

This means that the matrix B = llpfill(d+l)x(d+l) is tridiagonaL If we use the notation


bi =PL+1, ai =pt, Cj =PL-1

where a 0 = 0, c 1 = 1 and b 0 = k is the valency of r, then the matrix B will have the fol-lowing form:

0 1 0 k a 1 cz

b1 az •

B ---- bz •

Cd

0 ad

It should be mentioned that the condition for the matrix B to be tridiagonal can be taken as a definition of P-polynomiality.

The matrix B is usually presented in the following form:

{ * 1 Cz . . . Cj · · · Cd}

i(r) = 0 a 1 az · · · ai · · · ad

k b 1 b 2 · · · bi · · · *

and is called the intersection array of the DRG r. It is easy to see that Cj + ai + bi = k for 1 ~ j ~ d- 1 and cd + ad = k. Hence the middle row in the intersection array can be omitted. In this case the intersection array can be written as follows.

It follows from the general theory of association schemes that the algebra generated by the adjacem:y matrix A = A 1 of the graph r is isomorphic to the algebra generated by the matrix B consisting of the structure constants. On the other hand since r is connected, the matrix A generates the whole Bose-Mesner algebra of the scheme Y. This implies that the intersection array, which contains at most 2d independent parameters, determines the complete set of structure constants of the association scheme Y. Moreover the following recursive formulas hold:

Pf+1,j = (l!ci+1) (bj-1" PL-1 +(arak)· Pt+ ci+1" PL+l -bi-1 • Pf-1,j), (2.1)

where

Pk -pk ·pk 5:: ij - ji ' O,j = Ujk .

The numbers in the intersection array of a DRG, and hence of a DTG, satisfy a number of relations which are known as feasibility conditions. In this section the most significant such conditions are presented. The proofs of these conditions can be found in

294 A. A. IVANOV

Biggs (1974), Bannai & Ito (1984) and Brouwer et al. (1989).

Lemma 2.1.1. The following inequalities hold:

k > b 1 ~ b2 ~ ... ~ bd-1 ;

Lemma 2.1.2.

(i) kj • aj = O(mod 2);

(ii) n • k = O(mod 2).

As above, here kj = I rj (x) I for X E V (r), and

kj = PJj = (bo· b1 • · · · • bj_I) I (c1 • C2 • · · · • Cj).

Lemma 2.1.3. If Ci > bj then d < i + j. Lemma 2.1.4. If c 2 = 1 then a 1 + 1 divides k.

0

0

(2.2)

0

0

Lemma 2.1.5. The numbers Pt, 0~ i,j,k ~ d, which can be calculated by formula (2.1), arenonnegative integers. 0

Lemma 2.1.6. The adjacency matrix A of r has exactly d + 1 real eigenvalues So, 81, •.• , ed. Moreover, each Si is an eigenvalue of the matrix B. o

Recall that a vector ~ is called standard if its first nonzero component is equal to one. Let ~(8i) and ~(8i) be the right and the left standard eigenvectors of the matrix B corresponding to the eigenvalue ei.

Lemma 2.1.6. The multiplicity of 8i as an eigenvalue of the matrix A is a nonnegative integer equal to

The following lemma is known as the Krein condition.

Lemma 2.1.7. For any triple i,j,l suchthat 0~ i,j,l ~ d, the expression

d

qlj = L kv Vv(Oi) Vv(Oj) Vv(OJ), v=O

0

where v v(Oi) is an eigenvalue of the matrix Av, is nonnegative. 0

We note that these are the most important of the known feasibility conditions.


2.2. Reduction to primitive graphs

Let r be a G-distance-transitive graph and 1et (G, V(r)) be the corresponding DTR.

In this section we will study the situation when the perrnutation group (G, V(r)) is

imprimitive.

Let ßo, ~1 , ... , ~d be the comp1ete 1ist of 2-orbits of the group (G, V(r)) i.e. the orbits of G acting on the set V(r) x V(r). If the indices are ordered in the appropriate way then distance transitivity imp1ies

(x,y) E ~i ~ d(x,y) = i for 0~ i ~ d.

Let ri denote the graph with vertex set V(r) and edge set ~i· In accordance with the well-known condition for a perrnutation group to be primitive (see Sims (1967)), the group (G, V(r)) is imprimitive if and only if, for some 1 ~ i ~ d, the graph ri is disconnected. Since the graphs ri are defined in purely combinatorial terrns we can define primitive and imprimitive DRG's. Namely a DRG r is said to be primitive if all the graphs ri , 1 ~ i ~ d, are connected. Otherwise, we call r imprimitive.

Recall that a graph r is called bipartite if the set V(r) of its vertices can be divided into two parts in such a way that the two ends of every edge lie in different parts. If r is a bipartite DRG and i is an even integer, then the graph ri is disconnected and each of its connected components is contained in apart of r. It is well known that a graph is bipartite if and only if it contains no cycles of odd length. This implies that the bipartite DRG's are characterized by the property that ai = 0 for 0 ~ i ~ d.

A graph r is called antipodal if, for a vertex x e V(r) and for any two distinct vertices u, V E rd(x), the equality d(u, v) = d holds. For an antipodal DRG the graph rd is disconnected and the connected component containing x is just the set {x} u r d(x). Notice that each complete graph is both antipodal and primitive. Moreover the complete graphsexhaust the antipodal DRG's of diameter 1. The antipodal DRG's of diameter 2 are just the complete multipartite graphs. It is easy to see that a DRG for which kd = 1 is certainly antipodal. In the general situation the antipodal graphs are characterized by the property that P1d = 0 for 1 ~ i ~ d- 1.

It turnsout that an imprimitive DRG of valency k ~ 3 is either bipartite or antipodal. The theorem below was proved for DTG's in Smith (1971), but it can be easily genera1-ized to DRG's.

Theorem 2.2.1. Let r be an imprimitive DRG of valency k ~ 3. Then r is bipartite or antipodal (or both). 0

It should be emphasized that the property of a DRG to be bipartite and/or antipodal is deterrnined by its intersection array.

It is remarkable that starting with an imprimitive DRG a primitive one can be constructed. Two procedures for doing this are described be1ow.

296 A.A.IVANOV

Let r be a bipartite DRG with parts V 1 and V 2· Then V 1 and V 2 are connected cornponents of r 2. Let Lj denote the Subgraph of r 2 induced by vi for i = 1 or 2.

Lemma 2.2.2. The graphs 1:1 and q are distance regular of diameterd = [d 12] with the sarne intersection array

0

The graph :Ei (i = 1, 2) is known as the halved graph of r while r is said to be a bipartite doubling of the graph :E;.

Let r be an antipodal DRG of diameter d::?: 3. Let x denote the connected cornponent of r d which contains x. This connected cornponent will be called an antipodal

block of r. Let f be the graph whose vertices are the antipodal blocks of r, with two blocks x and ü adjacent if r contains an edge with one end in x and the other in u.

Lemma 2.2.3. The graph f is a DRG of diameterd = [d 12] with intersection array

{bo = k, b1, ... , bd; 1, c2, ... , cd},

where Ci= ci and bi = bi for 0 s; i s; d -1, and Cd = bd +Cd if d even and Cd =Cd is d ~- 0

The graph r is called the antipodal folding of r. In this case r is said to be an 1-fo/d

covering of r, where I is the cardinality of an antipodal block.

It is easy to show that if the halved graph :Ei of r is bipartite then :Ei is a cornplete - -

bipartite graph (which is also antipodal). If the folded graph r of r is antipodal then r is a cornplete graph and hence is primitive. This rneans that by at rnost the second step of bipartite halving and/or antipodal folding, we will corne to a primitive DRG. If these two procedures are applied in different orders then, in general, the results will be different.

Now let us consider the reconstruction problern by which an irnprimitive graph is obtained frorn its halved graph or folded graph.

In the case of bipartite doubling, the problern involves a study of the clique structure in the given halved graph. Namely the following lemma holds (see Remmeter (1986)).

Lemma 2.2.4. Let r be a bipartite DRG with parts V 1 and V 2 and let 1:1 be the halved graph of r having V 1 as the set of vertices. Suppose that 1:1 is not a cornplete graph. Then for every y E V 2 the Subgraph in Lt induced by r(y) is a clique of Lt . 0

So in order to reconstruct r frorn 1:1 one should find in the latter a certain family F of cliques. Then the vertices of r are the vertices of :E1 together with the cliques in F. The family F should possess a nurnber of properties. For instance I F I = I :E1 I , each clique in F has the same size k and each vertex of :E1 is contained in exactly k cliques frorn F. Here k is just the valency of the graph r.

DJST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA TION 297

If 1:1 is a complete graph then almost all information about the structure of r is lost

in 1:1 and we only have the following lemma (see Remmeter (1986)).

Lemma 2.2.5. If 1:1 is a complete graph Kn, then r is the incidence graph of a

2- (n,k, A.)-symmetric design, where A. = c2. 0

Now let r be an antipodal DRG of diarneter d ~ 3 and let r be its folded graph.

Then the mapping cp : x ~:X determines a covering of graphs in the usual topological

sense. It is weil known that the coverings can be studied in terms of subgroups of the fun

damental group of the graph.

Let !!;. be an arbitrary graph and v be a vertex of t'l. Then the fundamental group

7t(t'l, v) of !l with respect to v has as its elements all paths in !l with origin and terminal in

v. The multiplication and inverse operations are concatenation of paths and reversal of

orientation of a path, respectively. It is clear that the covering cp: r ~ r determines an

isomorphic embedding of 7t(r,x) into 7t(r,:X), so 7t(r,x) can be considered as a subgroup

of 7t(r,X). The index of this subgroup is equal to l, that is the size of an antipodal block of

r. Let ~(i) denote the subgroup of 7t(r,:X) generated by the paths of length at most i with

origin and tenninal both in :X. If C is a cycle in r of length at most d - 1 then, since cp is

an antipodal <:overing, q,-l (C) is the disjoint union of l copies of C. This implies that

7t(r,x) contains ~(d -1). Notice that d = 2J or 2J + 1 where J is the diameter of f. Using these arguments one can prove the following necessary conditions for the

existence of antipodal coverings (see van Bon & Brouwer (1988)). In what follows, if x,y

are vertices of a graph then C (x,y) is the union of vertices on the geodesics between x

andy, i.e. C(x,y) = {z I d(x,z)+d(z,y) = d(x,y)}.

Proposition 2.2.6. Suppose that !l is distance-regular of diameter d ~ 2 and has a

distance regular antipodall-covering of diameter 2J. Then for any two vertices u, v in !l

with d (u, v) = d, the subgraph induced by C (u, v) - { u, v } in !l is the disjoint union of I

subgraphs of t:qual size. 0

Proposition 2.2.7. Suppose that !l is distance-regular of diameter J~ 2 and has a

distance-regular antipodal 1-covering of diameter 2J + 1. Let u, v be vertices of !l with

d(u, v) = J and put E = { v} u (t'l(v) n t'lJ(u)). Then the collection of sets

C(u,w)- {u,w}, w e E, can be partitioned into I nonempty partssuchthat sets from dif

ferent parts are disjoint and all edges joining vertices in different parts are contained in

t'l(u). o

Let ~(d -·1) be the subgroup of 7t(r,:X) defined above, where d = 2d or 2d + 1. Then

~(d -1) ~ 7t(r,x) ~ 7t(f,:X). In some cases it is reasonable to study the covering 3 ~ r associated wilth the subgroup ~(d -1). If one can show that this covering is an isomor

phism then r has no antipodal coverings of diameter d. An advantage of this approach is

that the subgroup ~(d -1) is invariant under the action of the automorphism group of r. The latter implies that each automorphism of r can be lifted to an automorphism of 3.

This approach was used in lvanov & Shpectorov (1990) where it was proved that the

298 A A. IVANOV

DTG on 506 vertices related to M 23 has no antipodal coverings.

Let us consider in some detail the situation for imprimitive graphs in the distance transitive case.

Lemma 2.2.8. Let r be a bipartite G-DTG. Then L1 and ~ are isomorphic. Moreover Li is an H-DTG, where H is the subgroup of index 2 in G which preserves the partition of V (r) into bipartite halves. 0

The proof of the following lemma can be found in Ivanov & lvanov (1988).

Lemma 2.2.9. Let r be an antipodal G-DTG. Then r is an F-DTG where Fis the factor group of G over the subgroup N which preserves each of the antipodal blocks of r as a whole. Moreover, either N = 1 or N is an elementary abelian group of order pn

which acts regularly on each antipodal block of r and the action of G on N via conjugation is transitive on N# = N- ( 1 } . 0

Thus, starting with an imprimitive DTR of G we can always construct a primitive DTR of a group K. The group K can be obtained in at most two steps by consideration of subgroups of index 2 and/or factorization over normal subgroups whose properties are described in Lemma 2.2.9. Forthis reason the primitive DTR's are the most important. Nevertheless, it should be noticed that sometimes the problern of constructing all imprimitive graphs corresponding to a given primitive one is very nontrivial.

2.3 Relations on subdegrees

Let (G, V(r)) be a DTR of a group G. Let us consider some restrictions on the subdegrees of the group (G, V(r)), i.e. on the lengths of orbits of the stabilizer G(x) on V(r). Since r is a G-DTG, the orbits of G(x) on V(r) are just the subsets ri(x) for 0 $ i $ d. Hence the subdegrees of ( G, V (r)) coincide with the numbers ki , 0 $ i $ d, which can be calculated by formula (2.1) from the intersection numbers of the corresponding DTG.

The lemma below follows directly from the inequalities on the numbers bi and ci presented in Lemma 2.1.1.

Lemma 2.3.1. The sequence k 0 , k1, ... , kd is logarithmically convex, i.e. ki-1 • ki+l $ kt for 1 $ i $ d -1. 0

The next result can also be deduced from the inequalities in Lemma 2.1.1 (see Taylor & Levingstone (1978)).

Lemma 2.3.2. Let r be a DRG of diameter d ~ 3. Then

(i) the sequence ko, k1, ... , kd is unimodal, i.e. there are numbers h, 1 with 0 $ h $ l $ d such that

DIST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA Tl ON 299

1 < k 1 < · · · < kh = · · · = kz > · · · > kd ;

(ii) ki ~ kj in the case when 0 ~ i ~ j and i + j ~ d;

(iii) if ki=kj (O~i~j;i+j~d) then bi=Cj, bi+1=Cj-1····; in particular

ki+1 = kj-1;

(iv) if ki := ki+1 then ki ~ kj for all j, 0~ j ~ d;

(v) if ki_1 = k (3~ j~ d) then either k = 2 or r is a 2-fold antipodal covering of a DRG (i.e. j = d and kd = 1);

(vi) if k 2 = k then either k = 2 or r is a 2-fold covering of a complete graph (i.e. d=3,k3=1). 0

In some cases Lemmas 2.3.1 and 2.3.2 enable one to prove that the permutation group in question is not a DTR by analysis of its subdegrees. In fact a sequence of numbers should satisfy certain conditions to possess a reordering which is logarithmically convex. For instance the following proposition due to LA. FaradZev (see FaradZev & Ivanov (1988)) holds.

Lemma 2.3.3. If in the sequence a 1 ~ a 2 ~ · · · ~ an, the inequality ar < ai-2. ai+1 holds for some 2 ~ i ~ n -1, then it is impossible to reorder it so as to obtain a logarithmically convex sequence. o

An application of this lemma will be given in Section 6. 7.

It follows from Lemma 2.3.2(iv) that if three subdegrees are equal then they are maximal. In Section 3.7, as a consequence of so-called st-theorem (see Corollary 3.7.2), it will be shown that the numbers h and l in Lemma 2.3.2(i) satisfy the inequality l < 2· h. This, in view ofLemma 2.3.2(ii), implies that (/-h)~ d/3. So the following lemma holds:

Lemma 2.3.4. Let k 0 , k1, ••• , kd be the subdegrees of a DTR. Let m(ki) = I {j I kj = ki} I be the multiplicity of the subdegree ki and s be the number of different subdegrees. Then either m(ki)~ 2, or ki~ kj for all 0~ j~ d and m(ki)~ d/3 (in particular s ~ d /3). 0

Some sufficient conditions in terms of subdegrees for a permutation group to be distance transitive are also known. As an example we give the following simple lemma from Ivanov (1984).

Lemma 2.3.5. Let (G, V) be a transitive permutation group with subdegrees ko. k1o ... , k.r1 suchthat

ki + 1 > ki for 0 ~ i ~ d - 1 ;

300 A. A. IVANOV

kd-2. (k1 -1) < kd-1 + kd.

If, in addition, the graph r of the 2-orbit of ( G, V) with valency k 1 is connected, then it is aG-DTG. D

So the analysis of subdegrees of permutation groups is an effective tool for the classification of their DTR's. A method for the computation of subdegrees which is based on so-called marks of subgroups is known (see Ivanov et al. (1983 b), lvanov (1983)). A study of subdegrees based on this method forms a significant step in the classification of DTR's of groups Gwhich satisfy PSL 2(q) ~ G:::; PrL2(q) (see Section 6.5).

2.4. Multiplicity freeness

The permutation character of a DTR of G possesses the following very important property. Each irreducible character of G over Cis either not involved in this character or is invo1ved with multiplicity 1.

Let us consider a more general situation. Let (G, V) be a transitive permutation group, Y be the association scheme of 2-orbits of (G, V) and x be the permutation character of this group. So x is the function from G into C such that x(g) is equal to the number of elements in V fixed by the permutation g e G. It is well known that x can be expressed as a linear combination

X= L ei· $i, ie I

(2.3)

where the ei are nonnegative integers and { $i I i e I} is the set of characters of the irreducible representations of G over C.

The following proposition holds (see Wielandt (1964)).

Lemma 2.4.1. The Bose-Mesner algebra of the scheme Y is commutative if and only if ei e { 0, 1} for all i e /, i.e. if and only if x is multiplicity-free. D

If a permutation group (G, V) is a DTR then the corresponding scheme Y is metric, its Bose-Mesner algebra is commutative, and the following Iemma holds.

Lemma 2.4.2. The permutation character of a DTR is multiplicity free. D

Using this Iemma it can be shown that the order of the vertex stabilizer Hin a group G acting distance-transitively Oll r has fairly small index in G (see Saxl (1981)).

Lemma 2.4.3. If G acts distance-transitively Oll a graph r of diameter d and H = Gx for x e V(r), then

IG I:::; (d+l) IH 12 . D

Another relation between I G I and I H I will be presented in Section 3.7. Numerous and very important further applications of Lemma 2.4.2 are contained in Chapter 6 of the


survey.

2.5. DTG's and s-transitive graphs

In this section two important classes of intersection arrays are considered. A DTG with arrays from the first class will be called s-transitive. To a graph with intersection array from the second class a standard construction will be applied. This construction leads to a locally s-transitive graph. In both cases s;::: 2.

Let us start with some necessary definitions. Let r be a graph without vertices of valency one. Lets be a nonnegative integer. An s-path in r with origin x and terminal y is an ordered sequence W = (x =xo, x1o .. _., Xs =y) of vertices of r such that d(xi, Xi+l) = 1 for 0~ i ~ s -1 andxi :;txi+2 for 0~ i $', s -2. Let G ~ Aut(r). The group G is said to act s-transitively on r if G acts transitively on the set of all s-paths in r. In this case r is called an s-transitive graph. Let G act s-transitively on r, s ;::: 1. Then since r has no vertices ofvalency one, it is easy to see that G acts (s -1)-transitively on r. The following lemma is a partial converse of this claim.

Lemma 2.5.1. Let G act (s -1)-transitively on a graph r, s;::: 2. Then the action of G on r is s-transitive if and only if the elementwise stabilizer G(W) of an (s -1)-path

W = (Xo, X1, ... , Xs-1) acts transitively On the Set r 1 (ts-1)- {Xs-2}. 0

The 0- and I-transitive graphs are just the vertex-transitive and are-transitive graphs, respectively. It follows from the lemma below that the class of 2-transitive graphs is much more restricted.

Lemma 2.5.2. A group G ~ Aut(r) acts 2-transitively on r if and only if G acts 0-transitively and for x e V(r) the permutation group G(x/(x) is doubly transitive. 0

An action of Gon r is said tobe locally s-transitive if for any vertex x e V(r) the subgroup G(x) acts transitively on the set of s-paths in r with origin x. In this case r is called a locally s-transitive graph. The action of G on r is s-transitive if and only if it is locally s-transitive and vertex-transitive. Suppose that G acts on r locally s-transitively for s;::: 1 but not s-transitively. Then r is bipartite, and the orbits of G on V(r) are just the bipartite parts of r.

In what follows we will make use of the following lemma.

Lemma 2.5.3. Let r be a bipartite graph with bipartite parts V 1 and V 2- Suppose that G acts transitively on Vi for i = 1,2, and for a vertex x e V 1 the group G(x) acts transitively on the set of paths of length r with origin x. Then G acts locally s-transitively on r. Moreover, s ;::: r if r is odd and s ;::: r- 1 if r is even. o

Recently a number of very deep results concerning s-transitive and locally s

transitive graphs was obtained (see for example the survey Weiss (1981 (a))). These results have already played a significant role in the classification of DTG's (see Chapter 5 of the present article).

302 A. A.IVANOV

The applications of the theory of s-transitive and locally s-transitive graphs to the classification of DTG's are motivated by the following.

By definition a DTG is I-transitive. Let r be a G-DTG for which the following relations hold for some 2 ~ r ~ d:

Ci = 1 for 1 ~ i < r; ai = 0 for 1 ~ i ~ r- 1. (2.4)

Then for an r-path w = (xo, ... , Xr) of r the equality d(xo,Xr) = r holds, and for x,y of distance r there is a unique r-path with origin x and terminal y. This implies that there is a bijection between the set of r-paths in r and the pairs of vertices of distance r from one another. So we have the following:

Lemma 2.5.4. Let r be a G-DTG for which the relations (2.4) hold. Then G acts r-

transitively on r. 0

Now suppose that r is the maximal integer for which the relations (2.4) are valid. Then either ar # 0, or ar = 0 but Cr+1 # 1. An application of Lemma 2.5.1 gives the following.

Lemma 2.5.5. Let r be a G-DTG and r be the largest integer for which (2.4) holds. Then G acts s- but not (s + 1)-transitively on r, where s ~ r and

(i) if 0 < ar < k -1, then s = r; (ii) if ar = 0, 1 < Cr+1 < k, then s ~ r + 1. 0

Two extremal cases are not included in Lemma 2.5.5. The first one (ar =k -1) corresponds to the Moore graphs of diameter r and the second one (ar=O, Cr+1 =k) to the generalized (r + 1)-gons. All these DTG's are known (see Section 2.6). In particular r = 2, s = 2 or 3 in the first case, and (r + 1) e {3,4,6}, s = r + 2 in the second case.

Now suppose that for a G-DTG r we have the following

Ci = 1 for 1 ~ i ~ r ; ai = a 1 for 1 ~ i ~ r- 1 , a 1 # 0, r ~ 2. (2.5)

Since c 2 = 1 each clique (maximal complete subgraph) of r contains a 1 + 1 vertices, and any two distinct cliques have at most one common vertex. Let us consider a bipartite graph L1 whose first part V 1 consists of the vertices of r and whose second part V 2 consists of the cliques of r. A vertex x e V 1 and a clique y e V 2 are adjacent in L1 if y contains X. This graph d will be called incidence graph of r.

Let (xo, ... , x2r) be a 2r-path in the graph L1 with xo e V 1. Then x2r e V 1, and the vertices X 0 and X 2r are at distance r in r. Moreover, to any pair x,y of vertices in r which are of distance r from one another, a unique (2r)-path in L1 is associated. In view of Lemma 2.5.3 we have the following:

Lemma 2.5.6. Let r be a G-DTG, and suppose the relations (2.5) hold for some r ~ 2. Then G acts locally (2r -1)-transitively on d. 0

DIST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICATION 303

2.6. Moore graphs and generalized n-gons

Moore graphs and generalized n-gons play a significant role in nuroerous probleros related to the classification of distance-transitive graphs. Soroetiroes they arise theroselves as solutions to certain classifcation probleros and soroetimes as subgraphs of other DTG's. As was roentioned in the previous section, these graphs are extreroal froro the s

transitivity point of view. They will arise in the next section in a different context.

Traditionally the Mooregraphsand generalized n-gons are introduced as solutions to the following extreroal problero.

Recall that the girth of a graph is the nurober of edges in a shortest cycle. Let r be a connected graph of valency k and girth g. It is easy to see that the nurober of vertices in

such a graph is at least no(k,g) where

no(k,g) = 1 + k + k(k -1) + · · · + k(k -1)(g-3)12 if g is odd, and

no(k,g) = 1 + k + k(k -1) + · · · + k(k -l)(g-3)12 + (k -1)(g-2)12 if g is even.

It turns out that the minirouro n0(k,g) can be attained only in distance-regular graphs. Namely the following lemma holds:

Lemma 2.6.1. Let r be a connected graph of valency k, girth g and with exactly no(k,g) vertices. Then r is a DRG of diameterd = [g /2] with intersection array

{k, k -1, ... , k -1; 1, 1, ... , 1} if g is odd, and

{k, k-1, ... , k-1; 1, 1, ... , 1,k} ifg iseven. D

The graphs of valency k and girth g which have n0 (k,g) vertices are called the no(k,g )-graphs.

If g is odd then n 0(k,g)-graphs are known as Moore graphs. If g is even then n0(k,g )-graphs constitute a subclass in the class of generalized n-gons. This subclass is characterized by the property that the nurober of points on a line is equal to the nurober of lines through a point.

Even without the distance-transitivity assuroption, quite deep results conceming Moore graphs and generalized n-gons are available. For instance the following proposition holds (see Bannai & Ito (1984) for (i) and Feit & Higroan (1964) for (ii)).

Lemma 2.6.2 (i) Let r be a Moore graph of girth g and valency k;:::: 3. Then g = 5 andk E {3,7,57};

(ii) Let r be a generalized n-gon. Then n E {2,3,4,6,8, 12}. D

In the class of DTG's, a coroplete classification is obtained. Before formulation of the final result let us describe soroe exaroples.

Only two Moore graphs of valency k;:::: 3 are known.

304 A.A.IVANOV

The first one is the Petersen graph which can be described as follows. Let Q be a set of 5 elements. The vertices of the Petersen graph are the 2-element subsets of the set n, where two vertices are adjacent if the Subsets are disjoint. The Petersen graph has valency 3 and is a G-DTG for G = A 5 or S 5. The action is 2-transitive in the former case and 3-transitive in the latter case.

The second Moore graph is known as the Hofiman-Singleton graph. This graph is a 2-orbit of valency 7 of the permutation action of the unitary group PSU 3 (5) on the cosets of its subgroup A 7 . This graph is a G-DTG for G = PSU 3 (5) or PJ:.U 3 (5). The lauer is the extension of PSU3(S} by an automorphism of order 2. The stabilizer Gx of a vertex x is isomorphic to A 7 in the former case and S 7 in the lauer. The action is 3-transitive in both cases.

These two examples exhaust the distance-transitive Moore graphs. The existence problern for a Moore graph of valency 57 is still open but it is known that it cannot be a DTG (Aschbacher (1971)). Thus the following lemma holds:

Lemma 2.6.3. Let r be a DTG of valency k ~ 3 which is a Moore graph. Then r is either the Petersen graph or the Hoffman-Singleton graph. D

A generalized n-gon is an incidence system (P ,L,J) where P is a set of points, L is a set of lines and I is an incidence relation which satisfies the following conditions:

(i) each point is incident to exactly t + 1 lines;

(ii) each line is incident to exactly s + 1 points;

(iii) the bipartite graph with vertex-set P u L and edge-set I has diameter n and girth 2n.

In what follows the bipartite graph in (iii) will be identified with the corresponding generalized n-gon.

The known examples of generalized n-gons which are DTG's are related to the Lie groups of rank 2 (notice that s = t for distance-transitive generalized n-gons).

Let G be the Lie group A2(q), B2(q) or G2(q) where q = pm, p is a prime. Suppose that p = 2 for the groups of type B 2 and p = 3 for the groups of type G 2. Let B be a Borel subgroup in G, i.e. the normalizer of a Sylow p-subgroup in G. Then Bis contained in two maximal subgroups Pt and P 2 of G, which are known as parabolic subgroups. Let us consider the bipartite graph r with parts V 1 and V 2. where Vi is the set of cosets of Pi in G, i = 1,2. Two vertices from different parts are adjacent if the corresponding cosets have nonempty intersection. Then r is a generalized n-gon with parameters s = t = q suchthat n = 3,4 and 6 for the groups of type A 2 , B 2 and G 2 , respectively. Moreover r is an H-DTG, where His the extension of G by the so-called diagram automorphism which permutes the subgroups Pt and P2. As was already mentioned, the action of H on r is (n + !)-transitive. The graph r is also known as the graph of the parabolic geometry of the group G. The generalized n-gon related to a Lie group over GF(q) in the above sense will be denoted by lln,q·


The groups of type A 2 and B 2 are classical, and the corresponding generalized n

gons admit a clear geometric description. Namely, let (P,L,/) be the Desarguesian projective plane over the field GF(q). Then the graph r with vertex-set PuL and edge-set I is the generalized 3-gon related to the group A 2(q)=PSL 3 (q). Now let U be a 4-dimensional vector space endowed with a nonsingular symplectic form f Let us construct a bipartite graph r whose vertices are the lines of U together with the 2-dimensional isotropic subspaces W in U. A subspace W is isotropic if f(x,y) = 0 for any x,y E W. Two vertices of r are adjacent if the subspaces contain a common line. Then r is the generalized 4-gon related to the group B 2 (q) = Sp 4 (q ). This graph is a DTG if and only if q =2m.

The following Iemma is a consequence of results of Gleason (1956), Higman

(1964), Tits (1976, 1979) and Weiss (1979).

Lemma 2.6.4. Let r be a DTG of valency k ~ 3 which is a genera1ized n-gon. Then

r is isomorphic to An,k-1· D

2.7. The line graphs

The no(k,g )-graphs also arise in the classification of DRG's which are line graphs.

We Start with definitions. Let r be an arbitrary undirected graph. The line graph I (r) of r has the set of edges of r as its vertices, with two vertices of I (r) adjacent if the corresponding edges of r have a common vertex.

The following remarkable result conceming distance-regular line graphs was proved in Biggs (1974 b).

Lemma 2.7.1. If the line graph l(r) of a graph r is a DRG, then r is an no(k,g)graph and vice versa, i.e. the line graph of a no(k,g)-graph is distance-regular. Moreover I (r) is a DTG if and only if r is a DTG. D

In view of the results presented in the previous section, Lemma 2.7.1 gives a complete description of the DTG' s which are line graphs.

In some cases, the fact that the graph in question is a line graph can be deduced directly from its intersection array. For instance the following Iemma holds:

Lemma 2.7.2. Let r be a DRG for which C2 = 1, a 1 = (k /2)- 1. Then r is a line graph. D

If the hypothesis of the above Iemma is satisfied, then a vertex of r is contained in exactly two maximal cliques, each of size (k /2) + 1. If we take the set of these cliques as the vertices and join two cliques by an edge if they have a vertex in common, we obtain a graph for which r is the line graph.

306 A. A.IYANOY

2.8. Multiple metric structures

As we saw in Section 1, metricity (that is P-polynomiality) of an associatlon scheme (V, {Ri } 0~ i~ d) relates to a certain ordering of the basis relations. It is interesting to know how many metric structures a given association scheme can have. This question is rather important in regard to the determination of all G-distance-transitive graphs r such that the action of G on V(r) is similar to that of a given permutation group. A description of the association schemes having multiple metric structures was obtained in Bannai & Bannai (1980) (see also Section 3.4 in the book Bannai & Ito (1984)).

Theorem 2.8.1. Let Y =(V, {Ri } 0~ i~ d) be a metric association scheme with respect to the ordering R 0 , R 1, .•• , Rd and let the valency of the corresponding DRG be at least three. Suppose that Y has another metric structure. Then the new ordering in terms of the old ordering is one of the following:

(I) Ro,Rz,R4, ... , Rs,R3,R1;

(11) Ro,Rd,R1,Rd-1,Rz,Rd-2•Rd-3• · · ·;

(III) R 0, Rd, Rz, Rd-2· R4, Rd-4· ... , Rd-5· Rs, Rd-3• R3, Rd-1, R 1;

(IV) R 0,Rd-l>Rz,Rd_3,R4,Rd-S• ... , R 5,Rd-4•R 3,Rd_2,R 1,Rd.

In particular Y has at most two metric structures. 0

In fact, the paper Bannai & Bannai (1980) contains a more precise description. Namely, a set of relations on the intersection numbers is presented. The validity of these relations is necessary and sufficient for the existence of an additional metric structure of one of the above four types.

This result can be used in solving the problern of whether an automorphism of a group, acting distance-transitively on a graph, can be realized as an automorphism of the graph. This problern was considered in van Bon (1988), where the following very useful result was proved.

Lemma 2.8.2. Let r be a graph of diameter d on which G acts primitively and distance-transitively. Let H be the stabilizer of a vertex in this action and Iet cr be an automorphism ofthe group G. Then

(i) if cr centralizes Hand d ~ 3, then cr E Aut(r);

(ii) if cr norrnalizes Hand d ~ 5, then cr E Aut(r). 0

An application of Theorem 2.8.1 enables us to generalize Lemma 2.8.2. Let Y =(V, {Ri }0 ,.; i,.; d) be the association scheme corresponding to the distance-transitive action of Gon V(r), that is, {x,y} E Ri ~ d(x,y) = i, Os; i ::s; d. First of all, cr can act on the set Vif and only if Hcr is conjugate to HinG. If this is the case we can assume that cr normalizes H. Let cr act on V. Then, since cr normalizes G, cr maps every relation Ri

into some relation R cr(i)· So cr induces a permutation on the set of indexes 0, 1, ... , d,

and cr(O) = 0. It is easy to see that the following claims hold:

DIST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSJFICA TION

(1) o e Aut(r) if and only if o(1) = 1;

(2) if Pt are the structure constants of Y then pgf}foü) =Pt. 0 ~ i,j,k ~ d;

(3) the relations Ri and R cr(i) are isomorphic as graphs with vertex set V.

307

Suppose that o E Aut(r), i.e. that o(1) "# 1. Then it is easy to see that the ordering

R 0 , R a(l), ... , R a(d) determines an additional metric structure of the scheme Y and that

the graphs r ando(r) =(V, Rcr(l)) are isomorphic. So using Theorem 2.8.1 we can obtain rather strong restrictions on the parameters of the graph r. In this way one can prove the following proposition (van Bon (1989)) which can be considered as an extension of

Lemma 2.8.2 to imprimitive graphs.

Lemma 2.8.3. Let r be a graph of diameter d and valency k;:::: 3 on which G acts distance-transitively and let H be the stabilizer of a vertex. Let a be an automorphism of

the group G which normalizes H. Then c? e Aut(r). If o E Aut(r) and d "# 4, then either

d = 2 and r is a conference graph, or r is antipodal with blocks of size 2, and

o(1) = d -1. o

The situation described in Lemma 2.8.3 is realized in k-dimensional cubes. The case

d = 4 in this lemma leads to the possibility that r has the following intersection array:

{11(211+ 1), (11-1) (211+ 1), 112 • 11; 1, 11. 11(11-1), 11(211+ 1)}.

Lemmas 2.8.2, 2.8.3 can be applied to the study of distance-transitive representations of groups. Namely, in many cases, these lemmas enable one to consider the auto

morphism group of G rather than G itself (see Chapter 6 of the present survey).

2.9. Distance-transitive graphs and involutions

In van Bon (1988) the following very interesting situation was considered. Let G act distance-transitively on a graph r, and suppose the vertex stabilizer Hin this action coincides with the centralizer of an involution 't in G. If N G (H) = H, then the vertices of r can be identified with the conjugacy class of involutions in G which contains 't. This situation

is realized in the Johnson graphs J( n, 2), in Fischer's graphs related to 3-transpositions

in the groups F22, F23 and F24 and in certain graphs of parabolic geometries. In van Bon (1988) the following was proved.

Theorem 2.9.1. Let r be a graph on which G acts distance-transitively. Suppose that the vertex set V(r) is a conjugacy class of involutions in G, that G acts on r by conjugation and that there are elements in V(r) which commute in G. Take x,y e V(r) with

x adjacent to y. Then at least one of the following holds:

(i) r is a polygon or an antipodal 2-cover of a complete graph;

(ii) Gis a 2-group;

(iii) the order of xy is an odd prime. If a,b e V(r) with ab of order 2 then a and b

have maximal distance in r, and if a,b e V(r) then the order of abisnot 4;

308 A. A.IVANOV

(iv) the elements x and y commute, and if z e r2(x) then xz has order 2, 4 or an odd prime. Moreover, either 02(Cc(x)) #:- <x> or Cc(x) contains a normal subgroup generated by p-transpositions. D

In the study of DTG's on involutions, Theorem 2.9.1 sometimes enables one to determine the case in which two vertices (two involutions) are adjacent. In many situations it turns out that the vertex stabilizer Gx acting on r(x) has a nontrivial kerne!. In van Bon (1988) some interesting results concerning relations between kernels of Gx on rj(X) for different i arealso presented.

Lemma 2.9.2. Let r be a graph of diameterd on which G acts distance-transitively.

For x e V(r), let G~ denote the kernel of Gx on ri(x). If G~ *- 1 for some i ~ 1, then

G~ c G~-l c

G~ c G~+l c

D

It follows directly from this Iemma that for a subgroup K of Gx there are at most two different indices i with the property K = G~. This property is, in a certain sense, analogous to the unimodality property of the sequence k o, k 1> ••. , kd.

3. Bounding the diameter

3.1. Preliminary remarks

Historically, the firstproblern in the scope of the classification of distance-transitive graphs was the classification of graphs having a given small valency k. In the resolution of this problern the following question plays a principal role.

Question 3.1.1. Is the number of DTG's having valency k finite?

This question is equivalent to the existence of a function d(k) suchthat the diameter of a DTG of valency k does not exceed d (k ).

The cycle of length I is a distance-transitive graph of valency 2 and diameter [/ /2]. So for k = 2 the answer to Question 3.1.1 is negative. On the other hand, it is now known (c.f. Cameron et al. (1983) and Weiss (1985 a)) that for all k ~ 3 the answer is the affirmative. This result has very deep theoretical importance forms the basis for the local approach to the classification of DTG's.

In this section we will see step-by-step how the answer to Question 3.1.1 was carried out. The question is also meaningful for distance-regular graphs. One expects that in the nearest future an affirmative answer will be obtained to this case as well. The present situation in this area is presented in the penultimate section of the chapter.

DISTANCE-TRANSITIVEGRAPHSAND THEIR CLASSIFICATION 309

Westart with some general remarks. Let r be a DRG of valency k with intersection

array

Let us introduce the following notation: n (b,c) = # { i I bi = b, ci = c for 1 $; i $; d}. Put

n+ = I, n(b,c); no = I, n(b,c); b>c b=c

n_ = I, n(b,c); nmax = max n(b,c). b<c ~c

It is clear that d = n+ + n 0 + n_. By Lemmas 2.1.1 and 2.1.3 the following ho1ds.

Lemma 3.1.2.

(i) There are at most 2k pairs {b,c} such that n(b,c) # 0; in particular

d$; 2· k· nmax;

(ii) n_ $; n+ + no;

(iii) n 0 = n (b, c) for a certain pair { b, c } . 0

The standard approach to bounding the diameter of a DTG in terms of its valency k

is to first bound the order K of a vertex stabilizer in the automorphism group of the graph. Let us consider the relationship between k and K.

Let G = Aut(r) and K = I Gx I. Then in view of distance-transitivity, the value ki

divides K for all i , 1 $; i $; d; in particular ki $; K. On the other hand, since ki+l = ki • bi I Ci+l where bi and Ci+1 are integers less then or equal to k, the inequality ki+1 > ki implies ki+1 > ki(1 + 1 I k). So we have the following:

Lemma 3.1.3. The value of n+ is bounded by a certain function of K and k. For instance, n+ < loga(K lk) for a = 1 + 1 I k. 0

Lemmas 3.1.2 and 3.1.3 enable us to make the following two remarks.

Remark 3.1.4. To bound the diameter of a DTG (as well as a DRG) in terms of its valency k, it is sufficient to bound n max.

Remark 3.1.5. If the value of K = I Gx I for G = Aut(r) is bounded by a certain function of k, then the diameter of r is bounded by a certain function of k and n 0 .

In addition, it follows from the results presented in Section 2.2 that, from an imprimitive DRG of valency k and diameter d, a primitive one of valency k 1 $; k(k -1) and diameterd 1 ;::: (d- 3)14 can be constructed. So we can make the following remark.

Remark 3.1.6. If there exists a function d 1 (k) such that the diameter of a primitive DTG is bounded by d 1 (k) then there exists a function d (k) such that the diameter of any DTG ofvalency k is bounded by d(k).

310 A. A.IVANOV

3.2. The valencies 3 and 4

Bounding the diameter of a DTG of va1ency 3 formed the first step in the

classification of such graphs, which was obtained in Biggs & Smith (1971).

A bound for the order K of the vertex stabilizer in the automorphism group of a

DTG ofvalency 3 follows from the following very deep result due to Tutte (1947).

Theorem 3.2.1. Let 1 be a graph of valency 3. Suppose that G acts 1-transitive1y on

1 and that for X E V(l) the value K = I Gx I is finite. Then K divides 24 • 3. 0

Notice that in the hypothesis of the theorem it is not assumed that 1 is finite. It is

sufficient for the action of Gon 1 tobe 1ocally finite.

Thus to bound the diameter of a DTG of valency 3 it is sufficient to bound no in

terms of n+. Let us reproduce here the arguments from Biggs & Smith (1971) which

prove that no:::; n+ + 1.

By definition, ci = 1, bi = 2 for 1:::; i:::; n+ and Ci= bi = 1 for n+ + 1:::; i:::; n+ + n0 .

Suppose that d;::::2(n++l) and let x,y,ze V(f'), where d(x,y)=d(y,z)=(n++1),

d (x, z) = 2(n + + 1 ). It follows from the parameters of 1 that a path of length n + + 1 in 1 is

contained in a unique cycle of length 2(n+ + 1) + 1. Let us consider two such cycles

related to the paths joining x with y and y with z. These cycles intersect in vertex y. Since

the valency of 1 is three, they have a common edge incident with y. This gives immedi

ately two paths of length 2(n+ + 1) joining x and z. Hence n0 :::; n+ + 1. This fact, in view

ofTheorem 3.2.1, leads to the following.

Lemma 3.2.2. The diameter of a DTG of valency 3 is at most 14. 0

The bound in Lemma 3.2.2 is not at all superficial since a DTG of valency 3 and

diameter 8 exists.

Let us now turn to consideration of the next case, i.e. the case of valency four. Here

the situation is more complicated with regard to bounding the order of the vertex stabil

izer. In the late 60's and early 70's numerous attempts to generalize Theorem 3.2.1 to the

case of graphs having k;:::: 4 were made. But it turned out that already for k = 4 a direct

generalization was impossible. Namely, as was shown in Djokovic (1974), for any N there exist a graph 1 of valency 4 and a group G acting 1-transitively on 1 such that

I Gx I > N for XE V(1). Nevertheless, for all these examples the subconstituent ai<x) is

isomorphic to the group D s; in particular, it is not doubly transitive. By Lemmas 2.5.2

and 2.5.5, the girth g of 1 in this case is at most 4. If g = 4 then c 2 = 2, 3 or 4. In the

latter two cases d:::; 3, while in the former case Gx acts faithfully on l(x). If g = 3 then

a 1 = 1, 2 or 3. A direct construction shows that for a 1 = 3, 1 is the complete graph,

while for a 1 = 2, 1 is the octahedron. If a 1 = 1 then 1 is a line graph of a graph 11 having

valency 3, and G acts 1-transitively on 11 (11 is the incidence graph of 1). An application

of Theorem 3.2.1 shows that for x E V (1), the order of Gx divides 25 .

Thus, if G acts distance-transitively on 1 and ai<x) is imprimitive, then I Gx I

divides 25•

Now suppose that oi<x> is primitive, i.e. GI<x> ::A4 or S4. Then it follows from

Sims (1968), Quirin (1971) and Gardiner (1973) that I Gx I is bounded, without any

assumption of distance-transitivity.

Lemma 3.2.3. Let r be graph of valency 4, Iet G act 1-transitively on r and sup-

pose oi<x> is primitive. Then I Gx I divides 24 • 36• 0

So for k = 4, the value K = I Gx I is also bounded, and it is sufficient to bound no. A

bound was found in Smith (1973), Smith (1974 a). Herethenumber of cases tobe con

sidered is larger than in the case k = 3. For instance no = n(1, 1) or n(2,2). In rudimen

tary form, the paper Smith (1974 a) contains arguments which lie at the base of the sub

sequent results on diameter bounding for DTG's and DRG's.

3.3. The bipartite case

The first result of general type concerning the bounding of diameter was obtained

for the class of bipartite graphs. Here the situation is more simple since b; + c; = k. This

equality implies that no = n(k/2, k/2) and that there are at most k pairs {b,c} suchthat

n(b,c) ~0.

In Smith (1974 c) it was shown that the diameter of a bipartite DTG is bounded by a

certain function of k and K. Namely, the following propositionwas proved.

Proposition 3.3.1. If r is a bipartite DTG of valency k > 4, then n 0 S: 3 • n + + 2. o

The following result was obtained in Terwilliger (1982) and contains some impor

tant relations among the intersection numbers of a bipartite DRG.

Theorem 3.3.2. Let r be a bipartite distance-regular graph. Then, for any positive

integer n less then or equal to the diameter of r, Cn > 1 implies that there exists an

integer i (1 S: i S: n -1) for which Cn ~ c; + Cn-i· 0

It follows from the above Iemma that, for a bipartite DRG,

n(b,c) S: n(k -1, 1)

foreachpair (b,c}. Since n(k-1, 1)= (g -2)/2 where g is the girth of r, we come to

the following.

Corollary 3.3.3. Any bipartite distance-regular graph with at least one cycle is finite

with diameter d, valency k, and girth g satisfying d S: (k -1) (g- 2) I 2 + 1. 0

3.4. The Terwilliger bound

A principal role in the proof of Lemma 3.3.2 is played by the fact that the shortest

cycle in r has even length. This enabled P. Terwilliger to generalize his result 3.3.2 and

to obtain a more general bound on the diameter of DRG's. For the graphs to which it

312 A. A.IVANOV

applies, this bound is the best known.

The bound relies on inequalities involving certain intersection numbers. They are formulated in the following result (cf. Terwilliger (1983)).

Theorem 3.4.1. Let r be a distance-regular graph with the girth equal to 21 or to 21-1 anda1-1 ~ 2· c1-2. Then

__p__ > bi + Cj-1+1 Cl-1 - bi-1+1 -1 Cj -1 '

where 1 ~ j ~ i + 1 ~ d and p = max { c1, a1-1 } . 0

It is easy to see that if a1-1 ~ 2 • c1- 2, then the inequalities in Theorem 3.4.1 imply that n(b,c)~ n(b1>c1) + 2.

Theorem 3.4.1 can not only be applied when the girth g of r is equal to 2/, but also when g = 21- 1 and there are "many" cycles of length 2/. This theorem admits a further generalization, but before its formulation we introduce some additional notions.

Let a 1 ~ 0 and n(b 1 ,c 1) > 1. Then each edge {x,y} is contained in a unique clique of size a 1 + 2, which will be denoted below by cl {x,y}. Let W = (x o, x 1, ... , Xs) be a path in r. This path will be called geometric if cl {xi-1 ,Xj} ~ cl {Xj,Xj+1} for all 1 ~ i ~ s- 1. If, in addition, x 0 = Xs, then W is said to be a geometric cycle. It is clear that in this case s ~ 3. Finally, the length of the shortest geometric cycle of r is, by definition, the geometric girth of r and is equal to 2(r + 1) if Cr+1 > 1, and 2(r + 1) + 1 if Cr+1 = 1. It is clear that in the lauer case a,+1 > a 1. The introduced terminology is motivated by the fact that a geometric cycle of length s corresponds to a cycle of length 2s in the bipartite graph whose vertices are the vertices and cliques of r (i.e. in the incidence graph of r).

A generalization of Theorem 3.4.1 is presented below. Its proof is identical to that of Theorem 3.4.1, up to Substitution of the term "shortest cycle" by the term "shortest geometric cycle".

Theorem 3.4.2. Let r be a distance-regular graph. Let g be the geometric girth of r. If g = 2/-1, suppose that a1_1 ~ 2c1- 2. Then the inequality from Theorem 3.4.1 holds, where p = c1 for g = 2/ andp = max {c1, a1_1 } for g = 21-1. 0

The most restrictive diameter bound was obtained in the case when r contains a quadrangle. The following result is contained in Terwilliger (1985).

Theorem 3.4.3. Let r be a DRG which contains a quadrangle (x 0 ,x 1 ,x2 ,x3) such thatd(xo,Xz)=d(x 1,x3)=2. Then

Ci-bi~ ci-1 - bi-1 + a 1 + 2 for i = 1,2, ... , d -1.

Such a quadrangle definitely exists if a 1 (a 1 -1) < b 1 (c 2 -1). 0

Corollary 3.4.4. Let r be a DRG of diameter d and valency k. If a 1 (a 1 -1) < b 1 (c 2 -1) then d ~ k and, more precisely,


with equality if and only if equality holds in the inequality of Theorem 3.4.3 for all i. D

3.5. The Sims conjecture

It follows from Theorem 3.2.1 that if (G, V) isaprimitive permutation group which has a self-paired suborbit of length 3, then I Gx I is at most 48. In Sims (1967) it was shown that the condition that the suborbit be selfdual can be excluded. This observation Iead Charles Sims to the following conjecture, which is now widely known as the Sims conjecture.

Conjecture 3.5.1. There exists an integral function f possessing the following property. lf Gis a primitive permutation group on a finite set V, Gx is the stabilizer of a point x e V and k is the length of an orbit of Gx on V- {x }, then I Gx I ~ f(k).

Much work had been done since and this conjecture was finally proved in Cameron et al. (1982) using the classification of finite simple groups. The arguments in that paper show that one can take f(k) of the form exp(k 2 • 0 (k )).

In view of Remarks 3.1.5 and 3.1.6 we note that in order to bound the diameter of a DTG of valency k it is sufficient to bound no in terms of n+ in the class of the primitive DTG's. This wasdonein Cameron (1983).

Theorem 3.5.2. Let r be a primitive DTG of valency k > 2. Then n 0 ~ 6 • n + + 11.0

In the proof of this theorem the following arguments are used.

Let X e V(r), {y 1> Y2• ... , Yk} :::: r(x). For a vertex z e ri(X), put nX(z):::: (af(z), ... , ai(z)) where aj:::: d(z,yj)- d(z,x), j:::: 1, ... , k. In other words, the vector nX(z) describes the partition of the set r(x) with respect to z:

r(x):::: (r(X) (") ri-1 (z)) U (r(X) (") rj(Z)) U (r(X) (") ri+l (z)}.

The vector if (z) will be called the characteristic vector of the vertex z with respect to the vertex x. Then the following Iemma holds.

Lemma 3.5.3. Suppose that

then if(z) = if(w) for XE V(r), Z E ri(X), W E r(z). D

Let n be the Subgraph of r induced by the union of the subsets rj(X) for n+ + 1:::;; i:::;; n+ + no- 1. Then it follows from Lemma 3.5.4 that if z, w are contained in the same connected component of n, then a"'(z) = if(w). This implies, in particular, that n is disconnected. An analysis of the connected components of n under the assumption that n 0 > 6 • n + + 11 Ieads to a contradiction.

314 A. A.IVANOV

3.6. The case of infinite diameter

The following question is closely related to Question 3.1.1.

Question 3.6.1. What does an infinite, locally finite (i.e. of finite valency k) DTG

(or DRG) look like?

Aseries of infinite, locally finite DTG's can be constructed in the following way.

Let !1 = ß(k1>k2) be a tree whose vertices are divided into two parts !11 and !12 in such a way that the vertices in ßi have valency ki and there are no edges between the vertices of ßi, i = 1,2. It is easy to see that for any pair (k 1 ,k2 ) of positive integers there is a unique isomorphism class of trees possessing these properties. Let T(k 1 ,k 2 ) be the graph with vertex set !11 in which two vertices are adjacent if they are at distance two in !1. Then a direct check shows that T(kl>k 2) is an infinite DRG of valency k 1 • (k2 -1)

with the parameters ci = 1, ai = k2- 2, bi = (k 1 -1) (k2 -1) for all i ~ 1. The automorphism group of the tree !1 induces a distance-transitive action on the graph T(k 1 ,k2). So the latter is a DTG.

The graph T(k 1,k2) contains no geometric cycles and its connectivity is equal to 1. It is not hard to prove the following.

Lemma 3.6.2. Let r be a DRG of valency k and of infinite diameter. Then the fol-lowing conditions are equivalent:

(i) r does not contain a geometric cycle;

(ii) the connectivity of r is 1;

(iii) r is isomorphic to a graph T(k1ok2) for k = k1 · (k2 -1). 0

C. Godsil has posed a question concerning the existence of infinite, locally finite DRG's of connectivity greater than or equal to 2 (cf. Problem 11 in Unsolved Problems (1979)).

For the class of DTG's nonexistence was established in Macpherson (1982).

Theorem 3.6.2. Each infinite locally finite DTG is isomorphic to T(k 1,k2) for suit-able k1 and k2. 0

In the proof of this theorem Lemma 3.5.4 and certain results concerning cuts in graphs were used.

3.7. The st-theorem

In Ivanov (1983 a) the following assertionwas proved.

Theorem 3.7.1. Suppose that for a DRG r of valency k ~ 3 the relation (cs,as,bs) = (cs+t•as+t•bs+t) holds for some s > 1. Then t < s. 0

The paper lvanov (1983 a) was written independently of Cameron (1982) and Macpherson (1982), but some constructions in all of these papers are alike. For instance, in

Ivanov (1983 a), as weil as in the aforementioned papers, the characteristic vectors if(z) were introduced, and Lemma 3.5.3 was proved. The main idea of the proof of Theorem 3.7.1 is the following. Suppose that

Let x e V(r). Choose vertices y 1 and Y2 such that s + 1 ~ d(x,yi) ~ s + t- 1, i = 1,2, if(y 1 )~if(y 2), d(y 1,y2)=s. It tums out that vertices Y1oY2 with these properties exist. By Lemma 3.5.3 the shortest path joining y 1 and Y2 passes through either rs(x) or rs+r(X ). From the parameters of r we can calculate the nurober of paths between y 1 and y 2 having length s or s + 1. On the other hand, we can estimate the nurober of such paths which pass through r 8 (X). ln this way it is possible to ShOW that there are paths of length s or s + 1 passing through rs+t(x). The latter implies that t < s.

A very elegant proof of Theorem 3.7.1 was proposed by K. Nomura. This proof is contained in the survey paper Bannai & lto (1986).

Below, a few consequences of Theorem 3.7.1 are presented. These consequences generalize some results which were discussed above.

Corollary 3.7.2. For an arbitrary DRG r the inequality no ~ n+ holds. In particular the diameter of a DRG is bounded by a function of k and K, where K is the maximum of ki for 1 ~ i ~ d. 0

Corollary 3. 7 .3. lf the geometric girth of a DRG is g, then d ~ g • 4k-l. 0

Corollary 3.7.4. lf r is an infinite graph of finite valency k, then r is isomorphic to the graph T(k 1 ,k2) for some k1 and k2 suchthat k1 (k2 -1) = k. 0

Corollary 3.7.5. Let G be a group acting distance-transitively on a graph r and let H = Gx for x e V(r). Then

IGI < IHI 3 . 0

A.V. Ivanov has proposed to associate with a pair (u,x) of vertices in r, a matrix A(u,x) whose rows are the characteristic vectors if(zi), 1 ~ i ~ k, where { z 1 , ••• , zk} = r(u ). This matrix is called the characteristic matrix of the vertex u with respect to the vertex x. Investigation of properlies of these matrices enable him to obtain a nurober of new and quite strong feasibility conditions for intersection arrays. These conditions give in particular a diameter bound for certain classes of graphs. For instance the following theorem holds (see Section 4 in lvanov & lvanov (1988)).

Theorem 3.7.6. Suppose that in a distance-regular graph, the inequality n(b,c) > n(bb1) + 1 holds. Then 2~ b~ k/2, 2~ c ~ k/2, k- b- c ~ 1. 0

An exposition of the theory of characteristic matrices developed by A.V. Ivanov can be found in Farad.Zev et al. (1986), Ivanov & Ivanov (1988) and in Section 4.4 of the book Brouwer et al. (1989).

316 A. A lYANOY

In view of Theorems 3.3.2, 3.4.1 and 3.7 .6, it is natural to pose the following.

Conjecture 3.7.7. In a DRG the inequality n(b,c)'5: n(bl>l) holds for all pairs {b,c}. o

If this conjecture is true then the diameter of a DRG is bounded by a linear function

of its valency k and geometric girth g.

3.8. A theorem of Weiss

It follows from Corollary 3.7.3 that the diameter of a DRG is bounded by a function

of its valency k and geometric girth g. On the other hand, if [' is a DTG having geometric

girth g ~ 4, then its automorphism group acts locally t-transitively on the incidence graph ß of r, and t ~ g - 4 (cf. Lemma 2.5.6). Recall that the vertices of ß are the vertices and the cliques of [' with adjacency defined by inclusion. So to bound the diameter of [', it is

sufficient to bound the parameter t which describes the action of G on ß.

Let us first consider the situation when [' is triangle-free, i.e. when a 1 = 0. In this

case the geometric girth of [' coincides with its ordinary girth, and G acts s-transitively on [' for s ~ [(g -1)/2] (cf. Lemma 2.5.5). The result presented below is proved in Weiss ( 1981) modulo the classification of finite simple groups.

Theorem 3.8.1. Suppose that G acts s-transitively on [' and for x E V(l) the sub-group G x is finite. Then s '5:. 7. 0

It follows from the above theorem that g '5:. 16 for a 1 = 0. Hence we have:

Corollary 3.8.2. The diameter of a DTG having valency k and girth at least 4 is bounded by 4k+1. o

In the case when [' contains triangles an absolute bound on t is not yet known and we come to the following problern proposed to the author by E. Bannai.

Problem 3.8.3. Find an absolute bound on the geometric girth of distance-transitive graphs.

One can consider a more general problem.

Problem 3.8.4. Prove the existence of a constant c such that if G acts locally t

transitively on ß and Gx is finite for x E V(['), then t < c.

Locally !-transitive action of G on the bipartite graph ß, where ß is constructed from a distance-transitive graph, has the significant property that the girth of ß is strongly

dependent on t (for instance the girth of ß does not exceed 2t + 11). Forthis reason it is natural to consider the followiilg problern which has an intermediate position between the two previous ones.

Problem 3.8.5. Solve problern 3.8.4 under the additional assumption that the girth of ß does not exceed a linear function oft (say 2t + 11).

DJSTANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA TION 317

In Weiss (1985 a) it was proved that if d and G possess the properties stated in Problem 3.8.5, then I Gx I (and hence t) is bounded by a certain function of k. Here k is the maximum of the valencies of d (notice that in general d is not regular). To formulate this theorem of Weiss we need to introduce some definitions. Let U!:: V(ö), and i be a positive integer. Let Gj(U) denote the largest subgroup of Gwhich fixes each Vertex of r which is at distance at most i from some vertex in U. It is easy to see that for x e U the index of Gi(U) in Gx is bounded by certain function of k, i and I U I.

Below we present a consequence of the main result of the paper Weiss (1985 a) in the formulation due to Brouwer et al. (1989).

Theorem 3.8.6. Let d be a finite connected graph all of whose vertices are of valency at least two, with girth g ;;:: 3, and let G be a group of automorphisms of d which is locally t-transitive. If g ~ 2t + 11, then G 5(W) = 1 for every path Woflength 14. D

Notice that this result does not rely on the classification of finite simple groups.

To complete this section we formulate a consequence of Theorems 3.8.6 and 3.7.1 which was proved in Brouwer et al. (1989).

Corollary 3.8. 7. The diameter of a distance-transitive graph of valency k;;:: 3 is at most (k 6)! 4k. In particular, there are only finitely many distance-transitive graphs of a given valency k;;:: 3. D

3.9. On diameter of a DRG

In this section we give an account of recent progress in the bounding of the diameter of a DRG in terms of its valency.

All the methods of diameter-bounding which were discussed above are of combinatorial nature. These methods rely on an analysis of the combinatorial structure of the graph in question, in particular of its cycles. In addition to combinatorial methods, algebraic ones are known. These methods involve a study of the Bose-Mesner algebra of the considered graph.

The algebraic approach to the study of DRG's can be outlined as follows. Suppose we have an intersection array and that the existence problern of the corresponding DRG

is considered. Then by formula (2.1) we can calculate all the parameters pt, 1 ~ i,j,k ~ d. If these parameters are the structure constants of a semisimple algebra then they should satisfy certain conditions. The most important algebraic conditions are the integrality condition for m ('Ai) (Lemma 2.1.6) and the nonnegativity condition for the Krein parameters qt (Lemma 2.1. 7).

If the intersection array is essentially given, then a check of these conditions is reduced to a direct (but quite involved) computation. But in more important and interesting Situations (for instance in the bounding of diameter) these conditions are applied to an infinite class of intersection arrays.

318 A. A.IVANOV

The first result arising from application of the algebraic approach is the famous Feit-Higman Theorem (cf. Theorem 2.6.2 (ii)). This theorem can be considered as a result on the bounding of the diameter of DRG's from a certain special class. The result on Moore graphs (Theorem 2.6.2 (i)) was obtained in an analogous way. The first result on bounding the diameter of DRG's in a general situation through the use of algebraic methods was obtained in Ito (1982). In that paper it was proved that the diameter of a bipartite distance-regular graph of valency 3 is bounded. Moreover all such graphs were classified there. All distance-regular graphs of valency 3 were classified in Biggs et al. (1986) using both combinatorial and algebraic methods.

Recently E. Bannai and T. Ito using a combination of the algebraic and the combinatorial methods, have made significant progress in bounding the diameter of a DRG in

terms of its valency. Their results are presented in a series of papers, four of which are now available (cf. Bannai & Ito (1987 a,b, 1988, 1989)). Below we formulate the main results of these papers.

Theorem 3.9.1. For a DRG the value n 0 is bounded by a certain function of k. For instance no :5: 10 • k 2k. o

Theorem 3.9.2. Fora DRG the value n(k -1, 1) is bounded by a function of k and t wheret=d-n(k-1,1). o

Theorem 3.9.3. The diameter of a bipartite DRG is bounded by a certain function of its valency.

Theorem 3.9.4. The diameter of a DRG is bounded by a function of k and t where

t=d-n(b1,1)-n(1,bl). o

The following result from Boshier & Nomura (1988) should be mentioned.

Theorem 3.9.5. If n(k -1, 1) > 0 then n(k -2, 1) :5: 5. o

It follows from Theorems 3.9.1-3.9.5 that the diameter of a DRG of valency 4 is bounded.

In the opinion of E. Bannai and T. Ito, significant progress in the general problern of diameter-bounding can be expected, if abound for the case k = 5 can be found. The latter bounding is reduced to the following.

Problem 3.9.6. Find an absolute bound on n(2, 1) for DRG's ofvalency 5.

It is easy to show that n (2, 1) :5: 1 if n (3, 1) = 0, so the interesting situation is when n(3, 1) '# 0.

Problem 3.9.6 will be solved if the following conjecture can be proved.

Conjecture 3.9.7. Suppose that for a DRG r, Cj = 1 for 1 :5; i :5; s + 1 and as '# as-1·

Then a path w of length s in r is contained in a unique Subgraph L\(W) of r possessing the following properties:

(i) the diameter of L\(W) iss;


(ii) if x,y e A(W) then all paths of length d(x,y) and d(x,y) + 1 joining x and y are contained in A(W). D

It is clear that if a subgraph A(W) with these properties exists then it is distanceregular of valency as + 1. This conjecture is true if s = 2 (cf. Section 4.6.3 in Brouwer et al. (1989)).

A very interesting result concerning diarneter bounding for distance-regular graphs is proved in Godsil (1988).

Theorem 3.9.8. Let r be connected distance-regular graph of valency k > 2. Suppose that r is not a comp1ete multipartite graph. Let 9 be an eigenvalue of the adjacency matrix of r whose multiplicity is equal to m. If 9 ~ ±k, then the diarneter of r is at most 3m-4andk:s;(m-1)(m+2)/2. D

It follows from this theorem that there are finitely many distance-regular graphs (possibly none) with a given multiplicity m for a nontrivial eigenvalue. It is therefore of

interest to classify all DRG's for small values of m.

4. Distance-transitive graphs of small valencies

4.1. Trivalent and tetravalent graphs

As was mentioned above, the Iist of all trivalent DTG's (i.e. of valency 3) was obtained in Bigss & Smith (1971). After having bounded the diameters of these graphs (see Section 3.1), the intersection arrays corresponding to diarneters within this bound were enumerated by means of a computer. From these the intersection arrays satisfying the feasibility conditions were selected. The condition that ki should divide 48 (see

Lemma 3.2.1) was also used in this step. It turns out that there are 12 arrays possessing all these properties. Finally it was proved that every intersection array in this Iist corresponds to exactly one distance-transitive graph.

Information regarding the DTG's of valency 3 is presented in Table 1. Here each row corresponds to a DTG r of valency 3; each column corresponds to an isomorphism type of the vertex stabilizer Gx, x e V(f), in a group G such that r is a G-DTG. The groups G are indicated in the table.

320 A.A.IVANOY

Table 1

Gx

r z3 S3 S3 xZz s4 s4 xz2

K4 A4 s4

cube A4 xZ2,S4 S4 xZ2

dodecahedron As As xZz

K3,3 E 9.D4,E9.Z4 E9.Dg

3. K3,3 31+2D4 31+2.Ds

3'+2. z4

Petersen As Ss

Desargues AsxZ2,S5 Ss xZ2

Coxeter PGL2(7)

Heawood PGL2(7)

Biggs-Smith PSL2(17)

Tutte PGL 2(9) PrL2(9)

M!O

Poster Z 3.PGL2(9) Z3.PrL2 (9) z 3.M 10

Let us say a few words about the graphs in Table 1.

The structure of the graphs K 4 (the complete graph), K 3,3 (the complete bipartite

graph), the cube and the dodecahedron is evident.

The graph 3. K 3,3 is a 3-fold antipodal covering of the graph K 3,3 . For an arbitrary

k all graphs k. Kk.k can be constructed by the following procedure (cf. Gardiner (1974)).

Let TI= (P,L,I) be a projective plane of order k and (p,l) be a fiag of TI, i.e.

p e P , l e L, p II. Let us take for the vertices of r all lines nonincident with p and all

points nonincident with /. Then r is a k-fold covering of the graph Kk.k. The condition that TI be Desarguesian is sufficient but not necessary for r to be a DTG (see Section

6.8). There is a unique isomorphism class of projective planes of order 3 and up to iso

morphism there is a unique graph 3. K 3,3 . In the above table 31+2 denotes the unique

nonabelian group of order 27 and exponent 3.

The Petersen graph is the first member of the series of odd graphs Ok· The latter can

be defined as follows. Let Q be a set of cardinality 2k - 1. Then the vertices of 0 k are the

(k -1)-element subsets of Q. Two vertices are adjacent if their subsets are disjoint. The

Standard double covering 2. ok of ok is also a DTG. The Standard double covering 2. r of a graph r is defined by: V(2. r) = V(r) x (0, 1 }; (x, a)- (y, ß) if and only if x- y in r

DISTANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA TION 32!

and a ~ ß. The graph of Desargues is just 2. 0 3.

The graphs of Heawood and Coxeter admit natural descriptions in terms of the projective plane TI= (P,L,/) of order 2. The former graph is the incidence graph of TI. The vertices ofthe latterare the antiflags ofTI. Two antiflags (p1>/1) and (p2,/2) are adjacent if Pi does not lie on /3-i for i = 1 and 2.

The Biggs-Smith graph is a 2-orbit of the action of PSL2(17) on the cosets of its subgroup S 4 . An elegant description of this graph, as well as of the Petersen and Coxeter graphs, is contained in Biggs (1973).

The Tutte graph is the incidence graph of the generalized quadrangle related to the group B 2(2). Notice the sporadic isomorphisms B2(2)=S6, A6 =PSL2(9). This graph adrnits a 3-fold antipodal covering known as the Poster graph.

In subsequent sections we will return to the Iist of trivalent DTG's.

It should be emphasized that there is only one DRG of valency 3 which is not a DTG. This is the incidence graph of the generalized hexagon related to the group G2(2) (cf. Ito (1982) and Biggs et al. (1986)).

The classification of the tetravalent DTG's (i.e. of valency 4) was carried out in Smith (1973, 1974 a,b). The classification scheme is analogous to that in the trivalent case, only the number of possibilities to be considered is !arger and some intersection arrays satisfy all feasibility conditions but do not correspond to a DTG. The final list consists of 15 DTG's. This Iist can be found, for instance, in Brouwer et al. (1989).

4.2. A problern ofN. Biggs

The classification scheme which was applied to trivalent and tetravalent DTG's consists of the following two main parts: (a) bounding of diameter; (b) enumeration of all DTG's of given valency whose diameter does not exceed the obtained bound. Theseparts are quite independent. Namely, the first one merely ensures that the Iist obtained in the second one is complete. So in this scheme one can proceed to the second step before realization of the first step.

Since results on the bounding of diameters of DTG's of valency k ~ 5 were not available, the enumerationproblern for DTG's of small valency and small diameterwas quite popular in the mid-seventies (cf. Biggs & Gardiner (1974), Biggs (1976)).

In Biggs (1976), results of computer enumeration of DTG's of valency 5$ k $ 13 and diameter 3$ d $ 5 are presented. In that paper only the primitive DTG's are considered (they are called automorphic). In Biggs (1976) there were eleven intersection arrays for which the existence of corresponding DTG's could not be established nor disproved. This problern by N. Biggs attracted the attention of a number of mathematicians and soon the problern was completely solved. The Iist of the eleven aforementioned arrays, along with information about the corresponding graphs, is presented in Table 2. In

322 A. A.IVANOV

the last column of the table the minimal normal subgroup of Aut(r) is indicated.

Table 2

No. n d i(r) IAut(r)l

1. 231 3 ( 10,8,7;1,1,4} -2. 65 3 (10,6,4;1,2,5} + 24• 3· 52· 13 PSL2(52) 3. 210 3 ( 11,10,4; 1,1,5} -4. 208 3 ( 12,10,5;1,1,8} + 28 • 3· 52· 13 PSU3(4) 5. 88 3 ( 12, 10,2; 1 ,2,8} -6. 68 3 ( 12, 10,3; 1 ,3,8} + 26 • 3· 5· 17 PSL2(24) 7. 364 3 (12,9,9;1,1,4} + 26 • 36 • 52· 13 G2(3) 8. 1755 4 ( 10,8,8,8; 1,1,1,5} + 212. 33· 52· 13 2p 4(2)' 9. 315 4 ( 10,8,8,2; 1,1,4,5} + 28 • 33. 52· 7 h

10. 2925 4 ( 12,8,8,8; 1,1,1,3} + 212 · 33• 52·13 2p 4(2)'

11. 525 4 ( 12,8,6,4; 1,1,2,9} -

In Gordon & Levingston (1981) nonexistence of DTG's with intersection arrays No. 1, 3, 5, existence of DTG's with intersection arrays No. 4, 7, 8 and 10 and existence and uniqueness of DTG's with intersection arrays No. 2 and 6 were estab1ished. In Cohen (1981), nonexistence of DTG's (moreover, of DRG's) with intersection array No. 11, as well as existence and uniqueness of a DTG with array No. 9, was proved. The uniqueness of DTG's for arrays No. 4 and 7 was proved in Buekenhout & Rowlinson (1981). It was mentioned in the latter paper that graphs with arrays No. 2, 6, and nonexistence of DTG's with array No. 5 were proved independently in an unpublished paper by S. Doro.

An independent solution to the Biggs problern was obtained by Soviet mathematicians. Namely, in Zaichenko et al. (1980), a DTG with array No. 4 was constructed by means of a computer. In Zaichenko (1981) DTG's with arrays No. 6 and 9 were constructed by analogaus methods. In Ivanov (1981) a graph with array No. 2 was constructed by the method of transitive extension of permutation groups. Finally, in Zaichenko et al. (1982), the complete answer was announced.

Let us say a few words about the methods which were used for the enumeration of graphs with intersection arrays from Biggs's list.

I. The geometric method involves the construction of a graph with prescribed parameters from a certain known geometric object. For instance in Gordon & Levingston (1981) the graphs with arrays No. 2 and 6 were constructed in terms of inversive planes. In Cohen (1981) a graph with array No. 9 was described in terms of a vector space over the quaternions. In this construction a crucial role was played by the representation of Janko's group h as a group generated by quaternion reftections. The graphs No. 8 and


10 are, respectively, the point graph and the line graph of the generalized 8-gon related to the group 2 F 4 (2), while the graph No. 7 is the point graph (isomorphic to the line graph) of the generalized hexagon related to the group G2(3). The geometric description of graphs obtained in this way are very natural and beautiful, but certainly this method can not be considered as a universal one.

II. The local method involves a successive realization of the following steps.

1. Starting with a given intersection array, some upper and lower bounds on the order of the stabilizer Gx are deduced. The lower bounds follow from the transitivity of Gx on the sets ri(x) for 1 ~ i ~ d. To obtain upper bounds one can estimate the order of the subconstituent G~'(x) and the kernel of the action of Gx on ri(x) for suitable i.

2. Using the bounds obtained in the previous step, and the fact that Gx should have transitive representations of degrees ki = I ri(X) I, the list of all possible structures of Gx

as an abstract group is obtained.

3. The variants of the permutation action of Gx on ri(x) for 0 ~ i ~ d are described. This means that Gx is characterized as a permutation group on the set V(r).

4. The cellular ring of the permutation group (Gx, V(r)) is determined and all of its cellular subrings which are cells are enumerated.

5. The automorphism groups of the cells found in the previous step are determined and it is checked whether or not they give rise to a distance-transitive action.

All steps of this scheme were successfully carried out in Ivanov ( 1981 ). Some of the steps were carried out in Gordon & Levingston ( 1981) in a proof of nonexistence.

This method can be considered as a simplified model of the local approach to the classification of DTG's which will be discussed in the next chapter.

III. The global method. Since the number n of vertices of a graph is determined by its intersection array, abound on I Gx I implies abound on I G I. On the other hand, in some cases one can conclude that H '5J G ~ Aut(H), where H is a nonabelian simple group. This is true, in particular, wheneven n is divisible by at least two different primes, n is not a power of an integer m "# n and n is not equal to the order of a nonabelian simple group ( cf. Lemma 6.1.1 ). These results and some information concerning nonabelian simple groups sometimes enable one to identify H (and G) with a certain known group. Now we can search for r as a graph which is invariant under a primitive representation of G of degree n. This method was carried out in Zaichenko et al. (1980) and in Zaichenko (1981) for the construction of graphs. It also forms the basis for all uniqueness results concerning the graphs in question.

From this method the global approach to the classification of DTG's has evolved (see Chapter 6 of the present survey).

324 A. A.IVANOV

4.3. Valendes from 5 to 7

At first we should certainly bound the diameters of the considered graphs. By Corollaries 3.7.2 and 3.7.3 it is sufficient for this purpose to bound the geometric girth g of G or the order of the vertex stabilizer Gx.

Without loss of generality we will assume that g ;::: 5. Then it follows from Lemmas 2.1.4 and 2.5.4 that either

(a) a 1 = 0 and cr<x) is a doubly transtive group, or

(b) a 1 + 1 divides k.

For the values k in question we get k = 6 and a 1 = 1 in case (b ).

In case (a) the results presented in the survey Weiss (1981) give a bound on the order of Gx in all cases except when G[(x) contains a normal subgroup isomorphic to PSLn(q), n;::: 3, in its natural doubly-transitive representation of degree (qn -1) I (q -1).

Hence special consideration is necessary for the case when k = 7 and cr<x)::: PSL3(2). In this case G acts s-transitively on r for s = 2 or 3, and g ~ 8 by Lemma 2.5.4.

Now let k = 6, a 1 = 1. Then the incidence graph ~ of r has valency 3 and G acts edge-transitively on it. So we can apply the following result from Goldschmidt (1980).

Theorem 4.3.1. Let ~ be a graph of valency 3 and let G act edge-transitively on ~-

Then for a. e ~ the order of Ga divides 2 7 • 3. 0

So the diameters of the graphs in question are bounded and, in principle, we can apply the classification scheme which was used for k = 3 and 4. Nevertheless, a direct application of Corollaries 3.7.2 and 3.7.3 gives too highabound on diameter. Forthis reason we have used a modification of this scheme in which the bounding of diameter and enumeration of arrays are tied together.

The algorithm for the generation of all intersection arrays of DTG's with given valency k which was used in Ivanov et al. (1984) is based on the branch and bound method. The branching 1s carried out by altering the values of b 1,

c2, ... , C8 , b8 , ••• , bd_1 , cd. Suppose that the parameters in the first s -1 colurnns of the intersection array are fixed. We look over variants for the column number s by decrease of C8 from k to c8 _ 1 and for a given Cs by increase of bs from 0 to min {b8 _ 1, k-c8 _ 1 } (see Lemma 2.1.1). Foreach such variant we test the "local" feasibility conditions which are formulated in Lemmas 2.1.2- 2.1.5 andin Theorem 3.7.1, as well as some more specific conditions which were deduced from a study of the characteristic matrices of pairs of vertices. It is checked also that the value of ki is consistent with the bound on I Gx I. Since Theorem 3.7.1 and abound on K = maxki tagether imply abound on the diameter, only a finite number of variants need to be considered in this algorithm.

Let a variant satisfy all the aforementioned conditions. If bs ~ 0 then the process is continued, and for a fixed s colurnns we search the variants for the colurnn number s + 1. If bs = 0, then d = s and the intersection array is constructed completely. In this case we

test the "global" conditions formulated in Lemmas 2.1.6 and 2.1.7. If a constructed array corresponds to an imprimitive graph then we first construct the intersection array of the corresponding primitive graph and then test the "global" conditions for this array.

As a result the following theorem is proved.

Theorem 4.3.2. There are exactly 14, 24 and 14 isomorphism classes of DTG's of valency 5, 6 and 7, respectively. D

The Iist of DTG's of valencies 5, 6 and 7 can be found in Ivanov et al. (1984), FaradZev et al. (1986) andin Brouwer et al. (1989). Only one of the constructed graphs was not previously known. This is the antipodal triple covering of the graph on 330 vertices related to the group M 22· This graph gives an example of so called P-geometry (see Shpectorov (1985)).

4.4. Valendes from 8 to 13

Already for srnall values of k a bound on the order of the stabilizer Gx is controlled by the following two situations:

(1) the subconstituent ar<x) contains the altemating group Ak of the set r(x);

(2) G acts s-transitively on r for s ~ 4.

For the pairs (r,G) different from the cases (1) and (2) the bound on the order of Gx

can be essentially improved.

On the other hand, the characterization problern for the graphs satisfying (1) and (2) is very interesting. This problern was solved completely in the mid-eighties (see Sections 5.1 and 5.2 of the present survey).

Excluding Situations (1) and (2) from consideration enabled us to obtain reasonable bounds on I Gx I for 8 S: 5 S: 13. This was the starting point of the work Ivanov & Ivanov (1988).

The obtained bounds are presented in Table 3. We first make abrief remark. Let g

be the geometric girth of r, g ~ 5, and put r = [(g -1) /2]. Then k, = k 1 • b}-1 and hence b}-1 divides I Gx I. Notice that if b 1 is notapower of p then, to bound r, it is sufficient to show that I Gx I = pa· q for some integer q which satisfies (p,q) = 1. In this situation we will say that I Gx I divides poo • q.

Let r be a DTG of valency 8 S: k S: 13 and of geometric girth g ~ 5. Then a 1 + 1 divides k (Lemma 2.1.4) and we may assume that k > 2 · (a 1 + 1) (Lemmas 2.7.1 - 2.7.2). For the values k and a 1 in question, information about the order of Gx is presented in Table 3. Namely, if the Situations (1) and (2) are excluded, then I Gx I should divide at least one of the values presented in the corresponding entry. The meaning of the expressions containing oo is defined above.

326 A. A.IVANOV

Table 3

a1

k 0 1 2 3

8 29 32 72 200 3; 26 300

9 26 34 72 ; 28 34 200 32 ; 24 300

10 29 36 5 21034 5; 212 5

11 28 34 52 11 12 21o 35 52 112 2ll 34 52; 2734;2436 21035;2123

200 32 5 ; 28 300 5 13 28 300 13

The bounds presented in Table 3 are obtained using results from the following papers concerning edge-transitive actions of groups on graphs: Weiss (1979), Stellmacher (1984), Fan (1986).

In general the classification scheme for the considered values of k is analogous to that which was used for k = 5, 6 and 7. Nevertheless, a number of new "local" conditions (for instance Theorem 3.7.6) were checked. In fact, some of these conditions were discovered during the classification. Namely, for some initial portion of the intersection array the enumeration did notfinishin reasonable time. We then tried to prove that there is no DTG (or DRG) with such an initial portion. Sometimes this attempt led to a new feasibility condition.

In addition, during the classification of DTG's of valency 8 ~ k ~ 13, the method of computing the number of certain subconfigurations in the graph in question was used. This method is presented in A.V. Ivanov (1985). In some instances it was possible to complete the enumeration of the arrays using this method alone.

The developed technique can also be applied for k > 13. We stopped at k = 13 in order to repeat the classical result of Biggs (1976), including also the imprirnitive case, without any restriction on diameter. It is remarkable that all automorphic graphs of valency k ~ 13 appeared in Biggs (1976), either as known examples or as one of the eleven open cases with given intersection array.

4.5. Approach of A. Gardiner

All results on the classification of DTG's mentioned above were obtained with the aid of computer calculations. These calculations were essentially used during the enumeration of the feasible intersection arrays.

As a rule, the proofs obtained through the use of a computer are not perfect ones. The reason is that the reader has no opportunity to check all steps of the proof since the


computer programs were never published. So the classification results obtained without the use of a computer are perhaps of greater interest.

In Gardiner (1975) an independent computer-free classification of DTG's of valency 3 was obtained. The starring point for this work was the formulation of a complete list of possible isomorphism types of vertex stabilizer Gx. There are five such types (see Table 1). Roughly speaking the strategy is as follows.

Let G act distance-transitively on a graph r ofvalency 3, x e V(r), and suppose the abstract structure of Gx is known. For i an integer, we assume that the following things are known: (1) the parameters Cj, bj for 0 :>; j :>; i; (2) the permutation actions of Gx on rj(X) O:>; j :>; i; (3) the Subgraph of r induced by the Set:

i Ai= u rj(x).

j=O

Then the next step consists of enumerating all possibilities for the values of ci+l , bi+l

and for the action of Gx on ri+l (x). After that, the subgraph Ai(x) is extended to the subgraph Ai+l (x) in view of the fact that the latter subgraph should beinvariant under the action of Gx. In this step one also uses the fact that the subgraph Ai(y) is isomorphic to Ai(x) for all y e r1 (x). The fact that the diameter of a graph is bounded in terms of I Gx I (see Theorem 3.5.2, Corollary 3.7.2, Theorem 3.8.7) ensures that only a finite nurober of possibilities need to be considered. But in practice it turns out to be unnecessary to use the diameter bounds in essential form, since a bound always comes from consideration of the particular Situation.

It should be mentioned that the work Gardiner (1975) relies heavy on the classification of primitive permutation groups with a subdegree of 3 obtained in Wong (1967).

In a subsequent paper, Gardiner (1982), an analogous strategy for graphs of arbitrary valency is proposed. In this paper all DTG's of valency 4, 5 and 6 which contain triangles are classified. Afterwards, this strategy was realized in Gardiner (1985), where an independent computer-free classification of DTG's of valency 4 is given. In the papers Gardiner & Praeger (1985, 1986) an independent classification of DTG's of valency 5 and 6 is obtained in this manner.

Thus a computer-free classification of DTG's of valencies up to 6 is available. But we believe that in order to extend this classification to DTG's of higher valencies, some improvements to Gardiner's approach must be made.

4.6. A fixed points method

One of the possible modifications of Gardiner's approach is a fixed points method which was proposed in Ivanov (1984). This method was very useful in the classification of DTG's ofvalency 11 obtained in Ivanov & Chuvaeva (1985).

328 A. A.IVANOV

The method consists of the following. One constructs a graph r on which a group G acts distance-transitively and for which the vertex stabilizer Gx has prescribed isomor

phism type. One next considers subgroups F of Gx, andin each case

(a) the set !l(F) of vertices fixed by F is determined;

(b) the action of the normalizer NG(F) on !l(F) is described; and

(c) the subgraph of r induced by the set !l(F) is constructed.

Mainly we are interested in the maximal subset I.= I.(F,x) of !l(F) containing x

suchthat the subgraph induced by I. is connected. Let Gr; be the setwise stabilizer of I. in

N G (F) and (;[I.] be the action of G r; on I..

In the study of the action of NG(F) on !l(F) we shall use the following result from

Alperin (1965).

Lemma 4.6.1. Let {F 1, F 2· ... , Fm) be a maximal set of pairwise nonconjugate subgroups in Gx which are conjugate with F in G. Then NG(F) has exactly m orbits

!!..1, !!..2, ... , !lm on !l(F) and l!li I= [NG(Fi) :NG,<Fi)] for 1 ~ i ~ m. 0

As a direct consequence of this lemma we have the following:

Corollary 4.6.2. Suppose that F is conjugate in Gx to each subgroup F' ~ Gx to which it is conjugate in G. Then NG(F) acts transitively on !l(F). o

Corollary 4.6.3. Let F be conjugate in Gx to each subgroup F' ~ Gx for which

F' = F. Then NG(F) acts transitively on !l(F). 0

It is clear that transitivity of NG(F) on !l(F) implies transitivity of Gr; on I..

The situation when (;[I.] acts distance-transitively on I. is of particular interest.

Concerning this Situation, we have the following sufficient criterion which follows from

Corollary 4.6.2.

Lemma 4.6.4. Suppose that for each vertex y such that F ~ K = Gx n Gy the sub

group Fis conjugate in K to each subgroup F' ~ K to which it is conjugate in G. Then

G[I.] acts distance-transitively on I..

Corollary 4.6.5. Suppose that for each subgroup K such that F ~ K ~ Gx the subgroup F is conjugate in K to each subgroup F' ~ K to which it is conjugate in G. Then G[I.] acts distance-transtively on I.. o

Corollary 4.6.6. Suppose that F is a Sylow p-subgroup in Gx for some p. Then G[I.] acts distance-transitively on I.. o

Now let us turn to the DTG's of valency 11. In this case Gf(x) is one of the following groups:

Fi1, i e {1,2,5,10), PSL2(11), Mn, An, Sn,

where Fi1 is the Frobenius group with kernel of order 11 and with complement of order i.

In the classification we have distinguished the following three situations:

(1) Gf<x) is a Frobenus group. In this case it can be shown that I Gx I divides 22 • 52• 11. U sing this bound the corresponding intersecnon arrays were enumerated by the computer program described in Sections 4.3 and 4.4. The resulting list of graphs is the following: K 12, K 11 , 11 , 2. K 12 and 11. K 11,11·

(2) ci<x) contains the alternating group A 11 . All graphs of arbitrary valency corresponding to this case were classified (see Section 5.1).

(3) GI(x) is either PSL 2(11) or M 11 • Then Gx is isomorphic to one of the following groups: PSL 2(11), PSL2(ll) xAs. M 11 , M 11 xM10. In this case Gardiner's scheme was realized. In order to reduce the number of possibilities to be considered, the fixed points method was applied. For the subgroup F we chose the subgroups Z2 , Z2 x Z2, Q 8

and Q 8 x Q 8 , respectively. In all cases the hypothesis of Corollary 4.6.3 is satisfied, so I. is a vertex-transitive graph of valency 3. Moreover, in the case Gx = M 11 , the hypothesis of Corollary 4.6.5 is also satisfied so I. is a DTG in this case.

Let us consider in some detail the case Gx = PSL 2(11). Exactly this situation occurs in the exceptional graph on 266 vertices whose automorphism group is the sporndie simple group J 1. In this case the hypothesis of Corollary 4.6.5 fails for K = D 12 and K = z2 X z2. Nevertheless, a more detailed analysis of these Situations shows that G[I.] still acts distance-transitively on I.. Thus I. is one of the graphs in Table 1. Moreover a vertex stabilizer in G [I.] is isomorphic to S 3 , so I. is one of the following graphs:

K4' K3,3' Q3' 3. K3,3' D' 2. 03' 03.

Here Q 3 is the cube and D is the dodecahedron.

An analysis of the local structure of r Ieads to

Lemma 4.6.7. In the considered situation one of the following holds:

(i) I. is K 4 , ris K 12 and G = M 11 ;

(ii) I. is Q3, r is 2. K 12 and G = M 11 x Z2;

(iii) I. is the Petersen graph 0 3. 0

Concerning possibility (iii) the following result was proved in Ivanov (1987 c).

Lemma 4.6.8. Suppose r is a graph of valency 11, G acts vertex- and edgetransitively on r, Gx = PSL2(11), and for F = Z2 a connected component ofthe subgraph induced by !o.(F) is isomorphic to the Petersen graph. Then r is the graph related to J 1. 0

This Iemma was proved using group amalgams which are considered in detail in Ivanov & Shpectorov (1991b). Notice that initially it is not assumed in the Iemma that r is aDTG.

Thus in the classification of DTG's of small valency some effective classification methods are developed. These methods enable one to consider more general classification problems. So we can speak: about a local approach to the classification of

330 A. A.IYANOY

DTG's which is the subject of the next chapter.

5. A local approach to the classification of DTG's

5.1. The subconstituents Ak and Sk

In this section we will study the situation when G acts distance-transitively on a graph r of valency k and G_i(x) ::: Ak or Sk. As was mentioned in Sections 4.4 and 4.6 the

classification of such graphs r is extremely important for the classification of the DTG's of small valency. Surely this problern is of great independent interest. For k = 3, 4 and 5 the group Ak is isomorphic to PSL 2(k -1) so we will assume that k ~ 6.

Using some Standard methods from the theory of s-transitive graphs, it is easy to prove the following

Lemma 5.1.1. Let G act s-transitively (s ~ 2) on a graph r of valency k ~ 6 and sup-pose c_r<x>::: Ak or Sk. Then one of the following holds:

(i) s=2andG 1(x)=l;

(ii) s = 3, G1(x,y) = 1 andAk xAk-1 "51 Gx~ sk X sk-1;

(iii) s = 3, k = 7 and G 1 (x) = 1. 0

Case (iii) is due to the fact that S 6 has an outer automorphism. It is realized for instance in the Hoffman-Singleton graph.

In view of Lemma 2.5.5, if G acts distance-transitively on r and g is the girth of r then g ~ 6 in the case (i) and g ~ 8 in the cases (ii), (iii) of Lemma 5.1.1.

In Lemmas 5.1.2- 5.1.6 below it is assumed that the hypothesis of Lemma 5.1.1 holds, in particular k ~ 6. Notice that it is not assumed that G acts distance-transitively on r.

It is clear that if g = 3 then r is the complete graph Kk+1 and G = Ak+1 or Sk+1·

Lemma 5.1.2. (Cameron (1974)) Let g = 4. Then r is isomorphic to one of the fol-lowing graphs:

(1) Qk- the cube;

(2) Dk - the folded cube;

(3) Kk,k- the complete bipartite graph;

(4) 2. Kk+1 - the standard covering ofthe complete graph. o

Lemma 5.1.3. (Armanios (1981)) Let g = 5. Then either r is the Hoffman-Singleton graph or Gx acts intransitively on r 4 (x ). o

Lemma 5.1.4. (Praeger (1980)) Let g = 6 and G 1 (x) "#. 1. Then r is either the odd graph ok or its Standard double covering 2. Ok. 0

DISTANCE-TRANSITIVEGRAPHSAND THEIR CLASSIFICA. TION 331

Lemma 5.1.5. (Annanios (1981)) Let g = 6 and G 1(x) = 1. Then either k = 7 or Gx

acts intransitively on r3(x). D

Lemma 5.1.6. (Armanios (1981)) Let g > 6 and G 1 (x) * 1. Then either k = 7 or Gx acts intransitively on r4(x). D

By Lemmas 5.1.1 - 5.1.6, in order to obtain the complete list of the DTG's corresponding to the situation under consideration, one should study the cases k = 7. The DTG's of valency 7 were classified in Ivanov et al. (1984) (see Section 4.3). An independent description of DTG's with altemating or symmetric subconstituent was obtained in Yokoyama (1987). In the latter paper the case k = 7 was studied without the aid of a computer and a list of the groups acting distance-transitively on the corresponding graphs was also given. The final result is as follows:

Theorem 5.1.7. Let G act distance-transitively on a graph r of valency k ~ 6 and suppose Gf<x> = Ak or Sk. Then one of the following occurs:

(i) k = 7, r is the Hoffman-Singleton graph and G = PSU 3(5) or PLU 3(5);

(ii) k = 7, r is the Standard double covering of the Hoffman-Singletoll graph, G = PSU 3(5) x Zz, PLU 3(5) or PLU3(5) x Zz;

(iii) r is Kk+l and G = Ak+l or Sk+l;

(iv) r is Kk.k and G !:; Sk+l wr Zz;

(v) r is 2. Kk+l and G = Ak+l X Zz, sk+l or sk+l X Zz;

(vi) r is Qk and G = 2k A.Ak, 2k-l A.Sk or 2k A.Sk;

(vii) r is Dk and G = 2k-l A.Ak or 2k-l A.Sk;

(viii) risOk andG =A2k-1 orS2k-1;

(ix) ris 2. ok andG =Azk-1 xZz, s2k-1 orS2k-1 XZz.

Here PLU3(5) is the extension of PSU3(5) by an automorphism of order 2.

D

In Ivanov (1986 a) a uniform proof of Theorem 5.1.7 is proposed. This proof consists of the following four steps.

Step 1. Prove that r contains a family <1> of cycles with the property that each s-path is contained in a unique element from <\>. The realization of this step relies on a method of group amalgams and can be found in Chapter 3 of the survey Ivanov & Shpectorov (1991b) (cf. Theorem 3.1.2, Example 3.2.2).

Step 2. Treat the case when the length of a cycle from <1> is greater then g. In the situation under consideration this occurs only in 2. Kk+l and the covering of the Hoffman-Singleton graph.

Step 3. Determine up to isomorphism the amalgam A = ( Gx , G {x,y l , G { c l } where C E c1> and the edge (x,y} is contained in the cycle C. The methods for the realization of this step arealso considered in the aforementioned survey.

332 A. A.IVANOV

Step 4. Identify Gas a factor group of the universal closure of the amalgam A.

The advantage of this approach is that it can be applied to the case when a;<xl is an arbitrary doubly-transitive permutation group. The application scheme of this approach will be considered in Sections 5.5 - 5.7 below, where the general situation will be compared with the case a;<xl = Ak or Sk.

lf <1> is a collection of cycles in an s-transitive graph possessing the property stated in Step 1, then the elements of this family will be called apartments. Notice that this definition is different from that in lvanov & Shpectorov (1991b) since here we do not assume the validity of the exchange condition.

5.2. s-transitive DTG's for s ~ 4

An extremely important step in the classification of s-transitive DTG's was done by R. Weiss (1985 b) who classified these graphs for s ~ 4. The basis for this result is formulated in the following Iemma proved in Weiss (1981 b).

Lemma 5.2.1. Let G act s-transitively on a graph r with s ~ 4. Then PSLz(q) '.51 a;<x)::::; PrLz(q), where q = pm,p a prime. o

As a consequence of this Iemma the study of s-transitive graphs for s ~ 4 can be reduced to the study of the case where the projective linear subconstituents have dimension 2. In Weiss (1982) it was shown that in the considered situation r is locally isomorphic to a classical generalized n-gon for a suitable n. Namely the following Iemma was proved.

Lemma 5.2.2. Let r be a graph of valency k ~ 3 and Iet G act s-transitively on r, s ~ 4. Then s e { 4,5, 7} and there is a group F acting s-transitively on a generalized (t-1)-gon .:1 = .:11-t,k-1 suchthat G(x) = F(u) for u e V(L1). Moreover either t = s and F contains the corresponding Lie group, or t = 5, s = 4, k = 3 and F = PGL2(9). 0

It tums out that, besides two exceptions of valency 3, an s-transitive DTG for s ~ 4 is isomorphic to a classical generalized n-gon (see Weiss (1985 b)).

Theorem 5.2.3. Let r be a DTG of valency k ~ 3 which is s-transitive for s ~ 4. Then one of the following holds:

(i) r is the graph L1s-l,q where s = 4, 5 or 7, q = k -1, and q =2m for s = 5 and q = 3m for s = 7;

(ii) r is a triple antipodal covering of the graph ~. 2 (the Foster graph);

(iii) r is the Biggs-Srnith graph on 102 vertices with automorphism group PSL2(17). o

A crucial role in the proof of this theorem was played by the fact that if r is s- but not (s +!)-transitive DTG then the girth of r does exceed 2· s + 2. Already this condition on the girth of s-transitive graphs tumed out to be strong enough to complete the

DIST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFIC A Tl ON 333

classification. The list of exceptions is more extensive and interesting in this case. The classification was obtained in Weiss (1985 c).

Theorem 5.2.4. Let r be a graph of valency k ~ 3 and girth g. Let G act stransitively on r, s ~ 4, s ~ (g - 2) I 2. Let a = I V (r) I and F = [ G, G], the commutator

subgroup of G. Then either s = (g + 2) I 2 and conclusion (i) of Lemma 5.2.3 is valid or one of the following holds:

(i) k = 3 , s = 4, g = 8 , a = 30 and F = A 6;

(ii) k = 3, s = 4, g = 9, a = 102 and G:: PSL2(?);

(iii) k = 3, s = 4 or g = 10, a = 90 and F = 3· A 6;

(iv) k = 3, s = 5, g = 12, a= 243 and G:: Aut(PSL3(3));

(v) k =3, s =5, g = 12, a=486andG =Aut(GL3(3));

(vi) k = 3, s = 5, g = 12, a= 650 and G:: Aut(PSL2(25));

(vii) k = 4, s = 4, g = 10, a = 440 and F = M 12;

(viii) k = 4, s = 4, g = 10, a = 880 and F = 2· M 12;

(ix) k = 5 , s = 4, g = 9 , a = 17 442 and F = h; (x) k = 5 , s = 4 , g = 9 , a = 52326 and F = 3 x 13 •

In each case r is uniquely determined. D

Here 3 • A6 and 2 · M12 denote the nonsplit extensions of As and M12 by centers of order 3 and 2 respectively.

5.3. DTG's admitting elations

If G acts s-transitively on a graph r and s ~ 4, then for an edge {x,y} the subgroup G 1 (x,y) is non trivial. Here G 1 (x,y) denotes the largest subgroup of G which stabilizes each vertex which is adjacent either to x or to y. On the other hand, it is clear that if G 1 (x,y) = 1 then the Order of G X is bounded in terms of the valency k of r. A nontrivial element of G 1 (x,y) is called an elation. So a natural next step in the classification of s

transitive DTG's is the classification of graphs admitting elations.

The following lemma is a direct consequence ofresults from Weiss (1978, 1981).

Lemma 5.3.1. Let G act s-transitively on a graph r, 2 ~ s ~ 3, and suppose G 1 (x,y) # 1. Then PSLn(q) ~ ar<x>~ PrLn(q) for n ~ 3. D

Let us present the list of DTG's which ares-transitive for 2 ~ s ~ 3 and which admit elations. Here n is the parameter from Lemma 5.3.1 and g is the girth of r. A detailed account of the properties of these graphs can be found in Brouwer et al. ( 1989).

I. The incidence graph of points and hyperplanes of a rn-dimensional vector space over GF(q), n = m -1, s = 2, g = 4.

334 A. A.IVANOV

li. The incidence graph of rn-dimensional and (m +!)-dimensional subspaces of a (2m+ !)-dimensional space over GF(q), n = m + 1, s = 3, g = 6. This graph is referred to as the q-analog of the double covering of the odd graph.

III. The graph whose vertices are the maximal isotropic subspaces of a 2n

dimensional space equipped by a nonsingular form having Witt index n ; s = 2 , g = 4.

IV. The graph f'(M22) and its triple covering 1(3· M22). Here f'(M22) is the graph whose vertices are the blocks of the Steiner system S(5,8,24) containing neither of two fixed points; two vertices are adjacent if the blocks are disjoint. These graphs are DTG's

of valency 7 (see Section 4.4). They have, respectively, 330 and 990 vertices and intersection arrays

{7,6,4,4; 1, 1, 1,6} and

{7,6,4,4,4, 1, 1,1; 1, 1, 1,2,4,4,6,7}.

Here n = 3, s = 2, g = 5. The graph f'(M 22) was introduced in Biggs (1975) and the graph 1(3 · M 22 ) was constructed in Ivanov et al. (1984a).

The following theorem was proved in lvanov ( 1989).

Theorem 5.3.1. Let f' be a DTG which is s-transitive for s = 2, 3 and suppose Aut(f') contains elations. Then f' is one of the graphs in I-IV above. o

The proof of Theorem 5.3.1 can be carried out separately for different values of the girth g.

The case g = 4 was treated completely in Cameron & Praeger (1982) where the following result was proved.

Proposition 5.3.2. Let f' be a graph of girth 4, let G act s-transitively on f', s ~ 2, and suppose PSLn(q) ~ af<x) ~ Pf'Ln(q), n ~ 3, and G 1 (x,y) ;~; 1, then f' is isomorphic either to a graph of type I or to a graph of type III. 0

The case g = 5 under analogous assumptions was treated in lvanov (1986b, 1987b).

Proposition 5.3.3. Let f' be a graph of girth 5, let G act s-transitively on f', s ~ 2, and suppose PSLn(q) ~ af<x> ~ Pf'Ln(q), n ~ 3 and G 1 (x) ;~; 1. Then G 1 (x,y) ;~; 1, q = 2 and there exists a geometry G with the diagram

Pm: o--~p---o-------o- -o------o 1 2 2 2 2

(m + 2 vertices)

such that G acts flag-transitively on G. The graph f' is uniquely deterrnined by the geometry G. 0

DISTANCE-TRANSITIVE GRAPHSAND THEIR CLASSIFICATION 335

The graphs r(M 22 ) and r(3· M 22 ) satisfy the hypothesis of Proposition 5.3.3. The corresponding geometries can be described as follows. Let r be the graph r(M 22 ) or r(3 · M 22). Then the elements of G having type 0, 1 and 2 are the totality of vertices, edges and Petersen subgraphs of r. G is described by the diagram P 1.

The graphs of type I, II and III are also closely related to certain geometries.

The elements of the geometry An(q) are all proper subspaces of an (n + 1)dimensional vector space over the field GF(q); the type of an element is its dimension minus one (the projective dimension); the incidence of elements is deterrnined by inclusion. The geometry An(q) has the following diagram.

A : o-------o-------0- -0----0 n q q q q q

The elements of the geometry Dn(q) areallproper totally isotropic subspaces of an 2n-dimensional vector space over GF(q) equipped by a nondegenerate quadratic form of Witt index n. In this case the maximal totally isotropic subspaces have dimension n and are divided into two classes in such a way that only subspaces from different classes can have a subspace of dimension n - 1 in common. The elements of type (n - 1) and (n - 2) of the geometry Dn(q) are the two classes of maximal totally isotropic subspaces; for i < n - 2 the elements of type i are totally isotropic subspaces of dimension i + 1. The incidence of elements x and y of this geometry is defined as follows. If t(x) < n- 2 or

t(y) < n- 2 then incidence is deterrnined by inclusion; if t(x) = n- 1, t(y) = n- 2, then x and y are incident if they have a subspace of dimension n - 1 in common. The diagram of this geometry is presented below:

0

D : o-------on q q

/q -o------o

q~ 0

q

q

Let G be an arbitrary geometry and i,j be two types in this geometry. Let r;,j(G) denote the graph whose vertices are the elements of type i and j of the geometry G. Two vertices are adjacent if they are of different type and are incident in G; in particular r;.j(G) is a bipartite graph.

In terms of the introduced notation the graphs of type I, II and III are just

ro,n-1 (An(q)), r m-1,mCA2m(q)) and rn-2,n-1 (Dn(q)), respectively.

The geometries An(q), Dn(q), G(M 22) and G(3· M 22) are characterized in terms of their diagrams (the latter two in the class of ftag-transitive geometries).

336 A. A.IVANOV

Proposition 5.3.4. (Tits (1981)) Each geometry with diagram An, n:::: 3, is iso-

morphic to a geometry An(q) for suitable q. 0

Proposition 5.3.5. (Tits (1974), Timmesfeld (1983)) Each geometry with diagram

Dn, n:::: 3, is isomorphic to a geometry Dn(q) for suitable q. 0

Proposition 5.3.6. (Shpectorov (1985)) Each fiag-transitive geometry with diagram

P 1 is isomorphic either to G(M 22) or to G(3· M 22 ). 0

In view of these characterizations, to prove Theorem 5.3.1 it is sufficient to con

struct a geometry with the diagram An, Dn and P 1 from the graphrunder consideration.

Precisely this strategy is employed in Ivanov (1989). As elements of the geometry certain

subgraphs of rare considered. We define these subgraphs as connected components of

subgraphs induced by the vertices fixed by some special subgroups of the vertex stabil

izer. So it is just the fixed points method in the classification of DTG's considered in Sec

tion 4.6.

It should be mentioned that in the case g = 4 we have not carried the geometric

scheme of classification to fruition, but have instead used Proposition 5.3.2. At the same

time it is our opinion that an independent geometric proof of Proposition 5.3.2 would be quite interesting.

The fact that geometries with diagram pm arose in the classification of DTG's

which admit elations is quite intriguing. Up to now six examples of such geometries are

known. All the examples are related to sporadic simple groups. Besides the geometries

G(M 22) and G(3· M 22) with diagram P 1 mentioned above, there are geometries

G(M23), G(Co2) and G(J4) with diagram P 2 and G(F2) with diagram P 3 . A descrip

tion of these geometries and some of their properties can be found in lvanov & Shpec

torov (1988, 1989 a). The complete classification of geometries having diagram pm is

not yet available. This classification seems to be a hard problem. Nevertheless, it is

shown in Ivanov (1989) that the graph related to a geometry with diagram pm can be

distance-transitive only if m = 1. The lauer case is covered by Proposition 5.3.6.

5.4. A characterization of the odd graphs

We continue consideration of DTG's which ares-transitive for s:::: 2. After the case

of elations it is natural to study the situation G 1 (x,y) = 1 , G 1 (x) "# 1. This means that r

does not admit elations but Gx in its action on r(x) still has a nontrivial kernel. This

situation is realized in the odd graphs Ok andin the k-fold antipodal covers of the com

plete bipartite graphs Kk,k related to the Desarguesian projective planes. The odd graphs

were characterized in Praeger (1980) (see Lemma 5.1.4) under the assumption that the

girth of r is 6 and the subconstituent or<x) is either the symmetric or the alternating

group of degree k. In this section we present some methods which enable us to extend the

result of C.E. Praeger on a larger class of subconstituents.

So suppose that H = Gi<x> is a doubly transitive permutation group, G 1 (x,y) = 1,

G 1 (x) :# 1. In view of Propositions 5.3.2, 5.3.3 we will assume that g ~ 6. Our first goal is

to show that under certain assumptions on the group H the graph r contains 3-

apartments, i.e. a family K of cycles with the property that each 3-path is contained in a

unique cycle from the family. In lvanov & Shpectorov (1991b) this family is proved to

exist in the case when H is strongly doubly-primitive, i.e. when the point stabilizer in H

acts primitively on the remaining points and has no regular normal subgroups. Here we

give another scheme of proof.

For y e r(x) put M =GI,~>, L = G 1 (xl(y). Since G 1 (x) ..:;;! Gx,y• it is clear that

L ..:;;! M. Using some standard arguments from the theory of s-transitive graphs it is easy to

see that L is nontrivial. Let C=CG.(G1(x)) and K=cr<x>. Since G 1(x)'.5lGx,y•

G 1 (y) '.5l Gx,y and G 1 (x) n G 1 (y) = G 1 (x,y) = 1, the subgroup G 1 (y) is contained in C.

On the other band, it was mentioned above that G 1 (y) acts nontrivially on r(x ). Thus K

is a normal subgroup of H which is not regular. Let N be the minimal normal subgroup of

H. Then K ~ N and K > N if N is abelian. It is easy to check that the following equalities

hold:

HIK:: Gxl< C, G1(x) >:: Gx,y I< CG._,(G1(x)), G1(x) >::

=MI <CM(L), L >.

So we have the following:

Lemma 5.4.1. If G 1 (x,y) = 1 and G 1 (x)-:;:. 1, then the subconstituent H = Gi(x) con

tains a non-regular normal subgroup K such that H IK =MI <CM(L), L>, where

M =Gf.~>,L = G 1(xl(y>. o

ties: Hypothesis 5.4.2. Suppose that C contains a subgroup D =D(x) with the proper-

(a) D is normal in Gx;

(b) D acts transitively on r(x);

(c) D n Gx,y $. G1(y).

Let us indicate two cases when such a subgroup D exists:

Case 1. IfCM(L)= 1, then wecan putD = C.

Case 2. Let the minimal normal subgroup N of H be elementary abelian of order p n

and suppose that p does not divide I G 1 (x) I. Then we can take for D the Sylow p

subgroup of the preimage of N in Gx.

Now suppose that a subgroupD =D(x) possessing properties (a)- (c) is chosen. For

each vertex z e V(r) Iet us fix an element Kz e G which maps x onto z. Put

D(z) = g;1 D(x)g2 • Since D ..:;;! Gx the subgroup D(z) does not depend on the particular

choice of Kz·

338 A. A.IVANOV

It follows from properties (b), (c) that each orbit 0 in the action of Don r2(x) has

length k and I 0 n r(u) I = 1 for all u e r(x ). This fact enables us to define the family K

of3-apartments by the following rule: ( ... ,xi_1,xi-t.Xi,Xi+1,xi+2•···) e Kif and only if

Xi_2 andxi+2 are contained in the same orbit of D(xi) on r2Cxi).

It follows directly from the definition that the constructed family K possesses the

uniqueness and exchange conditions (cf. Delgado et al. (1985), Ivanov & Shpectorov

(1991b)). Existence of such a family imposes very strong restrictions on the structure of

the amalgam A={Gx,G{x,yl}. Namely the following Iemma holds (cf. Proposition

(3.6) in Delgado et al. (1985)).

Lemma 5.4.3. Let G act s-transitively (s ~ 2) on a graph r of valency k and suppose

that r contains a family K of 3-apartments which possesses the uniqueness and the

exchange conditions. Then there is a group F acting s-transitively on the complete bipar

tite graph Kk,k such that the amalgam A = { Gx , G {x,y 1 } is isomorphic to the amalgam

B = {F u , F {u. v 1 } where u, v are adjacent vertices of Kk.k· o Suppose that the minimal subgroup N of the subconstituent His a nonabelian group.

Since G 1 (x) '* 1, it follows from the above Iemma that F contains the direct product

N xN. This implies, in particular, that s = 3 and G 1 (x) is transitive on r(y)- {x} except

for the case k = 28, H = PrL2(8). In the latter case N is not doubly transitive.

Now Iet W = (x =x o. x 1o •.• , x1 =x) be an apartrnent from K of length l. It is clear

that l does not depend on W if G acts 3-transitively on r. Let us study the possibilities for

I.

Let G {WI and Gw be the setwise and the elementwise stabilizers of W in G, respec

tively. By definition Gw coincides with the elementwise stabilizer of any 3-path in W.

For 0 ~ j ~ I -1 put t1j = r(xi)- W. Let us define a relation - between !:1i and !:1j+2 by

the following rule. For u e t1j there is a unique cycle W 1 in K which contains the 3-path

(u, Xj,Xj+t.Xj+2). Ifv is the vertex adjacent toxj+2 in W 1o v *-Xj+lo then u- v. It is clear

that- establishes a bijection between t1j and !:1j+2 . If we extend the relation- by transi

tivity then we obtain a bijection between !:1i and t1j whenever W contains a path of even

length joining xi and Xj· In this case it is clear that an element from Gw acting trivially on

!:1i acts trivially on !:1i too. Suppose that G 1 (x) n Gx 2 '* 1. Then there is an element

h e Gw ~ G 1 (x) n Gx 2 which acts trivially on 1:10 but nontrivially on 1:11 , and we have

the following

Lemma 5.4.4. If G 1 (x) n Gx2 *- 1 for x 2 e r 2 (x) then the length of each apartrnent

from K passing through x and x 2 is even. 0

5.5. Perspectives in the classification of s-transitive DTG's

In this section we propose a classification scheme for s-transitive DTG's (s~ 2) with

the property G 1 (x) = 1. This scheme relies on the fixed points method and the application of group amalgams.

Let x e V(r), y,z e r(x), F = Gx.y,z· Let I: denote the connected component of the subgraph induced by the vertices fixed by F such that x e I:. Let M be the setwise stabilizer of I: in Na(F). Since G acts 2-transitively on r, the subgroup Mx induces on I:(x) a doubly transitive Frobenius group. We will partition the generalproblern into a nurober of subcases depending on the structure of the subgraph I:.

I. I:= r. In this case H = ai<x> is a doubly transitive Frobenius group and since G 1 (x) = 1, I Gx I = k(k -1). In particular, ki divided k(k -1) for a1l 1 S: i S: d. This enables one to obtain a bound on the diameter of r. For instance, if the girth of r is at least five then k3 S: k2 and d S: 9 by Corollary 3.7.2.

II. I: is a star. This case is characterized by the equality I:(y) = (x}. In this case H should satisfy certain additional properties. In fact, Iet 't e G {x,y} - Gx.y· Then in the considered situation the subgroups Gx.y,z and Gi,.,,z are not conjugate in Gx,y· This implies that the point stabilizer H., of y in the subconstituent H should have an automorphism 't whose square is an inner automorphism such that Hy,z is not conjugate to JP,.z in H,.

For many doubly transitive permutation groups such an automorphism 't does not exist For instance if H = S1 or A1 then it exists only if k = 7. In the latter case 't is an outer automorphism of S 6· Notice that the fact of whether or not I: is a star is determined by the structure of the rank 2 amalgam of the subgroups Gx and G {x.y 1.

As a first step in the classification of DTG's corresponding to this subcase one can consider the following problem, which has independent interest.

Problem 5.5.1. Describe the rank 2 amalgams corresponding to 2-transitive actions such that I: is a star.

There are infinite families of amalgams with the prescribed property, for instance Gx =AGLn(q), n :=:: 4, Gx.y = GLn(q) where 't is the contragredient automorphism. Nevertheless, we believe that this class of amalgams is not too large.

Below we list the known examples of s-transitive DTG's with G 1(x)= 1 and such that I: is a star. We also list the intersection arrays of the graphs as weil as the structure of subgroups G , Gx, Gx.y• G {x.y 1, Gx.y,z· For some graphs there are a couple of groups with the claimed properties. We give only one ofthem.

1. The Hoffman-Singleton graph: i(r)={7,6;1,1}; G=PI:U3(5), Gx=S7, Gx.y = S6, G{x.yl = Aut(S6), Gx.y,z =Ss. Notice that here s = 3 and 3-apartments exist

2. A bipartite covering of the complement to the Hoffman-Singleton graph: i{r)= {15,14,10,3; 1,5,12,15}; G =PW3(5), Gx =A7, Gx.y =PSL2(1), G{x.yJ =PGL2(7), Gx.y,z =A4.

3. The Higman-Sims graph: i(r)= {22,21; 1,6}; G =HS, Gx =M22 ,

Gx.y =PSL3(4), G{x.y) =PSL3(4). <'t>, Gx.y,z =24·As. Here 't is an automorphism of PSL3(4) which permutes the two classes of maximal parabolic subgroups.

340 A.A.IVANOV

4. The distance two subgraph in the Higman-Sims graph: i(I)= (16,15; 1,4}, G =Aut(M22), G" =24• S6, Gx,y =s6, Gtx,yl =Aut(S6), Gx.,y,z =s4 xS2.

5. A graph related to M 23 : i(I) = (15,14,12; 1,1,9}; G =Mn, G" =As. Gx,y:: 23 • PSL3(2), G (x.,y):: 23 • PSL3(2)· <'t>, Gx.,y,z:: 22• S4. Here 't is an automorphism of the group 23 • PSL 3(2) which permutes the two classes of complements isomorphic toPSL3(2).

In our opinion this list is rather curious and it should stimulate interest in the classification problem.

We will assume below that l: contains a cycle. Then F fixes a vertex in r(y)- (x} and the subgroups Gx.,y,z and Gi.y,z are conjugate in Gx.r So the element 't can be chosen in such a way that Gi,y,z = Gx,y,z· Then 't e M and M acts 2-transitively on l:. Our ensuing strategy will depend on the diameter of l:.

ill. The diameter of l: is at least 3. Then F stabilizes a vertex in r 3 (x ). This implies thatk3 ~ [G" :F] =k(k-1). So either the girth ofris4or k3 ~ k 2 and the diameterofr is at most 9.

N. The diameter of l: is 2. If the valency of l: is greater than 2, then l: is a nontrivial rank 3 graph with a doubly transitive subconstituent. Moreover this subconstituent is a doubly transitive Frobenius group, so the structure of l: is rather restrictive. This situation is realized in a DTG on 280 vertices with intersecnon array {9,8,6,3; 1, 1,3,8}, G = Aut(PSL3(4)), l: the Petersen graph (see Tchuda (1990)).

Now we shall assume that l: is either the pentagon or the quadrangle. So there is a family K of cycles in r (the images of l: under G) such that each path of length 2 is contained in a unique cycle from K. If K consists of pentagons then r is called a pentagraph (see Perkel (1979)) and if K consists of quadrangles then r is a rectagraph (see Neumaier (1982)). LetAdenote the amalgam of the subgroups Gx, G (x,y 1 and G (I:). Notice that this amalgam determines the length of l:.

V. Pentagraphs. In this case the existence of the amalgam A forces certain conditions on the structure of the subconstituent H.

If H = S~c or At then the amalgam A exists and it can be realized in a certain Coxeter group (see lvanov (1987 c)). But as follows from Lemma 5.1.3, the corresponding pentagraphs are not distance transitive for k ~ 6.

In Perkel (1979) the pentagraphs with PSL 2(q) ~ H, k = q + 1 were studied. The following result was proved in that paper.

Lemma 5.5.2. Let r be a pentagraph and G act 2-transitively on G. Suppose that H = Gf<"> contains a normal subgroup isomorphic to the natural doubly transitive representation ofthe group PSL2(q) of degree q + 1. Then q e (2,3,4,5,9}. D

If q is not contained in the above list then the amalgam A corresponding to a cycle oflength 5 does not exist (see Section 3.2 in Ivanov & Shpectorov (1991b)).


We believe that for other classes of doubly transitive groups (different from S1e and A~c) the existence of pentagraphs is more the exception than the rule.

Notice that there exists a pentagraph with subconstituent isomorphic to the representation of P SL 2 ( 11) with degree 11. This is the DTG related to the sporadic group J 1 (see Perkel (1980) and Section 4.6 ofthe present work).

V. Rectagraphs. The central example of rectagraphs is the k-dimensional cube Qk.

The automorphism group of the cube is the Coxeter group C~c. The latter is isomorphic to the semidirect product of the elementary abelian group E = 2k and the symmetric group S1c. Here E can be considered as the permutational GF(2)-module ofthe natural representation of S~c. The following result (see Neumaier (1980)) can be considered as a characterization of Q~c.

Lemma 5.5.3. Let r be a rectagraph of valency k. Then I V(r) I ~ 2k with equality attained only if r = Q~c. o

For each doubly transitive permutation group H of degree k there exist a group G acting 2-transitively on Qk suchthat the subconstituent is similar to H. Indeed, for Gone can take the semidirect product of E by the group H where the latter is considered as a subgroup of S~c.

Let r be an arbitrary rectagraph, G act 2-transitively on r and Iet A be the corresponding rank three amalgam. Then it follows from Lemma 5.5.3 that the universal group F(A) of the amalgam has order at most I Gx I· 2k and that the bound is attained if and only if F(A) acts 2-transitively on Q~e. So there are two types of amalgams: those which are embed in the group of the k-dimensional cube and those which do not.

As was indicated above, the class of embedded amalgams is rather !arge and examples exist for all subconstituents. Below we concentrate mainly on the embedded amalgams. W e believe that nonembedded amalgams are related, for the most part, to certain exceptional configurations. As an example we mention the group PSL3 (4) acting on the cosets of A 6 as a group of automorphisms of the Gewirtz graph.

Notice that if the amalgam A embeds in the group of Qk then the homomorphism F(A) ~ G induces a covering Qk ~ r. So the considered subproblern gives rise to a description of DTG's which are covered by the cube.

First Iet us consider the structure of the amalgams. Let H be a fixed doubly transitive permutation group. We are interested in groups acting 2-transitively on Qk and having H as a subconstituent. One of these groups is the semidirect product of E by H. Let us describe the other ones. The symmetric group S1c acting on E preserves two proper invariant subspaces E o and E 1 of dimension 1 and k -1, respectively. Let F be an overgroup of Hin S1e which contains H as subgroup of index 2. Then there are exactly three groups between E 1AH and E"AF. Two of them are EAH and E 1f...F. The third one will be called the diagonal subgroup.

Lemma 5.5.4. Let G be a group acting 2-transitively on Qk and having H as a sub-

342 A. A.IVANOV

constituent Then either G = EAH or G is the diagonal subgroup of the group EAF where Fis a subgroup of Sk containing H as a subgroup of index 2. 0

The next step is the description of all groups containing a given embedded amalgam. This problern can be reduced to a description of the invariant subspaces of W = E 1 1 (E 1 11 E o) where the lauer is considered as an H-module. If H = Sk or Ak then W is irreducible, but for other groups this may not be the case.

Fora realization of this step the paper Mortimer (1980) should be very useful. For almost all doubly transitive permutation groups H it is determined in that paper whether or not W is H-irreducible. Nevertheless, in many cases the complete list of invariant subspaces is unknown. As follows from Praeger (1988), the description problern for maximal invariant subspaces is equivalent to the classification of primitive permutation groups with regular abelian normal subgroup and doubly transitive subconstituent.

The invariant subspaces corresponding to DTG's should satisfy certain additional conditions. Up to now the case PSLn(q) <5JH ~ PrL2(q), n ~ 3, has been treated completely (see Ivanov (1988)).

Theorem 5.5.5. Let G be a group acting distance-transitively on a graph r and let G =NAH where N is an elementary abelian regular normal p-subgroup. Suppose that PSLn(q) ".5J H ~ PrLn(q), n ~ 3, and that H acts on r(x) as it acts on the set of points of the projective space PG (n - 1, q ). Then p = 2 and either

(i) q = 2 , r is the complete graph on 2n vertices (resp., a double cover of this graph), G = 2n APSLn(2) (resp., 2n+1 APSLn(2));

(ii) q = 4, r is the graph Her(n, 2) of Hermitian forms of dimension n over GF(22),

or one of the two distance-transitive antipodal covers of Her(3, 2). 0

The graphs Her(n,q) and their antipodal covers are described in Brouwer et al. (1989).

6. A global approach to the classification

6.1. A characterization of the Hamming graphs

In this chapter we discuss a classification scheme for DTG's which we call "a global approach". This scheme consists of (1) a description of the class of groups which can act distance-transitively on graphs; (2) a determination of the distance-transitive representations (DTR's) of the groups in this class; and finally (3) construction of the DTG's corresponding to each of the DTR's.

One of the advantages of this approach is that it enables one to make effective use of the classification of finite simple groups. Nowadays this approach is considered to be the most promising. In fact, it appears that the complete classification of DTG's will be obtained in this manner (see also the survey paper van Bon & Cohen (1988)).


In a rudimentary form this approach was already used in the solution of the problern

by N. Biggs (see Section 4.2). The first systematic result within this approach is the

classification of the DTR' s of the symmetric and altemating groups (Saxl ( 1981 ), Liebeck

et al. (1987) and lvanov (1986 c)). Nevertheless we stan with the paper Praeger et al.

(1987) where the class of groups which can have a primitive DTR was restricted essen

tially.

Suppose that G acts distance-transitively on a graph r and that the permutation

action of G on V (r) is imprimitive. Then by the results presented in Section 2.2, if we

consider in G a subgroup of index 2 and/or factorize over a normal p-subgroup, we

obtain a group acting distance-transitively on a primitive DTG. So in the first step one

can restriet oneself to primitive DTR's. The problern of determining those imprimitive

DTG's which correspond to a given primitive one will be discussed in Section 6.9. It

should be mentioned that sometimes this problern is rather non trivial.

So let G act distance-transitively on a graph r and suppose that Ginduces on V(r) a

primitive permutation group. An extremely important structural theorem about primitive

permutation groups was proved by M. O'Nan and L. Scott. This theorem was presented

in Cameron ( 1981 ). Nowadays it has numerous applications to the theory of permutation

groups and related topics. A corrected and improved version of the theorem, with a self

contained proof, was given in Liebeck et al. (1988).

Theorem 6.1.1. Any finite primitive permutation group is permutation equivalent to

one of the following types:

I. Affine groups;

Il. Almost simple groups;

III(a). Simple diagonal action;

III(b). Product actions;

III(c). Twisted wreath actions.

Let us give a brief description of these types.

0

Let X be a primitive permutation group on a finite set n of size n, and a be a point

in n. Let B be the socle of X, that is, the product of all minimal normal subgroups of X.

Then B = ym with m 2::: 1, where T isasimple group.

I. Affine groups. Here T = Zp for some prime p, and Bis the unique minimal normal

subgroup of X. In this case B is regular on Q of degree n = p m. The set Q can be

identified with B = pm so that Xis a subgroup of the affine group AGLm(p) with B the

translation group, andXa =X n GLm(p) is irreducible onB.

II. Almost simple groups. Here k = 1 , T is a nonabelian simple group and T ~ X ~ Aut T, Ta ~ 1.

III. In this case B = ym with m 2::: 2 and T is a nonabelian simple group. The struc

ture of such groups is much more complicated and we refer the reader to Liebeck et al.

( 1988) for details. Here we just mention that n = I T I m-l in the case III(a); n = n& (I > 1)

344 A.A.IVANOV

in the case III(b), where n0 is the degree of a primitive permutation group having type li or III(a); n = I T Im in the case III(c).

As we mentioned above the groups of type III are the most complicated. Nevertheless the complete classification of DTR's for precisely these groups was obtained in Praeger et al. (1987). Namely it was shown that a primitive group having type III(a) or III(c) cannot be a DTR. If a DTR has type III(b) then the corresponding DTG is either the Hamming graph H(d,q) or the complementary graph of H(2,q). Recall that the vertices of the Hamming graph H(d,q) are the words in a fixed alphabet having q letters, with two words are adjacent if they differ in exactly one coordinate. The automorphism group of H(d,q) is the wreath product Sq wr Sd of the symmetric groups. This graph is primitive if q;;:: 3. If q ~ 4 then Sq wr Sd has a regular normal p-subgroup (so it corresponds to type I).

Theorem 6.1.2. Let G act distance-transitively on a primitive DTG r of valency k ;;:: 2 and diameter d;;:: 2. Then one of the following holds:

(i) r is the Hamming graph H(d,q), d;;:: 2, q;;:: 5, or the complementary graph of H(2,q);

(ii) G is almost simple;

(iii) G is affine. 0

As a consequence of this theorem the classification of primitive DTG's was reduced to consideration of the almost simple and affine cases. In the former case one uses the classification of the finite simple groups in order to study DTR's of the known nonabelian simple groups and of their extensions by outer automorphisms.

6.2. The symmetric and alternating groups

As was mentioned in the previous section the first systematic result within the global approach to the classification of DTG's is the classification of DTR's of the symmetric and altemating groups.

The main examples of graphs admitting distance-transitive action of the symmetric group Sn are the Johnson graphs J(n,k), k ~ n/2. lf Qn is a fixed set of cardinality non which Sn acts in a natural way, then the vertices of J(n,k) are the k-element subsets of Qn· Two vertices are adjacent if the intersection of the corresponding subsets has size k- 1. If n = 2k + 1 then the association scheme of J (n,k) has an additional metric structure corresponding to the odd graph Ok+l· If n = 2k then the graph J(n,k) is antipodal. The folded graph of J(2k,k) is called the even graph and is denoted by Ek. The vertices of Ek can be considered as partitions of Qn into two parts of equal size. For small values of n there areadditional examples of DTR's of Sn.

A very important step in the description of DTR's of the symmetric group was accomplished in Saxl (1981) where the multiplicity-free permutation representations of

Sn for n > 18 were described.

Theorem 6.2.1. Let Sn act multiplicity-freely on cosets of a subgroup H. If n > 18 then one of the following holds:

(i) An-k X Ak ~ H ~ Sn-k X sk for 0 ~ k < n 12;

(ii) n = 2k , Ak x Ak < H ~ (Sk wr S 2);

(iii) n = 2k , His a subgroup of (S 2 wr Sk) of index at most 4;

(iv) n = 2k + 1, H fixes a point in Qn and acts as in (ii) or (iii) on the remaining points;

(v) An-k xHk~ H~ Sn-k xHt. where k = 5,6 or 9 and Hk is isomorphic to F~, PGL 2 (5) or PrL 2(8), respectively. o

Since aDTR is multiplicity-free (see Section 2.4), to classify the primitive DTR's of the symmetric group it is sufficient to consider the action of Sn on k-element subsets of Qn (case (i) of the above theorem) as well as on partitions of Qn into r subsets each of size m (where n = mr); and to investigate the case of small n (n~ 18). Moreover in consideration of the partitions we can assume that either r = 2 (case (ii) of Theorem 6.2.1) or m = 2 (case (iii)).

The classification of DTR's of Sn was obtained independently in Liebeck et al. (1987) and Ivanov (1986 c). The concrete problems solved in that papers are slightly different. Namely, in Liebeck et al. (1987) the primitive graphs which admit action of groups G satisfying Ak ~ G ~ Aut(Ak) were described. This result comprises a part of the almost simple case in Theorem 6.1.2. In lvanov (1986 c) all DTR's (including imprimitive ones) of the symmetric groups were classified. This work was stimulated by a question proposed in Biggs (1979). Despite the difference in the aims, the arguments in these papers are rather similar.

In view of the known examples of DTR's of the symmetric groups described above, to study the cases (i) and (ii) in Theorem 6.2.1 one should describe all metric structures of the Johnson scheme J(n,k) and of the folded scheme of J(2k,k). This problern is rather technical since the structure constants of J(n,k) are well known (see Theorem 2.8.1). In case (iii) one should prove that there are no corresponding metric structures. By Lemma 2.3.2 (ii) it is sufficient to consider only 2-orbits corresponding to the minimal and second to minimal subdegrees. Here the central argument is to show that certain structural constants are nonzero. It turns out that a metric structure exists only for k = 3. This exception is due to the existence of an outer automorphism of the group S 6 • In the study of the small values of n some ad hoc arguments were used. In Ivanov (1986 c) computer calculations based on algorithms described in lvanov et al. (1982) were applied at this stage. For the description of imprimitive representations detailed information about the primitive ones, as well as results presented in Section 2.2, were used.

From the combined results ofthe papers Liebeck et al. (1987) and lvanov (1986 c) it is easy to deduce the complete description of DTR's of groups satisfying An ~ G ~ Aut(An).

346 A. A. IVANOV

In view of Lemmas 2.8.2, 2.8.3 and the fact that Aut(An) =Sn for n "# 6, it is reasonable to expect that to each DTR of a group under consideration, a DTR of Sn is associated. We begin with a description of the exceptional situations and then formulate the result for the symmetric groups. The cases of primitive and imprirnitive representations of Sn will be described separately.

Theorem 6.2.1. Let An, n;;:: 5 act distance-transitively on a graph r, and let H be the stabilizer of a vertex x E V(r) in this action. If Aut(r) does not contain Sn then one of the following holds:

(i) r is the complete graph K 1s with either n=7, H=PSL3(2), or n = 8 , H = 23 • P SL 3 (2), respectively;

(ii) n = 9, H = PrL 2 (8), r is a graph of valency 56 on 120 vertices or its complement, Aut(r) = Ot (2);

(iii) r is the dodecahedron or the icosahedron. 0

Theorem 6.2.2. Let Aut(A 6) act distance-transitively on a graph r, and let H be the stabilizer of a vertex. If the action of S 6 on r is not distance-transitive then r is one of the following:

(i) the complete bipartite graph K 6,6 , H = S 5;

(ii) the incidence graph of the generalized 4-gon related to the group Sp 4 (2) = S 6 ,

H=S4 xS 2 ;

(iii) the line graph of the graph in (ii), Ha Sylow 2-subgroup in Aut(A 6 ) of order

(iv) a rank 4 graph of valency 5 on 36 vertices, H = F~. 0

Theorem 6.2.3. Let Sn , n;;:: 4, act prirnitively and distance-transitively on a graph r and let H be the vertex stabilizer. Then r is one of the following:

(i) the complete graph Km where n = m, H = Sn-1> or (n,m) = (5,6), H = Ft or (n,m) = (6, 10), H = (S 3 wr S 2), respectively;

(ii) the Johnson graph J(n,k), 1 < k < n/2, H = Sn-k x Sk;

(iii) the odd graph ok+l ' n = 2k + 1' k;;:: 2' H = sk+l X Sk;

(iv) the even graph Ek, n = 2k , k;;:: 4, H = (Sm wr S 2);

(v) the complementary graph of J(n, 2), n;;:: 6, H = Sn_2 x S 2: (vi) the complementary graph of Ek, k = 4,5, H = (Sk wr S 2).

In each case the action of Ak is also distance-transitive. o

Theorem 6.2.4. Let Sn , n;;:: 4, act distance-transitively on an imprirnitive graph r and let H be the stabilizer of a vertex. Then r is one of the following:

(i) the bipartite doubling 2. Km of the graph Km where n = m, H = An-l, or (n,m) = (5,6), H = D 10· or (n,m) = (6, 10), H = (S 3 wr S 2)+, respectively;

(ii) the Johnson graph J(2k,k), k;;:: 3, H = Sk x Sk;


(iii) the bipartite doubling 2. ok+l of the graph ok+b k ~ 2' n = 2k + 1,

H = (Sk-l xSkt;

(iv) the subgraph induced by the vertices at distance two from a fixed vertex in the Hoffman-Singleton graph, n = 7, H = PGL 2 (5);

(v) the incidence graph of the 2-(15,7,3) design or of its complement, n = 7, H = PSL 3(2), or n = 8, 23 • PSL 3(2), respectively;

(vi) the line graph of the Petersen graph, n = 5, H = D s;

(vii) the octahedron, isomorphic to 1(4,2). The action of An is distance-transitive only in the cases (ii) and (iv.) 0

Using Section 5 in Ivanov (1986 c) and the results from van Bon & Brouwer (1988)

the following result can be proved.

Theorem 6.2.5. Let G act distance-transitively on an antipodal graph r and let N be the kernel of the action of G on the folded graph. Suppose N :F- 1 and An !5: GIN !5: Aut(An)· Then one of the following holds:

(i) N = Z 2 , G ~ Zz xAn and r admits a distance-transitive action of Sn or An;

(ii) N = Z 3 , n = 6, 7 , G contains the nonsplit central extension of An by Z 3 and r is an antipodal triple cover of one of the following graphs:

(a) the complete graph K 6;

(b) the complete bipartite graph K 6,6;

(c) the complementary graph of J(n, 2), n = 6 or 7;

(d) the incidence graph ofthe generalized 4-gon related to Sp 4 (2). 0

6.3. The linear groups

After studying the symmetric and alternating groups it is natural to study the case of linear groups, that is, of groups G satisfying PSLn(q) ~ G !5: Aut(PSLn(q)).

Here the main examples are the Grassman graphs which are also known as qanalogs lq(n,k) of the Johnson graphs. Let Vn(q) be an n-dimensional vector space over the field GF(q). Then the vertices of lq(n,k) are the k-dimensional subspaces in Vn(q);

two vertices are adjacent if the intersection of the corresponding subspaces has dimension k -1. The group PSLn(q) acts distance-transitively on this graph and the full automorphism group of lq(n,k) is PrLn(q) if n :F- 2k and Aut(PSLn(q)) otherwise. If n = 2k + 1 then lq(n,k) is the halved graph of a bipartite graph known as the q-analog of the double covering of the odd graph and is de'loted by Ok(q). The vertices of this graph are the k-dimensional and the (k + 1 )-dimensional subspaces of Vn (q ). Two vertices are adjacent if the corresponding subspaces are disjoint and have different dimension. The group Aut(PSLn(q)) acts distance-transitivcly on thc graph Ok(q) while PrLn(q)

preserves the parts of the bipartition.

348 A.A.IVANOV

For small values of n there are other examples of DTR's of linear groups. There are lots of exceptional examples for n = 2 and small q.

The multiplicity-free permutation representations of linear groups for n ~ 8 are classified in lnglis et al. (1986) (see also Inglis (1986)).

Let PSLn(q) ~ G ~ Aut(PSLn(q)) and let a be the inverse-transpose automorphisll2 so a induces a duality on V= Vn(q). For an arbitrary subgroup K of the group G let K denote the complete preimage of K in <rL(V), a>. Each element in the coset rL(V) a will be called a graph automorphism. Notice that if Z is the complete group of scalar transformations of V then <rLn(q), a> I Z = Aut(PSLn(q)).

Theorem 6.3.1. Let H be a maximal subgroup of a group G such that PSLn(q) ~ G ~ Aut(PSLn(q)) and H does not contain PSLn(q). If the representation of G on the cosets of His multiplicity-free and n ~ 8 then one of the following holds:

(i) G ~ PrLn(q) and His the stabilizer of a proper subspace of V;

(ii) G contains a graph automorphism and H is the stabilizer of a pair { U, W} of subspaces with dim U = 1, dim W = n -1;

(iii) G contains a graph automorphism, n = 2m and H is the stabilizer of an rndimensional subspace of V;

(iv) G contains a graph automorphism, n =2m+ 1 and H is the stabilizer of a pair { U, W} of subspaces of V where U < W, dim U = m and dim W = m + 1;

(v) n =2m and SLm(q 2) ~ H; (vi) n =2m and SL2m(q) ~ H;

(vii) q is a square and SLn(q 'h) ~ H; (viii) q is a square and SUn(q'h) ~ii.

Conversely, if His a maximal subgroup of G as in (i)-(vüi) and, in addition, G contains PGLn(q) in the cases (v), (vii) and (viii), then G acts on the cosets of H multiplicity-freely. D

In the proof of this theorem a result by Aschbacher (1984) on the structure of maximal subgroups in classical groups was applied. In that paper in a linear group G acting naturally on Vn(q), eight classes of subgroups (the classes (C 1), ... , (C 8)) were described. The subgroups in these classes are the stabilizers in G of certain natural structures defined on Vn(q). These structures are subspaces, decompositions of Vn(q) into direct sums and tensor products of subspaces, etc. It was shown that if H is a maximal subgroup of G which belongs to neither of these classes then the socle Z of H is a nonabelian simple group. Moreover, Z acts absolutely irreducibly on Vn(q) and His contained in Aut(Z). In the proof of Theorem 6.3.1 the subgroups in the classes (C 1), ... , (C 8) and the case of a nonabelian socle were considered separately. In the latter case some information regarding irreducible representations of nonabelian simple groups was used.

In Inglis (1986) the subgroups listed in Theorem 6.3.1 were studied from the

DIST ANCE-TRANSITIVE GRAPHS AND THEIR CLASS!FICA Tl ON 349

distance-transitive point of view. It was shown that for n;;::: 8 only those subgroups in (i) correspond to DTR's. These DTR's surely give rise to Grassman graphs.

An independent classification of DTR's of the linear groups of dimension n;;::: 8 was obtained in van Bon & Cohen (1988). Here Theorem 6.3.1 was also taken as the point of departure. In consideration of the cases (vi), (vii) and (viii), the fact that H = Cc('t) for a suitable involution 't was used. An application of Theorem 2.9.1, concerning distancetransitive graphs on involutions, enabled a very elegant and unified proof in these cases.

The primitive DTR's of the linear groups in dimension 2 ~ n ~ 7 were completely described in van Bon & Cohen (1989). In consideration of the case n = 2, weil known information about maximal subgroups in the groups PSL2(q) was used. In the case 3 ~ n ~ 7 the primitive multiplicity-free permutation representations were first determined. Here the arguments arerather similar to those in Jnglis et al. (1986).

For the case n = 2 an independent classification which covers the imprimitive case as weil, was achieved in FaradZev & Ivanov (1990). Here the analysis relies on information about subdegrees of the primitive representations of the groups PSL2(q). These subdegrees were calculated in Tchuda (1986). It foilows from these calculations that the number of different subdegrees is always bounded by a constant which does not depend on q. On the other band, if we exclude the natural doubly transitive representations then the ranks will grow at least linearly in q. Using this fact in conjunction with Lemma 2.3.4, it is easy to show that if q is large enough then each primitive DTR is just the natural doubly transitive representation. The treatment for smail q was achieved using computer programs for the construction of DTR's from character tables. Theseprograms are described in Section 6.7 of the present survey.

There is an infinite farnily of imprimitive DTR's of linear groups in dimension n = 2. This farnily can be described as foilows. Let W be the set of nonzero vectors of V 2 (q) and let (q -1) = r· s, where r and s are integers, r;;::: 1, s;;::: 1, and s divides (q -1) I (2,q -1). LetS be the subgroup of order s in the multiplicative group of the field GF(q). Let us define an equivalence relation - on W by the rule: x- y if and only if x = ay for a e S. Let ll(q,r) be the graph having W I- as its vertex set, with two vertices adjacent if the corresponding classes contain vectors x,y satisfying x 1 Y2 -X2Yl eS. Then ll(q,r) is an r-fold antipodal cover of the complete graph Kq+l (cf. Brauwer et al. (1989)). The group PrL2(q) acts distance-transitively on ll(q,r) if and only if either r = 2 or the following three conditions hold:

(a) r is a prime number;

(b) the order m of Aut(GF(q)) is divisible by r- 1;

( c) if n is the smallest integer dividing m such that r divides (p n- 1 ), then n = r - 1. Here p is the characteristic of the field GF (q) and q = p m.

We next formulate the result which give the description of DTR's of the linear groups.

350 A. A.IVANOV

The following proposition follows from results in Inglis (1986) and van Bon & Cohen (1988 a) for n;;::: 8 andin van Bon & Cohen (1988 b) for 3:::;; n:::;; 7.

Theorem 6.3.2. Let G be a group satisfying PSL(n,q) '51 G:::;; Aut(PSL(n,q)), n;;::: 3. Suppose that G acts distance-transitively on a graph r having diameter at least 2. Then one of the following holds:

(i) r is a Grassman graph; (ii) n = 3 and r is the line graph of the incidence graph of the Desarguesian projec

tive plane PG(2,q);

(iii) r and G correspond to one of the five exceptional situations listed in Table 4, where F = Aut(r), H is the stabilizer of a vertex in F and if the diameter of r is 2, only one of r and its complement is listed. o

Table 4.

(n,q) F H IV(I)I i(I) Name

(3,4) PSL3(4).D12 PSU(3,2). D 12 280 (9,8,6,3; 1,1,3,8} affine subplanes in PG(2,4)

(3,4) PSL3(4). 22 Aut(A6) 56 ( 10,9; 1,2} Gewirtz (4,2) Ss S6 x2 28 { 15,8 ; 1,6} complement of

1(8,2) (4,2) Ss Ss xS3 56 ( 15,8,3 ; 1,4,9} 1(8,3)

(4,3) PG0+(6,3) PSp(4,3). 22 117 (36,20; 1,9) non-isottopic lines

For the case n = 2, we have the following result due to van Bon & Cohen (1988) and FaradZev & lvanov (1990).

Theorem 6.3.3. Let G act distance-transitively on a graph r, where PSL 2(q) ~ G:::;; PrL 2(q), q;;::: 5. Then one of the following holds:

(i) r is the graph Kq+l;

(ii) r is the graph 2. Kq+l and G contains a subgroup of index 2;

(iii) r is the graph f}.(q,r) where r;;::: 2, r divides (q -1) I (2,q -1), and either r = 2 or the aforementioned condition (a)-(c) are satisfied.

(iv) the distance-transitive representation (G, V(r)) appears in Table 5. 0


Table 5.

q G H IV(f)l i(f) p.b.a r

5 PGL 2(5) s4 5 {4; 1} p. Ks PGL 2(5) Dl2 10 {3,2; 1,1} p. Petersen graph 0 3

PGL 2(5) A4 10 { 4,3,1 ; 1,3,4} b.a 2·Ks PGL 2(5) Zs 12 {5,2,1; 1,2,5} a. icosahedron PGL 2(5) Ds 15 { 4,2,1 ; 1,1,4} a. 1ine graph of 0 3

PGL 2(5) D6 20 {3,2,2,1,1 ; 1,1,2,2,3} b.a Desargues graph

7 PSL 2(7) s4 7 {6; 1} p. K1 --

PSL2(7) A4 14 {12,1; 1,12} a. 7 o K 2

PGL 2(7) s4 14 {3,2,2; 1,1,3} b. Heawood graph PGL 2(7) D16 21 {4,2,2; 1,1,2} p. 1ine graph of

PG(2,2)

PGL 2(7) Dl2 28 {3,2,2,1 ; 1,1,1,2} p. Coxeter graph

8 PrL 2(8) Z9AZ6 28 {27; 1} p. Kzs PrL 2(8) F~ 36 {14,6; 1,4} p. 1(9,2)

PrLz(8) Z9A.Z3 56 {27,10,1 ; 1,10,27) a. Gosset graph

9 PIL2(8) Ss 6 {5; 1} p. K6 PrL 2(9) Ss 12 {6,5; 1,6} b. K6,6 PIL 2(9) As 12 {5,4,1 ; 1,4,5} b.a. 2·K6 PIL2(9) S4 xZ2 15 {6,4;1,3} p. 1(6,2)

--Mto E9AE4 20 {18,1; 1,18} p. 10 o K 2

PrL 2(9) S4 xZ2 30 {3,2,2,2; 1,1,1,3} b. Tuttes 8-cage PrL 2(9) fixZz 36 {5,4,2; 1,1,4} p. PrL 2(9) SDI6 45 { 4,2,2,2; 1,1,1,2} p. gen. 8-gon

11 PSL 2(11) As 11 { 10; 1} p. Ku PGL 2(11) As 22 {5,4,3 ; 1,2,5} b. 2-(11,5,2)-design

16 PrL2(16) A 5 ·Z2 xZ2 68 (12,10,3; 1,3,8} p. Doro graph

17 PSL 2(17) s4 102 {3;2,2,2,1,1,1; p. Biggs-Smith graph 1,1,1,1,1,1,3}

19 PSL 2(19) As 57 {6,5,2; 1,1,3} p. Perkel graph 25 PIL2(25) PGL2(5)xZ2 65 { 10,6,4; 1,2,5} p. 1ocall y Petersen

6.4. The classical and exceptional groups of Lie type

In a linear group defined on Vn(q) every maximal parabolic subgroup is the stabilizer of a certain proper subspace of Vn(Q) and vice versa. This implies that each maximal parabolic gives rise to a DTR. For other groups of Lie type the main examples of DTR's also arise from representations over maximal parabolics. For example, let G be the classical group, corresponding to a form f defined on Vn(q), and let W be a maximal totally isotropic subspace of Vn(q) with respect to this form. Then the stabilizer of W in Gis a

352 A.A. IVANOV

maximal parabolie subgroup whieh gives rise to a DTR of G. Corresponding DTG's are just dual polar spaee graphs. But some other maximal parabolies do not eorrespond to DTR's. So the problern of deseribing DTR's of groups of Lie type over maximal parabolie subgroups arises naturally. The eomplete solution to this problern is given by Theorem 10.9.4 in Brouwer et al. (1989). The result is as follows:

Theorem 6.4.1. Suppose that G is an ordinary or twisted finite Chevalley group and P is a maximal parabolie subgroup of G. Then the aetion of Gon the eosets of Pis a DTR of G if and only if one of the following eonditions hold:

(i) G is a linear group;

(ii) G is not oftype E 6 , E 1 , E 8 or 2 E 6 , and P eorresponds to a terminal node in the Dynkin diagram of G;

(iii) G is of type E 6 or E 7, and P eorresponds to a solid node as indieated in the associated diagram below

0

E6 : *-------o-------o-------0-------*

0

I E7: o-------o-------o-------0-------o-------*

0

In partieular it follows from this theorem that if G is a group of Lie type and neither of its maximal parabolies produees a DTR, then Gis either of type E 8 or 2 E 6 •

The proof of Theorem 6.4.1 relies on a correspondence between the association scheme of a Lie type group G acting on the cosets of its parabolic P and the association scheme of the Weyl group of G.

So the study of DTR's of the Lie type groups is reduced to a consideration of nonparabolic subgroups. A result conceming multiplicity-free permutation representations of the classical Lie type groups of dimension at least 13 is proved in Inglis (1986).

Theorem 6.4.2. Let Go be one of PSp(n,q), PSU(n,q'h) or PO.(n,q) with n;;::: 13 and let G be a group with Go ~ G ~ Aut(G 0). Let V be the natural projective module for Go (i.e. V= Vn(q)). Suppose that His a maximal subgroup of G with Go :t H suchthat the representation of Gon the cosets of His multiplicity-free. Then one of the following holds:

(a) (i) His the stabilizer of a totally singular subspace U of V. Moreover if n =2m and G 0 is Po+ (2m, q) then either G contains a graph automorphism or d ~ m !2;

(ii) Gois PSU(n,q'h) or PO(n,q) and His the stabilizer of a non-singular subspace ofV;

(iii) n =2m, Go is PSp(2m,q) and H is the stabilizer of a non-singular 2-dimensional subspace of V;

(iv) n =2m, q is even, Gois PSp(2m,q) and H t?POE(2m,q);

(v) n =2m, q is a square, Gois PSU(2m,q'h) and H t?PSp(2m,q'h);

(vi) n = 2, q is odd, Gois PSp(2m,q) andH t?PSU(m,q);

(vii) n =2m, Gois pQH'" (2m,q) and H t?PSU(m,q);

(b) (i) Go =PO(n,q) and H is the stabilizer of an anisotropic non-singular 2-dimension subspace of V;

(ii) n = 4m , G 0 = PSp (4m,q) and His the stabilizer of a pair { U, U j_) of nonsingular 2m-dimensional subspaces of V;

(iii) n =2m, Go= pQ+(2m,q) and His the stabilizer of a pair { U, W) of maximal totally singular subspaces of V with V = U $ W;

(iv) n =2m, q is a square, Go= PSp(2m,q) and H t?PSp(2m,q'h);

(v) q is a square, Go= pQE(n,q), H t~PQii(n,q'h) and e = + if n is even;

(vi) q is odd and a square, G = PSU(n,q'h) and H t?PO(n,q'h);

(vii) n=4m, Go=PSp(4m,q)andH t;::PSp(2m,q 2);

(viii) n =2m, Go= PQE(2m,q), H t?P01i(m,q 2) and ö = e if m is even;

(ix) n = 14, q = 2, G0 = pQ+(14,2) and H t?A 16·

Conversely, if G and H are as in (a) and if G contains po<-Y" (2m,q) in (a) (iii), then G acts multiplicity-freely on the cosets of H. o

This theorem comprises a very important source of information for the classification of DTR's of classical groups. But the work hereisstill in progress.

Classification of the DTR's of the exceptional groups of Lie typeisalso in progress. In the work by Cohen et al. (1989) the proof of the conjecture formulated below is reduced to consideration of a few concrete pairs G,H.

Conjecture 6.4.3. Let G 0 be an exceptional group of Lie type, and suppose Go $;1 G $ Aut(Go) where G acts primitively and distance-transitively on a graph r. Let H be the stabilizer of a vertex in this action. Then one of the following holds:

(i) His maximal parabolic subgroup, satisfying the conditions in Theorem 6.4.1;

(ii) Go= G2(q), H = SU 3(q). 2 for q = 2,3,4,8 and r isarank three graph;

(iii) Go= G2 (4), H = J 2 and r has rank three. 0

354 A.A.IVANOV

Notice that all rank three representations of the Lie type groups are known (see

Kantor & Liebler (1982) for the classical groups and Liebeck & Saxl (1986) for the

exceptional groups).

6.5. The sporadic groups

We begin this section with a list of all known DTR's of groups satisfying

T ~ G :S;; Aut(T), where T is a sporadic simple group.

DISTANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICATION 355

Table6.

T T(x) IV(I)I i(I)

Mu M10 11 {10;1}

PSL2(11) 12 {11;1}

A6 22 {20,1 ; 1,20}

32• SD16 55 {18,8; 1,4}

M12 Mu 12 {11;1}

Aut(A6) 66 {20,9; 1,4}

M22 M21 22 {22; 1}

24.A6 77 {16,15; 1,4}

A1 176 {70,51; 1,34}

23 .PSL3(2) 330 {7,6,4,4; 1,1,1,6}

M23 M22 23 {23 ; 1} PSL3(4). 2 253 {42,20; 1,4}

24.A7 253 { 112,75; 1,60}

As 506 {15,14,12; 1,1,9}

M24 M23 24 {23 ; 1}

M22·2 276 { 44,21 ; 1,4} 24 .PSL4(2) 759 {30,28,24; 1,3,15}

M12·2 1288 {495,288; 1,180}

Jl PSL2(11) 266 { 11,10,6,1 ; 1,1,5,11}

h PSU(3,3) 100 (36,21 ; 1,12} 21+4 .As 315 {10,8,8,2; 1,1,4,5}

HS M22 100 (22,21; 1,6} PSU(3,5). 2 176 (175;1}

PSU(3,5) 352 {175,72,1; 1,72,175}

McL PSU(4,3) 275 {112,81; 1,56}

356 A. A.IVANOV

Table 6. (continued)

T T(x) IV(r)l i(r)

Ru 2F 4(2) 4060 {1755,1024; 1,780}

Co 2 PSU(6,2). 2 2300 {891,512; 1,324}

Sz G2(4) 1782 {416,315; 1,96} 3. PSU(4,3) 22880 {280,243,144,10; 1,8,90,280}

Fi22 2. PSU(6,2) 3510 {693,512; 1,126} PQ7(3) 14080 { 10920,2511 ; 1,8680}

Fi23 2. Fi22 31671 {3510,2816; 1,351} Pnt(3). S3 137632 { 109200,22599; 1,86800}

Fi24 ' 2. Fi23 306936 {31671,28160; 1,3240}

In all cases except T = h , IV (r) I = 315, the action of T on r is already distancetransitive.

The following partial results on the classification of DTR's of sporadic groups are available.

In Liebeck & Saxl (1986) the rank 3 representations of sporadic groups are classified.

Theorem 6.5.1. Let G be a group satisfying T ~ G ~ Aut(T), where T is a sporadic simple group. Then all rank 3 representations of G are those presented in Table 6. o

A description of the DTR' s of the nonabelian simple groups of orders up to 109 ,

obtained in Farad.Zev & Ivanov (1988), gives the following:

Theorem 6.5.2. Let G be a group, satisfying T ~ G ~ Aut(T). Then

(i) for T=M 11 ,M12 ,M22,M23 ,J1,h,HS, McL and Ru, the list of DTR's in Table 6 is complete;

(ii) forT= J 3 , He and O'N there are no DTR's of G. 0

An independent computer-free proof of the fact that neither He1d's group nor its automorphism group have DTR's was obtained in van Bon et al. (1989).

In van Bon (1988) the following proposition is proved using the methods presented in Section 2.9.

Theorem 6.5.3. Let r be a graph on which G acts primitively and distancetransitively with vertex stabilizer H. Suppose that G has a normal sporadic simple subgroup and that H = Cc(t) for some involution 't E Aut(G). Then G, r is one of the

DISTANCE-TRANSIT!VE GRAPHSAND THEIR CLASSIFICATION 357

following pairs of groups and graphs.

(i) M 22 ~ G ~ Aut(M 22) and r is the 2-residue of S(5,8,24), a graph on 330 ver-tices;

(ii) G = Aut(J 2) and r is the near octagon associated with J 2• on 315 vertices;

(iii) F ~ G ~ Aut(F) where Fis one of Fi 22 , Fi23 , Fi 24 and r is the Fischergraph on 3-transpositions or its complement D

Nowadays all maximal subgroups of sporadic groups except for some small subgroups of the Monster and Baby Monsterare known (cf. Kleidman & Liebeck (1987)). So a prospective classification of DTR's for these groups seems tobe well within reason.

6.6. The affine case

In this section we discuss the present state in the classification of the affine DTG's.

Let r be a DTG of affine type. This means that there is a group G which acts distance-transitive1y on r and that G contains an elementary abelian subgroup N acting regularly on V(r). As was mentioned above, the Hamming graph H(n,q) is an example of an affine graph if q = 2, 3 or 4. Other known farnilies of affine graphs can be constructed as follows.

Let X be a set of n x m matrices over GF (q) which is closed under addition, and let r be a positive integer. Let r = r(X,r) be the graph with vertex set X in which two vertices x,y e X are adjacent in r if x - y is a matrix of rank r. This construction gives us three farnilies of affine DTG's.

Bilinear forms graphs Hq(n,m). Here Xis the set of all n xm matrices over GF(q)

and r = 1.

Hermitianforms graphs Her(n,q). Here m = n, r = 1 and q is a square. Xis the set of all Hermitian rnatrices, that is, A E X for A = ilaijilnxn if and only if aij = a}i for 1 ~ i,j ~ n, where a is the involutory automorphism of GF(q).

Alternating forms graphs Alt(n,q ). Ht:re again m = n and r = 2. X is the set of all altemating matrices, that is, A E X for A = llaijlln xn if and only if aij = -aji for 1 ~ i,j ~ n.

Let r be a graph in one of the above constructed farnilies and let G be the automorphism group of r. Then the regular elementary abelian subgroup N of G is just the additive group of the matrices in X. It is remarkable that the group G itself is isomorphic to a maximal parabolic subgroup in a Lie type group L, which corresponds to a DTR of L.

Namely, let L be a Lie type group and let P be a maximal parabolic subgroup of L such that the action of L on the cosets of P is a DTR of L. Let .:E be a DTG corresponding to this action, d be the diameter of .:E, and e be a positive integer. Let~= ~(L,P, .:E,e) be a graph having .:Ed(x) as its set of vertices, where x is a vertex of .:E, andin which two vertices in ~ are adjacent if they are at distance: e in .:E. Then the following proposition holds

358 A. A. IVANOV

(see Section 9.5 in Brouwer et al. (1989)).

Proposition 6.6.1. (i) If PSLn(q) '5JL, Pis the stabilizer of a d-dimensional subspace of Vn(q), l: is the Grassmangraph and e = 1, then t1(L,P,l:,e) is isomorphic to Hq(n -d,d);

(ii) if P SU u(q) '.5J L , P is the stabilizer of a maximal totally isotropic subspace, l: is the dual polar space graph of type 2Au_1 (q) and e = 1, then t1(L,P, l:,e) is isomorphic to Her(d,q);

(iii) if PSO!tJ(q) '.5J L, P is the stabilizer of a maximal totally isotropic subspace, l: is the dual polar space graph of type Dd(q) and e = 2, then t}.(L,P, l:,e) is isomorphic to Alt(d,q).

Moreover, in each of the above cases, P induces a distance-transitive action on t1(L,P,l:,e). D

The relationship between the graph Her(d,q) and the dual polar space graph of type 2Au_1(q), established in Proposition 6.6.1 (ii), was used in a crucial way in Ivanov & Shpectorov (1989, 1991a) during the characterization ofHer(d,q) by its parameters in the class of distance-regular graphs.

Now let us turn to the general classification scheme for affine DTG's.

Let r be an affine DTG on which G acts distance-transitively, and let N be an elementary abelian regularnormal subgroup of G. Let x be a vertex of r and H = Gx be the stabilizer of x in G. Since for any y e V(r) the subgroup N contains a unique element which maps x onto y, we can identify V(r) with the elements of N. Surely in this case x corresponds to the identity element. On the other hand, N is elementary abelian of order pm for some prime p and integer m ~ 1. So N is an rn-dimensional vector space over GF(p) and His a subgroup of GLm(p). Let us deduce from distance-transitivity some conditions on H. The arguments presented below are probably due to J. van Bon & A.M. Cohen.

As above we assume that x e V(r) corresponds to the zero vector in N. Let ye r(x)andl=l(y)={z lze V(r),z=a·yforae GF(p)} bethelineofNcontaining y. Suppose that l Q; {x} u r(x). Let z E l \ ( {x} u r(x)) and z = y 1 + Y2 for Yi E l n r(x). Then it is clear that z e r2(x). On the other hand, the stabilizer of l in H acts semiregularly on /- {x} and hence k2 = k 1. Now by Lemma 2.3.2 (vi) we have the following:

Lemma 6.6.2. Using the above notation, either

(i) I is a clique of r,

(ii) d ~ 2, or

(iii) d = 3 and r is a 2-fold covering of a complete graph. D

Since r is connected, each vertex z e V(r) is a linear combination of vectors in r(x). By Lemma 6.6.2, if d > 3 then z can be expressedas a sum of certain vertices in r(x). Now it is easy to see that the number of summands in the shortest such expression

of z is less than or equal to the number of edges in a shortest path joining z with x. So we have following.

Corollary 6.6.3. If d > 3 then. d ~ m = dim(N). o Finally, by distance-transitivity the nurober of orbits of H on V(r) is equal to d + 1.

This implies

Corollary 6.6.4. If d > 3 then the number of orbits of H on the set of nonzero vec-tors of N is less than or equal to the dimension of N over GF (p ). o

By this corollary we conclude that we should study the subgroups of GLm(p) with m or fewer orbits of the underlying vector space. This condition tums out to be rather restrictive and it forms a good starting point for the classification. The prirnitivity condition has a clear interpretation in such terms. Namely, r is primitive if and only if His irreducible onN.

The classification of affine DTG's along these lines is currently being pursued by J. van Bon in Amsterdam. 1>

The complete classification of affine DTG's is now available for d = 2, i.e. the affine rank three graphs are classified in Liebeck (1987).

An alternative approach to the classification of affine DTG's with the property that o_i<x> is a linear group in the natural doubly transitive representation was used in Ivanov (1988) (see Theorem 5.5.5 of the present swvey).

6.7. Construction of DTR's of a group from its character table.

The aforementioned methods for describing DTR's are effective only if the groups under consideration are sufficiently large. Indeed, specialtreatmentwas required for the altemating groups An for small n and for the groups PSL 2 (q) when q is small. The classification of DTR's for the exceptional Lie type groups was reduced to consideration of certain concrete small cases. Some specific situations can also arise in consideration of the classical groups of small dimension over small fields. Finally there is almost no chance of finding a unified approach in the: case of the sporadic simple groups. On the other hand, there are a lot of exceptional DTR's for small groups. Because of these exceptions the classification of DTG's is both nontrivial and very interesting.

If the number of conjugacy classes of the group G is small, then a unified procedure for the deterrnination of DTR's of G can be proposed. The starting point of this procedure is the character table of G. On the basis of this procedure a computerprogram was produced, see FaradZev & Ivanov (1988). In this section we present a brief description of this procedure and its results.

1) J. van Bon, Affine Distance-Transitive Groups, Ph.D. Thesis, CWI, Amsterdam, 1990.

360 A.A.!VANOV

Let G be a finite group. Let a 1 ,a 2 , •.. , a8 be representatives of the conjugacy classes of G, <\ll ,q,2 , .•. , 4>s be the irreducible characters of G over q; and T = ll$j(ai)ll8 xs

be the character table. We will assume below that a 1 = 1 is the identity element of G and q,1 is the trivial character of G. The procedure for determining the DTR's of Gis divided into three main steps.

I. Find alt characters of DTR' s of G. Let X be the character of a DTR of G. Then since x is the character of a generously transitive representation of G, the following conditions hold:

(1) x is multiplicity-free, i.e. x = L 4>i, where 1 s;;; { 1,2, ... , s }. ieJ

(2) the trivial character 4>1 is involved in x. i.e. 1 E J;

(3) if Gli and $j are algebraically conjugate over {!, then I {i,j} n J I is even;

( 4) if i e J then Gli is the character of a representation of G over IR.

The latter condition is equivalent to the following

i E J ==:> n($i) d~f I~ I L 4>i(a2) = 1 , ae G

where n($i) is known as the Frobenius-Schur index of the character Gli·

Since x is a permutation character, it should possess certain additional properties. For a E G let <a> denote the cyclic subgroup generated by a. Put o(a) = I <a> I, c(a)= ICc(a)l,n(a)= INc(<a>)l.

Lemma 6.7.1. Let x be a permutational character of G. Then the following condi-tions hold:

(i) I G I I x(1) isapositive integer;

(ii) x(a) is a nonnegative integer for each a E G;

(iii) if <a>::;; , then x(a)~ x(b);

(iv) let p be a prime and pm be the highest power of p which divides I G I I x(1),

and let a be an e1ement of G suchthat o(a) = pm and x(a) > 0. Then x(a) divides n(a)

and n(a) I x(a) divides I G I I x(l). o

Let H be a subgroup of G such that the permutation character of G acting on the cosets of H is equal to X· Then it is clear that I H I = I G I I x(l). Moreover, for any integer e we can calculate the number of elements a E H suchthat o(a) = e. Consideration of the possibilities for the structure of minimal normal subgroups of H leads to some additional and more complicated conditions on x (see FaradZev & lvanov (1988) for details). It should be mentioned that x does not determine Hup to conjugacy or even up to isomorphism. For instance the Mathieu group M 23 contains two different subgroups 24 • A 7 and P SL 3 ( 4). 2, corresponding to the same character. Each of these subgroups determine a DTR of M 23 (see Table 6 in Section 6.5).

II. Derermine corresponding subdegrees. Let (G, 0) be a permutation representation of G with associated character x found in the previous step. Let H be the stabilizer of a point and 01>02, ... , Or be the orbits of H on 0 of respective sizes ni = I Oi I for 1 :S: i :S: r. The goal of this step is to determine all possibilities for the array n 1 ,n2, ... , nr of subdegrees.

Without loss of generality we may assume that 1 = n 1 :S: n 2 :S: • • • :S: nr. The following lemma is obvious.

Lemma 6.7.2. (i) r is equal to the number of irreducible components of x; (ii) ni divides I H I for 1 :S: i :S: r;

r

(iii) L ni =x(1)= 101. i=1 D

Many conditions on the subdegrees n l>n 2 , ... , nr are especially known in the primitive case (cf. FaradZev et al. (1991)). We present here just one ofthese conditions.

Lemma 6.7.3. Let (G, 0) be a primitive group. Let L\ be graph whose vertices are indexed by the numbers n 1 ,n2, ... , nr; the vertices ni and nj are adjacent if ni q = nj t for some positive integers q,t, suchthat q,t < n2. Then L\ is connected. D

The form of the character x also implies some conditions on the subdegrees.

Lemma 6.7.4. If a e G with x(a) > 1 then there is a subdegree ni, i ~ 2, suchthat ni· o(a) divides IH I. D

III. Reconstruct the intersection array. Suppose that (G, 0) is a DTR of G having subdegrees n 1 ,n2, ... , nr. Our goal here is to reconstruct the intersection array of the corresponding DTG. First we reorder the sequence n 1 ,n2, ... , nr to obtain a logarithmically convex sequence k 0 ,k 1, .•• , kd. For the existence of such a reordering certain conditions must hold (see Lemma 2.3.3).

When a logarithmically convex sequence ko,kb ... , kd is found, we determine the possibilities for intersection numbers Ci, 1 :S: i :S: d, bi, 0 :S: i :S: d -1, satisfying the relations ki • bi = ki + 1 • Ci+ 1 for 0 :S: i :S: d- 1. Next we check the feasibility conditions for the intersection array (bo,b 1, ... , bd_1; c 1,c 2 , .•• , cd}. The most restrictive condition is the following. From the intersection array we can calculate the eigenvalue multiplicities mi , 0 :S: i :S: d. These multiplicities should be equal to the dimensions of the irreducible components of the character X· This means that, up to reordering of the indices, we have mi = Xi(l).

If the constructed intersection array satisfies all feasibility conditions then, as a rule, it in fact corresponds to a DTR. In our calculations we found only one counterexample in the primitive case. This counterexample is well known in the literature; it is a "pseudo doubly transitive representation" of M 22 of degree 56.

362 A. A IVANOV

The algorithm whose ideas we have just presented above, was implemented by computer program. By means of this program all DTR of the nonabelian simple groups of order up to 108 were classified in FaradZev & Ivanov (1988). For the alternating groups we obtained a proof independent of the result in Liebeck et al. (1987) in the case of small n. Calculations with the groups PSL 2(q) for small q form part of the complete treatment of this series (see Section 6.3). Results for the sporadic groups are presented in Table 6 above. The DTR's of the other nonabelian simple groups of order up to 108 is given in Table 7 below. Presently this calculation has been completed for the nonabelian simple groups of order up to 109• But when the number of real irreducible representation becomes !arge (say !arger then 40) it is impossible to realize this algorithm in its present form since too many possibilities must be considered on the first step.

DISTANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA Tl ON 363

Table7.

G H 1\-'(I)I i(I)

PSL3(3) 32• SL2(3) 26 {24,1; 1,24}

32• SDt6 39 {36,2; 1,36}

PSU3(3) 3. 32.4 56 {27,10,1 ; 1,10,27}

3. 32.4 56 {27,16,1; 1,16,27}

4.S4 63 { 6,4,4 ; 1,1,3}

PSL3(4) A6 56 {10,9; 1,2}

PSU4(2) 24.As 27 {10,8; 1,5}

s6 36 {15,8; 1,6}

3. 32• SL2(3) 40 {12,9; 1,4}

32• 3. SL2(3) 40 {12,9; 1,4} 2. 24.32 45 { 12,8; 1,3}

PSU5(5) A1 50 {7,6; 1,1}

M10 175 { 12,6,5; 1,1,4}

5.52.4 252 { 125,52,1 ; 1,52,125}

5. 52.4 252 {125,72,1; 1,72,125}

PSL3(5) 52• SL2(5) 62 {60,1; 1,60}

52• (4* GL2(3)) 155 { 150,4; 4,150}

PSp4(4) 22• 24• SL2(4) 85 {20,16; 1,5}

PSL2(16). 2 120 {51,32; 1,24}

(As xAs). 2 136 {60,35; 1,28}

PSp6(2) PSU4(2) 56 {27,10,1; 1,10,27}

PSU4(2) 56 {27,16,1; 1,16,27}

25 • s6 63 {30,16; 1,15}

As 72 {35,16,1; 1,16,35}

As 72 {35,18,1 ; 1,18,35}

PSU3(3). 2 120 {56,27; 1,24}

26 .PSL3(2) 135 {14,12,8; 1,3,7}

364 A.A.IVANOV


G H IV(r)l i(r)

PSL3(7) 72• 2. PSL2(7) 114 {112,1; 1,112}

PSU4(3) 34.A6 112 {30,27; 1,10}

PSU4(2) 126 {45,32; 1,18} PSL3(4) 162 {56,45; 1,24}

3!+4. GL2(3) 280 {36,27; 1,4}

G2(3) PSU3(2) 351 {126,80; 1,45}

3!+2• 32• GL2(3) 364 {12,9,9; 1,1,4}

PSp4(S) s!+2. 4As 156 {30,25; 1,6} 53. (2xA 5). 2 156 {30,25; 1,6}

PSU3(7) 71+2. 24 688 {343,150,1; 1,150,343} 71+2• 24 688 {343,192,1; 1,192,343}

PSL4(3) PSU4(2). 2 117 {36,20; 1,9}

34• 2(A4 xA4). 2 130 { 48,27; 1,16}

PSL 5 (2) 26• (S3 xPSL3(2)) 155 {42,24; 1,9}

24.A7 248 {240,7; 1,240}

PSU5(2) 21+6. 3!+2. 2A4 165 {36,32; 1,9} 3xPSU4(2) 176 { 40,27 ; 1,8}

24+4. (3 xAs) 297 {40,32; 1,5}

2p 4(2) 2. [28]. 5. 4 1755 {10,8,8,8; 1,1,1,5} 22• [28]. S3 2925 { 12,8,8,8; 1,1,1,3}

PSL3(9) 34• SL2(9) 182 {180,1 ; 1,180}

PSU3(9) 32+4. 40 1460 {729,328,1 ; 1,328,729} 32+4. 40 1460 {729,400,1 ; 1,400,729}

PSU3(11) 11 1+2• 20 2664 { 1331,610,1 ; 1,610,1331} 11 1+2• 20 2664 { 1331,720,1 ; 1,720,1331}

3D4(2) 2!+8 • PSL2(8) 819 { 18,16,16; 1,1,9} 22• [29]. (7xS3) 2457 {24,16,16; 1,1,3}

D!ST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA TION 365

6.8. Imprimitive graphs

Now let us turn to the second step in the realization of the global approach to the classification of DTG's, namely to the problern of constructing imprimitive graphs corresponding to given primitive ones. This problern is reasonable as well in the class of DRG's.

Thus let r be a given DRG and suppose that we are interested in the bipartite DRG's whose halved graphs are isomorphic to r, as well as in the antipodal graphs whose folded graphs are isomorphic to r. As follows from Lemmas 2.2.4, 2.2.6 and 2.2.7, the problern can be reduced in the case of bipartite doubling, to the study of cliques in r while for antipodal coverings, the structure of geodesics joining the vertices at maximal distance plays a crucial role. Notice that the structure of cliques and the structure of geodesics are both interesting and important for the characterization of graphs in terms of their parameters.

The structure of cliques in DRG's from known infinite families was studied in Remmeter (1984, 1986, 1988 a). Within these investigations a new farnily of DRG's was constructed in Brouwer & Remmeter (1992). Let us say a few words about this new family.

Let Bd(q) and Cd(q) be distance-regular graphs of dual polar spaces of the corresponding types. It is known that these graphs have the same set of par~eters and in fact are isomorphic if and only if q is even. For an arbitrary graph 3 let 3A denote the graph whose vertex set is that of 3 and in which two vertices are adjacent in 3 if they are at distance 1 or 2 in 3. It was shown in Ustimenko (1991) (see also lvanov, Muzichuk & Ustimenko (1989)) that the graphs Bd(q) and Cd(q) are DRG's, and that the former is isomorphic to the halved graph 1!2Dd+l (q) of the dual polar space graph of type Dd+l (q). If q is odd then the graphs Cd(q) are new examples. The graph Bd(q) admits a bipartite doubling to the graph Dd+l(q). It was shown in Brouwer & Remmeter (1992) that the graph C d(q) also admits a bipartiw doubling. The bipartite graphs arising in this way have parameters equal tothat of the Dd+l (q) graphs and for odd q they form a new farnily of DRG's.

For other graphs of sufficiently large diameter from the known families of DRG's, it was shown in Remmeter (1984, 1986, 1988 a) that bipartite doublings exist only in those cases which were previously known.

The antipoda1 coverings of the classic:il DRG's were studied in van Bon & Brouwer (1988). No new examples were found and for the case of diameter at least 3 only one question remained unanswered. This was the question of existence of DRG's for two given intersection arrays. The graphs with these arrays are antipodal coverings of a graph on 506 vertices related to M 23. A negative answer to this existence problern was given in lvanov & Shpectorov (1990).

Thus, in the case of sufficiently large diameter, estab1ishing existence of DRG's from a description of the bipartite doublings and antipodal coverings seems rather

366 A. A. IVANOV

optimistic. If the diameter becomes small the situation changes dramatically. The reason is that in this case the structure of a graph can be lost completely in its folded or halved graph.

This is the case for instance if ~ is a bipartite DRG of diameter 3. The halved graph r of ~ is just the complete graph and it "remembers" almost nothing about ~- To find all the bipartite doublings of r = Kn one should describe up to isomorphism all 2-(n,k, A.)symmetric designs for different k and A. (see Lemma 2.2.5). It is clear that without any additional assumptions this problern is too hard. On the other band, if we are interested in DTG's alone, then a complete description can be obtained. In fact, let ~ be a bipartite DTG of diameter 3 and let G act distance-transitively on ~- Then the stabilizer in G of the bipartite parts of ~ induces a doubly transitive group on each part. Using the list of all doubly transitive permutation groups, the complete classification of these graphs ~ can be obtained (see Kantor (1985), Ivanov & Ivanov (1988)).

Another situation of this type is the classification of the antipodal coverings of complete graphs. The graphs ~(q,r) mentioned in Section 6.3 are examples of such coverings. Other known examples can be constructed from regular two-graphs, from generalized quadrangles, from Moore graphs and from projective planes with specific polarities. A detailed treatment of these coverings is contained in Hensel (1988). Apparently, significant progress in their classification can be achieved in the class of DTG's.

Now let us turn to the antipodal coverings of the complete bipartite graphs.

A graph ~ is an l-fo1d antipodal covering of the complete bipartite graph Kk.k if and only if it has the following intersection array: {k,k-1,k-t, 1; 1,t,k-1,k}, where t = k ll. In this case ~ is said to be of type l. Kk.k· A nurober of combinatorial objects were found which are equivalent to the graphs l. Kk,k· The following proposition was formulated in Ivanov et al. (1984).

Proposition 6.8.1. A graph of type l. Kk,k is equivalent to a matrix S having k rows and (k-1) l = k(k-1) I t colurnns with entries from the set I= { 1,2, ... , klt = l} and which possesses the following properties:

(i) for any distinct rows s 1, s 2 and integers i,j E /, the nurober of colurnns having i on the intersection with the row s 1 and j on the intersection with the row s 2 is equal to t- 1 ifi = j and t if i '# j;

(ii) there is a partition of the set of columns into classes of l columns each with the property that colurnns in the same class have pairwise distinct components and any two columns in different classes have exactly t equal components. D

A graph of type l. K~c,k can be reconstructed from S in the following way. Put W={wi ll:Si:Sl}, U={uiJ 11:5j:Sl}. Let V(~)=WuUuR, whereR is the setof columns of the matrix S. The vertex wi is adjacent to Ujr if i = r, and Ujr is adjacent to q E R if the j-th component of the column q is equal to r.

As a nontrivial example we mention a DTG of type 3. K 6, 6 having 3· Aut(A 6 ) as the automorphism group. This graph was characterized in Lemma 6.2.5 (ii) (b) and its

DIST ANCE-TRANSITIVE GRAPHS AND THEIR CLASSIFICA TION

matrix S is the following:

1 3 2 3 1 2 3 2 1 2 1 3 2 1 3 1 3 2 2 3 1 2 1 3 3 2 1 3 2 1 3 2 1 1 2 3 3 2 1 1 3 2 3 2 1 2 1 3 1 2 3 2 1 3 2 1 3 1 3 2 3 2 1 3 1 2 1 3 2 3 2 1 1 3 2 2 1 3 2 3 1 1 3 2 1 3 2 2 1 3

367

Later, in Farad.Zev et al. (1986), an equivalence between the graphs l. Kk.k and the resolvable transversal designs RT(k,t,l) in the sense ofHanani (1974) was established.

Proposition 6.8.2. Let~ be a graph of type l. Kk.k, V(~)= V 1 u V 2 be the bipartition of V(~). Let us construct a scheme T = (B, G u P) by the following rule: B =V 1 is the point set of the scheme; G is the set of antipodal blocks contained in V 1, P is the set of neighbors of vertices in V2. Then T is aresolvable transversal design RT(k,t,l). The parallel classes of proper blocks P in the design T are the antipodal blocks contained in V 2. If p 1 and p 2 are blocks from distinct parallel classes of P then I p 1 f1 p 2 I = t. Conversely, if T is a RT(k,t,l) where k = t·l and any two proper blocks from different parallel classes have exactly t points in common, then an I. Kk,k graph can be constructed fromT. D

Finally, in Aldred & Godsil (1988), the following description of the antipodal coverings of the complete bipartite graphs was proposed.

Proposition 6.8.3. A graph of type l. Kk.k is equivalent to an array T(h,i,j), 1 :s;; h,j :s;; 1, 1 :s;; i :s;; k -1, of subsets of a k-set with the following properties:

(i) For each j, 1 :s;; j :s;; n, T(h,i,j) is a resolvable 2-(k,t,t-1)-design, where I. t = k.

I (ii) L. I T(h,i,j) n T(h',i',j) I = t, i "I= i'.

j=1

(iii) Foreach quadruple x,y,j,j' satisfying 1 :s;; x,y :s;; k, 1 :s;; j,j' :s;; land x :F- y, j :F- j', we have x e T(h,i,j) and y e T(h,i,j') for exactly t pairs h,i. o

Using any one of Propositions 6.8.1, 6.8.2 and 6.8.3, one can characterize the following two extremal cases.

Case 1: l = 2. The graphs of type 2. KA~k are equivalent to the Hadamard matrices of order k. Let ~ be a graph of type 2. Kk.k and x e V(~). Then each vertex y e ~2 (x) determines a partition of the set ~(x): the vertices adjacent to y and the nonadjacent vertices. If y,y' are antipodal vertices from ~2(x), then y and y' determine the same partition of ~(x). Let y 1 ,y2, ... , Yk-1 be a set of representatives of the antipodal blocks from ~2 (x). Let us construct a matrix M = llmijllkxk by the following rule. The rows are indexed by the vertices Z1 ,z2, ... , Zk Of ~(X); if either j =kOr d(Zj,Yj) = 1 u:s;; k -1) then mij = 1,

368 A. A. IVANOV

otherwise mij = -1. Then M is a Hadamard matrix of order k. This implies, in particular, that if k > 2 then 4 divides k.

lf ~ is a DTG of type 2. Kk.k then the automorphism group of the corresponding Hadarnard matrix M acts doubly transitively on its rows and columns and M is Mequivalent to its transpose. All matrices possessing these properties are known (see Theorem 7.6.5 in Brouwer et al. (1989)).

Case 2: l = k. A graph of type k. Kk.k is equivalent to a projective plane of order k

with a distinguished flag. This equivalence was established in Gardiner (1974).

Let 1t be a projective plane of order k and (p,l) be a flag where p is a point and l is a line. Let ~ be the subgraph of the incidence graph of 1t induced by the points which do not lie on l and by the lines which do not pass through p. Then ~ is a graph of type k. Kk,k and the structure of 1t can be reconstructed from A

If 1t is Desarguesian then ~ is a DTG. The classification problern for planes corresponding to distance-transitive graphs of type k. Kk.k is very interesting and nontrivial. A number of necessary conditions for the existence of such a plane are given in Gardiner (1974). As a result of these conditions the plane must be a translation and dual translation plane, i.e. it must be coordinatized by a semifield. This semifield should also possesses certain properties. In Chuvaeva & Pasechnik (1990) it was proved that these conditions are also sufficient for distance-transitivity of the corresponding graph. Before presenting this result we recall some definitions.

A finite semifield Q is a finite algebraic system containing at least two elements. Q

possesses two binary operations, addition and multiplication, which satisfy the following axioms.

Al. n is a group with respect to addition with a neutral element 0.

A2. If ab = 0 then either a = 0 or b = 0.

A3. a(b +c) =ab+ ac, (a +b)c = ac + bc.

A4. There is an element 1 E n such that la = a 1 = a.

Here a,b,c are arbitrary elements of n. Suppose that Q and 3 are two semifields. An isotopism from Q to 3 is a triple

(a,ß,y) of non-singularlinear transformations from n to 3 (both considered as vector spaces over their prime subfields) such that xa* yß = (.xy)Y for all x,y E n. Here * denotes the multiplication in 3. An isotopism of n onto itself is called the autotopism. An anti-automorphism of n is an involutory permutation e of its elements, such that (X +yl = yE +XE and (.xy)E = yE XE.

Now we can formulate the main result of Chuvaeva & Pasechnik (1990).

Theorem 6.8.4. Distance-transitive graphs of type k. Kk,k are in one-to-one correspondence with nonisotopic semifields n of order k with the properties:

(i) n possesses an anti-automorphism;


(ii) the autotopism group of the projective plane 7t coordinatized by Q with respect to points 0, U and V is transitive on the points of lines OU, OV and UV distinct from 0, u~~ o

Notice that Theorem 6.8.4 implies, in particular, that DTG's of type k. Ku exist only if k is a prime power.

A nontrivial class of semifields satisfying the conditions in Theorem 6.8.4 is that constructed by Albert (1958). These semifields are known as twisted fields.

Just recently the complete classification of the DTG's of type k. Ku was obtained in Liebler (1991).

Theorem 6.8.5. Suppose r is a DTG of type k. K~c,k and a is the associated projective plane. Then 7t is either Desarguesian or it is coordinatized by a twisted field of Albert. o

In Hensel (1988) it is shown that starring with a graph k. K~c,k one can sometimes construct a graph l· K~c,k where l is a proper divisor of k.

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1. Introduction

ON SOME LOCAL CHARACTERISTICS OF DISTANCE-TRANSITIVE GRAPHS

A.V. Ivanov

The definitions, notations and preliminary information conceming distance-regular graphs (DRG) and distance-transitive graphs (DTG) can be found in the papers [1], [2],

[3], [5].

Among the problems involving classification and construction of DTG's a traditional one is the problern of constructive enumeration of DTG's of a given valency k. At the first stage of such an enumeration the intersection arrays satisfying certain necessary conditions (the feasible arrays) are searched. Usually this stage consists of the enumeration of all possible ways to construct an intersection array i (r) with a fixed beginning. Some known combinatorial conditions enable one to reduce the number of possibilities to be considered. These conditions, in particular, imply that the sequence {b;} be nonincreasing and the sequence { ci} be nondecreasing. For each intersection array which is subsequently constructed, certain stronger algebraic conditions are then checked. These conditions are formulated in terms of eigenvectors of a tridiagonal matrix of order d + 1 consisting of the numbers from the intersection array i (r) [1], and they require a large amount of computation. The second stage, namely the construction of the DTG's with the given intersection array i (r), is a separate problem.

In certain cases (see, for example [1]) the nonexistence of a graph with given inter

section array is proved by arguments which are ad hoc and cannot be applied to other sets of parameters. In [9] an attempt to work out some general methods for constructing DTG's and for proving their nonexistence was made. But in the practical application of these methods, certain special and quite delicate investigations of the possible automorphism groups of these graphs play a significant role. These investigations usually rely on some information about the structure of concrete primitive permutation groups.

The purpose of the present paper is to approximate the "global" property of a DTG (i.e. the transitivity of its automorphism group on pairs of equidistant vertices) by means of certain "local" characteristics which describe the embeddings in r of certain smaller configurations (subgraphs). In some cases the existence of such embeddings follows directly from the form of the set of parameters i (r). The characteristics are related by certain arithmetic relations which depend only on the parameters of the graph and can be calculated by a partially complete array i (r,1. In this case the existence of solutions to these relations is a necessary condition for the existence of a DTG with the corresponding parameters. This enables one to use said solutions at the first stage of the enumeration problem. These solutions can be used to detail the structure of a possible graph and to simplify its construction at the second stage.

379

380 A. V.IVANOV

The general method for generaring the relations is presented in Section 2. In the next two sections the particular cases are studied in details. These are the embeddings of complete graphs with edges of the same colour (Section 3) and the embeddings of nondegenerate cycles of ordinary edges (Section 4).

The technique described in this paper was successively used in the construction of the possible sets of parameters i (r) of the DTG's of valency 1<$. 13 [5], [6].

2. Embeddings of Subgraphs

For a graph G we denote by V ( G ), E ( G) and Aut ( G) its vertex set, edge set and full automorphism group, respectively. Besides simple undirected graphs we will consider complete coloured graphs. For these graphs E (G) is a map defined on the set V (G) x V (G) and taking values in N 0 = {0, 1,2, · · · }. This map assigns to an edge { v,u} its colour, denoted by col (v,u). We assume that col (v,u) = 0 if and only if v = u. It is natural to represent the map E (G) by the matrix lleijllnxn of ordern = I V (G) I for which eij=co[ (u;,Uj) for U;, Uj E V(G).

Let r be a DTG of diameter d with intersection array i (1). A complete coloured graph (which will also be denoted by n can be associated with r in a natural way. This coloured graph has the same vertex set as r and col(v,u)=d(v,u), where d(v,u) is the distance between V and u in the original DTG r.

Foravertex v E V(l) the set V(r) is divided into subsets f';(v), 0~ i ~ d, suchthat f';(v) consists of the vertices which are adjoined to v by edges of colour i. It follows from distance-transitivity that each vertex u e 1 1(v) is adjoined via edges of colour i to exactly p); vertices from the set f'j(v). The value pg, which is equal to I f';(v) I, is denoted by k;, 0 ~ i ~ d. So kt = k is the valency of the DTG r.

As was already mentioned, in some cases the form of the intersection array implies that r contains a certain subgraph G. Let us give a few such examples.

Example 1. Let pfj be nonzero. Then r contains a triangle with edges of colour i, j, l.

Example 2. Let p~; = 0 for 1 ~ i ~ d -1, i.e. 1 is an antipodal graph [1], [2]. Then r contains a complete graph on (kd + 1) vertices with all edges of colour d.

Example 3. Let c 2 = 1. Then r contains a complete subgraph on (a 1 + 2) vertices with all edges of colour 1.

Example 4. [6] If (c,_1 ,a,_1 ,b,_1)-# (c,,a,,b,) and c, = 1, then r contains a nondegenerate cycle of length 2r + 1. Here, by a nondegenerate cycle, we mean a complete coloured graph in which the edges of colour 1 form a cycle and col (v,u) is equal to the number of edges in a shortest path which connects v and u and contains only edges of colour 1.

ON SOME LOCAL CHARACTERISTICS OF DIST ANCE-TRANSITIVE GRAPHS 381

Example 5. [2] Let c 1 = c2 = C3 = 1, a 1 = 0, a2 = 2. Then r contains a Petersen subgraph.

These examples, the list of which can certainly be extended (see for example [2]), stimulate interest in the existence problern for a DTG r with a given intersecnon array i(I) and which contains subgraphs isomorphic to a given graph G.

Let r be a DTG of diameterd with intersection array i(I) (we consider ras a complete coloured graph) and let r contain G as a subgraph. It should be mentioned that a necessary condition for these assumptions is the validity of the following inequalities (triangle inequalities) on the elements of the matrix E (G): I eiJ -eik I ~ e1k for 1 ~ i,j,k ~ n

where n = I V(G) I.

Let G denote the set of subgraphs of r which are isomorphic to G_ To each element G' of G an injective map «1> of V(G) into V(I), which preserves colours of edges, can be associated- Let us fix a vertex v e V(I). Then for a subgraph G' e G, a map 'II of V(G')

into the set D = {0, 1, · ·- ,d} can be defined as follows: 'II: u H d (v,u) for u e V(G').

The composition '1'«1> of the maps «1> and 'II is a function defined on the set V (G). This function will be called the position of the graph G in the graph r with respect to the vertex v and the subgraph G'. The function '1'«1> can be presented in the form of a matrix

A = A ('JI«l>) = llaijllnxn whose entries are

{eiJ if i =!:-}

(2.1) aij = 'l'cl>(ui) if i =}.

Let A be the set of all possible positions of G in r, Le. the set of all functions from V(G)intoD; lAI =(d+1)n_

The group Aut (r) acts on the set A in a natural way. Positions which lie in the same orbit under this action will be called equivalent. Subgraphs G 1 and G 2 are said to have the same position with respect to the vertexvif G 1 = «1> 1 (V(G)), G2 = «1>2(V(G)) and the positions z 1 =w1 c1>1(V(G)), z2 ='J12«l>2(V(G)) are equivalent. Equivalence implies the existence of an element g e Aut(G) suchthat

'1'1 «1>1 (u) = '1'2 «1>2 (uK)

for all u e V(G)_

It should be mentioned that not all positions from A can be realised in the graph r_ In fact, a position '1'«1> for example determines the subgraph generated by the set V ( G') u { v } up to isomorphism_ At the same time the colour of an edge in the graph r is equal to the distance between the corresponding vertices in the original DTG_ Hence for a triple {ui,UJ,v} which consists of a pair Uj,Uj e V(G') and the vertex v, the following inequality ( triangle inequality) holds:

382 A. V.IVANOV

I col (v,ui)- col (v,uj) I :5: col (ui,uj)·

By the definition of 'I' we have \jf(Ui) = col (v,ui) = aii· So the above inequality can be rewritten in the following form:

(2.2) I aii- ajj I :5: aij, 1 :5: i,j :5: n.

Moreover, it is clear that if a position '1'<1> is realised in r then the diagonal of the matrix A ('1'<1>) contains at most one zero element. This follows from the fact that <1> cannot map more than one vertex from V ( G) to v.

Let A' be the subset of A consisting of all positions '1'<1> for which there is at most one zero on the diagonal of the matrix A ('1'<1>) and for which the inequalities (2.2) are satisfied. It is easy to see that each equivalence class is either contained in A' or disjoint from A'. Let us choose in an arbitrary way one representative from each equivalence class from A. LetZ= Z (G) be the complete collection of such representatives. In what follows we will assume that Z = {zi I i e I}, and to a position zi e Z the subset Gi of G

consisting of all subgraphs which realise Zi is associated.

Let Aut(zi) be the automorphism group of the position zi = 'l'i t\li, i.e. the subgroup of Aut(G) which consists of all automorphisms of G which preserve the function Zj. Let { Oij I j e J (i)} be the set of orbits of Aut(zi) on V (G) and put Sij = I Oij I. It is clear that the function zi is constant on each orbit Oij· This constant will be denoted by z(Oij)·

Let Oij be an orbit of Aut(zi) on V (G) with z(Oij) = l. Let us fix a vertex u e r 1(v) and let Gij denote the subset of Gi consisting of all subgraphs G' such that u e V (G') and <l>i1(u) E oij· The cardinality of the set Gij will be denoted by Xij· Since r is distance-transitive the numbers {Xij} do not depend on the particular choice of the vertices V E V (r) and U E r/(v ).

Let w e V (G). The set A' contains a position ziw suchthat zi)t) = col (w,t) for all

t e V (G). It is easy to see that positions Ziw and Zip are equivalent if and only if the ver

ticeswand p are contained in the same orbit of Aut(G) on V(G). Without loss of generality we can assume that Ziw e Z. It is clear that Aut (ziw) coincides with the stabilizer of

the vertex w in the group Aut(G).

Let us consider a subgraph G' e G which realises a position zi e Z, zi = 'l'i <l>i, and let u e V(G') be a vertexsuch that cpj1(u) e Oij· Lethe Aut(r) map u onto v. Then h maps G' onto some other subgraph G" e G which realises a position which is equivalent to the position ziw for w e Oij· Forthis reason the set V(G') ofvertices of G', along with

the set V (G) of vertices of G (notice that cpj1 (V (G')) =V (G)), will be considered simultaneously under the two positions zi and ziw.

Now consider the orbits of Aut(zi) on the set V(G). Let p be the number of those

orbits which do not contain the vertex w and let g = Ziw(Oiwp), t e D. Let r(i,j,p,t)

denote the number ofvertices S in QiwP SUChthat Zj(S) = t:

Forw e V(G) and/ e D weput

z~ = {zi 1 Zj e z, zj(w) = /}.

So Z~ is the subset of Z consisting of all functions whose value on the vertex w e V (G) is equal to l. For Zj and w E V(G), the index jw will be assigned to the orbit oij" of

Aut(zi) on V(G) which contains the vertex w.

It should be mentioned that the complete collection of possible positions, as well as the parameters Sjj, r(i,j,p,t), do not depend On the graph r but are determined by the graph G and the value of d.

Theorem 2.1. Let r be a DTG which contains G as a subgraph. Then the following assertions are valid:

(i) Xij k1 I Sij is an integer for Zj E Z, V' j and l E Zj(Oij);

(ii) Xjj kz I Sij = Xiskt I Sis for Zj E Z where t = Zj(Ois), l = Zj(Oij); (2.3)

(iii) !, r(i,jw,P,t)Xij" =p~1 xi"p for w e V(G), for any p such that Z;E Z~

w ~ Oi"p• 0~ l, t~ d, g = ziw(Oi"p);

(iv) if r(i,jw,P.t) ~ P~t for 0 ~ l,g,t ~ d, then Xij" = 0.

Proof. By construction, I Gi I = Xij k1 I sij for I= zi(Oij)· This immediately implies (i). In addition, the nurober I Gi I can be obtained in other ways by consideration of vertices from different orbits of Aut(zi). This leads to (ii).

Let us fix the vertices u e r 1(v) and w e V(G). Recall that the index iw is assigned to the position zi" such that zi" (t) = col (w,t) for t e V (G). For a position zi e Z~ (here

1 ~ i ~ IZ~ I) and an edge eh of colour g = Zi" (Oi"p) connecting u with a vertex

uh E rz(V) (here 1 ~ h ~ P~z), we put

fhi(l,g,t,w,p)= I{G' I G'e G,u,uh e V(G')<pj1(u)=w, cpj1(uh)e Oip} I.

It is clear that

fhi(l,g,t, w',p) = fhi(l,g,t, w",p)

for w',w" e Oij. In view of this fact we will write fhi(l,g,t,jw,p). Let us construct a matrix F(l,g,t,jw,p)=llfhi(l,g,t,jw,P)II of size p~1 X IZ~I(1~h~p~1,1~i~ IZ~I). Notice that because of the transitivity of Aut (r) on the set of edges of colour g, and from the definition of the numbers (Xij ), each row sum of this matrix is equal to xi"p. Thus we

have the following:

384

(2.4) I, fhi(l,g,t,jw,P) =xi"p. 2je z~

A. V. IYANOY

Now let us sum the elements in the i-th column of the matrix F(l,g,t,jw,p). By means of this summation we will calculate the number of subgraphs of r isomorphic to G which realise the position Zj E z~ and suchthat <l>i1(u)=w, w E oij"· Each of these

graphs will be met exactly r(i,jw,P,t) times via edges of colour gwhich adjoin u E rl(v) to vertices in rr(v ). In this way we obtain the following expression for Xij":

p~ (2.5) Xij" = I, fhi(l,g,t,jw,P) I r(i,jw,p,t)

h=l

It follows from (2.4) that the sumofall elements of the matrix F (l,g,l,jw,p) is equal to P~r Xi"p. This fact and equality (2.5) imply that

Thus we have proved (iii).

The values of some variables Xij can be determined directly. For instance, if for some t the inequality r(i,jw,p,t) > P~r holds, then Xij" = 0. In fact, if Xij" '# 0 then it follows from the definition of the map Zj, and from the definition of the number r(i,jw,p,t), that the vertex u = <l>i(w) is adjoined by edges of colour g to at least r (i,jw,p,t) vertices from the set r 1(v). But in the graph r there are only P~r such edges, a contradiction. This implies (iv). So the theorem is proved completely. D

The relations (2.3) hold for an arbitrary subgraph G of the DTG r. If Gis a concrete graph then usually some additional restrictions can be imposed on these relations. The first of these concern the possible values of the parameters Xi"p. This will be demonstrated in subsequent sections of the paper.

Let us make some remarks conceming a strategy into which relations (2.3) can be incorporated.

1. It is clear that, for an arbitrary subgraph G of the DTG r with intersection array i (r), it is sufficient to include in a system S of the form (2.3) only the relations (2.3 iii -iv) expressed for a set w 1, w2, · · · of vertices of V (G) which lie in different orbits of Aut(G) on V (G). Indeed, since all arguments carried out concern the action of Aut(G) on V(G), relations corresponding to vertices from the same orbit coincide. This means that the system S can be regarded as a set {S 1 ,S 2 , · · · ) of subsystems. Here the subsystem Si consists of the relations (2.3 ii), and relations (2.3 i, iii-iv) expressed for a vertex from the i-th orbit of Aut(G) on V (G). If the i-th orbit of Aut (G) on V (G) contains more than one vertex then the subsystem Si should also contain some relations of the form (2.3 ii)

expressed for vertices which are in the same orbit of Aut(G) on V(G) andin different


orbits of the automorphism group Aut(zj) of the position Zj· Since the set of variables of the subsystems (2.3 ii) are mutually disjoint, these subsystems can be solved independently. Afterwards, the solutions should be compared via the relations (2.3 ii) which were not inserted in any of the subsystems Si e S.

2. Suppose that certain variables Xij (1 ~ i ~ I Z I) are determined in the solution. Then from the relations (2.3 ii) the values Xit, for all t = 1,2, · · ·, can be determined. This enables one to organise the solution effectively in a recursive way. Indeed, one can write down and solve the relations of type (2.3 iii-iv) for different w e V (G) and for I increasing from 1 to d. If the set of possible values of the variables Xij" for Zi e Z~ is

found, then by means of the relations (2.3 ii) the set of possible values of any variable Xij, j = 1,2, · · ·, such that zi(Oij) ~ zi(Oij) =I, can be found. So when we consider the

relations for Z~, it is sufficient to search only for possible values of the variables Xij"

such that Zi(Oij) ~ zi(Oij). For the remaining variables the sets of possible values were

found earlier.

3. Since, for the recursive solution to the system (2.3), we need at each moment only the sets Z~, it is unnecessary to write down the whole setZ of all possible positions if there is a way to obtain Z~ by special methods. It is obvious that

d

Z= U Uz~ weV(G) 1=0

where the intersection Z~ n Z~ is generally nonempty. So considering all subsets Z~ we will not loose any of the information contained in (2.3). Finally, it should be mentioned that if it is possible to write down the sets Z~,Z~, ... ,Z~, then it is useful to number the elements of these sets independently. In the event that an element zi e Z is contained simultaneously in Z~ andin Z{:, where p '#I, then the numbers assigned to zi in these sets can differ. If necessary, it is easy to change from one numeration to another. In what follows, if the numeration corresponding to the set Z~ is used, then the top index I, w will mark the corresponding positions (zf·w) and variables (xf:j).

The ideas presented in 1 - 3 above enable one to organize the calculations involving relations (2.3) in such a way that, in some cases, the nonexistence of an integer solution to the system (2.3) can be recognised even before the system is written down completely. This fact enables one to use Theorem 2.1 during the search for possible intersection arrays of a DTG (see [5]). Specifically, we write down the system (2.3) for a partially constructed array i (r) and determine its solutions. If there are no nonnegative integer solutions then we need not continue the construction of the array i (r) since, in any case, the necessary conditions formulated in Theorem 2.1 will not be satisfied. To use Theorem 2.1 in this manner, it is necessary to have an algorithm for calculating the intersection numbers which appear in the system (2.3) corresponding to a partially constructed intersection array. Such an algorithm is given by the following Iemma which will be formulated without proof.

386 A. Y.IYANOV

Lemma 2.2. [8] The intersection numbers satisfy the following recursive formulas:

1 1 {1ifi=l PiO=POi= Oifi::;:.l;

C[ if i = [-1

a1 if i =I PI -pl _

i 1 - 1i- b1 if i =I+ 1

0 otherwise

It shou1d be mentioned that some recursive formulas for computing all intersection numbers from the intersection array were known earlier (see for example [1]). But the advantage of the formulas presented in Lemma 2.2 is that they can be used also for a partially constructed intersection array i (r). In this case if g is the length of the constructed portion of the intersection array i(r), then the numbers Pt can be calculated for all indexes i,j, I which satisfy the inequality i + j +I :;;; 2g + 1. The latter inequality ensures that a path of length j from a vertex u E r 1(v) to some vertex in the set ri(v) will not contain a vertex which is of distance more than g from v. On the other hand, the number of paths satisfying that condition can be calculated from the initial "length g" portion of the intersection array. If in addition bg = 0, i.e. g = d, then the presented formulas can be applied for all indices 0:;;; i,j,l:;;; d.

The construction of the setZ and the determination of the numbers r(i,j,p,t) is significantly easier if Aut(G) acts transitively on the set of vertices and on the sets of edges with the same colour. In the remairring sections of the paper we will consider two such cases. In this situation, because of the transitivity of Aut ( G) on V ( G ), the index w in the notation Z~ will be omitted.

3. Embeddings of Coloured Cliques

Let G be the clique (complete subgraph) K;:, with m (m~ 3) vertices whose edges are of colour n (1:;;; n:;;; d). Since in this section we will work with cliques with different numbers of vertices, the pair (n,j), 1 :;;; j :;;; m, will be assigned to the variables introduced above.

The index i * will mark the position zt E Z (n,m) such that 'Vi* $i• = n for each vertex w E V (K;:,). It is clear that if G = K;:, then Aut(G) =Sm- the symmetric group acting on m points. Now it follows immediately that subgraphs G 1. G 2 E G(n,m) realise equivalent positions if and only if, for each 0:;;; i :;;; d, there are the same number of vertices in the subgraphs G 1, G 2 adjoined to v in the graph r by edges of colour i. This enables us to use the following simple method for constructing the set Z 1(n,m), o::; I:;;; d.

ON SOME LOCAL CHARACTERISTICS OF DISTANCE-TRANSITIVE GRAPHS 387

Recall that the index w in the notation Z~ above is omitted in view of the transitivity of Aut (K;:,) on V (K;:,).

With each position zf(n,m) E Z 1(n,m) we associate a vector ri(I,n,m) = (a~) with 2n + 1 integer coordinates, where

a}= l{u I UE V(K;:,),Zj(U)=l+j-(n+1))1.

Then the following conditions hold:

2n+1 . L aj =m; j=1

(3.1) (X~+1 ~ 1;

a~ a~ ~ 0 ~ lp -q I::;; n.

The first condition ensures that graph G' = cp(K;:,) has exactly m vertices, the second one- the presence of the vertex u E V(G') in the graph r 1(v), and the third one- the absence in G' of vertices w 1 , w 2 E V ( G') such that

W1 E rl+p-n-1 (V), W2 E n+q-n-1 (V). (If SUCh vertices W1, W2 exist, then in view Of the obvious inequality

col(w~ow2)~ 1(/+p-n-1)-(l+q-n-1)1 = lp-ql

the distance between these vertices is greater than n.)

If we obtain nonnegative integer solutions to (3.1), then we enumerate all possible vectors { ( a})). Each of these vectors uniquely determines a position of the graph K;:, in r with respect to the vertex v.

It should be mentioned that for G = K;:, each orbit of Aut (zi) is characterised by the value t = zi(Oi1), and the action of Aut(ziJ on V(K;:,) has only one orbit which does not

contain w E V (K;:,). This orbit will be marked by the index Pm· Now it is clear that the values SiJ• r (i,j,pm, t) can be easy calculated from the Coordinates of the vector ai(l,n,m ):

.. -{ <X~-I+n+1 if 1/-t I::;; n where t = zi (Oij)

s,, - 0 otherwise ;

r(i, j, Pm• t) = Sjj- Ölt,

where ölt is the Kronecker symbol.

As an example let us write down the vectors ai(l, 1,3) corresponding to the positions zf(l,3) E 2 1(1,3), o::;; l::;; d, i = 1,2, · · · ,5. At the same time we enumerate the orbits of the action of Aut (zf(l,3)) on V (K;:,). Such an orbit is empty if aj(l, 1,3) = 0.

388 A. V.IVANOV

aW, 1, 3)

~ 1 2 3

1 2 1 0 2 1 2 0 3 0 3 0 4 0 2 1 5 0 1 2

The value Xi,.p,. (n,m) for the unique orbit which does not contain w e V (K~) can be easily obtained if one takes an edge of colour n adjoining the vertex v to a vertex in rn(v) and recalls that there is only one orbit (say the orbit number /) of Aut(zj") on V (K~) suchthat z(Ott) = n.

(3.2)

Proposition 3.1.

Xi,.p 3 (n, 3) = P:!n;

Xi,.p,.(n,m)=Xtj*(n,m-1) form> 3. 0

Notice that form > 3 there is a spectra of possible values for xi .. p,. (n,m). These values arise during the recursive solution to the system of the form (2.3) (see below) for m=3,4, ···.

A position zq(n,p) e Z(n,p) is said tobe embedded in a position zi(n,m) e Z(n,m) if the function zq(n,p) is the restriction of the function zi(n,m) to a suitable subset of cardinality p. In this case we write zq(n,p) c zi(n,m).

The following assertion is obvious.

Proposition 3.2. Foreach number j orbit of the action of Aut(zi(n,m)) on V(K~) we have

(3.3) Xij(n,m) = 0 if Zq(n,p) c zi(n,m) and Xqs(n,p) = 0

for some number s orbit of Aut(zq(n,p)) on V(K;). 0

For the balance of this section it is assumed that the relations (3.2) and (3.3) are added to the system (2.3). In this manner it is possible to organise the testing of the necessary conditions for the existence of a DTG containing the coloured cliques in a recursive way. For this purpose one should solve the systems of the form (2.3) for m = 3,4,5, · · ·, using the relations (3.2) and (3.3) and substituting the solutions obtained at the previous step. Then all solutions to the system (2.3) obtained for varying m can be ordered in the form of a tree U. In this tree each vertex a(Xij(n,q)) e V(U) corresponds to some solution (Xij(n,q)), 1::.;; i::.;; IZ(n,q) I, j = 1,2, · · ·, and vertices a(Xij(n,q)) and ß(Xij(n,p )) are adjacent if and only if q = p + 1 and (Xij(n,q)) is a solution to the system (2.3) obtained from Substitution of (Xij(n,p )) in the relations (3.2), (3.3).

Now the existence in the tree U of a terminal vertex corresponding to a solution (Xij(n,m)), suchthat xt r = 0 is a necessary condition for the existence of a DTG. Let us find all such terminal vertices in the tree U and all paths connecting these vertices to the root. These paths describe the possible embeddings in r of the coloured triangles, tetrahedrons , · · · , maximal coloured cliques.

The methods described in this section were first applied to the pararneters ofDTG's from Biggs' list [1]. Recall that in [1] the problern of constructive enumeration for DTG's of diameter d ~ 5, k ~ 13, and with primitive automorphism group was con

sidered. This problern was reduced to the existence problern for graphs with 11 possible sets of parameters. The complete solution to Biggs' problern was obtained independently by two groups of mathematicians (see [9] and the references in [7]). It turned out that for

four sets of parameters DTG's do not exist, and for seven sets such graphsexist and are unique. In spite of the fact that, at the moment when the methods presented in this paper became available Biggs' problern had already been solved, the application of these methods gave some good results. In a unified manner the nonexistence of three DTG's was proved. Also, for five existing DTG's the subgraphs induced by the vertices which are at distance at most d - 1 from a fixed vertex were described up to isomorphism.

Biggs' list contains the parameters <k,b 1 ,b2; c1>c2,c3 > = <10,8,7; 1, 1,4>. If a DTG with this pararneters exists then k1 = 10, k2 = 80, k3 = 140. Let us write down the equations for n = 1, m = 3, which describe the embeddings of triangles of ordinary edges. We will use the enumeration of the positions (z!(1,3)) and of the orbits of Aut(zi(1,3)) on the corresponding sets as in the example given above.

By (2.3 iv) or (3.3) we have

(3.4)

By (3.2) Xiwp, = a 1 = 1. This, along with relation (2.3 iii) for I = 0, implies that

xg,2 = 5, while relation (2.3 ii) implies that xb = 1. Let us write down the set of relations (2.3 üi) for l = 1; in view of (3.4):

xb =1

xb+xb =1

xl.2 +2xb =8. These equations imply that xb xl.2 = 0, x!.2 = 4. This agrees with the value of xb obtained above. Since xl.2 and xb are known we obtain from (2.3 ii) that xi,2 = 0, xi2 = 1. Now let us consider the equations (2.3 iü) for I = 2

390 A. V. IVANOV

2x2 2 1.2 +x2,2 =1

xi2 + 2xh +x~.2 = 2

x~.2 +2xl2 =7.

Using these equations and the values of xf.2 and xh obtained above, we have xb = 0, x~.2 = 1, xl2 = 3. Now it follows from (2.3 ii) that

xl2 =x~.2k2/(2k3) = 1 x 80/ (2 x 140) = 217.

The absence of an integer solution to the system (2.3) proves the nonexistence of the graph in question.

In an analogaus way, for the same parameters n and m, the nonexistence of a graph with parameters <13,10,7;1,2,7>, <10,5,4,2;1,2,2,10> and with, respectively, 144 and 96 vertices can be proved. For the set of parameters <12, 10,2;1,2,8> from Biggs' list (a graph on 88 vertices) the system has a unique solution for n = 3, m = 3 and has no nonnegative integer solution for n = 3, m = 4. This proves the nonexistence of such a graph.

4. Positions of Nondegenerate Cycles

An ordinary cycle C which is a subgraph of r will be called nondegenerate if the length of a path connecting two vertices in the cycle which lies entirely within the cycle does not exceed the distance between the vertices in r.

It is proved in [6] that the condition

(4.1) (cr-1 ,ar-1 ,b,_1) '1:- (c"a"b,), c, = 1

is sufficient for the existence in r of a nondegenerate cycle of length l = 2r + 1. U sing the methods developed in [6] it can be shown that the condition

(4.2) (c1>a1>b1) = (c,_l>ar-l>br-1) '1:- (c"a,,b,), c, ~ 2

is sufficient for the existence in r of a nondegenerate cycle of length l = 2r. In what follows it is assumed that either (4.1) or (4.2) is satisfied. We will consider nondegenerate cycles of length 2r + 1 and 2r, respectively. Certainly the graph can contain nondegenerate cycles of other lengths.

Following the methods of Section 2, let us consider as G the set of all subgraphs of r which are isomorphic to the cycle C1 of length land which are nondegenerate in r. First we construct the set Z1 = Z1(CI) of alldifferent positions of C1 in r. For increasing l, the cardinality of the set Z1 grows very quickly, and it is impossible to write down the set Z 1 by band. For the construction of Z1 the algorithm described below can be used. The basis for the algorithm is the following metbad of coding positions of cycles.

Let G' be a subgraph from G which realises the position zl E Z1,

u E rl(v), 4>-1(u) E Oij· Starting at the vertex u let us define a vector a 1 = a1(i,j) as

follows. We fix one of the two orientations of the cycle. Then the coordinates of the vector correspond to the arcs of the (directed) cycle G'. Moreover the first coordinate corresponds to an arc incident to u, and nl and <J.f+1 correspond to arcs which have a common vertex. The vector a1 is defined as follows. The s-th coordinate a! corresponding to the arc (us_1 ,us) is equal to d (us, v)- d (us_1, v ). It is clear that the Coordinates can take only three possible values: 1, 0 and -1. The vector a 2 = n2(i,j) is defined analogously via the inverse orientation of the cyde. Notice that one of the vectors a 1 and a2

can be obtained from the other by means of the following procedure. One multiplies all the coordinates by -1 and then reverse their order. This procedure will be called inversion. The larger of the vectors a 1, a 2 with respect to the lexicographic order will be called the code of the position zf related to the j-th orbit of the action of Aut(zf) on V (Ce). It is easy to show that the sum of the coordinates of the vector n(i,j) is equal to 0 and that the position z! can be uniquely reconstructed from this vector.

Thus, to construct the sets Z 1 of all different positions of C1 in r one should enumerate all vectors of length l whose coordinates are -1, 0, + 1 such that the sum of the coordinates is equal to 0 and such that the inversion does not produce a 1exicographically larger vector.

If a is a vector satisfying the above conditions, then the vectors <J.p which can be obtained from a by means of cyclic translation by p vertices ( 1 :5: p :5: l - 1) followed by, if necessary, inversion for maximization, also satisfy these conditions. This fact simplifies the enumeration problern for such vectors.

Moreover it is easy to show that si is equal to the number of coincidences of the vector n(i,j) with the vectors ap(i,j), (1 :5: p :5: /), obtained by cyclic translation of the code n(i,j) by p elements, and perhaps by inversion for maximization.

The numbers r(i,j,p,t) are also determined by the code n(i,j) = (ns). Let n = zi.., (Oi..,p)· Let us determine the numbers

Now

. n . n ß'n = L <J.s, '),}n = L <J./+1-s ·

s=1 s=1

if ßin =')..in and t = Zj(Oij) + ßin

if ßin "# /..in and t = zi(Oij) + ßin or t = Zi(Oij) +/..in

otherwise.

The following arguments, which are carried out under the assumption that either relation ( 4.1) or ( 4.2) is satisfied, enable one to find values for some of the variables in the system (2.3) and to obtain upper bounds for others.

Each graph G' E G which contains the vertex v realises a position zi..,. This immedi

ately implies the following.

392 A. V.IVANOV

Proposition 4.1. Suppose that the parameters of the DTG r satisfy either (4.1) or (4.2). Then the equality XiwP = yk, I kt holds for f= Ziw (Oiwp) and

{ a - a 1 for odd l

y = c:(c,-1) foreven land r > f c,(c,-1)/2 forevenlandr=f.

Here p is the integer assigned to the orbit of Aut (ziw) on V (Ce) under consideration. 0

Let the DTG r contain nondegenernte cycles of length l and set ß = (ß;), ßi E {0, ±1}, 1 ~ i ~ n, where n = [l/2] + 1. Let us consider the set P = P(u,l, ß,m) of all distinct paths of length m, 0~ m ~ n, which originate at the vertex U E rt(V) and which satisfy the following property.

over

Let P E P, P =<Po, p 1, ... ,pm). Then it is assumed that

i Po= u, PiE rr,(V) where ti = 1 + L ßs,

s=1

0~ i ~ m, Pi+1 E r1(pi), 0~ i ~ m-1, Pi ~Pi+2• 0~ i ~ m-2.

It is clear that I P(u, l, ß, 0) I = 1.

Proposition 4.2. IP(u,l,ß,i) I= IP(u,l, ß,i-1) I x (yi-ei) for each 1 ~ i ~ n. More-

{Ci if ßi = 1

Yi = ai if ßi = 0

bi if ßi=-1,

where

i

ti = 1 + L ßs s=1

and

The proof is carried out by induction on the length of the path. Each of the paths contained in P(u,l, ß,i -1) can be extended in Yi ways to a path of length i and the value Yi is related to ßi· If i ~ 2 and ßi-1 + ßi = 0, then for one of the paths, Pi= Pi-Z and this path should be excluded. This proves the proposition. o

Proposition 4.3. Let the parameters of a DTG r satisfy either (4.1) or (4.2). Let a.1

be the code of a position z; related to the i-th orbit of Aut(z;) on V (C') and let a.2 be the

ON SOME LOCAL CHARACTERJSTICS OF DIST ANCE-TRANSITIVE GRAPHS

inverse ofthe code a1. Let n = [1 /2] + 1, 1 = zi(Oij), u E r1(v). Then

Xij5. h· min(l P(u, 1, a1,u)l, I P(u, 1, a2,n) I)

where

-{ 112 if a1 = a2

h - 1 otherwise .

393

Proof. Foravertex u E r 1(v) 1et us consider the set G' of subgraphs of the graph r realising the position zi = 'l'i <Pi such that for any G' E G', we have u E V (G') and cpj1(u) E Oij· It is obvious that each graph from the set G' contains as a subgraph a path from the set P(u,1, ai ,n), i = 1,2. Let us show that if (4.1) or (4.2) is satisfied, then distinct graphs from G' contain distinct paths :from P(u,1, ai,n), i = 1,2. In fact, suppose that for i = 1 or 2 some path PE P(u,1,ai,n), P = (po.PI> ... ,pn), Po= u, is contained in two graphs G 1 ,G 2 E G'. Since d(u,pn) = [(l-1)/2] it follows from either re1ation (4.1) or (4.2) that the shortest path between u and Pn is unique. The equality G 1 = G2 follows immediately. If a 1 = a2 then P(u,1, a 1 ,n) = P(u,1, a2 ,n), and each graph GE G' contains two paths from P(u,1, a 1 ,n). This enables us to include the multiplierhin the bound for Xij· The proposition is proved completely. o

The following remarks concern the application of (2.3), under the restrictions (4.4), to the determination of the possible positions of the cycles C1 of length 1 in a DTG r.

If e > 5 then the set of possible positions Z 1 is too large. This implies that the system (2.3) contains many variables and, as a consequence, almost always has a nonnegative integer solution. In this respect the question about the effectiveness of the necessary conditions arises.

Experience of regarding the application of conditions (2.3) to the case of nondegenerate cycles for the classification of DTG's of valency up to 13 (see [5], [6]) shows that the effectiveness of these conditions depends on the initial portion of the intersection array i (r). In some cases application of (2.3) becomes effective given as few as 5-6 columns of the array i (r), while without conditions (2.3) enumeration may have to be carried to the level of 18-20 columns. (In the latter case, the computational time is about 100-150 times as great). But counter-examples arealso known. Indeed, it is sometimes the case that the time required for generaring and solving the system (2.3) is prohibitive, due to the existence of too many solutions. In such situations, it may be true that the initial portion of the intersection array possesses a small number of possible completions, and that these possibilities can be eliminated by means of some simple tests.

The application of (2.3) in the case of nondegenerate cycles of length more than 7 seems to be unreasonable since the efforts required do not seem to yie1d positive effects.

394 A. V. IVANOV

References

1. N.L. Biggs, Automorphic graphs and the Krein condition, Geom. Dedic., 5 (1976),

117-127.

2. A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance Regular Graphs. Springer Ver

lag, Berlin, 1989.

3. LA. Farad.Zev, M.H. Klin, M.E. Muzichuk, Ce/lu/ar rings and groups of automor

phisms of graphs, [In this volume]

4. A.A. Ivanov, On 2-transitive graphs of girth 5, Europ. J. Comb., 8 (1987),

393-420.

5. A.A. Ivanov, A.V. Ivanov, LA. Farad.Zev, Distance-transitive graphs ofvalency 5, 6

and 7. Zhurnal Vichisl. Mat. i Mat. Fizik., 11 (1984), 1703-1718. [In Russian,

translated in: U.S.S.R. Comput. Maths. Math. Phys., 24 (1984), 67-76.]

6. A.A. Ivanov, A.V. Ivanov, Distance-transitive graphs of valency k, &;;k:<;;13. In:

Algebraic, Extremal and Metric Combinatorics 1986, London Math. Soc. Lect.

Note Ser. 131, pp. 112-145, 1988.

7. L.A. Kaluznin, V.L Suschanskii, V.A. Ustimenko, Use of a computer in the theory

of permutation groups and its applications, Kibemetika, 6 (1982), 83-94. [In Rus

sian]

8. P. Terwilliger, Distance-regular graphs and generalizations. Thesis, Univ. of Illi

nois, Urbana, 1982.

9. V.A. Zaichenko, A.A. Ivanov, M.H. Klin, Construction and investigation of certain

automorphic graphs, In Methods and programs for solution of optimisation prob

lems on graphs and networks, pp. 48-50, Novosibirsk, 1982. [In Russian]

ACTION OF THE GROUP M 12 ON HADAMARD MATRICES

I. V. Chuvaeva & A.A. Ivanov

1. lntroduction

During recent years, interest in geometric characterizations of finite simple groups and, in particular, of sporadic groups has grown significantly [4], [5], [20]. In this respect

a systematic investigation of the primitive representations of these groups is useful. In

[ 18], [ 19] the lattices of 2-closed overgroups of the primitive representations of the nonabelian simple groups of order less then 106 were studied with the help of computer calcu

lations. These papers contain a number o:f interesting results presented in the form of

computer output. So the problern to find a dear combinatorial-geometric interpretation of these results has arisen. As the simplest example of such an interpretation, a description

of the 2-orbits of the primitive representation of the group M 12 of degree 220 in terms of

the Steiner system S (5,6, 12) is obtained by I. V. Chuvaeva in [7]. In an analogous way,

but with more involved constructions, it turns out to be possible to describe all primitive

representations of the group M 12, with the exception of the representation of degree 144 on the cosets of a subgroup isomorphic to L2(11). Some fragments of this description can

be found in [7]. The difference between the~ primitive representation of M 12 on the cosets

of L2(11) and all of its other primitive representations is mainly due to the fact that

L2(11) is the only proper subgroup of M 12• up to conjugacy, which acts primitively in both representations of M 12 of degree 12. For this reason, the description in terms of the

Steiner system S (5, 6, 12), which was very simple for other primitive representations,

turnsouttobe too complicated in the case of the representation of degree 144.

It is known that M 12 has two nonsimilar transitive representations of degree 144.

Both of these representations are obtained by consideration of the actions of M 12 on the

cosets of certain subgroups isomorphic to L 2 ( 11 ). One of the representations is primitive

while the other one is imprimitive. These representations will be denoted by 1t1 and 1t2,

respectively. The representations 1t1 and 1t2 give rise to two primitive representations of the group Aut(M u) on the cosets of subgroups isomorphic to PGL2(11). Specialfeatures

of the representations 1t1 and 1t2 were known earlier (see, for example, [5]).

The main goal of the present paper is to obtain a combinatorial interpretation of the

representations 1t1 and 1t2. The exposition of the results in this paper follows the genetic

principle. At first we were not able to find an appropriate language for the description of these representations. Consequently we constructed the representations 1t1 and 1t2 in the

form of transitive extensions of the group L2(11). The construction of these extensions is

of independent interest, since here we have two different extensions with the same degree of the same abstract group. The algorithm for transitive extensions which we have used

is presented in Section 2 while the main steps in the realization of the scheme in the

situation considered are discussed in Section 3.

395

396 I. V. CHUV AEV A AND A. A. IV ANOV

While the representations 1t1 and 1t2 were almost completely constructed in the form of transitive extensions, it turns out that the extension of the group L2(11) to the representation 1t2 corresponds to the extension of the Paley design P (11) to the Hadamard matrix H (12) by means of well-known methods. This enables us to understand all steps in the procedure of the transitive extension and also to obtain a suitable description of the representation 1t1 (see Section 4).

In Section 5, a bipartite graph .1. with parts .1.1 and .1.2 of equal size, which admits M 12 as an automorphism group, is constructed. The action of M l2 on .1-i in this case is equivalent to the representation 1ti, i = 1,2. It is shown that .1. is an edge- but not vertextransitive regular biprimitive graph of parabolic type (see definitions in [ 17]) as weil as distance-regular (see definitions in [2]) with intersection array ( 12,11,10,7; 1,2,5,12} 1).

2. The General Scheme of Transitive Extension. The Fixed Points Method

Let H be a primitive permutation group on a set n and X E n an element fixed by all permutations from H. A group (G, Q) is said tobe a transitive extension of the group H if (G, Q) is transitive and the stabilizer G (x) of the element x coincides with H. The transitive extension problern is that of finding all the transitive extensions of a given group.

The transitive extension problern is too general and it is unlikely that there exists an effective algorithm for its solution. Nevertheless, if (G, Q) possesses certain additional properties the situation becomes more hopeful. We will assume that the group (G, Q) has a connected symmetric 2-orbit. The proposed scheme of transitive extension consists of the following two steps.

1. To find the V-ring of the group (G, Q) as a subring of the V-ring of the group (H, Q) and to construct the coloured graph r of the V-ring (see definitions in [10]).

2. To find an automorphism 't of the graph r such that x ~ = y, y ~ = x for y E n- { x} where the pair (x,y) is contained in a connected symmetric 2-orbit of the group G.

A simple example of the realization of this scheme is given in [ 18], page 20. In [ 16] the method of transitive extension was used in the construction of an automorphic graph (distance-transitive graph with primitive automorphism group) on 65 vertices. The existence problern for this graph was posed by N.L. Biggs. In that case, some of the calculations were carried out by means of a special computer program. A program which applies to the general situation is not yet available.

1) After the preparation of a previous version of the paper it became known to the authors that the graph .1. was independently constructed by D. Leonard. This fact is mentioned in (3] where this graph is characterized in terms of its intersection array.

ACTION OF THE GROUP M 12 ON HADAMARD MATRICES 397

A significant improvement is given by the so called fixed points method, which consists in the following. For some subgroups F of H we determine the set !1(F) = {z : z e n, zl = z 'rt f e F} of elements fixed by F, reconstruct the normalizer NG(F) and find the action of the lauer on the set !1(F). The theoretical basis for this method is the following lemma proved by J. Alperin [1], which generalizes some classical results of C. Jordan and W. Manning (see Section 2 in [21]).

Lemma 1. Let (G, Q) be a transitive permutation group, G (x) = H be the stabilizer of an element x e n and F be a subgroup of H. If H, in its action on the set M = {g - 1 F g ~ H, g e G } via conjugation, has r orbits M 1, M2, ... , M, then the normalizer NG(F) in its action on the set !1(F) ~ n of elements fixed by F has exactly r orbits !11, !12, ... , 11,. Moreover, I !1i I = [NG(Fi) : NH(Fi)] for Fi e Mi, 1 ~ i ~ r. 0

A subgroup F of H is said to satisfy the strong conjugacy condition if each subgroup F' of H whose action on Q is similar to that of F on Q is conjugate with F in H. In the case of strong conjugacy, Lemma 1 ensures the transitivity of the action of the normalizer NG(F) on the set !1(F), and to reconstruct this action one should find a transitive extension of the group (NH(F), !1(F)). The lauer group has a smaller degree than the initial one since I !1(F) I "# In I if F is nontriviaL So the existence of a transitive extension of the group (NH(F), !1(F)) is a necessary condition for the existence of a transitive extension of the group (G, Q). If a transitive extension of the former group is found then it can be used during the implementation of Steps 1 and 2 of the proposed scheme. In fact, if we know the V-ring of the group (NG(F), !1(F)), then it is easy to reconstruct the subgraph of r induced by the set !1(F). Information about this subgraph is useful in consideration of the sets of fixed elements of other subgroups of H and also in the construction of the whole graph r. The information about NG(F) is useful in the second step. Indeed, if F < H (y) and the pair (x,y) is contained in a selfdual 2-orbit of the group (NG(F), !1(F)), then NG(F) contains an element 't suchthat x't = y, y't = x. If the action of NG(F) on !1(F) is reconstructed then we know the action of't on !1(F), and to realise the second step it is sufficient to extend this action to the whole set n. It is clear that if F is the largest subgroup which fixes !1(F) elementwise, then, up to multiplication by elements of F, there is at most one extension of 't from !1(F) to n.

In some problems on transitive extensions, the initial group (H, Q) is not defined completely. For example, we may know only the abstract structure of H or the number and lengths of its orbits on n. In this situation, in order to use the proposed scheme of transitive extension, it is necessary first to reconstruct the group (H, Q). The fixed points method can also be used.

398 I. V.CHUVAEVAANDA. A. !VANOV

3. Transitive Extensions of the Group L2(11)

We start with the known fact (see [5], [13]) that each of the two representations of the group M 12 on the cosets of subgroups isomorphic to L2(11) has rank 5 and its subdegrees are 1, 11, 11, 55, 66.

Let (H,O.) be a permutation group of degree 144 which is isomorphic to the group L2(11) and which has 5 orbits O.i, 0~ i ~ 4 of lengths 1, 11, 11,55 and 66, respectively. Our next goal is to findalltransitive permutation groups (G,O.) in which the stabilizer of an element x e 0. is a group (H, 0.) with the above properties.

To realize the scheme of transitive extension proposed in Section 2, we should first find all permutation groups (H, 0.) with the prescribed properties. For this purpose we will use the lattice of subgroups of the group L2(11), presented in Fig. 1 (see [8]).

Fig. 1.

We will consider the group L2(11) as the automorphism group of the Paley design P (11). Recall the relevant definitions (see [12], [14]). Let GF (q) be the Galois field of order q, q = 4m- 1, and Q be the set of nonzero squares of the field GF (q). The Paley design P (q) is the incidence system S = (V,B, cl>), where V and Bare the sets of elements and blocks on which some bijection maps ~ and 11 onto GF (q) are defined. The incidence relation ci> is defined by the following: cl>= {(v,b) e V xB I ~(v)-'ll(b) e Q}. The pairs (v,b) which are contained in ci> are called flags and those which are not contained in ci> are called antiflags. It is known that P (q) is a symmetric 2-(q, (q-1)/2, (q-3)/4) design. We will identify the Paley designP(q) with its (1,-1) incidence matrix P(q) = IIPijll such that Pij = 1 if (vi, Vj) e ci> and Pij = -1 otherwise. As was already mentioned, the group L2(11) is the automorphism group ofthe Paley design P (11) (see [14]).

In Table 1 an interpretation of certain subgroups and transitive representations of the group L2(11) in terms of the Paley design P (11) is presented. The first column contains the identification of F, the second one- its index, the third one- the Iist of elements of P(11) fixedunder the action of Fand, in the last column, the sets on which L2(11) acts in a manner similar to its action on the cosets of F.

ACTION OF THE GROUP M12 ON HADAMARD MATRICES 399

F [L2 (11): F] The elements fixed by F Cosets of F in L2 (11)

As 11 V E V V

Af 11 beB B

A4 55 (v,b) E 

D10 66 (v,b) E 'P 'P D12 55 {Vj,Vj}

or {bi,bj} WorU D(, 110 a pair (Vj, vj), i ::f:. j Df,' 110 a pair (bi,bj), i ::f:. j

Z3 220 a submatrix

[ ~ ~] Zz 330 a submatrix

[ I -1 -1] -1 1 --1 -1 -1 1

Table 1.

Remark 1. A subgroup Z 2 fixes in P ( 11) the submatrix presented in Table 1 which can be extended to a submatrix of the following form.

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

Remark 2. Since A. = 2 for the design P (11), there is a bijection between the set W

of the unordered pairs of elements and the set U of unordered pairs of blocks.

It is known that the design P (11) admits a polarity. If I! is such a polarity then <!J.,Lz(11)> = PGLz(ll) = Aut(Lz(ll)). Since VJ.l = B, ßll =V, the subgroups A 5 and AS' are conjugate in Aut(L 2 (11)). The subgroups D(, and D(,' are also conjugate in PGL 2(11).

It follows from Fig. 1 that Lz(ll) has two conjugacy classes of subgroups of index 11, two such conjugacy classes of index 55 and only one such dass of index 66. This implies that up to similarity there are exactly four representations of degree 144 with orbit lengths 1, 11, 11, 55,66:

400 I. V. CHUVAEVA AND A. A. IVANOV

ilo={x},Q1=V, ilz=B, !13=W, !14='1'; (1)

no = {x}, n1 = v, nz =B, !13 =, !14 ='I'; (2)

no = {x}, n1 = nz = v, !13 = w, !14 ='I' ; (3)

(4)

We will show that the representations 3 and 4 do not have a transitive extension. This is easily done using the fixed points method.

Let us try to reconstruct the coloured graph r of the V-ring of the group (G, Q) with basis graphs {rdf=O corresponding to the orbits {Odf=o of the group (H,Q). In the graph r an edge (y,z) has colour i if and only if it is an edge of the graph r;, 0~ i ~ 4. The following arguments can be applied to both Case 3 and Case 4.

The subgroups A 5 and A 5' are not similar in their actions on the set Q since I ~(A 5) I = 3 while I ~(A 5') I = 1. For this reason, the subgroups A 5 and A 5' are not conjugate in G and by Lemma 1 the subgraphs induced by the vertices fixed by these subgroups are vertex-transitive. Let ~(A 5) ={X, v', v"}, v' E !11, v" E ilz, where v' and v" correspond to an element v of the design P (11). Now it is easy to show that the subgraph induced by the set ~(A 5) is isomorphic to the graph in Fig. 2. Here the number i indicates that the edge corresponds to the graph r;. Now Iet us try to reconstruct the subgraph induced on the vertices fixed by the group D 1o. All subgroups D 1o are conjugate in L 2 (11), so the normalizer Nc(D 10 ) acts transitively on the set ~(D w)={x, v', v",<l>}, where <\> = (V, b) E '1'. U sing the structure of the Subgraph ~(A 5) we can draw a fragment of the subgraph ~(D 10) (see Fig. 3) and it is easy to see that this fragment cannot be extended to a transitive graph. Thus it is proved that the representations 3 and 4 have no transitive extensions.

Fig. 2. Fig. 3

For the representations 1 and 2, transitive extensions were constructed and it was proved that, in each case, the extension is unique. To apply the fixed points method we

ACTION OF THE GROUP M 12 ON HADAMARD MA TRICES 401

first have to reconstruct the action (NG(F), d(F)) and the subgraph induced on the set d(F) for certain subgroups F of H. The most significant information is obtained from consideration of the subgroups Z 2 and Z 3.

For representation 1, we have ld(Z3) I = 6 and the action of (NG(Z3), d(Z3)) is similar to the wreath product Z2 wrZ3, while ld(Z2)1 =20 and the action of (NG(Z2), ~(Z2)) is similar to the action of Ss on the cosets of a subgroup D 6 • In this case r 1 and r 2 from a pair of dual 2-orbits, all 2-orbits are connected and hence the transitive extension is a primitive permutation group.

For representation 2, we have I ~(Z 3) I = 9 and (N G (Z 3 ), ~(Z 3)) is similar to the action of the group D 6 x D 6 on the edges of the complete bipartite graph K 3,3; I ~(Z3) I = 16 and the factor group NG(Z2)/Z2 is isomorphic to a semidirect product of the elementary abelian group of order 16 by the group S 3. In this case all 2-orbits are selfdual but the graphs r 1 and r 2 are disconnected. Each of these graphs is a union of 12 disjoint complete graphs on 12 vertices.

For representation 2, an automorphism which permutes the vertices of an edge (x,y)

of the graph r3 of valency 55 was constructed. If Z2 is a subgroup of the elementwise stabilizer of the edge (x,y) then the V-ring of the group (NG(Z2), d(Z2 )) is known and (x,y) is contained in a symmetric 2-orbit of this V-ring. We have constructed the automorphism 1: in NG(Z2 ). The general scheme for this construction is the following:

1) find the orbits of the elementwise stabilizer of the edge (x,y) on the set Q and to determine the action of 1: on these orbits;

2) determine the action of't on the set ofvertices fixed by Z2;

3) extend the action of't to the whole set Q;

4) prove that 1: is unique up to multiplication by elements of the stabilizer of the edge (x,y).

In this way it was proved that the transitive extension of representation 1 is similar to the representation 1t2 of the group M 12 on the cosets of a maximal subgroup L2(11).

By similar arguments it was shown that representation 2 leads to the representation 1t1 of the group M 12 also on the cosets of a subgroup L2(ll).

4. An Interpretation in Terms of the Hadarnard Matrix H (12)

A general disadvantage of the method of transitive extension for the construction of combinatorial objects is the lack of homogeneity of the description obtained. In particular, the vertex x has a special position. In some cases it tums out to be possible to find post jactum a homogeneaus description of the constructed object. For the representations 1t1 and 1t2 such a description was obtained in terms of the Hadamard matrix of order 12.

402 I. V. CHUV AEV A AND A. A. IV ANOV

Let us first give some definitions. Let M = llm;}n xn be a square (1,-1)-matrix. Let us define a coloured graph r(M) with five colours whose vertices are the entries of the matrix M, where edge { (i,j), (s,t)} has colour k in accordance with the following rule:

k = 0 if i = s, j = t, k = 1 if i = S, j '# t,

k = 2 if i '# s, j = t, k = 3 if i '# s, j '# t, m;j mir msj msr = I,

k = 4 if i '# s, j '# t, mij m;1 msj ms1 = -1 .

As above, Iet P (11) = IIPijll, I :s; i,j :s; 11, be the Paley design over GF (11). Let us extend this design to a Hadamard matrix H (12) = llhijll, 0 :s; i,j :s; 11, by supplying it with an additional all 1 's row and column. In this case vertex x corresponds to the element hoo, vertex v; E V to the element hOi and the element bi E B to the element h;o.

Using the method of transitive extension, it was possible to show that the coloured graph of the V-ring of the representation 7tz coincides with the graph r(H (12)). The latter enables a clearer understanding of the steps in the transitive extension procedure. In the action on the matrix H ( 12), the elements fixed by the subgroups A 4 , D 10, Z 3, Z 2 form submatrices of the following form:

A4:M1=[~ ~] D10:Mz=[~ -~] 1 1 1 1

1 1] 1 1 -1 -1 1 1 Zz: M4 = 1 _ 1 1 _ 1 1 1 1 -1 -1 1

Moreover, the subgraphs induced by the sets of fixed vertices are isomorphic to the graphs r(M;), 1 :s; i :s; 4.

In the new terrninology the automorphism 't can be regarded as an automorphism of the Hadarnard matrix H (12), i.e. as a permutation on the set of rows and colurnns allowing multiplication of certain rows and colurnns by -1.

The representation 1t1 can also be described in terms of the matrix H (12). To obtain such a description we should study the cliques in the graph r4 of the V-ring of the representation 1tz. It follows immediately from the definition of the graph r(H (12)) that each clique in the graph r 4 contains at most 12 vertices. Since r(H (12)) is vertextransitive, it is sufficient to describe the cliques passing through the vertex x (which corresponds to the element hoo of the matrix). lf a clique contains 12 vertices, the other 11 vertices of the clique lie in different rows and colurnns of the matrix P (11). This means that such a clique defines a one-to-one correspondence cp between V and B. But the sets V and B are marked by the elements of the field GF (11). Hence <P maps GF (11) onto

ACTION OF THE GROUP M12 ON HADAMARD MA TRICES 403

itself. The following conditions are necessary and sufficient for «1> to define a clique of the graph r4:

1) a- cp(a) is not contained in Q for each a E GF (11);

2) for any distinct a,ß E GF (11) exactly one of the values a- cp(ß) and ß- cp(a) is contained in Q.

The graph r 4 is edge-transitive, hence the number of cliques passing through X is equal to the product of the number of cliques for which cp(O) is equal to a fixed element of the set GF (11)- Q by the cardinality of the latter set. Now it can be checked directly that there are exactly two maps satisfying the above conditions in addtion to cp(O) = 0. The first one is defined by cp1 (a) = a; and the second one by cp2(a) = c (a)a where c (a) = 5 if a E Q and c (a) = 9 otherwise (all calculations are modulo 11).

Thus each vertex of the graph r 4 is contained in exactly 12 cliques of size 12, and the total number of cliques is 144. Now it is easy to see that the action of M 12 on the set of cliques is transitive and coincides with the representation 1t1 of the group M 12 on the cosets ofthe subgroupL2(ll).

5. ABipartite Graph of Valency 12

Let us consider the bipartite graph ~ with parts ~1 and ~2· Here ~2 is the set ofvertices of the graph r 4 of the V-ring of the representation 1t2 and ~1 is the totality of cliques of the graph r4; the incidence relation corresponds to the inclusion of vertices into cliques. It follows from Section 4 that ~ is a regular graph of valency 12 and that the group G = M 12 acts edge-transitively on ~. Moreover, the action (G, ~i) is similar to the representation 1ti· It follows directly from the lattice of subgroups of the group L2(ll) that the stabilizer in G of an edge of ~ is isomorphic to the group F~1 • Let us show that G acts transitively on the pairs of nonadjacent vertices from different parts. Forthis purpose it is sufficient to show that the stabilizer G (x) acts transitively on the 132 cliques which do not contain x. Let x correspond to the element hoo of the matrix H (12) and let K be a clique which does not contain x. Then K contains certain elements hiQ, hoj of the matrix H (12) for i,j E { 1,2, ... , 11 }. The subgroup G (x,K) which fixes x and preserves K as a set, fixes also the vertices hiO and hoj· The stabilizer of the clique KinGis isomorphic to L2(ll) and it acts transitively on the vertices of K. But the group L 2(11) has only one representation of degree 12 and the elementwise stabilizer of a pair of elements in this representation is isomorphic to Zs. Hence IG(x,K)I :S; 5. Since IG(x)l = 132· 5, we conclude that G (x) acts transitively on the 132 cliques which do not contain x. Now it is easy to see that ~ is a distance-regular graph (see definitions in [2], [10]) with the following intersection array.

404 l. V. CHUVAEVA AND A. A. IVANOV

12 1 11 2 10 12

66 13Z 77

The automorphism group of the graph A is isomorphic to the group Aut(M 12) which contains M 12 as a subgroup of index 2. The group Aut(M 12) preserves the parts of the graph and acts primitively on both A1 and A2 . Moreover the stabilizer of any vertex of the graph is isomorphic to the group PGL 2(11). Thus A is a biprimitive edge- but not vertex-transitive graph of parabolic type (see definitions in [17]). No other graphs with these properties are known to us.

6. Some General Properties of the Graph r(H)

In this section we will prove some properties of the coloured graph r(H), where H

is an arbitrary Hadamard matrix of order n > 2. It is known [12] that in this case n = 4m, m > 1.

Lemma 2. Let H = llhijlln X n be a Hadamard matrix of order n = 4m. Then r(H) is the coloured graph of a cellular ring W (H) of rank 5.

Proof. Let us show that in the graph r(H) the nurober Y!j of triangles with edges of colours i and j and with a common edge of colour l (see Fig. 4) does not depend on the common edge and is a function of m.

Fig. 4.

The values Y!j can be calculated directly (see Table 1). In these calculations it should be noted that if the matrix H is converted to normal form, i.e. if its first row and column contain only 1's, then for a vector (s,t,a,ß), 2~ s,t~ 4m, s "#t, a,ßE (1,-1}, there are exactly m columns k, 1 ~ k ~ 4m, such that hsk = a, hrk = ß (see Fig. 5).

ACTION OF THE GROUP M12 ON HADAMARD MATRICES

11 .•. 1111 ... 111·1 ... 1111 ... 11

~ 1 1 ... 1 1 1 1 ... ., 1-1-1 ... 1 1-1-1 ... -1 -1

t 1 1 ... 1 1-1-1 ... -1-11 1 ... 1 1-1-1 ... -4-1

m m m m

Fig. 5.

X 1 ..... 1 ..... 1

1. .... 1 .... -1 y

1 ..... 1 .... -1 z

X 1 ..... 1 ..... 1

1 .... -1. ... -1 !I

1 ..... 1 ..... 1 z

Fig. 6.

405

Let us illustrate the calculation of r!j by the example y~3 • We assume that the matrix H is in normal form. Without loss of generality we can also assume that x corresponds to the entry hoo of the matrix. Then there are two possible positions of x, y and z in the matrix H (see Fig 6). If y is a fixed entry then there are 2m - 1 possible ways to choose the row of z and m - 1 ways to choose the column of z in both situations. Hence 'Y~3 = (2m-1) (m-1) + (2m-1) (m-1). The other values yfj can be calculated in an analogous way (see Table 2). Since these values depend only on m, the graph r(H) is the coloured graph of a cellular ring W (H) of rank 5. o

406 I. V. CHUVAEVA AND A. A. IVANOV

1 i j

0 1 2 3 4

0 1 0 0 0 0

1 0 4m-1 0 0 0

0 2 0 0 4m-1 0 0 3 0 0 0 8m2 - 6m + 1 0 4 0 0 0 0 8m2 -2m

0 0 1 0 0 0 1 1 4m-2 0 0 0

1 2 0 0 0 2m-1 2m

3 0 0 2m -1 4m 2 -6m +2 4m 2 - 2m

4 0 0 2m 4m2 -2m 4m 2 -2m

0 0 0 1 0 0 1 0 0 0 2m -1 2m

2 2 1 0 4m-2 0 0 3 0 2m -1 0 4m 2 -6m +2 4m 2 - 2m

4 0 2m 0 4m 2 -2m 4m 2 -2m

0 0 0 0 1 0 1 0 0 1 2m-2 2m

3 2 0 1 0 2m-2 2m

3 1 2m-2 2m-2 4m 2 -6m+4 4m2 -4m

4 0 2m 2m 4m 2 -4m 4m 2 - 2m

0 0 0 0 0 1 1 0 0 1 2m -1 2m-1

4 2 0 1 0 2m -1 2m -1 3 0 2m -1 2m -1 4m 2 -6m+2 4m 2 -4m+1 4 1 2m-1 2m -1 4m 2 -4m+1 4m 2 -2m

Table 2.

It shou1d be mentioned that the ring W (H) is pseudo-affine and hence amorphic (see definitions in [11]). This means, in particular, that an arbitrary merging of the graphs

ri, 1 ::::: i ::::: 4, leads to some cellular subring W' of the ring W (H). In particular, the

graphs r4, rl + r3, r2 + r3 are strongly regular with the same parameters

v = 4(2m)2 , k = 2(2m)2 - 2m, A. = 1.1 = (2m)2 - 2m but they arenot generally isomorphic.

Since A. = 1.1 these strongly regular graphs are so-called (v,k, A.)-graphs. It is weil known (see, for example, [6]) that, from a (v,k, A.)-graph, a certain 2-(v,k, A.) design can

be constructed. At the sametime it is shown in [13] that the (1,-1)-incidence matrix of a

2-(v,k, A.)-design with parameters v = 4t2 , k = 2t2 - t, A. = t 2 - t is a Hadamard matrix of

ACTION OF THE GROUP M 12 ON HADAMARD MATRICES 407

order 4t2 • This means that starting from a Hadamard matrix of order 4m it is possible, by consideration of the coloured graph r(H) and the subrings of its cellular ring W (H), to obtain three Hadamard matrices of order ( 4m )2 which are in general nonisomorphic. To each of these matrices we may apply the same procedure. Thus for an arbitrary Hadamard matrix we obtain an infinite family of amorphic cellular rings of rank 5 and correspondingly of (v,k, A.)-graphs.

Remark 3. In [13] a correlation between the action of the group M 12 on H (12) and

of its action on a Hadamard matrix of order 144 was mentioned. As was shown above, this correlation is of a general nature.

Remark 4. In general, the ring W (H) is not isomorphic to a subring of an affine ring of larger rank. For example, this is the case for the ring W (H (12)). This fact follows directly from the description of the cliques in the graph r 4 obtained in Section 4.

Remark 5. If the matrix His constructed from a (v,k, A.)-graph d by the above procedure, then the graph r 4 can be described in terms of subrings of the symmetric cartesian square of the graph d (see [9]).

Remark 6. Not all cellular rings with the parameters presented in Table 2 can be constructed from a Hadamard matrix. The minimal counterexample is related to a strongly regular graph with parameters v = 16, k = 6, A, = J.L = 2 [15].

References

1. J.L. Alperin, On the theorem of Manning, Math. Z. 88 (1965) 434-435.

2. N.L. Biggs, Algebraic Graph Theory, Cambridge Tracts in Math. 67, Cambridge University Press, Cambridge (1974). Zbl284.05101

3. A.E. Brouwer, On the uniqueness of a regular near octagon on 288 vertices (or the semibiplane belanging to the Mathieu group M 12), Math. Centre Report ZW196, Amsterdam (1983). 12 pp.

4. F. Buekenhout, Diagrams for geometries and groups, J. Combinatorial Th. (A) 27 (1979) 121-151.

5. F. Buekenhout, Geometries for the Mathieu group M 12 , pp. 74-85 in: Combinatorial Theory, Lecture Notes in Math. 969 (ed. D. Jungnickel & K. Vedder), Springer, Berlin, 1982.

6. P.J. Cameron and J.H. van Lint, Graphs, Codesand Designs, London Math. Soc. Lecture Notes 43, Cambridge Univ. Press, Cambridge (1980).

7. LV. Chuvaeva, On some combinatorial objects which admit the Mathieu group M 12

as an automorphism group, pp. 47-52: Methodsfor Complex System Studies, Moscow, VNIISI, 1983. [In Russian]

408 I. V. CHUVAEVA AND A. A. IV ANOV

8. L.E. Dickson, Linear groups with an exposition of the Galois field theory, Dover, New York (1958).

9. LA. FaradZev, Cellular subrings of the symmetric square of a cellular ring of rank 3. [In this volume]

10. LA. Faradzev, M.H. Klin and M.E. Muzichuk, Cellular rings and groups of auto

morphisms of graphs. [In this volume]

11. Ja.Ju. Gol'fand, A.V. Ivanov and M.H. Klin, Amorphie cellular rings. [In this volume]

12. M. Hall, Jr., Combinatorial Theory, Blaisdell, Waltham, Mass. (1967).

13. M. Hall, Jr., Group properties of Hadamard matrices, J. Austral. Math. Soc. 21 (1976) 247-256.

14. D.R. Hughes, A combinatorial construction of the small Mathieu designs and groups, Annals ofDiscr. Math. 15 (1982) 259-261.

15. Q.M. Hussain, On the totality of solutions for the symmetric incomplete block designs: A. = 2, k = 5 or 6, Sankhya 7 (1945) 204-208.

16. A.A. Ivanov, Construction by computer of some new automorphic graphs, pp. 144-146 in: Aerofisika i prikladnaya matematika ( Aerophysics and applied

mathematics), MFTI, Moscow, 1981. [In Russian]

17. M.E. Iofinova and A.A. Ivanov, Bi-primitive cubic graphs, [In this volume]

18. A.A. lvanov, M.H. Klin and LA. FaradZev, The primitive representations of the

nonabelian simple groups of order less than 106, Part I, Preprint, The Institute for System Studies, Moscow (1982). [In Russian]

19. A.A. Ivanov, M.H. Klin and LA. FaradZev, The primitive representations of the nonabelian simple groups of order less than 106 , Part JJ, Preprint, The Institute for System Studies, Moscow (1984). [In Russian]

20. S. V. Shpectorov, A geometric characterization of the group M 22 • [In this volume]

21. H. Wielandt, Finite Permutation Groups, Academic Press, New York (1964).


CONSTRUCTION OF AN AUTOMORPHIC GRAPH ON 280 VERTICES USING FINITE GEOMETRIES

FL. Tchuda

1. Introduction

Nowadays, much information about different classes of distance-regular graphs is available (see survey [2]). The majority of these graphs were constructed using classical geometries over finite fields. As a rule one obtains in this way an infinite series of graphs. The classification problern for distance-transitive graphs admitting a transitive representation of a linear group became realisable. In the resolution of this problern some serious difficulties are expected regarding sporadic representations, i.e. those which do not belong to an infinite series. The elirnination of distance-regular graphs which are not distance-transitive is also a significant step in the solution of this problem. These observations stimulate an interest in searching for special constructions which give a simple and beautiful description of certain distance-transitive graphs. The necessity of such constructions also arises in the interpretation of graphs which were discovered by means of a computer.

The present paper contains a construction of a distance-transitive graph on 280 vertices which is invariant under a primitive representation of the group PSL3(4) of degree 280. The vertices of the graph are all embeddings of the affine plane of order 3 in the projective plane of order 4. The existence of this graph was mentioned in [1], but we have not found a description of the graph in the literature.

The author is grateful to M.H. Klin for proposing the problern and for continually stimulating interest in its solution.

The definitions and prelirninary results concerning distance-transitive graphs

(d.t.g.'s), automorphic graphs (a.g.'s) and finite planes can be found in [1] and [5],

respectivel y.

2. A model of the projective plane of order 4

LetAdenote the affine plane of order 3 containing 9 points and 12 lines and Iet H be the collineation group (the automorphism group) of the plane. The set of points of the plane A is decomposed into 4 farnilies of pairwise parallel lines. Any point of the plane lies on exactly one line from each parallel family.

409

410 F. L. TCHUDA

The points and the lines of A will be called old points and old lines respectively. In terms of A we will define an incidence system with new points and new lines.

The set of new points consists of 9 old points and 12 old lines. On this set of points we define new lines of two types. Each line i of the first type contains 5 new points. One of these points corresponds to an old point x and the other 4 points correspond to 4 old

lines passing through x. Among the five points on a line I of the second type there are three old collinear points (they determine an old line /) and two new points corresponding to two other old lines in the family of parallels containing / .

. • • . . • • . . .

• • . • . • • . • . . . • . .

• • . . . • . . . . • • . . . • . . . .

• . . • • • • • . . . ..

• • • . . • • • . . • • • . .

• • • .. • • • . . .. • . .

• • • . . • • • . . • • • . .

Fig.l. The incidence matrix of the defined system is given in Fig. 1. A direct check which can be simplified by use of the transitivity of H on points and lines, shows that the new incidence system is a model of the projective plane of order 4. This model will be denoted below by P.

It is easy to see from Fig. 1 that

1) the plane A is embedded in P (see the intersection of 9 former rows and 12 latter columns of the matrix);

2) the plane P admits a polarity 1t with 9 absolute points (the matrix is symmetric);

3) each automorphism of A can be extended to an automorphism of P.

3. Description of the graph in terrns of the affine plane of order 3.

First let us solve the inverse problem. Namely, Iet us construct the affine plane of order 3 using the projective plane of order 4.

Let X = {x 1 , x 2 , x 3} be an arbitrary triple of new non-collinear points. This triple determines a triple Y = {y 1 , Y2, Y3} of new lines which pass through two points from X.

Let us consider the set of all points using the lines y 1 , y 2 , y 3 , where the points x1, x2, X3 are deleted. In this way we obtain a set of 9 new points. The procedure

described above will be called the deleting procedure. This procedure goes back to [3]. If

CONSTRUCTION OF AN AUTOMORPHIC GRAPH ON 280 YERTICES USING FINITE GEOMETRIES 411

we apply the procedure to the set of three new points corresponding to a parallel family of old lines, then we obtain the set M of old points. In this case a new line has intersection with M of cardinality three if and only if this line is of the second type. The intersectians of M with the lines of the second type form the system of lines in the plane A. It is easy to see from Fig. 1 that by means of the deleting procedure, the plane A can be obtained in exactly four ways.

The group G of collineations of the projective plane acts transitively on the triples of noncollinear points. This implies that the deleting procedure always Iead to an embedding of the affine plane of order 3 in the projective plane P. The set of points of such an embedding will be called an affine nine. Let Vr be the set of all affine nines. By calculating the nurober of triples on noncollinear points of P, and keeping in mind that each affine nine can be constructed in four ways, we have I vr I = 21 • 20 · 16/3! • 4 = 280.

Taking vr as the set of vertices, we define two vertices to be adjacent if and only if the affine nines corresponding to these vertices have one-element intersection. Our goal is to prove that this graph r is automorphic. For this purpose we need a precise description of its vertices. Notice that each automorphism of P, and hence of A, is an automorphism of the graph r. For this reason it is natural to describe Vr as a union of orbits of the induced action of the automorphism group H of the affine plane A on Vr.

4. A decomposition of the graph. The proof of distance-regularity

Since the automorphism group G of the plane P acts transitively on the set of unordered triples of non-collinear points, G acts transitively on vr. Let us consider the decomposition DM(r) of the graph r with respect to the Vertex M, i.e. the partition of vr into layers rj(M). Here C(M) is the set of vertices of r which are at distance i fromM, i = 0, 1 , · · · , d, where d is the diameter of r. Since His a subgroup of the stabilizer of the vertex M in the transitive group (G , Vr) it is convenient to describe the layers of the decomposition as unions of certain orbits of the group H on vr. Information about these orbits is presented on Fig. 2 in terms of the old points and lines.

412 F. L. TCHUDA

N Tripies Cardinality Nines Cardinality Designation

- 0 0 0

1 ----- 12. 2/J!:It 0 0 0 ~ ro - 0 0 0 . . . ~ 2 o--.--o 9·8/2:'36 9 f1 . . .

o---o---

J ---. 9· .... 2:72 . . . ~ 72 r2

... X 9. 8. 6/2.:2~6

5 ~ 12 .g. 't/31 :72

kt1 r;l 72 3 . 0 . 6 -o-- 9. 8. 6/2.=216 . . .

0 . 0

7 . . . 9·8·6/3!=72 . 0 . ~ r_/ 72

6 + 9·4-·6=216 . 0

I 0

9 . . 9·8 ·3/2~08

Ff • . 54 r._

10 I 12.2. 9/2=108

Fig. 2.

In the column "triples" of Fig. 2 the representatives of ten orbits of the group H on the set of triples of non-collinear points of the plane P are indicated. From each such triple, by means of the deleting procedure, an affine nine can be constructed. A schematic picture of this nine is given in the column "nines" opposite the representative of the orbit of the initial triples. One should notice that some orbits of nines are constructed from different orbits of triples. In Fig. 2 information about the cardinalities of orbits of triple and nines and the notation for orbits of nines is also given. Each of the six orbits of nines can be described in a formal way. For example, a nine from 1 2 consists of five old points lying on two parallel lines and 4 old lines passing through an odd number (1 or 3) of these points. As was already mentioned, each layer of DM([") consists of certain orbits of the group (H, V["). It can be checked that d = 4 and that 1 0 (M) = {M}, 1 1 (M) = 1 1 ,

i2(M) = 12, 13(M) = 1 3' u 1/', 14(M) = 14.

CONSTRUCTION OF AN AUTOMORPHIC GRAPH ON 280 VERTICES USING FINITE GEOMETRIES 4I3

It follows from the description of DM(r) given above, and from the transitivity of Aut (r), that the invariants a 1 , b 1 , c 1 ,az, bz, c3, a4, c4 of r exist. In addition we can define the numbers a 31 , b 31 , c 3 1 , a 3", b 3", c 3" as invariants of the sublayers r 3 1

and r3" respectively.

The values of all these parameters of r can be calculated directly from the system of representatives of the orbits ri(M). Namely, for each of the six representatives one finds its 9 neighbours in r and determine in which orbits these neighbours are contained. It turns out that

so r is a distance-regular graph with the intersection array presented on Fig. 3.

1 8 0 9

Fig. 3.

1

aQ

Notice that we have already proved that the stabilizer of the vertex M in Aut (r) acts transitively on ri(M), i = 1, 2, 4. So it is sufficient to prove that the stabilizer acts transitively on the set r3(M). This could be done using the interpretation of affine nines as absolute points of polarities of the projective plane. But we prefer another, more combinatorial way, whose style is quite close to the above constructions.

5. The Levi graph and its subgraphs

Let us consider the Levi graph L(P) of the projective plane P. The vertices of this graph are the points and the lines of the plane P. A point and a line are joined by an edge in L(P) if and only if they are incident. The graph L(P) has no other edges. It is clear that L(P) is a bipartite graph of valency 5. The set of points (resp., lines) of the plane P will be called the left (resp., right) part of the graph L(P). We are interested in subgraphs of a special kind in the graph L(P).

An induced Subgraph rl of r is said to be a matehing if rl is a regular graph of valency 1. Notice that this definition is slightly different from the generally accepted one where a matehing is assumed to be a subgraph.

A subgraph of L(P) which is a matehing with 9 edges will be called an affine matehing if the vertices of the left part which are incident with these edges form an affine nine. It turns out that for each affine nine there is a unique affine matching. To prove this claim it is sufficient to check it for a particular affine nine, say for M. Let us consider a matehing consisting of the edges (x, i), where i e M. We will show that there is no other matchings having 9 edges of this form. For each vertex y from the right part let us

414 F. L. TCHUDA

calculate its multiplicity with respect to M as the nurober of vertices from M which are adjacent to y. Each vertex fromM = {i, x e M) has multiplicity 1 with respect to M. All other vertices from the right part of L(P) have multiplicity 3. Hence only M can form a matehing with M.

So there is a bijective correspondence cp between the elements of the set vr and the set of affine matchings in L(P). For XE vr let X denote the right part of the affine matehing corresponding to X.

Notice that X can be easy reconstructed from X since it consists of all vertices from the right part whose multiplicity (with respect to X) is equal to 1.

Let 1t be the polarity of the plane P such that M is the set of absolute points of 1t (see Section 2). Notice that Aut (L(P)) =GA.< 1t >.In view of the fact that 1t maps the affine matehing M u M onto itself, we conclude that 1t acts as a permutation on the set of all affine matchings. Using the bijection cp we can assume that 1t acts on the set vr. This action will be denoted by ft. For X E vr we have xit = j(1t.

6. The group Aut (r)

Since 1t is induced by a permutation 1t on the set of vertices of L(P) which permutes the parts of the graph, 1t preserves the cardinality of intersections of the affine nines. In particular 1t e Aut(r). At the same time M1t = M, so 1t preserves each layer in the decomposition DM(r). Since H <l HA.< 1t > the automorphism 1t either stabilizes orbits of H on vr or permutes these orbits.

Using the description of the action of 1t given above, it is easy to find the image Xit for each Xe vr. Forthis purpose one should consider the multiset of 45 right ends of the edges of the graph L(P) whose left ends are contained in X. After that one should select all elements with multiplicity 1 in this multiset and find their images under the action of the polarity x.

After producing these constructions for the representatives X; of all orbits of the group H on Vr we have the following:

Xf E r; for i = 0, 1 , 2, 4; (X 3 ')it E r3", (X 3 ")it E r3 '.

Hence 1t permutes the orbits r 3, and r 3" and < H ' 1t > acts transitively on the Set r 3 (M). So r is a d.t.g.

The primitivity of the group Aut (r) follows, for example, from the inequality ad "#= 0. It is known [6] that ad = 0 for each imprimitive d.t.g.

It was shown in [4] using a computer that I Aut(r) I = 241920. So since Aut(r) ::1 G A. < 1t >, we have Aut(r) = G A. < 1t >.

CONSTRUCTION OF AN AUTOMORPHIC GRAPH ON 280 VERTICES USING FINITE GEOMETRIES 4I5

References

1. N.L. Biggs, Automorphic graphs and the Krein condition, Geom. Dedic., 5 (1976), 117-127.

2. A.M. Cohen, A synopsis of known distance-regular graphs with Zarge diameter, Math. Cent. Afd. Zuivere Wisk., 1981, W168, 33 pp.

3. W.L. Edge, Some implications of the geometry of the 21-point plane, Math. Z., 87 (1965), 348-362.

4. A.A. Ivanov, M.H. Klin and I.A. FaradZev, Primitive representation of the nonabelian simple groups of order less then 106, Part 2, Preprint, VNIISI, 1984 [In Russian]

5. F. Karteszi, Introduction to Finite Geometries, Akademiai Kiado, Budapest, 1976.

6. D.H. Smith, Primitive and imprimitive graphs, Quart. J. Math. Oxford (2), 22 (1971), 551-557.


Part 3. Amalgams and Diagram Geometries

1. Introduction

APPLICATIONS OF GROUP AMALGAMS TO ALGEBRAIC GRAPH THEORY

A.A.lvanov & S.V. Shpectorov

We start with an example. Let r be a graph and G be a subgroup of its automor

phism group. The group G is said to act s-transitively on r if it acts transitively on the set

of paths of length s in r. Below, for a I-transitive group G, s will be the largest integer suchthat G acts s-transitively. In [Tut] the following theoremwas proved.

Theorem 1.1. Let r be a finite connected trivalent graph and G be a I-transitive

automorphism group on r. Then s ~ 5 and the order of the stabilizer in Gof a vertex of r is3·2s-l. 0

In view of this theorem the following questions rnay arise. Since there are no s

transitive trivalent graphs for s > 5, is it the case that there are only a few such graphs for

s = 4 and 5? Is it possible to classify all such graphs?

In fact only a few 4- and 5-transitive trivalent graphs are known. For instance the main result of the paper [Con] is just a construction of a new trivalent 5-transitive graph. Nevertheless, the answers to the above questions are negative. To see this let us reformu

late our conditions in terms of group amalgams.

It should be noticed that our understanding of the term "amalgam" differs slightly from that of [Kur], [Ser]. For us an amalgam is just a collection of groups with common identity element and with multiplications and operations of inverse coinciding on inter

sections. The groups which constitute an amalgam are called members of the amalgam

and elements of the members are called elements of the amalgam. The number of members in an amalgam is by definition the rank of the amalgam.

Letrand G be as above. Let X be a vertex of r and {x,y} be an edge incident to x. Let H and K denote the stabilizers in G of the vertex x and of the edge {x,y}, respectively. We will consider some properties of the amalgam A = {H,K}.

First of all, by I-transitivity of G, the edge {x,y} can be inverted, so I K: H n K I = 2. Since r is a trivalent graph I H : H n K I = 3. It is a little bit harder to see that the elements of A generate the whole group G. This fact follows from the connectivity of r. Finally, since G acts faithfully on r, the subgroups Hand K have no com

mon nontrivial normal subgroups. An amalgam possessing the latter property is called simple. So we have the following:

417

418 A. A. IVANOV AND S. V. SHPECTOROV

(1) A = {H,K} isasimple amalgam and the index of H n KinHand K is equal to 3 and 2, respectively;

(2) the finite group G contains the amalgam A and is generated by the elements of A.

The graph r is not even mentioned in conditions (1) and (2). Nevertheless, it can be reconstructed up to isomorphism. Let us define a graph r(G, A) by the following rule. The vertices of this graph are the cosets of H in G and the edges are the cosets of K in G. A vertex and an edge are incident if the corresponding cosets have nonempty intersection. (Throughout the paper we will consider right cosets only.) Since the index of H n K is 2, r(G, A) is in fact an undirected graph without loops but, possibly, with multiple edges. If we assign to each vertex v (to each edge e) of r the set of elements of G which map x onto v (respectively, {x,y} onto e), then we obtain an isomorphism of r onto r(G, A).

Thus a description of the pair (G,A) is equivalent to a description of the pair (G, r). Now let (G, A) be an arbitrary pair satisfying conditions (1) and (2). Then we can define the graph r = r(G, A) as above. In view of (2) r is connected. By (1) r is trivalent and G acts on it 1-transitively by right translations.

So for any A and G satisfying (1) and (2) there is a graph r = r(G, A) for which the hypothesis of Theorem 1.1 is true. This implies, in particular, that the parameter s defined above can be calculated purely in terms of the amalgam A. Thus the classification of all connected s-transitive graphs for a concrete value of s is equivalent to the classification of all pairs (G,A) satisfying (1) and (2) with an additional restriction on the order of the groupH e A.

Is it possible to carry out such a classification? In fact condition (2) implies only that the amalgam Ais finite. So the classification of the pairs (G, A) can be divided into two steps. The first step consists of determining all finite amalgams A which satisfy condition (1) (and, perhaps, an additional condition on the order of H). The second step consists of finding, for each particular amalgam A, all groups which satisfy (2). These two steps arerather different, so we will consider them separately.

It follows from Theorem 1.1 that the orders of members of A are not too !arge. Namely, the order of H divides 48. This means that there are only finitely many such amalgams and it is possible to describe all of them. In fact up to isomorphism there are exactly 7 such amalgams (cf. [DjMi]).

Regarding the second step, the situation is completely different. Condition (2) means that G is a finite factor group of the universal group of the arnalgam A. The universal group 6 of A has all elements of A as generators and all equalities abc = 1 which arevalid in members of A, as relations. If A = {H,KJ then the universal group of A coincides with the free product of H and K with joint subgroup H n K. Moreover, there are infinitely many finite factor groups of 6 which contain A. This shows that the

APPLICATIONS OF GROUP AMALGAMS TO ALGEBRAIC GRAPH THEORY 419

classification problern for the pairs (G, A) is close to intractable.

Let us summarise our observations.

1) Some of the natural conditions on r and on G can be reformulated in terms of the amalgam A related to the action of Gon r.

2) The above properlies determine the amalgam A up to a finite number of possibil-

ities.

3) A determination of all G and r, corresponding to a given amalgam A, is equivalent to a determination of all factor groups of the universal group of A.

We presented here this rather well-known example in order to demonstrate some

specific features of the method which will be called below the amalgam method. A typical application scheme of amalgams to algebraic graph theory can be described as fol

lows.

Let r be an undirected connected graph and G be a group of its automorphisms. Suppose that our goal is to describe all such pairs (I',G), satisfying certain additional conditions. We propose the following general approach to this problem.

1. Define in r a certain collection ß 1 ,ß2, ... , ßn of subgraphs. Let X 1 ,X 2• ... ,Xn denote their stabilizers in G.

2. Using the conditions on (I',G), characterize up to isomorphism all possibilities for the amalgam A, having the subgroups X 1 ,X 2, ... , Xn as its members.

3. Foreach A determine the universal group 6 of the amalgam A.

4. Determine all possibilities for G as a factor group of 6. 5. From the group G and the amalgam A, embedded into it, reconstruct the graph r. Let us make the following remarks. First, not every condition on (I',G) can be

reformulated completely in terms of amalgams. But what is necessary isthat those conditions which can be reformulated be still rather restrictive. The additional conditions can be used later in steps 4-5. For instance such a situation takes place in the case of the distance-transitivity condition on the graph r. Secondly, the purpose of the present survey is just to demonstrate the possibilities of the method of amalgams. So we will primarily restriet ourselves to the case when ~~ 1 is a vertex of r and ß2 is an edge incident to ß 1• In this case step 5 can be realized in a trivial way, namely I'= I'(G, {X ~oX2 }).

We conclude this section by establishing the following terminology and notation.

If the contrary is not stated explicitly, all graphs in question are assumed to be locally finite, undirected, connected, without loops and multiple edges. For a graph r let V(I'), E (I') and Aut(I') denote its vertex set, edge set and full automorphism group, respectively. A path of length s in r is a sequence (xo,x 1, ... ,xs) of vertices, suchthat {xi,Xi+1 ) e E(I') for 0~ i ~ s -1, Xi .exi+2 for 0~ i ~ s- 2. If x 0 =xs then the path is called a cycle. The length of a shortest eycle of r is said to be the girth of r. For x,y e V (I') let d(x,y) denote the distance between these vertices. By definition d(x,y) is

equal to the number of edges in a shortest path joining x and y. The diameter of r is the maximum of the distances between pairs of vertices. For x E V (I') put rj(X) = {y I y E V(r), d(x,y) = i}. The set rl (x) will be denoted by r(x). A graph r is said to be regular if k = I r(x) I does not depend on the particular choice of x. In this case k is called the valency of r.

Let G ~ Aut(r) and U ~ V (r). The stabilizer of the subset U in G is the largest subgroup of G which preserves U as a whole. The pointwise stabilizer of U in G is the larg

est subgroup which maps each vertex of U by itself. The stabilizer will be denoted by G { U} and the pointwise stabilizer by G (U). Instead of G( {x,y, ... }) we will write simply G (x,y, ... ). If H ~ G { U}, then H u denotes the permutation group induced by H on U. It is clear that Hu = HIH(U). Fora positive integer i we put

Gi(x) = 11 G (y). d(x,y)~ i

In addition, for {x,y} E E(r) we put Gi(x,y) = Gi(x) 11 Gi(y). The permutation group G (xl<x) will be called a subconstituent of Gon r.

2. A vertex and an edge

In this section we concem ourselves with three subjects related to rank 2 amalgams arising from I-transitive actions on graphs. First we give a brief account of results on the determination of the structures of these amalgams. In particular, these results deal with bounding the order of a vertex stabilizer in terms of the valency of the graph. In the second subsection we consider a particular question arising in the classification of amalgams of rank 2. The point is that there can be distinct amalgams with pairwise isomorphic members. The possibilities for such amalgams are described in a lemma due to D. Goldschmidt. We will illustrate the application of Goldschmidt's lemma on a number of examples which will be used in the subsequent exposition. In the last subsection we

concem ourselves with Serre's paper "Arbres, amalgams and SL2"· This paper provides good insight into the description problern of all pairs (G, r) corresponding to a given rank 2 amalgam.

2.1. Bounding the vertex stabilizer

Let G be a group acting faithfully and 1-transitively on a graph r. In accordance with the amalgam method, we consider the structure of an amalgam A0 consisting of the

subgroups H = G(x) and K = G{x,y}, where x E V(r) and {x,y} E E(r). So Ao is a finite simple amalgam and I K : H 11 K I = 2.

Remark. Since we intend merely to illustrate the amalgam method, we will not consider the case of locally I-transitive action. For this reason we will not have the

APPLICA TIONS OF GROUP AMALGAMS TO ALGEBRAIC GRAPH THEOR Y 421

opportunity to mention the specialists who were involved in the study of the more general situation.

The history of the investigation of the amalgams Ao goes back to the aforementioned theorem of Tutte. This theorem in fact restricts the structure of the amalgams in the case k = 3 where k = I H : H n K I is the valency of r. Afterwards, a number of attempts were made to bound the orders of members of the amalgam purely in terms of the parameter k. But it turned out [Djo] that even in the case k = 4, if Hinduces on the cosets of H n K (that is on r(x)) the dihedral group D g, then it is impossible to bound the order of H. For other transitive groups of degree 4 such a bound exists. In view of this fact, abound on the order of His now considered in terms of the subconstituent Hr(x)

rather then in terms of k.

The majority of papers on this subject contain a treatment of the case when the subconstituent is a doubly transitive permutation group. Note that the subconstituent is doubly transitive if and only if G acts 2-transitively on r. The complete list of doubly transitive permutation groups is a consequence of the classification of finite simple groups. So we take for the subconstituents the known doubly transitive groups. To deal with some of these cases, special and rather involved methods were developed. Here we follow only the main line of this work.

The first result is due to [Thm], [Wie], [Gar]. In the form presented below it was proved in [Weil].

Theorem 2.1.1. Let G be a 2-transitive automorphism group of r. Then G 1 (x,y) is a p-group for some prime number p. o

As a consequence of this theorem it is easy to prove that either G 1 (x,y) = 1 or the subconstituent H r(x) has a p- local point stabilizer, that is G (x,y l(x) has a nontrivial nor

mal p-subgroup. The first possibility surely gives us a bound on the order of H. So let us turn to the second possibility.

The known doubly transitive permutation groups having a p-local point stabilizer are the groups PSLn(q), PSU3(q), Sz(q), 2G2 (q) and their extensions by outer automorphisms, as well as some affine type groups. A permutation group is said to be of affine type if it has an elementary abelian regular normal subgroup. A detailed analysis of all possibilities was done mainly by R. Weiss (cf. the survey [Wei2]). The final result is as follows.

Theorem 2.1.2. Let G act 2-transitively on r and let G 1 (x,y) be a nontrivial pgroup. Then PSLn(q)-5, Hr(x) '5, PrLn(q), n~ 2, k = (qn-1)/(q-1) where q isapower

~~ 0

Thus one should consider the case of projective subconstituents in dimension n ~ 2.

The analysis can also be carried out in terms of the parameter s which determines the degree of transitivity of G on r. Let us show first that for each doubly transitive group there is a group acting 3-transitively on a graph and having the given doubly

422 A. A. !VANOV AND S. V. SHPECTOROV

transitive permutation group as a subconstituent.

Example 2.1.3. Let D be a doubly transitive permutation group on a set n of cardinality k. Let us consider two copies D' and D" of this group acting on the sets Q' and Q", respectively. Let r be the complete bipartite graph with parts Q' and Q". So r is iso

morphic to Kk.k· Let G be the extension of the direct product D' x D" by an involutory automorphism 1: which permutes D' and D". Then it is easy to see that G acts 3-transitively on r and that for X E V (I) the permutation group G (x)f<x) is similar to D.

It follows from the next proposition that the study of s-transitive graphs for s ?: 4 also Ieads to the case of projective subconstituents.

Theorem 2.1.4. [Wei3] Let G act 4-transitively on a graph r of valency k?: 3. Then PSL 2 (q)::; Hf<x)::; PrL 2 (q). D

The possibilities for the amalgam { G (x), G {x,y}} in the case of projective subconstituents in dimension 2 were described in the paper [Wei4]. But before formulating this result let us recall some definitions.

A finite bipartite graph is said to be a generalized n-gon, n?: 2, if it possesses the following properties:

(a) any two vertices are joined by a path of length at most n;

(b) there is at most one path of length less than n joining any two vertices;

( c) each pair of vertices is contained in a cycle of length 2n.

The vertices in one part of a generalized n-gon are called points and in the other part they are called lines. We will be interested primarily in the vertex-transitive generalized n- gons. In this case the graph in question is regular, i.e. the nurober of lines through a point is equal to the nurober of points on a line. The regular generalized n-gons can be characterized as the bipartite graphs of valency k having diameter n and girth 2n. We will understand the term "generalized n-gon" mostly in this restricted sense. If we do not wish to specify n, we will speak simply of generalized polygons.

Let us give some examples. A generalized 2-gon is just the complete bipartite graph Kk,k for some k. The class of generalized 3-gons of valency k coincides with the class of projective planes of order k - 1.

The generalized 3-gons, related to the Desarguesian projective planes, and some other known examples of I-transitive generalized polygons can be obtained from Lie type groups of rank 2 by means of the following procedure.

Let L be a finite Lie type group of rank 2. Let P 1 and P 2 be two maximal parabolic subgroups of L each containing a given Borel subgroup. Let .1 = .1(L) be the bipartite graph whose vertex set is the union of the cosets of Lover subgroup P 1 and P 2 ; two vertices are adjacent if the intersection of the corresponding cosets is nonempty. The graph obtained in this way is a (not necessarily regular) generalized n-gon. The value of n is determined by the condition that 2n is the order of the Weyl group of L.

If L is one of the Lie type groups A2(q), B2(2m) and G2(3m), then L admits a socalled diagram automorphism 't which pennutes the parabolies P 1 and P 2. In each of these cases i = <L, 't> acts 1-transitively on !J..

The generalized polygons related to the Lie type groups A 2 (q ), B 2 (q) for q = 2m, and G2(q) for q =3m are called the classical generalized polygons and they are denoted by !J.3,q. A4,q and ll{,,q· Notice that l!.n,q is a regular generalized n-gon. In each of the above cases the group i acts (n + 1)-transitively on !J.n,q·

Now we areready to fonnulate the main result of [Wei4].

Theorem 2.1.5. Let G act s-transitively, but not (s + 1)- transitively, on a graph r of valency k, where s ~ 4, k ~ 3. Then s e { 4, 5, 7} and there is a group L acting s

transitively on a classical (t -1)-gon !J. = !J.t-1,k-1, suchthat for u e V (!J.), {u, v} e E(!J.), the amalgarns {H,K} and {L(u),L{u,v}} are isomorphic. Moreover, either t =s and L contains the corresponding group of Lie type, or t = 5, s = 4, k = 3 and L = PGL 2 (9). D

For a long time bounding the order of a vertex stabilizer in the case of projective subconstituents in dimension n ~ 3, was an open problem. In that case s = 2 or 3. In [Wei5] abound on the orderwas obtained in the case when s = 3 and the characteristic of the field is at least 5. In [Troll this result was extended to the case s = 2. Namely, it was proved that if the characteristic of the field is at least 5, then G2{x) = 1. Just recently [Tro2] the final result was announced. It tumed out that G 5 (x,y) = 1 in all cases. If the field contains more than 2 elements then this bound can be improved. Notice that if the field contains just 2 elements then a number of examples with rather complicated structures of H are known. These examples arise from sporadic simple groups acting on the so-called P- geometries.

2.2. Goldschmidt's Iemma

Nowadays results similar to Theorem 2.1.5, i.e. describing a large class of amalgams up to isomorphism, are quite rare. Solutions to concrete problems by means of the amalgam method usually consist of a step-by-step reconstruction of the corresponding amalgams. In the case of a rank 2 amalgam {H,K} this means a reconstruction of the groups H and K up to isomorphism and an identification of H n K as subgroups of H and K. If the structures of groups H and K are rather complicated, even the first step is difficult enough. In addition, if the groups H and K are reconstructed up to isomorphism and the subgroups of H and K which correspond to H n K are determined, there can be a number of nonisomorphic amalgams. The spectrum of these possibilities is described by a lemma proved in [Gol].

The amalgam {H,K} is determined by a pair of embeddings of B =H n K into H and K, respectively. Let a pair ((\ll,(h) determine a fixed amalgam {H,K}. Put At =NAut(H)($1(B)), At =$!(At), where $f(h)=$1 h$11. Let a subgroup A2 be

424 A. A. IV ANOV AND S. V. SHPECTOROV

defined analogously with respect to K and C\12- Notice that A 1 and A2 are subgroups of Aut(B).

Lemma 2.2.1. The isomorphism classes of amalgams {H,K} are in a bijective

correspondence with the double cosets of A 1 and A 2 in A = Aut(B ). o The correspondence stated in Lemma 2.2.1 can be defined as follows. If (C\11 .C\12)

defines a concrete amalgam {H,K}, then any other amalgam is described by a pair

(C\11, 'VC\12) where 'V is an automorphism of the group B. The amalgams determined by the pairs (C\11, 'VC\12) and (C\lt. 'V'C\12) are isomorphic if and only if 'V and 'V' are in the same double coset of A 1 and A 2 in A. It is clear that we can consider the group 0 = Out(B) of outer automorphisms of B and the images 0 1 and 0 2 of A 1 and A 2 in this group instead

of the groups A 1, A 2, A themselves.

Let us illustrate Goldschmidt's Iemma by a few elementary examples. First we present a simple example leading to two different amalgams. Notice that here the index of H n Kin K is greater than 2.

Example 2.2.2. Let H = S6, K =L3(2) X 2 and H n K = s4 X 2. The subgroups H

and K contain a unique dass of subgroups S 4 x 2 up to conjugacy in their automorphism groups. On the other hand I 0 1 I = I 0 2 I = 1 and I 0 I = 2. So by Goldschmidt' s Iemma there are up to isomorphism exactly two amalgams {H,K}. 0

Example 2.2.3. Let H = Sk or Ak and H n K = Sk-1 or Ak-1, respectively. Then if the condition I K : H n K I = 2 holds, it is easy to obtain the list of possibilities for the group K. For instance, if H = Sk then either K = Sk-1 x 2 or k = 3 and K = Z4, or k = 7 and K = Aut(S 6). In all cases it is easy to check that 0 = 0 2• and hence for each particular group K there is exactly one amalgam {H,K}. 0

In a certain sense Goldschmidt's Iemma formalizes our intuitive understanding of how to construct amalgam {H,K} from the groups Hand K. On the other hand, the fact that H and K are groups does not play any role in the proof. So it is easy to generalize this Iemma to the case of amalgams of higher rank or even to more general classes of objects. Namely, Iet Hand K be two amalgamsandin Hand K Iet two subamalgams be chosen.

Suppose both these amalgams are isomorphic to an amalgam A. By definition an automorphism of an amalgam is a permutation of its elements which preserves all members of the amalgam and which, upon restriction to any one of its members, produces an automorphism. As wasdonein Goldschmidt's Iemma, we can define the groups 0, 0 1 and 02. Moreover, the number of different unions of the amalgams Hand K with intersection isomorphic to the amalgam A coincides with the number of double cosets of 0 1 and 0 2 in 0. The proof is given in [Goi].

A more detailed consideration of amalgams having rank greater than 2 will be given

below. At present we would like to illustrate how Goldschmidt's Iemma can be applied to this situation.


Example 2.2.5. The group A 7 contains a rank 3 amalgam consisting of subgroups H 1 = A 6, H 2 = (S 4 x S 3 t and H 3 = P SL 3 (2) with the following properties. For Hij=HinHj, H123=H 1 nH2nH3, we have Hij=S4 for 1:5;i,j:5;3, i '# j ; H 123 = D s. This amalgam arises from the action of A 7 on the exceptional C 3-

geometry related to this group. We claim that up to isomorphism this amalgam is determined by the above properties. Let us indicate the rnain steps in the proof of this claim.

Step 1. For each i, 1 :5; i :5; 3, Hi contains, up to conjugacy in its automorphism group, a unique subamalgam {S4,S4} whose members have a common subgroup iso

morphic to D s.

Step 2. Since S 4 has no outer automorphisms, the arnalgam { H 2 ,H 3 } is determined in a unique way.

Step 3. Let us glue H 1 to the amalgam { H 2 ,H 3 } via the subamalgam {S 4 ,S 4}. The "outer automorphism group" of the latter amalgam has order 2. Namely there is an automorphism which acts trivially on the first member and acts on the second member as the center of a Sylow 2-subgroup acts via conjugation. So I 0 I = 2. On the other hand 0 1 is also of order 2 since there is such an automorphism in S 6 :5; Aut(H 1 ). In this way we obtain the result. D

It should be mentioned that in the reconstruction of amalgams of high rank, Goldschmidt's lemma is not the most convenient instrument. There are arguments of another type which sometimes enable us to avoid the first step, i.e. the reconstruction of the members of the amalgam up to isomorphism.

2.3. Serre's paper

Now we will briefly present what is known about steps 3 and 4 of the amalgam method (described in the introduction) when the rank is equal to 2. The original and most fundamental paper on this subject is the paper [Ser] and we will follow here its content.

It can be shown that the universal group 6 of the arnalgam {H,K} coincides with the free product of these groups with common subgroup H n K. In particular, if H > H n K and K > H n K then 6 is an infinite group. In [Ser] an action of 6 on an infinite tree was described. This enabled Serre to give very simple and clear proofs of a number of classical results concerning the properties of 6.

In our particular case the situation can be described as follows.

Suppose that H = G(x) and K = G {x,y} arise from a group G acting 1-transitively on a graph r. Let G' be another group which contains the amalgam Ao = {H,K} and suppose that G' is generated by the elements of this amalgam. Suppose that there is a homomorphism q, of G' onto G which is identical on the elements of the amalgam A0 .

Such a homomorphism q, will be called an Ao-homomorphism. It is clear that each Aohomomorphism induces a covering r(G',Ao) ~ r in the obvious topological sense.


Let w = (x 0 =x,x 1' ... 'Xn) be path oflength n in r, which originates at X. The path W is said tobe simple if Xi *Xi_1 for 1::::; i::::; n. Let f be a graph whose vertex set is the set of all simple paths in r with origin x. Two paths are adjacent in f if the difference of their lengths is equal to 1 and one of them is a subpath of another. It is easy to see that r is a tree and that if ~ is the mapping which assigns to each path its terminal vertex, then ~ is a covering.

Each automorphism from G can be lifted to an automorphism of f. This lifting is unique in the following sense. Let g e G and suppose that g maps u onto v where u,v e V(r). Let us pick preimages ü e ~-1 (u) and ii e ~- 1 (v). Then there is a unique automorphism g = g(ü, ii) of f which maps ü onto ii and sarisfies the equality g~ =~·In particular, if we fix a vertex i e ~-1 (x) then the set {g(i,i) I geH} forms a subgroup which stabilizes ii and is isomorphic to H. In an analogous way the subgroup K can be extended to a subgroup which stabilizes the edge {i ,ji}. Here ji is the unique vertex such that ji e f(i) and ~(ji) = y. To simplify the notation we denote these subgroups also by H

and K. The subgroups H and K together generate in Aut(f) a group G which acts 1-transitively on f. Since f is a tree it does not admit a nontrivial covering. So G in fact coincides with the universal group of the amalgam {H,K}.

The relationship between coverings of graph and morphisms of groups generated by the same amalgam {H,K} tums outtobe very useful in certain situations.

Example 2.3.1. As we saw in the introduction, the property of s-transitivity a trivalent graph is precisely that property for the corresponding amalgam. It is clear that the property of 2-transitivity for a graph is again that property for the amalgam. From the theory presented above, we have the following result which is not immediately obvious. For any s and any fixed valency, s-transitivity of the graph is a property of the amalgam { H,K}. Indeed, the action of H on the set of paths of length s in r with origin x is similar to its action on the set of paths of length s in f with origin i.

The following example is due to Djokovic [Djo]. 0

Example 2.3.2. Let N be the kemel of the natural homomorphism of G onto G. The intersections of N with H and K are both trivial. This implies that N acts freely on f, so semiregularly on both V (f) and E (f). As was shown in [Ser], a group which acts freely on a tree is a free group. So it contains infinitely many subgroups of finite index which are normal in the group G. Hence the classification problern for all factor groups (or even for all finite factor groups) of the group G is rather intractable. Perhaps it is possible to classify the minimal factor groups is some cases. But to the best of our knowledge there has not yet been found a reasonable way to handle this problem. o

It follows from the above examples that there is no hope of classifying the stransitive graphs without additional assumptions (conceming for instance the girth of the graphs), but the classification problern for s-transitive amalgams is rather reasonable.

Example 2.3.3. Some sporadic simple groups contain amalgams {H,K} arising from actions of certain classical groups on their natural geometries. So these sporadic groups are automorphism groups of objects which are "locally isomorphic" to classical geometries. This fact enables one to construct the sporadic groups in terms of the corresponding classical geometries and to obtain certain characterizations of these groups. Such constructions were carried out for J 3 in terms of PG (2,4) [Wei6], forM 12

in terms of PG(2,3) [Wei7] and forM 22 in terms of PG (3,2) [StWe]. D

3. A vertex, an edge and a cycle

As we saw in the previous section, the action of the stabilizer of a vertex on the set of paths originating at that vertex can be described in terms of the corresponding rank 2 amalgam. But this amalgam does not control which of the paths are cycles and which are not. Indeed, the graph corresponding to the universal group of the amalgam is a tree and it contains no cycles. On the other hand, the factors of the tree should contain some cycles. Below we will show that in terms of the rank 2 amalgam alone it is possible to formulate certain conditions on the lengths of some cycles. But if we want to guarantee existence of cycles of prescribed length, we should bring into play amalgams of rank at least 3. Before increasing the rank of the amalgams, Iet us discuss a very important class of cycles.

3.1. Apartments

In a restricted sense apartments are just the cycles of minimallength in the generalized polygons. Let ~ be a generalized n-gon. Let ~ be the universal cover of ~. i.e. ~ is a tree. Then the preimage of each cycle of ~ is an infinite path (a track) in the tree ~- Let T be the collection of all tracks which are preimages of shortest cycles of ~- As a direct consequence of the axioms of a generalized n-gon, the following conditions are satisfied.

The uniqueness condition. Each (n + 1)-path in the tree ~ can be extended in a unique way to a track in T.

The exchange condition. Let (Xn,Xn-1> ... ,Xt.Xo.Y1, ... ,yn) and <xn,Xn-1• ... ,X1,Xo,z1, ... ,Zn) be two 2n-paths in~ which have an n-path in common, i.e. z 1 "# y 1 • Suppose that each of these paths can be extended to a track in T. Then the path <Yt.Yn-t. ...• Yt.Xo,Zt. ... ,Zn) can also be extended to a track in T (seeFig. 1)

428 A A IV ANOV AND S. V. SHPECTOROV

Fig. 1.

Now suppose that an arbitrary graph contains a farnily T of paths (cycles or infinite tracks) which satisfy the uniqueness and exchange conditions. Then the elements of this farnily will be called apartments. When we wish to specify the value of n we will speak of n-apartments. Notice than there can be a few families of apartment. If G is a group acting 1-transitively on r then we will be interested only in the families of apartments which areinvariant under G. If r contains a family T of apartrnents then the preimages of the paths from T form a family of apartments in the covering tree t of r. Analogously, if f is a farnily of apartments in f which is preserved by the group (;, then their irnages in r form a family of apartments. This irnplies, in particular, that existence of apartments is not a property of r but a property of the corresponding amalgam { H, K } . It is clear that if G is an s-transitive group of automorphisms of r then there are no t-apartments for t:::; s - 1. The most interesting Situation is when G acts s-transitively on r and r contains a family of (s- 1)-apartrnents.

The existence of apartments is a very strong condition. In many cases it enables one to reconstruct the amalgam { H,K} up to isomorphism. Let r = t be a tree and T be a farnily of n-apartments in r, n:?: 3. The farnily T enables us to construct a generalized n

gon from r. Let-be the relation on V(r) defined as follows:

d(x,y) = 2n and the unique path joining x - Y ~ x with y is contained in a track from T .

Let us consider the transitive closure of the relation -, that is the minimal equivalence relation on V (r) which contains -. Let .1 be the graph whose vertex set is the set of equivalence classes of the vertices of r. Two classes are adjacent if they contain adjacent vertices of r. Then the following proposition holds.

Theorem 3.1.1. [DGS] The graph .1 is a generalized n- gon. If, in addition, a group G acts s-transitively on r, s = n + 1, and preserves T, then G induces an s-transitive action on .1 and the rank 2 amalgams corresponding to the actions of G on r and .1 are isomorphic. o

APPLICA TIONS OF GROUP AMALGAMS TO ALGEBRAIC GRAPH THEORY 429

This very beautiful result explains in a certain sense why the rank 2 amalgams related to the actions of the Lie type groups on generalized n-gon exhausts the amalgams corresponding tos-transitive actions on graphs with s:;:: 4. Namely, Theorem 3.1.1 shows a way to prove Theorem 2.1.5: Consider the universal cover of the graph in question and look for apartments in this cover.

The generalized 2-gons are, in a certain sense, degenernte objects. Nevertheless, Theorem 3.1.1 is still meaningful for n = 2. In this special case, which was not considered in [DGS], weillustrate applications ofTheorem 3.1.1.

Let F be a permutation group on the set Q. This group will be called strongly dou-bly primitive if

(a) it is doubly transitive;

(b) the stabilizer F ( a) of a point a E n acts primitively on n - { a};

( c) F ( a) contains no normal groups acting regularly on Q - { a}. Notice that doubly primitive groups are precisely the groups satisfying (a) and (b).

The class of strongly doubly primitive groups contains the groups Sn and An for n :;:: 6, the groups Sp 2n (2) in their doubly transitive representations, the Mathieu groups, and some doubly transitive groups of affine type.

Theorem 3.1.2. Let G act 2-transitively on r, G 1 (x,y) = 1, G 1 (x) :F 1, and Iet the subconstituent G (x)r(x) be a strongly doubly transitive permutation group. Then G acts 3-transitively on r and r contains 2-apartments.

Let us indicate the main steps in the proof of this theorem.

Step 1. By means of a standard argument in the theory of s-transitive graphs, it is easy to show that if y E r(x) then G 1 (x) induces a nontrivial action on r(y)- {x}. Since G (yl(y) is strongly doubly primitive and G 1 (x) <l G (x,y), the action of G 1 (x) on r(y) - {x} is transitive and nonregular. This implies in particular that G acts 3-transitively on r.

Step 2. Letz E r(y)- {x}. The set of vertices fixed by the subgroup G (z) n G 1 (x) forms an imprimitivity block of the group G (x,yl(y)--{xJ. Since G(yl(y) is strongly doubly primitive, we conclude that G (z) n G 1 (x) fixes only z in r(y)- {x}.

Step 3. Put C=Cc(x)(G1(x)). Then C is normal in G(x). In addition G 1 (x,y) = G 1 (x) n G 1 (y) = 1 and G 1 (x), G 1 (y) <l G (x,y). Hence G 1 (y) <l C. Since G 1 (yl(y) is non trivial, C acts transitively on r(x).

Step 4. The subgroup C commutes with G (z) n G 1 (x) and acts transitively on the set r(x). So for each vertex u E r(x) there is a unique vertex V E r(u)- {x} suchthat G(v)nG 1(x)=G(z)nG 1(x).

Step 5. Let (xo,XI,Xz,X3) be an arbitrary 3-path in the graph r. As was shown above, there is a unique way to extend this path to a path ( · · · ,x _1 ,x 0 ,x 1 ,x 2, · · · ) possessing the property that G (xi-2) n G 1 (xi) = G(xi+2) n G 1 (xi) for all i. It follows


directly from the definition that the set of these paths satisfies the exchange condition. So these paths are apartments. 0

Example 3.1.3. Let r be the odd graph Ok of valency k. This means that V (r) consists of all (k -1)-element subsets of a fixed set Q of cardinality 2k- 1; two vertices are adjacent if the corresponding subsets are disjoint. Then the group G = Aut(r) coincides with the symmetric group S (Q) = S 2k-t of the set n. Let x E V (r) correspond to a subset 11 of Q. Then G(x)=S(11)xS(Q-11):=Sk-t xSk. Let I: be the subset of Q

corresponding to a vertex adjacent to x. Then I Q- (11 u I:) I = 1. So r(x) is in a bijective correspondence with the elements of the set Q- 11, and the normal subgroup Sk_1 in G(x) acts trivially on r(x). In an analogous way, one can see that the set r 2 (x) is indexed by the pairs (a., ß) where a E Q- ~ and ß E 11. This implies, in particular, that G 1 (x,y) = 1. Finally, since Sk is strongly doubly primitive for k ~ 6, we see that the hypothesis of Theorem 3.1.2 is satisfied for these k. So at least for k ~ 6 the graph Ok

contains apartments. It can be shown that the apartments have length 6 and that they can be described as follows. Let Q = ~1 u ~2 u 2 where I ~1 I = I ~2 I = k- 2, 121 = 3. Then the six vertices of the odd graph corresponding to the subsets 11i u { a}, i = 1, 2, a E 2, form a cycle which is an apartment. Now by Theorem 3.1.1, the amalgarn { G (x ), G {x,y}} is isomorphic to the analogous amalgarn arising from the group (Sk wr S 2) acting on the complete bipartite graph Kk,k· o

Remark 3.1.4. The conclusion of Theorem 3.1.2 concerning the existence of apartments is true for a larger class of subconstituents. In a more general situation the main step of the proof is to show that all orbits of the subgroup C (x) = CG(x)(G 1 (x)) on r(x)

have length k, and for y E r(x) all vertices of r(y)- {x} areindifferent orbits of C (x).

The apartments are the paths ( · · · ,xo,Xt. · · ·) suchthat Xi-2 and Xi+2 are in the same orbit of C (xi) for all i. These arguments can be carried out if C (x) n G 1 (x)-# 1, i.e. when the center of G 1 (x) is trivial. But if this is not the case we cannot prove the result. On the other hand, we know of no exarnples when G 1 (x) =t= 1, G 1 (x,y) = 1 and the apartments are absent. So we come to the following conjecture.

Conjecture 3.1.5. Let G act 2-transitively on a graph r, G 1 (x) =t= 1, G 1 (x,y) = 1. Then r contains a family of 2-apartments. 0

Let us turn now to I-apartments. Here Theorem 3.1.1 is meaningless since the notion of generalized 1-gon does not exist. Moreover, for n = 1 the exchange condition is a direct consequence of the uniqueness condition. Let us discuss what is happening with regard to the uniqueness condition.

Let G act 2-transitively on r and suppose that r contains a family of I-apartments. Then each 2-path can be extended in a unique way to an apartment. Let (y,x,z) be a 2-path. Then G (x,y,z) should fix a Vertex u E r(y)- {x}, which is the continuation of the path (y,x,z) in the corresponding apartment. Since G (x,y) acts transitively both on r(x)- {y} and on r(y)- {x }, we have the equality G (x,y,u) = G(x,y,z). On the other hand there is an element 't in G {x,y} which maps z onto u. So 1: normalizes G (x,y,z ). It is


easy to see that the existence of an element in G {x,y} - G (x,y) which normalizes

G (x,y,z) is equivalent to the existence of a vertex u with the prescribed properties. If

such an element 't exists and if, in addition, G(x,y,z) does not fix a vertex in

r(x)- {y,z }, then the connected component of the subgraph fixed by G (x,y,z) forms an apartment. So we have the following.

Proposition 3.1.6. Let G act 2-transitively on a graph r and lety,z E r(x). Then Iapartments exist only if there is an element in G {x,y} - G (x,y) which normalizes

G (x,y,z). If G (x,y,z) does not fix vertices in r(x)- {y,z }, then this condition is also

sufficient. 0

Notice that if G 1 (x) "#I then G 1 (x) acts fixed-point-freely on r2(x), so there is no

I-apartment in that case. If G 1 (x) = I then usually I-apartments exist. A typical example is the k-dimensional cube. Here the apartments are the shortest cycles.

At the same time there are examples of 2-transitive graphs with G 1 (x) = I which

have no I-apartments. All examples which are known to us are distance-transitive graphs. They are the Hoffman-Singleton graph, the double cover of the complementary graph of the Hoffman-Singleton graph, and some graphs related to the sporadic simple

groups HS, M 22, M 23 (see Section 5.5 in [Ivn2]). This list is rather interesting and it Ieads to the following problem.

Problem 3.1.7. Describe the rank 2 amalgams corresponding to 2-transitive actions

with G 1 (x) = I and without apartments. 0

3.2. Lengths of apartments

In this subsection we consider some conditions on the lengths of apartments which

follow from the structure of the rank 2 amalgam consisting of a vertex and an edge stabilizer.

Let C be a cycle in r passing through the edge {x,y} such that its stabilizer G { C}

acts 2-transitively on this cycle. In other words, if l is the length of C then G { C} I G (C) = D21· It is easy to see that if Cis a I-apartment and G acts 2-transitively on r then this condition is always satisfied.

We are interested in possible values of l for a given amalgam {G(x), G{x,y}} = {H,K}.

Let

te (HnG{C})-G(C);

a e (K n G{C})- G(C).

Then 't2, a 2 e G (C) and the element ta induces a translation on one edge along the cycle C. This implies that (tai e G (C). The elements 't and a induce automorphisms of the group G (C) and, modulo the inner automorphisms, these automorphisms do not depend


on the particular choice of 1: and a. Let m be the minimal number with the property that (1:a)m induces on G (C) an inner automorphism. Then m also does not depend on the particular choice of 1: and a. This immediately implies the following.

Proposition 3.2.1. m divides l. 0

Below we will illustrate this proposition by a number of examples.

Example 3.2.2. Let G (x) = G (xpx) = Ak where k is the valency of the graph. First let us describe all the possibilities for the amalgam (H,K}. Since G (x,y) = Ak-l if k -# 7 we have only the following two possibilities: K := Sk-1 and K = Ak-1 x 2. These possibilities correspond to two amalgams A1 and A2, respectively. If k = 7 then we have two additional amalgams corresponding to K = PGL2(9) and K = M 10· By Proposition 3.1.6, in the case of amalgams A1 and A2, the graph r contains I-apartments. If Cis such an apartment, then G (C) = Ak-l· Let 1: and a be as defined above. Then for the amalgam A1 ,

the automorphisms of G (C) induced by 1: and a are both conjugations by suitable elements from Sk-I· Hence 1:0 induces an inner automorphism and m = 1. In the case of the amalgam A2 , the automorphism induced by 1: is the same as in the previous case, but a can be chosen in such a way that it commutes with G (C). So m = 2 in this case and the length of an apartment is even.

Notice that the amalgams A1 and A2 are both realized in certain groups acting 2-transitively on the k-dimensional cube Qk. Indeed, Aut(Qk) = 2k J...Sk and this group contains a subgroup N = 2k-l J...Ak. The factor group Aut(Qk) IN is elementary abelian of order 4, so there are exactly three subgroups of index 2 in Aut(Qk) which contain N. One of these subgroups stabilizes the parts of Qk while two other groups act 2-transitively and are generated by the amalgams A1 and A2, respectively. o

Example 3.2.3. Let G 1 (x) = 1 and PSL2(q) $ H $ PrL 2 (q), k = q + 1 > 4. Then G (x,y) is a Borel subgroup, i.e. the norrnalizer of a Sylow p-subgroup from Ho =PSL2(q) where q isapower of p, and G(x,y,z) is the normalizer in G(x,y) of a Cartan subgroup of Ho n G (x,y). Since q > 3, the subgroup G (x,y,z) does not fix a vertex in r(x)- (y,z}. By Proposition 3.1.6 this gives us apartments in r. o

The following very interesting result was proved in [Perl].

Theorem 3.2.4. Suppose that in the Situation described above r contains apartments of odd length. Then q E ( 4, 5, 9}. 0

A simple proof of this theorem can be carried out using Proposition 3.2.1. Let F = G (x,y,z) n HO· Then Fis a cyclic group of order (q -1) I (2,q -1). The element 1:

inverts F, that is r = r 1 for f E F. On the other hand, the element (J either commutes with F or acts on F in a manner similar to the way the field automorphism of order 2 acts on the multiplicative group of GF(q). Since the apartments are of odd length, the elements 1: and a induce conjugate automorphisms of G (C) = G (x,y, z ). Now a direct check shows that this is possible only for the values of q listed in the theorem.

For the values of q listed in Theorem 3.2.4 and only for these values do there exist triply transitive permutation groups of degree q + 2 whose point stabilizers contain PSL 2(q) as a normal subgroup. This situation also corresponds to apartments of odd length, namely of length 3. For q = 2, 3,4 and 5, graphs with apartments of length 5 exist. We will discuss these examples in the next subsection.

Example 3.2.5. Let r be the incidence graph of a Desarguesian projective plane and G be a 4-transitive subgroup of the automorphism group of r. This implies that G contains a subgroup Go of index 2 such that PSL3(q) ~Go~ PrL3(q), and that each element from G - Go permutes the parts of r.

The length of an apartment in r is equal to 6 and hence m divides 6. Let us study in which situations m can be less than 6. It follows from the inequality m '# I = 6 that for an apartment C, there are elements in G (C) <a,'t>- G (C) which commute with G (C).

The pointwise stabilizer of the apartment C in the full automorphism group

PrL3(q) is a group F which is the extension of the Cartan subgroup by the field automorphisms. So in the general case G (C) =Gon F. We will assume below that C is fixed by the Standard Cartan subgroup, i.e. by the subgroup of diagonal matrices.

Put F 0 = F n PSL3(q). The elements of the normalizer of F o in Aut(r) have the form a = gfsv where g is an element of the Cartan subgroup, f is a field automorphism, s is a permutation matrix and v is either the trivial or the contragredient automorphism. We

of course exclude the case q = 2 when the subgroup F o is trivial.

First Iet us separate out the cases when m is not divisible by 2. If this is the case, then there exists an automorphism a of the presented form, such that s = 1 and v '# 1, which commutes with G (C) and hence with F 0 • Let X be a diagonal matrix with deterrninant 1 and let a, ß, y be the values on the diagonal of X. Then the diagonal values of xa are a-p', ß-p', y-P'. By assumption, a commutes with F 0 , so xa = AX, where A = AE is a

scalar matrix of order 3. Since the pair (a,ß) is arbitrary, we conclude that Ais trivial. So for each element a of the field GF(q) we have the equality a-p'-l = 1, i.e. pr + 1 is

divisible by q- 1. Now it is easy to see that this is possible only if q ~ 3, or q = 4 and Pr =2.

In an analogaus way one can study the case when m is not divisible by 3. For an appropriate choice of a the matrix Xa has the following diagonal elements: yP', CJ!'', ßP'. Since a and y are arbitrary elements of the field, the parameter A. can also be arbitrary. This is possible only if q = 2 or q = 4.

So we have shown in particular that if q '# 2 and m = 1, then q = 4. Moreover, in the latter case Go cannot be equal to PrL3(q) since, otherwise, m would be divisible by 3. Using this fact it can be shown that there are exactly two amalgams for which m = 1. It is remarkable that both of them are realized on 4-transitive graphs related to the sporadic simple group J 3 and to its triple cover [Wei6]. The length of an apartment in each graph ~~ 0

In view of this example the following problern is rather interesting.

Problem 3.2.6. Calculate the values of m for the groups acting (n + 1)-transitively on the classical generalized n-gons of type B 2 and G 2. D

The previous examples show that the parameter m controls the existence of stransitive graphs with a given amalgam {H,K} and with a given length of apartments.

3.3. Amalgams of rank 3

In the previous section we discussed how to reconstruct the amalgam Ao = {H,K} in certain situations and how to deduce from the structure of this amalgam information about lengths of apartrnents. Now suppose that the amalgam Ao is known and that the length l of apartments is prescribed. With an apartment C a subamalgam consisting of the subgroups H 0 = G (C) • <'t>:::;; H and K o = G (C) · <cr>:::;; K can be associated. The group L = G { C} is generated by its subgroups Ho and K o, and L I G ( C) = D 21· So the additional relation for L has the form ('toi = c where c is a suitable element of G (C). If we determine c then the rank 3 amalgam A = { G (x), G {x,y}, G { C}} will be determined. For small values of l the universal group of this amalgam can be finite. If this is the case, then we are in a position to classify the graphs in which we are interested. Notice that the arnalgam Adetermines the length of apartments.

Let us find conditions which the element c should possess. If 't and cr are both involutions then each of them should invert c and, in addition, c should induce on G (C) the same automorphism as ('tcr)1• Even under these conditions it is not easy to find all possibilities for c in each particular Situation. In order to avoid these difficulties we choose a more convenient point of view. Namely we consider the kernel of the natural homomorphism of the universal group of the amalgam {H 0 ,K 0 } onto the group L.

So, let Go be the universal group of the rank 2 amalgam A0 . Then Go acts on the covering tree of the graph r. The subgroup i of 6 0 which is generated by the amalgam {H 0 ,K 0 } acts 1-transitively on the track C which covers the apartment C. Hence L = i I G (C) = <~,cr> = D oo and i coincides with the universal group of the amalgam {Ho ,K o } . If N is the kernel of the natural homomorphism of G onto G, then the kernel of the homomorphism of i onto L is just No = N n i. The subgroup N 0 is normal in i and its intersections with Ho and K o are both trivial. Since G (C) is also normal in i, the subgroup No centralizes G(C). In addition the image of No in L contains the unique subgroup of index l in theinfinite cyclic group <'tcr>. Let F be the full preimage in i of this subgroup of index l. Now we are ready to formulate a precise condition on the subgroup


N 0• The amalgams A are in bijective correspondence with the subgroups N O• which possess the following properties:

F = G (C) x N 0 and N 0 is invariant under 't and a.

Remark 3.3.1. If the center G (C) is trivial, then No coincides with the centralizer of G (C) in F. In particular the amalgam Ais uniquely detennined. o

Remark 3.3.2. If the amalgam Ao possesses nontrivial automorphisms, then some amalgams A corresponding to different subgroups No may be isomorphic. 0

Remark 3.3.3. If m divides 1 then F = G (C) CF(G (C)), but not always does G (C) have in F a direct complement invariant under 't and a. So the above conditions on No are more restrictive than the condition that m divides 1. 0

Remark 3.3.4. One should understand that the divisibility condition and the conditions on the subgroup N 0 are only necessary for the existence of a graph with the prescribed properties. In fact, the universal group of the amalgam A may be trivial. 0

Now let us consider some examples.

Example 3.3.5. Let H = G (x)r(x) = Sk. As was shown in Example 2.2.3 there is a serial amalgam Ao = {Sk,Sk_1 x2} as well as two exceptional amalgams {S3,Z4} and {S7,Aut(S6)}. The second exceptional amalgam is 3-transitive and it will be considered in the next example. Herewe restriet ourselves to the serial amalgam Ao.

This amalgam admits 2-apartments and G (C) = Sk-2· So we can assume that t,a are involutions commuting with G ( C); in particular m = 1. In addition the center of the group G (C) is nontrivial only if k = 4. So for k :#- 4 there is a unique amalgam A = {H,K,L}. The subgroup No corresponding to this amalgam is generated by (tcr)1•

Let us detennine the universal group G of this amalgam. We assign the numbers from 1 to k to the vertices of r(x) and assume that C contains vertices with numbers 1 and 2. Then we can choose t tobe the transposition (1,2). Let a 1 =a, a2 =t and for 3 ~ i ~ k, let ai denote the element of H which induces on r(x) the transposition (i -1, i). It is easy to see that { ai} 1 s i s k forms a Coxeter system for the group G, and hence G is a factor group of the Coxeter group with diagram

(1)

This Coxeter group is finite only if l ~ 4, or l = 5 and k ~ 4. It is possible also to determine the graphs corresponding to concrete groups G and to their factor groups.

The case 1 = 3 corresponds to the complete graph on k + 1 points; if 1 = 4 we have the k-dimensional cube Qk and the folded cube Dk. If l = 5, k = 3, then we have the dodecohedron and the Petersen graph; for 1 = 5, k = 4, we have two graphs, one related to the group H 4 = (SL 2 (5) * SL 2 (5)) · 2, the other arising from a factor group of H 4 isomorphic to (L2(5) x L2(5)) • 2. Here * stands for central product.

In the case k = 4, besides the amalgam corresponding to No = <(tcri >, we have a second amalgam corresponding to N 0 = <(tcri c > where c is the unique nontrivial element of the subgroup G ( C) = Z 2 . lt follows from a result by M. Ronan [Ron] that the universal group of this amalgam is infinite for I ;::: 6. For small values of I the universal groups were found by coset enumeration. These calculations show that G is trivial for I= 3, G = PGL 2 (7) for I= 4, and G = (PGL 2 (11) x S3)+ for I= 5 (the latter fact is due to L. Soicher). 0

Example 3.3.6. Now Iet k = 7, H = S7 and K = Aut(S 6). As was mentioned above this amalgam is 3-transitive. Moreover, it admits 3-apartments and G (C) = F~ for such an apartment C. Here F~ stands for the Frobenius group of order 20. Since this group is perfect, the elements 't and cr commute with G (C). This implies that m = 1 and, since the center of G (C) is trivial, for each I there is exactly one rank 3 amalgam A. Regarding the universal group of these amalgams, we have the following result obtained by coset enumeration. For I = 4, 6, we get a group of order 2, for I = 3, 7 the group is trivial, and for I = 5 we get the group P'LU 3 (5) acting on the Hoffman-Singleton graph. Finally for l = 8, the algorithm did not finish in a reasonable amount of time. It is natural to conjecture that the universal group is infinite for I;::: 8. 0

Example 3.3.7. Now Iet us turn to the case H = G(xpx> =Ak. Here for each k, k '# 7, we have the following two amalgams of rank 2: {At.Ak-1 x 2} and {At.Sk_1 } (see Example 3.2.2). If k = 7 we have the following two additional amalgams: {A 7,PGL 2 (9)} and {A 7,M 10 }.

Let us first consider the serial amalgams {At.Ak_1 x 2} and {At.Sk-1 }. The parameter m for these amalgams is equal to 2 and 1, respectively. If k '# 5 then the center of G (C) = Ak_2 is trivial. So in each case we can construct a unique amalgam A = {H,K,L}, with I even in the first case.

In order to deterrnine the universal groups of these amalgams, we use the following arguments. Let Ao = {H,K) be the original rank 2 amalgam. Then there is an involutory automorphism <jl of A0 suchthat the amalgam A0' = {H',K'), consisting of the subgroups H' = H · <$> and K = K • <<jl>, is just the amalgam {St.Sk_1 x 2} from Example 3.3.5. This implies that the universal group G0 ' of the amalgam A0' acts on the tree f which corresponds to the universal group G0 of the amalgam A0 . Moreover, it is clear that G0

has index 2 in Go'· Without loss of generality we may assume that the system of apartments of G0 remain invariant under G0 '. Indeed, for k "?. 5 this system of apartments is unique. In the remaining cases the existence of a couple of such systems is compensated for by automorphisms of amalgams.

The stabilizers L and L' of a track C in the groups in question, act 2-transitively on this track and L has index 2 in L '. So for k '# 5 the subgroup No, corresponding to the unique amalgam {H,K,L} with given value of I, is normal in L' and deterrnines an amalgam {H',K',L') corresponding to the same l. It is also clear that the latter amalgam coincides with the amalgam from the Coxeter group. Now it is easy to identify the universal

group of the amalgam {H,K,L} with an index 2 subgroup in the corresponding Coxeter

group.

Now Iet k = 5. Then G (C) = Z3 is abelian of order 3 and the subgroup F 0 is iso

morphic to z3 X z. It is clear that both 't and cr invert z in this group. The actions of

these elements on G ( C) depend on the structure of the corresponding rank 2 amalgams.

In the amalgam {As,A4 x2} the element t inverts G(C) and cr commutes with this

group. In the amalgam { A 5 ,S 4} both of the elements invert G ( C). So in the former case

there is exactly one amalgam {H,K,L} and this amalgam is embedded in the correspond

ing Coxeter group. In the second case, taking into account automorphisms of the rank 2

amalgam, we obtain up to isomorphism exactly two amalgams. One of these amalgams is

also contained in the Coxeter group. The universal group of the second one was treated

by computer. For l = 3 we clearly get the trivial group, for l = 4 we get the group

PGL2(11) and for l = 5 we get the group PSL2(31) (see [Per2]).

The ama1gams {A7,PGL2(9)} and {A7,M10} are embedded in the amalgam

{S7,Aut(S6)}.

The results considered in the above two examples can be formulated as follows.

Theorem 3.3.8. Let r be a 2-transitive graph of girth l and valency k > 3 and sup

pose that for X E V (r) we have G (x) = 1 and G <xPx) = sk or Ak. Then

(i) if l = 4, either

(a) r is one ofthe following: Qk, Dk> or Kk+l,k+l minus a matching, or

(b) k=4,G(x)=S4, G=PGL2(7),or

(c) k = 5, G (x) =As, G =PGLz(11);

(ii) if l = 5, either

(a) r is related to a factor group of the Coxeter group Hk or of its subgroup of

index 2, or

(b) k = 4, G(x) = S4, G = (PGL2(11) xS3t, or

(c) k=5,G(x)=As, G=PSL2(31). D

In view of the above arguments, the proof of the theorem is reduced to the case

when apartments are not cycles of minimallength. Consideration of this case Ieads to the

complete bipartite graphs minus a matching.

In the following example we continue the treatment of graphs with strongly doubly

primitive subconstituents, started in Theorem 3.1.2 and Proposition 3.2.7.

Example 3.3.9. Let G(x)r<x) ~ S" or A", G1(x) =/:- 1, G1(x,y) = 1 and l = 6.

By Theorem 3.1.1 the amalgam {H, K} arises from a group acting 2-transitively on

ß = Kk,k· Let F = Aut(Kk,k)· Then F ~ S" wr S2 and F contains A" x A" as a

subgroup of index 8. Since F / (Ak x Ak) ~ D8 there are exactly 3 index 2 subgroups in

F. Let Dt, D2, D3 be these subgroups. Then one of them, say D3, is just the stabilizer

of the parts of ß. So D1 and D2 act 3-transitively on ß, and the stabilizer of a vertex

induces Sk on the vertices adjacent to this one. So there are exactly three isomorphism classes of rank 2 amalgams possessing the prescribed properties. Theseareamalgams arising from F, D1 and D2 • It is easy to see that in each case the pointwise stabilizer of an apartment has trivial center. So there are three isomorphism classes of rank 3

amalgams with l = 6. D

To calculate the universal groups of these rank 3 amalgam we use the following result due toP. Terwilliger [Ter].

Theorem 3.3.10. Suppose that G acts 3-transitively on r and that the girth of r is 6. Then lrl ~ 12· Okl. D

Here 2· Ok denotes the double cover of the odd graph Ok. The automorphism group of 2· ok is isomorphic to s2k-1 X 2. There are three subgroups in Aut(2· Ok) which act 3-transitively on 2· Ok and have Sk as subconstituent. By Theorem 3.3.10 they are the universal groups of the corresponding rank 3 amalgams.

One of the possible directions is the enumeration of amalgams A and their corresponding universal groups. The most interesting case is that of amalgams corresponding to finite but nontrivial universal groups. For cubic 4- and 5-transitive graphs this problern was treated in [Big]. The known examples arise for small values of /. So we come to the following conjecture.

Conjecture 3.3.12. If A0 is an amalgam corresponding to an s- but not (s + !)transitive action, and I ~ 2s + 1, then the universal group of the amalgam A0 is infinite. D

The following result from [DeWe] proves this conjecture in a very important particular situation.

Theorem 3.3.13. Let G be a finite group with a BN-pair of rank 2 and let 'V be the associated generalized n-gon, so that B = G aß for some edge ( a, ß} and N I(N nB) '=D2n. Choosexa E Ga n N -Gß, Xß E Gß n N -Ga, and let h = (XaXßt

(so h E B). Let <X IR> be a representation for the free amalgamated product Ga* 8 G ß and let G[v] =<X IR, ((x aXß)n h-1 )v = 1> for each v ~ l. Then G[ll = G, but G[v] is

infinite for all v ~ 2. D

Another direction for further investigation is to extend the methods and results presented above to the case of locally s-transitive actions.

3.4. Maximal subgroups of the O'Nan-Sims group

In this section we discuss an unexpected application of the technique presented above. We show how Theorem 3.3.8 can be used in the determination of the maximal subgroups of the O'Nan-Sims sporadic simple group O'S.

The general scheme for the description of the maximal subgroups of a group consists of two main steps. In the first step the local subgroups, that is the normalizers of p-

subgroups (p prime) are classified. The methods for treating this step areweil developed and rely on the determination of conjugacy classes of elementary abelian subgroups. The second step is the description of non-local maximal subgroups. This step requires the determination of conjugacy classes of subgroups which are direct products of isomorphic nonabelian simple groups. The methods for treating this step depend heavily on the particular type of nonabelian simple group. For instance the classes of subgroups isomorphic to A 5 can be determined by means of the character table of the original group.

One of the central problems in the description of the maximal subgroups of the group O'S is to determine the classes of subgroups isomorphic to A6. PGL2(ll) and PSL 2(31). The latter two groups arise in Theorem 3.3.8. The group A6 has some relation to this theorem as weil. Namely A 6 would have arisen in the theorem if the case 1 = 3 were included in it.

On the base of Theorem 3.3.8 we can formulate the following.

Proposition 3.4.1. Let G be a group containing an amalgam {L,M} isomorphic to {A 5 ,S 4 } (the intersection of the members is isomorphic to A 4 ). Let F be a subgroup of order 3 in Ln M. Then <NL(F), NM(F)> I F = D2k for some k and we have the following:

(i) if k = 2 then <L,M> = S 5;

(ii) if k = 3 then <L,M> =A6;

(iii) ifk = 4 then <L,M> = 24 • A5, 25 • A5 or PGL2(1l);

(iv) if k = 5 then either <L,M> = PSL2(31) or <L,M> contains a section iso-morphic toA5 xA5. D

The classification of A 5 subgroups in O'S, together with an analysis of its local subgroups, shows that O'S does not contain a section isomorphic to A5 xA 5. In addition, the structure of the normalizer of an element of order 3 in O'S is not too complicated, and for each particular subamalgam {As,S4 } one can calculate the value of k. So the classification of subgroups isomorphic to A 6 , PGL2(11) and PSL 2(31) is reduced to the determination of the subamalgam { A 5, S 4}. The latter tums out to be a reasonable problern.

For the other types of subgroups arising in O'S, with the lone exception of PSL2(3),

the classification can also be reduced to the determination of certain subamalgams. This project was realized in [ITS] where all maximal subgroups of O'S and Aut(O'S) were determined. This problern was solved independently in [Wil], [Yos].

References

[Big] N.L. Biggs, Presentations for cubic graphs, pp. 57-63. In Computational Group Theory, ed. M.D. Atkinson. Acad. Press, 1984.

440 A A. IV ANOV AND S. V. SHPECTOROV

[Con] M. Conder, A new 5-are-transitive eubie graph, 1. of Graph Theory, 11 (1987),

303-307.

[DOS] A. Delgado, D.M. Goldschmidt & B. Stellmacher, Groups and Graphs: New Results and Methods. Birkhauser Verlag, Basel, 1985.

[DeWe] A. Delgado & R. Weiss, On eertain eoverings of generalized polygons. Bull.

London Math. Soc. 21 (1989), 209-234.

[Djo] D.Z. Djokovic, Automorphisms of regular graphsandfinite simple group amal

gams, pp. 95-118. In Algebraic Methods in Graph Theory, vol. 1, 1981, Amster

dam: North-Holland.

[DjMi] D.Z. Djokovic & G.L. Miller, Regular groups of automorphisms of eubie graphs, J. Combin. Theory B29, (1980), 1955-236.

[Gar] A. Gardiner, Are transitivity in graphs, Quart. J. Math. Oxford, (2) 24 (1973), 399-407.

[Gol] D. Goldschrnidt, Automorphisms of trivalent graphs. Ann. Math., 111 (1980), 377-406.

[Ivn1] A.A. Ivanov, Graphs of girth 6 and a eharaeterization of the odd graphs, pp.

92-107. In problems in Group Theory and Homological Algebra. Yaroslavl, 1986. [In Russian]

[Ivn2] A.A. Ivanov, Distanee-transitive graphs and their classifieation. [In this

volume]

[ITS] A.A. Ivanov, S.V. Tsaranov & S.V. Shpectorov, Maximal subgroups of the

0' Nan-Sims sporadie simple group and its automorphism group, Soviet Math. Dokl. 34 (1987), 534-537.

[Kur] A.G. Kurosh, Group Theory. 3rd ed. Moscow, Nauka, 1967. [In Russian]

[Perl] M. Perkel, Bounding the valeney of polygonal graph with odd girth. Canad. J.

Math. 31 (1979).

[Per2] M. Perkel, A eharaeterization of PSL(2, 31) and its geometry. Canad. J. Math. 32 (1980), 155-164.

[Per3] M. Perkel, A eharaeterization of J 1 in terms of its geometry, Geom. Dedic. 9 (1980), 289-298.

[Pra1] C.E. Praeger, C.E. Symmetriegraphsand a eharaeterization of the odd graphs. Lecture Notes Math. 829 (1980), 211-219.

[Pra2] C.E. Praeger, Primitive permutation groups and a eharaeterization of the odd graphs. J. Combin. Theory (B), 31 (1981), 117- 142.

[Ron] M.A. Ronan, On the seeond homotopy group of eertain simplicial eomplexes and some eombinatorial applieations, Quart. J. Math. Oxford (2), 32 (1981), 225-233.


[Ser] J.-P. Serre, Arbres, Amalgams, SL2, Asterisque 46, Soc. Math. de France, 1977.

[StWe] G. Stroth & R. Weiss, Modified Steinberg relations for the group J 4• Geom.

Dedic., 25 (1988), 513-525.

[Ter] P. Terwilliger, Distance-regular graphs and (s,c,a,k)-graphs, J. Combin.

Theory (B), 34 (1983), 151-164.

[Thm] J.G. Thompson, Bounds for orders of maximal subgroups, J. Algebra, 14

( 1970), 135-138.

[Troll V.I. Trofimov, Graphs with projective suborbits, Izvestiya AN SSSR, Ser. Mat.

55 (1991), 890-916. [In Russian]

[Tro2] V.I. Trofimov, Vertex stabilizers of graphs with projective suborbits, Doklady

AN SSSR, 315 (1991), 544-546. [In Russian]

[Tut] W.T. Tutte, Afamily of cubical graphs. Proc. Cambridge Phyl. Soc., 43 (1947),

459-474.

[Weil] R. Weiss, Elations of graphs, Acta Math. Acad. Sei. Hungar. 34 (1979), 101-

103.

[Wei2] R. Weiss, s-transitive graphs, pp. 827-847. In Algebraic Methods in Graph

Theory, vol. 2, 1981. Amsterdam: North-Holland.

[Wei3] R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica, 1 (1981),

309-311.

[Wei4] R. Weiss, Groups with a (B,N)-pair and locally transitive graphs. Nagoya Math.

J., 74 (1982), 1-21.

[Wei5] R. Weiss, Symmetrie graphs with projective subconstituents, Proc. Amer. Soc.,

72 (1978), 213-217.

[Wei6] R. Weiss, A characterization and another construction of Janko's group h, Trans. Amer. Math. Soc., 298 (1986), 621-633.

[Wei7] R. Weiss, A characterization of the group M 12 , Algebra Groups Geom. 2

(1985), 555-563.

[Wie] H. Wielandt, Subnormal subgroups and permutation groups, Lecture Notes, The

Ohio State Univ., Columbus, 1971.

[Wil] R.A. Wilson, The maximal subgroups of the 0' Nan group, J. Algebra, 97

(1985), 467-473.

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Fac. Sei. Univ. Tokyo, Sect. lA, Math., 32 (1985), 105-141.

A GEOMETRIC CHARACTERIZATION OF THE GROUP M22

S.V. Shpectorov

1. lntroduction

In this paper, a geometry is understood tobe a quadruple (r, I, ~, t) where r is the element-set of the geometry; I is a reflexive, symmetric relation (incidence relation) on r; ~ is the set of types of elements of the geometry and t is a function from r onto ~ which associates with each element its type. A geometry is assumed to satisfy the following.

Axiom. The restriction of t on each maximal subset of pairwise incident elements is a bijection.

An analogous notion of geometry appeared in papers by J. Tits where Lie type groups were characterized as automorphism groups of so called spherical type buildings. Recently, interest in geometries increased significantly due to their relations to the sporadic simple groups. Initially, this activity was stimulated by papers of F. Buekenhout. At present many of the sporadic simple groups are realized as automorphism groups of certain geometries. In this respect the geometric approach gives a hope for a unified description of the finite simple groups (see [4] for a historical survey).

Let us recall some definitions. Let (r, I, ~, t) (or simply r) be a geometry. The geometry r is said to be connected if the graph with vertex-set r and edge-set I is connected. The set ~ is called the type of the geometry. The rank of the geometry is by definition the cardinality of ~- An arbitrary set of pairwise incident elements of r is called.fiag. Let F be a flag. Then t(F) is the type of F while the cardinality of t(F) is the rank of F. The following significant construction is related to the notion of flag. If F is a flag of a geometry r then the residual geometry (r F , I F , ~F , tp) can be defined. Here ~F = ~- t(F); rF = (v e r 1 (~p): F u {v} is a flag}; Ip and tp are the restrictions on r F of I and t respectively. It is easy to see that residual geometry is a geometry.

It was Tits who firstly noticed that the properties of a geometry are essentially determined by its residual geometries of rank 2. Let us define a diagram over the set of types ~ as a mapping D defined on the set of 2-element subsets of ~ such that D ( ( i , j}) is a class of geometries having type ( i , j}. A geometry (r, I , ~, t) is said to be described by a diagram D over ~ if each residual geometry r F with I ~F I = 2 is contained in D (~F ). For i '# j let D r( ( i , j}) denote the totality of residual geometries r F with

443

444 S. V. SHPECTOROV

/:,.p = { i , j}. It is clear that a diagram D over /:,. describes r if and only if D ( { i , j}) ::2 D r( { i , j}) for all i ""j from /:,.. For this reason D r will be called the minimal diagram of the geometry r.

In practice, diagrams are usually represented by means of graphs. The vertices of such a graph are in a correspondence with the types from /:,.. For each i ""j e /:,., the vertices corresponding to i and j are joined by an edge of special kind which symbolizes the dass D ( { i , j} ). Throughout the paper the empty edge

• • denotes the dass of generalized 2-gons (any two elements of different types are incident), the edge

-denotes the dass of generalized triangles (projective planes) and the edge

p -denotes the dass consisting of a single geometry, namely the geometry of vertices and edges (left and right types respectively) of the Petersen graph with the natural incidence relation.

An automorphism of a geometry is a permutation a of its elements such that if v, v 1 , v2 e r and (v 1 , v2) e I then t(v 0 ) = t(v) and (vY , v~) e /. The group of all automorphisms of r is denoted by Aut (r). A group G :5: Aut(r) is said to be flagtransitive if it acts transitively on the set of maximal flags of r. Geometries admitting a flag-transitive automorphism group will be called the flag-transitive geometries. These geometries are dosely related to groups. In addition they are very natural from the diagram point of view, since, in the minimal diagram of such a geometry, each edge denotes a dass consisting, up to isomorphism, of a unique rank 2 geometry. Moreover, it is easy to see that a disconnected flag-transitive geometry splits into a union of isomorphic connected flag-transitive geometries.

Let a group G act flag-transitively on a geometry r (the action is not assumed to be faithful) and F be a maximal flag in r. Then one can consider the amalgam Ap consisting of the stabilizers of the elements of the flag. The precise definition of amalgam will be given in Section 6; in Sections 1-5 it is sufficient to consider amalgam as a collection of subgroups of a certain group. It is obvious that up to isomorphism the amalgam Ap does not depend on the particular choice of F. So to each group G and geometry r a new object is associated. This object (i.e. the amalgam) plays an intermediate role in the transition from diagram and geometry to group and vice versa.

As was shown by A.A. lvanov, the sporadic simple Mathieu group M 22 and the group M 22 (triple cover of M 22 ) can be represented as flag-transitive automorphism

A GEOMETRIC CHARACTERIZATION OF THE GROUP M22 445

groups of geometries described by the diagram

p

These geometries have a natural interpretation in terms of two distance-transitive graphs (DTG's) on 330 and 990 vertices which were constructed in [2]. Namely the elements of first, second and third types of the geometries are respectively the vertices, the edges and the Petersen subgraphs in the corresponding DTG's. In the present paper the converse problern is solved.

Theorem. If G is a flag-transitive subgroup of Aut (1), where r is a connected geometry described by the diagram

p

then either r is the geometry related to M 22 and G = M 22 or Aut (M 22); or r is the A A ------ -----geometry related to M 22 and G = M 22 or Aut (M 22). Here Aut (M 22) is an extension of

M 22 by an outer automorphism.

In the proof of the theorem we will use essentially group-theoretical notions, notation and methods. The majority of them can be found in [1]. Let us recall some standard notation. If His a group then 02(H) is the maximal normal 2-subgroup of H; 0 2(H) is the subgroup of H generated by the elements of odd order from H; Z(H) is the center of H and ct>(H) is the Frattini subgroup of H. If His a 2-group then 0.1 (H) is the subgroup of H generated by all its elements of order 2 and 0.1 (H) is the subgroup generated by all squares in H.

The group generated by the elements x, y, · · · is denoted by <X, y, · · · >. For an elementary abelian 2-group of rank n > 1 the notation E 2• is used. Finally if a group H acts on a set Q and flo !::: Q then H (flo) is the elementwise stabilizer and H { flo} is the setwise stabilizer of flo in H.

The content of the paper is as follows. In Section 2 with each geometry r a certain graph S is associated. In terms of this graph the problern can be reformulated in graphtheoretical language. In Section 3 the cardinality of the vertex-set of S is shown to be bounded. In Section 4 some elementary properties of the action of G on the graph S are studied and, in particular, the order of the vertex stabilizer is bounded. In Section 5 the precise structure of two subgroups contained in the amalgam Ap is determined. Section 6 is a kind of introduction to the final part of the paper. In this section the definitions of amalgam and universal group are given, their properties are illustrated, and one significant proposition is proved. In Section 7-8 all possibilities for the amalgam Ap are determined and in Section 9 the corresponding universal groups are calculated. This


completes the proof of the theorem.

2. A graph related to the geometry

W e Will USe the folllowing notation. lf r iS a geometry Of type .1 and i E .1, then ri is the set of elements in r having type i. For a residual geometry r F we will write r F.i

instead of (r F )i.

Let a geometry r be described by the diagram

p

over .1 = { 1 , 2, 3} (types increase from left to right).

(2.1) 1. Ifv E r2, then I r(v},l I = 2. 2. If U, V E r2 and r(u},l = r(v},b then U =V.

3. lf U, V E r3 and r(u},2 = r(v},2• then U =V.

Proof Let V E r2 and w E r(v},3· Since r(v} is a generalized 2-gon, we have r(vJ,l ~ r(wJ,t ~ r(wJ,l· But r(wJ is the geometry of vertices and edges of the Petersen graph. So I r(v}.l I = 2.

Let u' V E r2 and r(u},l = r(v},l = u. Let us choose w E u. By definition r(w} is a projective plane, hence there ist E r(u},3 (") r(v},3· Let us consider r(t}· This is the geometry ofvertices and edges of a graph which has no multiple edges, so u = v.

Now let u' V E r3 and r(u},2 = r(v},2 = u. Choose w E u and t E r(w},l· Since r(wJ is a generalized 2-gon, r(wJ.3 ~ r(tJ,3· But r{t} is a projective plane and r(t,u},2 = r(t,v},2· This implies that u =V. 0

Lemma (2.1) enables us to associate with the geometry r a graph E by the following rule. Let V(E) = r 1 and E(E) = r 2 . In view of (2.1, 1-2) we obtain an undirected graph without loops and multiple edges. It follows directly from the definition that the vertices and the edges of E which are incident with a fixed element of type 3 of the geometry form a subgraph isomorphic to the Petersen graph. A subgraph obtained in this way will be called marked. It follows from (2.1.3) that for different elements of type 3 the corresponding marked subgraphs are distinct. For this reason the geometry r can be reconstructed from the graph E and its set of marked subgraphs.

The set of marked subgraphs will be denoted by U and the subset of U consisting of all subgraphs containing a fixed vertex v will be denoted by Uv. If \1' is a graph, v E V(\1') and n isanatural number, then \f'n(v) (respectively, \fln(v)) will denote the set of vertices of \1' which are at distance exactly n (respectively, at most n) from v.

Now we can express the main property of the family of marked subgraphs.

A GEOMETRJC CHARACTERIZA TION OF THE GROUP M22 447

(2.2) For v e V(E), the set E1 (v) provided with the set of lines {'P1 (v): 'P e Uv} is a projective plane. 0

Assertion (2.2) means that through each pair of adjacent edges exactly one marked subgraph passes, and that any two marked subgraphs which have a common vertex, have also a common edge which is incident to this vertex. The marked subgraph passing through the edges (v, x) and (v, y) will be denoted by (v; x, y).

Notice that it also follows from (2.2) that if u and v are vertices of E which are at distance 2 from each other, then Uu r1 Uv :1:- 0.

(2.3) The valency of Eis 7.

Proof. A line in the projective plane E1 (v) contains three points. Hence E1 (v) is of order2andiE1(v)l=7. 0

It should be mentioned that each projective plane of order 2 is isomorphic to the plane over the field of two elements. The automorphism group of this plane is isomorphic to the group L3(2).

To conclude this section we present a simple Iemma which follows immediately from the connectivity of the Petersen graph.

(2.4) The geometry r is connected if and only if the graph Eis connected. o

3. Bounding the number of vertices

Let d denote the diameter of the graph E. For 1 ~ i ~ d, Iet bi =max {I Ei+l(v) rl E1(u) I: v e V(E), u e Ei(v)} and Ci= min { I.S';-l(v) n .S\(u)l :V E V(E), u E Ei(v)}. The following simple Iemma generalizes some well-known facts for distance regular graphs to the graph E.

(3.1) 1. bi +Ci~ 7

2. If I~ i < j ~ d then bi ~ bj and Ci~ Cj.

Proof Let I Ei+l (v) r1 E1 (u) I = bi. Then I Ei-l (v) r1 E1 (u) I~ Ci. Hence bi +ci~ I E1(u) I =7. Now let 1~ i < j~ d, v e V(E) and u e Ej(v). Let us choose we Ei(u)riEj-i(v). Then Ei-l(w)r~E 1 (u)~Ej_1 (v)r~E 1 (u) and Ei+l (w) rl E1 (u);;;;;? Ej+l (v) rl 8 1 (u). So bi ~ bj and ci ~ Cj. 0

It is also easy to see that if v e V(E) then I Ei(v) I ~ I Ei-l (v) I • bi_1 / ci for 2~ i ~ d. For this reason, if E is connected, the parameters bi and Cj can be used for bounding the cardinality of V(E).

(3.2) b2 ~ 4.

Proof If u e E2(v) then a marked subgraph 'P passes through u and v. Since 'P has diameter 2 we have 'P1 (u) ~ E2(v). So I E3(v) r1 E1 (u) I~ 4. 0

S. V. SHPECTOROV

(3.3) If u e E4(v) then there is 'P e Uu suchthat 1. 'P1 (u)!::: E4(v) and 2. I 'P1 (u) n E3 (v) I~ 2.

Proof Let w e E2(v) n E2(u). Let us choose a marked subgraph 'P containing u

and w and a marked subgraph X containing w and v. As was mentioned in Section 2, the subgraphs 'P and X have in their intersection an edge (w, x). Since both 'P and X are of diameter 2, x e E2(v). So 'P1 (u) n 'P1 (w) :;o!: 0 and 'P1 (u) n 'P1 (x) :;o!: 0. But 'P1(w)n'P1(x)=0, since 'P has no triangles. Thus I'P1(u)nE3(v) 1~2. It is also clear that 'P 1 (u) !::: E4 (v ). 0

As an immediate consequence of the previous proposition we have the following.

(3.4) c4 ~ 2.

(3.5) b 5 ::::; 1.

0

Proof Let u e Es (v) and suppose to the contrary that w 1 :;o!: w2 e E1 (u) n E6(v ). Let us choose ae E1(u)nE4(v). Put X=(u;w 1 ,w2), E>=(u;a,w1) and A = (u; a, w2). In addition, let 'P be the marked subgraph whose existence is guaranteed by Lemma (3.3) applied to a. Since a marked subgraph has diameter 2, each of the subgraphs E> and A should intersect 'P in an edge joining a to some vertex from E4(v) n 'P1 (a). By Lemma (3.3) there is only one such vertex x. So 'i'1 (a) n 81 (a) = 'P1 (a) n A1 (a) e E4(v). This means that the intersection of E> and A contains a pair of adjacent edges. Hence E> = A =X and (u , a) is contained in X. We have proved that E1 (u) n E4 (v) c X 1 (u ). But the valency of X is equal to 3, hence I E1 (u) n E4(v) I ::::; 1. On the other hand, I E1 (u) n E4(v) I~ es~ c4 ~ 2 and the obtained contradiction proves the Iemma.

Now we are able to prove the main result of this section.

(3.6) If Eis connected, then I V(E) I ~ 1898.

0

Proof. Using the bounds (3.2) and (3.4) for v e V(E), we have the following: I Eo(v) I= 1, I E1(v) I =7, I E2(v) I::::; 42, I E3 (v) I::::; 168, I E4(v) I::::; 336, I Es(v) I ::::; 672. For larger indices, by Lemmas (3.4) and (3.5), the cardinality of each subsequent layer is at mosthalf of the previous one. Hence I V(E)- E5(v) I ::::; 672. o

4. Bounding the order of the vertex stabilizer

Let us summarise the known properties of the Petersen graph which will be used below. Let H::::; Aut(P) act edge-transitively on the Petersen graph P and Iet v e V(P).

Pl.H=A 5 orSs.

P2. H(v) acts 2-transitively on P 1 (v).

P3. If H(P 1 (v)) :;o!: 1, then H = S s and H(P 1 (v)) acts serniregularly on V(P)- P 1 (v)

with the orbits P 1 (u)- {v}, u e P 1(v).


P4. If g e Aut P fixes elementwise a path of length 3 in P, then g = 1.

Recall that an automorphism of a projective plane which fixes two lines pointwise is trivial. The dual proposition is the following. An automorphism which fixes setwise each line passing through at least one of two chosen points is trivial.

Let us return to the graph 8. Let G be an automorphism group of 8 which preserves the set U.

(4.1) Ifv "# w e 8 1 (u), then G(81(v)) n G(81(w)) = G(82(u)).

Proof Let g e G(81(v)) n G(81(w)). LetAdenote the subgraph of 8 induced by the vertices fixed by g. If 'I' is a marked subgraph which contains either the edge (u, v) or the edge (u , w ), then the intersection 'I' n A contains a path of length 2. This implies that 'I'g ='I'. Hence g stabilizes the lines passing through at least one of two points of the projective plane 8 1 (u).

Now let 'I'= (u; v, w) and a e 8 1 (u)- \{11 (u). It is easy to see that both the intersection of (u ; v , a) with A and the intersection of (u ; w , a) with A contain paths of length 3. By P4 the element g fixes them pointwise and hence g fixes pointwise two lines in 8 1 (a). Soge G(81(a)). Finally, if b e \{11 (u)- {v, w}, then we have g e G(81(v)) n G(81(w}}S: G(81(v)) n G(81(a}}S: G(81(b)). D

In what follows it is assumed that the geometry r and the graph 8 are both connected. Let v e V(8).

(4.2) G(83 (v)) = 1.

Proof. It is sufficient to show that G(8\v)) = G(84 (v)). Let u e 8 3(v) and w e 8 2(v) n 8 1(u}. By (3.2) there are a "#b e 8 1(w) n 8 2(v). By (4.1) we have

G(8\v}}S: G(81(a)) n G(81(b}}S: G(81(u)). o (4.3) I G(82 (v)) I S: 2.

Proof. Firstlet us show that G(82 (v)) n G(81(u)) = 1 for u e 8 2 (v). Suppose to the contrary that 1 "# H = G(82(v)) n G(81(u)). Then in view of (4.2), H $ G(81(w))

for some w e 82(v). Let us choose a e 8 1 (u) n 8 1 (v) and b e 8 1 (w) n 8 1 (v). By (4.1)

a "# b. Let us consider a path (v, a, c, d, b) of length 5 in (v; a, b). Applying (4.1)

successively we have HS:G(81(v))nG(81(u))nG(81(b}}S:

G(81 (v)) n G(81 (c)) n G(81 (b )) S: G(81 (v )) n G(81 (d)) S: G(81 (w)). This contradiction proves that H = 1.

Now we can finish the proof of the lemma. As was shown above, G(82(v)) acts faithfully on 8 1 (u). Let \{1 be the marked subgraph passing through v and u. Since 'PS: 8 2 (v), the subgroup G(82(v)) fixes the line '1'1(u) pointwise. Now we observe that an automorphism of the projective plane of order 2 which fixes three points not on a single line is trivial. This means that G(82(v)) acts semiregularly on 8 1 (u)- '1'1 (u). Let a e 81 (u)- \{11 (u), b e 8 1 (u) n 8 1 (v) andX = (u; a, b). ThenX 1 (b) S: 8 2 (v). Now by P3, we have I aG(S2(v)) I S: 2. o

450 S V. SHPECTOROV

5. The structure of groups constituting Ap

Recall that the geometry r and the graph .E are assumed to be connected. Let a group G:::; Aut r acts ftag-transitivityon r. Then G acts faithfully on .E and G preserves

the set U.

The goal of the present section is the determination of the structure of groups constituting the amalgam Ap for some maximal fiag F of the geometry r. In terms of the graph .E, the amalgam Ap consists of the stabilizer of a vertex v, the stabilizer of an edge (v, x) and of the stabilizer of a marked subgraph 'P passing through (v, x). The property of fiag-transitivity is equivalent to the property that G acts transitively on U and 1-transitively (transitively on the paths of length 1) on 'P. Put H = G(v), K = G {v, x} and L = G {'P}. In addition put H 0 = G(.E1(v)), Z = G(.E2(v)) and L 0 = G('P). The available information concerning the intersections of the subgroups H , K and L can be expressed in the form of the following Iemma.

(5.1) The item corresponding to row A and colurnn B in the table below contains the value [A : A n B ].

H K L

H 1 7 7

K 2 1 3

L 10 15 1

Proof The nurnbers 7 in the first row are the numbers of points and lines in the projective plane .E1 (v). The numbers 10 and 15 in the third row are the numbers of vertices and edges of the subgraph 'P, which is isomorphic to the Petersen graph. The index I K : K n H I is equal to 2 since the edge (v , x) is incident to exactly two vertices: v and

x. Finally IK: K n LI is the number of marked subgraphs containing the edge (v, x).

Since each marked subgraph containing (v,x), contains v, this number is equal to the number of lines passing through a given point in the projective plane of order 2. o

(5.2) H I Ho =L3(2).

Proof. It is clear that H I H 0 is contained in L3 (2). Since the unique subgroup of L3 (2) acting 2-transitively on the points of the corresponding projective plane is L 3 (2) itself, it is sufficient to show that H I H 0 acts 2-transitively on .E1 (v). Since H acts transitively on Uv there exists h e H such that (v ; a , b )h = (v ; c , d). So without loss of generality we can assume that (v ; a , b) = (v ; c , d) = 'P. Now the claim of the Iemma follows directly from Property P2. o

(5.3)L1Lo=S5.

Proof Since L acts 1-transitively on 'P, it follows from P1 that LI Lo = A 5 or S 5. Suppose that LI Lo =A5. We will show that in this case Ho= 1. Let w e .E2 (v) and X be the marked subgraph passing through v and w. Since X 1 (v) ~ .E1 (v ), it follows from P3

A GEOMETRIC CHARACTERIZA TION OF THE GROUP M22 451

that H 0 acts trivially on X, i.e. Ho ~ G ( w ). The latter implies that Ho= G(3 1 (v))~ G(32 (v)) and hence Ho= 1. Now it follows from (5.2) thatH=L3(2). In this case H n L = S 4 and direct calculations show that I Lo I = 4. So Lo = Z(L ). On the other hand Lo :5 L n Hand the center of S 4 is trivial, a contradiction. D

In what follows we will use extensively some information about subgroups, modules and extensions of the groups L 3 (2) and S 5. In all cases the properties can be checked directly so we formulate them without proof.

Recall that the group L 3 (2) has exactly four irreducible modules over the field of two elements. Their dimensions are 1, 3, 3 and 8. One of the two 3-dimensional modules is called natural. The action of L 3(2) on the nonzero vectors of this module is similar to its action on the set of points of the projective plane. Another 3-dimensional module is dual to the natural one. The action of L3(2) on its nonzero vectors is similar to the action of L 3 (2) on the lines of the projective plane.

(5.4) H 0 I Z = E 8 is the natural module under the action of H I HO·

Proof. Let us choose w1 '# W2 E ::::1 (v). PutX = (v; w1 'w2) and Iet e and A be the marked subgraphs containing (v, w 1) and (v, w2), respectively, with each distinct from X. Put R = H 0 (X u e u A). This subgroup fixes pointwise two lines in the plane 3 1(w 1). Hence R ~ G(31(w 1)). Analogously R :5 G(31(w2)). By (4.1) we have R =Z. Now it follows from P3 that I Ho :Ho(~) I = 2 for ~ e Uv. Forthis reason Ho I Z is an abelian group of order :5 23, and the lines of 3 1 (v) are in correspondence with the hyperplanes of Ho I Z. Since H acts transitively on Uv, we have that either Ho I Z = E 8 is the natural module under the action of H I H 0 , or I Ho I Z I = 2. But in the latter case an element x e H 0 - Z acts nontrivially on all X e Uv. By P3 the element x has only orbits of length 2 on 3 2(v). Thus the image of x in G(u) I G(31 (u)) ~ L 3(2) has cyclic type 1· 23

for u e 3 1 (v ). This contradicts the fact that each element of order 2 in L 3 (2), in its action on the points of the projective plane, has cyclic type 13 • 22 • D

For the subsequent analysis we need the following standard Iemma.

(5.5) If D ~ H n K and D , then D = 1.

Proof. Since G acts faithfully on E, it is sufficient to show that <H,K> = G. Let Go= <H, K > and E:o be the subgraph of 3 induced by the orbit of Go on V(E:) which contains v. It is easy to see that the valency of 3 0 is equal to 7. Since 3 is connected we have E0 = E. Finally G0 contains the stabilizer of a vertex, hence G 0 = G. D

Recall that the group S 5 has, up to isomorphism, exactly three irreducible modules over the field of two elements: one I-dimensional and two 4-dimensional. One of the 4-dimensional modules is the natural module for A 5 = L 2 ( 4). This module will be called "natural". It is characterized by the property that the centralizer of an element of order 3 in S 5 is trivial. In another 4-dimensional module this centralizer has dimension 2. The lauer module is the natural module for the group S 5 = 04 (2).

(5.6) Lo is elementary abelian and H 11 L has inside Lo two 2-dimensional main factors and at most one I-dimensional factor.

Proof By (5.4) H 11 L has inside 0 2(H 11 L) either one or two I-dimensional factors. Since I L : H 11 L I = 10, we have (H 11 L) I L 0 = Z 2 x S 3. This proves the second claim of the lemma. In addition LI L 0 acts nontrivially on Lo I (Lo). Hence the rank of Lo I (Lo) is greater or equal to 4 and therefore I (Lo) I ::; 2. If (Lo)-# I, then (Lo) = Z, contrary to Z <l L. o

(5.7) H = Z x 0 2(H) and 0 2(H):: E 8 A, L3(2).

Proof. Let us start with the first claim. If Z = I then there is nothing to prove. Suppose that Z = Z 2. Then I H 0 I = 24 . If H 0 is nonabelian then it is extraspecial. But there is no extraspecial group of order 24 . If H 0 is not elementary abelian then the existence of the row I < !11 (H 0 ) < !11 (H 0 ) < H 0 of characteristic subgroups precludes the existence of a 3-dimensional main factor of H inside H 0 , a contradiction to (5.4). Thus Ho is elementary abelian. Let S be a Sylow 2-subgroup of H contained in H 11 K. Let us consider the action of H I H 0 on H 0 . If, under this action, H 0 is an indecomposable module then it can be shown that Z = Z(S). In this case Z <l (H 11 K) NK(S) = K. As this contradicts (5.5) we have that the module H 0 is decomposable. Let E be the 3-dimensional submo-

- - - --dule of H 0 and H = H I E. Then we have that Z = 0 2 (H) and H I Z = L3 (2). Thus either - - -H=Z2xL3(2) orH=SL2(7). IfH=SL2(7) then Ql(S)=Ql(Ho)<l <H,K>, a contradiction to (5.5). Hence H =z2 xL3(2) and we have that H = Z 0 2 (H) = Z x 0 2 (H). So the first claim of the lemma is proved. The second claim follows from the fact that a nonsplit extension of the natural module by L3(2) has 2-rank 3, while 0 2(H) has 2-rank:?: 4 by (5.6). o

Let us fix the notation E = 0 2 ( 0 2 (H) ). It follows from above that E = E 8 is the natural H I H 0 -module. Let J be an irreducible L-submodule of L 0 . Since Z <l L, J is not I-dimensional. So by (5.6) J is the natural LI Lo module.

(5.8) (i) If Z -# 1 then Lo = Z J is an indecomposable module;

(ii) L splits over Lo.

Proof We will use the fact that S 5 splits from its natural module. In particular one can assume that Z -# 1. Since the centralizer in the natural module of an element of order 3 from S 5 is trivial, then Z $ J. This implies the claim (i). Put L =LI J. It is easy to see that either 0 2(L) = S L2(5) or L =A 5 A.Z4 or L = S 5 xz2. But it follows from (5.7) that H 11 L :: E 4 x S 3. This is possible only if L:: S 5 x Z2. Now claim (ii) of the lemrna follows from the fact that a complement to L 0 can be found in the preimage of a subgroup of L isomorphic to S 5. o

6. Amalgams

This section is a preliminary to the final part of the paper. Here again G denotes an

arbitrary group, r denotes an arbitrary geometry, and so on.

An amalgam is a collection of pairwise intersecring subgroups such that, for all

groups A and B from this farnily, the restrictions to A r1 B of the group operations coincide. The groups constituting an amalgam are called members of the amalgam, while the

elements of these groups are called elements of the amalgam. An essential example of an

amalgam is a collection of subgroups of a given group.

Let A be an amalgam. A group G is said to be an A-group if A is a collection of

subgroups of G. Let G 1 and G2 be A-groups. A homomorphism $: G1 ~ G2 will be called A-homomorphism if the restriction of $ to the set of elements of A is the identity

map. The universal group of an amalgam can be defined in terms of generators and rela

tions in the following way. The generators are all elements of the amalgam, the relations

are all relations of the form a · b = c which hold in members of the amalgam. For each

A-group G there exists a unique A-homomorphism of the universal group into G.

Not every amalgam is embeddable in a group. Here is a simple example. Let A be

the amalgam which consists of the subgroups < a1, a2 >, < a2, a3 > and < a1, a3 > of the group < a 1 , a 2 , a 3 > = Es. Let us adjoint to A the group generated by elements a 1 a 2 and a 3, which is isomorphic to S 3 . The constructed amalgam is not embeddable in

any group since, in the universal group, the equality a2 = a3 holds.

Let A be an amalgam and G be an A-group. Let A be the set of members of the amalgam A. Put G I A = u G I A (here G / A is the set of (right) cosets of A in G). Two

Aeö cosets are said to be incident if they have a nonempty intersection. The type of a coset g A E G I A is the subgroup A as an element of A. As above, a flag is a set of pairwise incident elements of G I A.

The following simple lemma shows the significance of the above construction.

(6.1) If r is a geometry, G is a group which acts flag-transitively on r and F is a

maximal flag in r, then GI Ap is a geometry isomorphic to r. A flag in G I A is called standard if all cosets which are contained in the flag have

nontrivial intersection. It is easy to show that a maximal standard flag has type A and that all standard flags of given type form an orbit under the action of G. So the "nongeometricity" of GI A and its "non-flag-transitivity" are related to the existence of nonstandard flags. A nonstandard flag consists of at least three elements. Hence if I A I :5: 2 then G I A

is a flag-transitive geometry, while if I A I = 3 then GI Ais still a geometry which may not be flag-transitive. If I A I ~ 4 then it is possible that GI A is not a geometry. Let us give some examples.


1. Let G = E 4 and Iet the amalgam A consist of the three subgroups of order 2 in G. Then G I A contains nonstandard flags and the action of G is not flag-transitive.

2. Let A be an amalgam and G I A contain nonstandard flags. Let R be a group. Then (G xR) I (Au {R}) is not a geometry. In fact GI Ais embeddable in a natural way into ( G x R) I ( A u { R } ). Let us show that a nonstandard flag from G I A cannot be

extended to a flag by a coset of R. Indeed, all cosets of the initial flag are contained in G and each coset from (G xR) IR intersects Gin exactly one element.

(6.2) (Tits [3], page 5). Let G be a group and A, B, C be subgroups in G. Then the

following two conditions are equivalent

1. G I {A , B , C) contains no nonstandard flags.

2. A f1 B C = (A f1 B) (A f1 C).

(6.3) Let G act flag-transitively on a geometry r and Iet F be a maximal flag of r. Let G be an Ap-group such that there is an Ap-homomorphism of G into G. Then G I Ap is a geometry and G acts flag-transitively on it. If the rank of r is at least 3, then the minimal diagram of G I Ap coincides with that of r.

Proof It is necessary to show that G I Ap has no nonstandard flags. Let Ap = { G 1 , · · · , G,} and suppose that F 1 is a flag of GI Ap. Without loss of generality

we can assume that F1 = {g1 G1, · · ·, gs Gs} and that g1 = 1. Let G 1 f1 gi Gi = uj(G 1 f1 Gj). By (6.2) there are no nonstandard flags of rank 3 in GI Ap since there are no such flags in GI Ap. Hence

uk(G 1 f1 Gk) f1 Ut(G 1 f1 Gt) = G 1 f1 gk Gk f1 gl Gt '# 0. Notice that

A' = { G 1 f1 G2, · · · , G 1 f1 G,} is the amalgam Ap• for the flag F' = { G 2 , · · · , G,}

of the geometry r(Gd (here Gi = 1· Gi is an element of r). Since Gis flag-transitive on

r. the subgroup G1 is flag-transitive on r(Gd and hence r(Gtl =G11A' contains no

nonstandard flags. This implies that

G1f1g2G2f1 ··· f1g5 Gs=u2CG1f1G2)f1 ··· f1U5 (G1f1Gs)=P0,

i.e. F 1 is a Standard flag. The second claim of the Iemma follows from the fact that, by

(6.1), the amalgam Ap determines residues of all nonempty flags. D

Lemma (6.3) is applicable, in particular, when G is the universal group of the amalgam Ap. Notice that in this case the geometry GI Ap is connected. In fact, 6 is gen

erated by a set of subgroups such that each of the subgroups fixes an element of the flag

F. This means that G preserves the connected component of r which contains F.

7. Reduction to the case Z = 1

Let us return to the situation considered in Sections 2-5. Here we assume that Z :p 1 and that G is the universal group of the amalgam { H , K , L } .

Put T = L 11 K and S = T 11 H. It is easy to see that T is a Sylow 2-subgroup of both L and K, while S is a Sylow 2-subgroup of H and I T : S I = 2. Moreover, H 11 K and H 11 L are preimages of the two maximal parabolic subgroups of H I Ho = L3 (2) which contain S!Ho. Put H1=0 2(H), S1=S11Ht. Lt=02(L)St. K 1 = (K 11 H 1 )(K 11 L 1) and G 1 = < H 1 , K 1 , L 1 >. It follows from (5. 7) that H 1 is isomorphic to a maximal parabolic subgroup of L4(2).

(7.1) S 1 <l T.

Proof. We have I K : H 11 K I = 2 and I H 11 K : H 1 11 K I = 2. This implies that 0 2 (K) = 0 2(H 111 K) and hence 0 2(02(H 111 K)) <l T. In addition J <l T. But it can be checked directly that S 1 =1 02(02(H 111 K)). 0

It is clear that I H : H 1 I = 2 = I S : S 1 I.

(7.2) IL :L 1 I =2andZ$ L 1 •

- - - -Proof. Put L = L I J. We already know that Z = Z (L) and that L = Z 2 x S s. It fol-

lows from (7.1) that SI 11 02(i)~ [T. T] '# 1. Hence I L: LI I~ 2. Now the claim of the Iemma follows from the fact that L = Z L 1. 0

PutT 1 = T 11 L 1.

(7.3) IK:K1 1 =2andZ$ K1.

Proof. Since K = T 0 2(H 11 K), we have K 1 = T 1 0 2(H 1 11 K), 0 2(H 11 K) = 0 2(H 111 K), and T 1 11 0 2(H 111 K) is a Sylow 2-subgroup of 0 2 (H 1 11 K). Now the claim of the Iemma follows from (7 .2). 0

(7.4) IG:G 1 1=2andZ$ G 1.

Proof. Let z be a nonidentity element of Z. It is clear that z normalizes each of H 1 , K 1 and LI· In addition z 2 = 1 and G = < z , H 1 , K 1 , L 1 >. Hence I G : G 1 I ~ 2. Since G is the universal group, to prove the Iemma it is sufficient to construct a group which contains the amalgam and such that z E < H 1 , K 1 , L 1 >. Let {; be the universal group of the amalgam { H 1 , K 1 , L 1 } . Since z normalizes each of the subgroups H 1 , K 1 and L 1 , this element induces an automorphism of the group {;. Hence we can construct the semidirect product {; A. < z > which possesses all of the prescribed properties. o

Now we are in a position to prove the rnain result of the section.

(7.5) G 1 acts flag-transitively on r. Proof The subgroup S is the stabilizer of a maximal flag F. In view of (7.4) each

coset of S in G has nonempty intersection with G 1. Hence G 1 acts transitively on the maximal flags of r. 0

8. Reconstruction of the amalgam AF

First we consider the case Z = I. Let r' be another geometry, Iet G' ~ Aut(r') and assume the pair (r', G') possesses all the properties prescribed for (r, G). Let H', K' and L' be the subgroups, corresponding to H, K and L, respectively.

By (5.7-8) we have H' = H =Es A.L3(2) andL' =L = E 16 A.S 5·

(8.1) The amalgams {H, K, L} and {H', K', L'} are isomorphic.

Proof. Put T' = L' 11 K' and S' = T' 11 H'. Since in the group S 5 all subgroups isomorphic to Z2 x S 3 are conjugate, we can find an isomorphism cii : L ~ L' such that cji(L 11 H) = L' 11 H' and cji(S) = S'. We will construct an isomorphism of the arnalgams {H, K, L} and {H', K', L') as an extension of <i). Notice that <i)(T) = T' since T' = NL'(S') and T = NL(S).

Thus c1> I L = <i). Let us extend c1> to H. Let D be a complement to E in H and R = D 11 (H 11 L). Let R' = cl>(R). PutE'= 0 2(H'). By (5.4) E' R' = H' 11 L' is the normalizer in H' of a 2-dimensional subspace of E'. Then R' is contained in a complement D' to E' in H'. Forthis reason c1> IH can be defined in the following way. If e E E and x E D then cl>(e x) = cl>(e) x', where x' is an element from D' suchthat cl>(vx) = cl>(v( for every v E E. Now it is sufficient to extend c1> to K. We already know the action of c1> on L 11 K and on H 11 K. In addition it is shown that K = (L 11 K) (H 11 K) and K' = (L' 11 K') (H' 11 K'). Let us choose an element x of order 3 in H 11 K and an element y in (L 11 K)- H such that [x, y] =I. Let Q = 0 2 (0 2(H 11 K)) and Q' = cl>(Q). It follows from (5.7) that Q <1 K, Q' <1 K' and CK'(Q') = Z(Q'). Since [x, y] = 1 the elements x and y induce commuting automorphisms on Q. But then cl>(x) and c~>(y) induce commuting automorphisms on Q'. So [cl>(x), c~>(y)] e Z(Q'). Since I <x > I = 3 and I Z(Q') I = 2 we have [cl>(x), cl>(y)] =I. Now it is easy to check thatthe map of Konto K' can be defined in the following way. lf s ES, ee {0, 1, 2} and ÖE {0, 1}, then cl>(s· xE· y 5 ) = cl>(s) cl>(x)E $(y)5. This completes the proof ofthe lemma. D

Now we areready to consider the general situation.

(8.2) AF is isomorphic either to the amalgam in M 22 (isomorphic to that in M 22 ) or to the amalgam in Aut (M 22) (isomorphic to that in ~ 22).

Proof If Z = 1 then AF is isomorphic to the amalgam in M 22 by (8.1). Suppose that Z ~ I. Then in terms of the previous section, it is sufficient to show that the amalgam {H, K, L} can be reconstructed in a unique way from the amalgam {H 1 , K 1 , L 1 }. To reconstruct the amalgam, it is sufficient to define the action of z on the groups H 1 , K 1

andL 1.

We know that z acts trivially on H 1. Suppose again that x is an element of order 3 in H 1 11 K 1· Put C = CK, (x). It is easy to show that I C I = 4. Since

[z, x] = 1, [z, K 1 ] ~ 1 and K 1 = (H 1 11 K 1) C, the element z induces on C the unique nontrivial automorphism. This implies that the action of z on K 1 can be reconstructed in a unique way. Now it is sufficient to notice that L 1 = < H 1 11 L 1 , K 1 11 L 1 >. D

9. Proof of the theorem

(9.1) The groups M 22 and kt{M 22) are the universal groups of the corresponding amalgams.

A ~ -

Proof. Let G = M 22 or Aut(M 22) and Iet G be the universal group of the corresponding amalgam. We know that there is an Ap-homomorphism of 6 onto G. On theotherhand 16 I~ 1898· I H I< 2· 990· I H I =2· I GI by(3.6). HenceG =6.

Now we can finish the proof of the theorem. If G is an arbitrary group acting flagtransitively on a connected geometry r with the diagram

p

then G is a factor-group of one of the universal groups mentioned above. In this way we obtain, other than the universal groups themselves, the groups M 22 and Aut(M 22) acting on the geometry related to the distance-transitive graph on 330 vertices.

Acknowledgement

This work was carried out under the guidance of Professor A.l. Kostrikin to whom the author is highly indebted. The author also received significant help from A.A. Ivanov, who took part in discussions on almost all themes mentioned in the paper.

References

1. D. Gorenstein, Finite Groups, Rarper and Row, New York, 1968.

2. A.A. Ivanov, A.V. Ivanov, LA. FaradZev, Distance-transitive graphs of valency 5, 6 and 7, Zumal Vicisl. Mat. i Mat. Fiz. 24 (1984), 1704-1718 [In Russian, translated in USSR Comput. Maths. Math. Phys. 24 (1984), 67-76].

3. J. Tits, Buildings of Spherical Type and Finite RN-pairs, Lect. Notes Math., 386, Springer-Verlag, 1974.

4. J. Tits, Buildings and Buekenhout geometries, In Finite Simple Groups II, ed. M. Collins, Acad. Press, New York, 1981, pp. 309-320.

BI-PRIMITIVE CUBIC GRAPHS

M.E. Ioftnova & A.A. Ivanov

1. lntroduction

A graph is called bi-primitive if it is bipartite and its automorphism group acts edge-transitively, preserves parts and acts primitively on each part. In this paper the biprimitive cubic graphs are classified.

All graphs in question are assumed to be finite, connected, and without loops or multiple edges. Mainly we'll consider undirected graphs. If r is a graph then V (r), E (r) and Aut (r) denote its vertex set, edge set and automorphism group respectively. In this paper the edge- but not vertex-transitive regular graphs are investigated. By definition, if r is such a graph then Aut (r) acts transitively on E (r) and intransitively on V (r).

It is well known that in this case r is a bipartite graph with parts V 1 and V 2 of the same size and Aut (r) acts transitively on each of these parts. This class of graphs was introduced by J. Folkman in [8], who has constructed the first examples of such graphs and has posed a number of problems conceming these graphs (see also [23]). The answers to most of these problems are now known (see [15]). In earlier papers on edgebut not vertex-transitive regular graphs, certain combinatorial constructions played a most significant role. Such graphs were essentially constructed and the properties of their automorphism groups were checked by some heuristic methods. A first attempt to formulate the property of a graph to be edge- but not vertex-transitive in pure group theoretical termswas made in [17]. But in regard to sufficiency conditions for the existence of such graphs, both group theoretical and combinatorial arguments were involved. Using certain methods for the investigation of bipartite graphs by means of group theoretical constructions (see [10], [20]), we were able to use just the group theoreticallanguage in the investigation of edge- but not vertex-transitive graphs (see Section 2). This was the starring point of the present paper.

lf r is an edge- but not vertex-transitive regular graph then using the covering procedure given in [2], one can obtain an infinite farnily of such graphs having the same valency. This implies that classifying the edge- but not vertex-transitive graphs is intractable. So it is reasonable to study edge- but not vertex-transitive graphs with certain additional properties.

It is natural to assume primitivity of the action of Aut(r) on the parts Vi , i = 1 , 2. In this case we'll use the term "bi-primitive graph". It is shown in Section 2 that, from a primitive permutation group with an antisymmetric orbit possessing certain additional properties, a bi-primitive graph can be constructed (see Lemma 2.5). This shows that the class of bi-primitive graphs is also too large. A graph obtained by the construction described in Lemma 2.5 has the property that the stabilizers H 1 and H 2 of two adjacent

459

460 M. E. IOFINOV A AND A. A. IV ANOV

vertices are conjugate in Aut(r). If H 1 and H 2 are not conjugate in Aut(r) then such a bi-primitive graph will be called a graph of parabolic type. The known infinite farnilies of such graphs arise from the parabolic geometries of the Chevalley groups B 2(q), q "# 2k

and G2(q), q "# 3k [22]. This is just the motivation for the introduced terminology. Some

bi-primitive graphs of parabolic type play a significant role in diagram geometries (see

[5], [6], [19]). So bi-primitive graphs of parabolic type form an interesting class of

objects which should be investigated in detail.

The cubic graphs were studied intensively during recent years. For example, histori

cally the first classification result in the theory of distance-transitive graphs was the classification of cubic distance-transitive graphs, obtained by N. Biggs and D. Smith [3]. The complete description of cubic graphs for which the automorphism group acts primi

tively on the vertex set and transitively on the edge setwas obtained by W. Wong in [24]. The stabilizers of vertices and edges in the edge-transitive cubic graphs were described by D. Goldschmidt in [10].

As was mentioned in [9], the classification of cubic bi-primitive graphs is a next important step in this direction. As far as we know, no significant progress was obtained in such a classification. The main result of this paper is the following.

Theorem 1.1. Up to isomorphism, there are exactly five bi-primitive cubic graphs. They have 110, 126, 182, 506 and 990 vertices and their automorphism groups are PGL 2(11), G2(2), PGL2(13), PSL 2(23) and Aut(M !1}, respectively. All these graphs

are of parabolic type. 0

2. A Formulation in Terms of Group Amalgams

Let r be a bipartite graph with parts V 1 and V 2· Let Aut-(r) denote the maximal subgroup of Aut (r) which preserves each of the parts. It is clear that Aut-(r) has index

at most two in Aut(r). A group G ~ Aut-(r) is said to act bi-primitively on r if G acts

transitively on E (r) and primitively on V; for i = 1, 2. In this case Aut- (r) also acts bi

primitively on r. If, in addition, Aur (r) = Aut(r) then r is a bi-primitive graph; other

wise r is vertextransitive and Aut-(r) has index 2 in Aut(r).

Let G be a finite group and let H 1 , H 2 be subgroups such that I H 1 I = I H 2 I . Let r(G, H 1, H 2) be the bipartite graph whose vertices are the cosets of H 1 and H 2 in G, i.e. V(r(G, Ht.H2))= {H;g I g E G, i = 1, 2}; verticesH1 g 1 andH2 g2 are adjacent if and only if H 1 g 1 n H 2 g 2 "# 0.

It is easy to see that the edges of the graph r(G, H, H 2) are in one-to-one

correspondence with the cosets of H 1 n H 2 in G, and hence G acts edge transitively on

r(G,H 1 , H 2). It is known that if G = <H 1 , H 2 >, then r(G, H 1, H 2) is a connected graph of valency k = I H 1 : H 1 n H 2 I = I H 2 : H 1 n H 2 I. In this case, if a subgroup

N ~ H 1 n H 2 is normal in both H 1 and H 2 then it is normal in the group

BI-PRIMITIVE CUBIC GRAPHS 461

G = <H 1 , H 2>, and N acts nivially on the graph r(G, H 1, H 2). For this reason it is

assumed below that there are no nonnivial subgroups normal both in H 1 and H 2.

It is easy to see that the following Iemma holds.

Lemma 2.1. Let G :5 Aut(f) act edge- but not vertex-transitively on r, and Iet

H 1 , H 2 be the stabilizers in G of two adjacent vertices. Then r is isomorphic to

r(G, H 1> H 2).

A niple (G,H1,H2) will be called a doser of the amalgam

H 1 ~ H 1 n H 2 ~ H 2· It should be noted that our terminology is different from that

used in [10], [20]. By an embedding of a dosure (G, H 1, H 2) into a dosure

( G ', H~, H~ ), we mean an embedding cp of G into G' such that cp(Hi) ~ H: for i = 1 , 2.

Notice that <1> induces an embedding of the amalgam H 2 ~ H 1 n H 2 ~ H 2 into the

amalgam H~ ~ H'1 n H~ ~ H~.

Two dosures (G, H 1 , H 2) and (G', H~, H~) are called equivalent if the graphs

r(G, H 1> H 2) and r(G ', H~, H~) are isomorphic. Let us consider the equivalence dass

of dosures corresponding to a graph r. It is easy to see that this dass contains a maximal

element. In fact, each dosure in this class is embeddable in (G-, HJ., H2 ), where

c- = Aut-(r) and HJ. , Hi are the stabilizers in G- of two adjacent vertices. Notice that

G- contains the 2-dosure of G considered as permutation group of the set

V (r( G, H 1, H 2)) (for the definition of 2-closure see [ 18]).

Lemma 2.2. Let (G, H 1, H 2) be embeddable in (G', H'1, H~) and let Hi = G n H:

for i = 1, 2. Then the dosure (G, H 1, H 2) is equivalent to (G', H~, H~) if and only if

I c' : G I = I H~ n H~ : H 1 : n H 2 I.

Proof. It is clear that the equality I d : G I = I H~ n H~ : H 1 n H 2 I is equivalent

to the equality IG' :H~ nH~ I= IG :H1 nH2I and that G n (H~ nH~)=H 1 n H2.

This implies that if I c': G I = IH~ n H~ : H 1 n H 2 1, then G acts edge-transitively on

the graph r(G ', H~, H~ ). In this case the closure (G, H 1, H 2) is equivalent to the closure

(G', H~, H~) by Lemma 2.1.

Finally, if (G, H 1, H 2) is equivalent to (c', H~, H~) then

IG :H1 nH2 1 = IE(r(G,Hl>H2)1 = IE(r(G',H~,H~)I = IG' :H~ nH~ I. o

Lemma 2.3. Let the closure (G, H 1 , H 2) be maximal in its class. Then the follow

ing two conditions are equivalent.

(i) The graph r(G, H 1, H 2) is vertex-transitive;

(ii) there is a group D containing G as a subgroup of index 2 such that

(Hf, H~) = (H 2· H 1) for some element d E D.

Proof (i) ~ (ii). If r(G, H 1, H 2) is vertex-transitive then it follows from the max

imality of (G, H 1, H 2 ) that G has index 2 in Aut(r(G, H 1> H 2 )) = D. The edge

transitivity of G forces the existence of an element d in D with the prescribed property.

(ii) :::> (i). It is easy to see that d acts on r(G, H 1> H 2) and that it permutes the parts V 1 and V 2. So D = <G, d > = Aut(r(G, H 1, H 2)) acts vertex-transitively. 0

Let ~ be a directed graph. Let us define an undirected graph r by the rule: V (r) = V (M x { 1 , 2}, E (r) = { { (x, 1) , (y, 2) } I (x, y) e E (~) } . The graph r is called the standard double covering of the graph ~. The procedure of standard double covering can be also applied to an undirected graph. In this case, one should consider the edge {x, y} as the pair of directed edges (x, y) and (y, x).

Lemma 2.4. Let the closure (G, H 1, H 2) be maximal in its class. Then the following two conditions are equivalent.

(i) The graph r(G, H 1, H 2 ) is the standard double covering of an undirected graph ~ and G acts transitively on both E (11) and V(~);

(ii) Aut(r(G, H 1, H 2)) = G x <'t>, 1:2 = 1.

Proof (i) => (ii). Let 't be the map which interchanges the vertices from different parts corresponding to the same vertex from V(~). It is easy to see that 't e Aut(r(G, H 1, H 2 )). Now condition (ii) follows from the maximality of(G, H 1· Hz).

(ii) => (i). Notice that 1: e G and that 't interchanges the parts of r. Let us define a graph ~ by the following rule. As the vertex set of 11 we take one part of the graph r, vertices x and y are adjacent in r if and only if {x,y T} e E (r) (in this case {x T, y} e E (r)

also). It can be checked directly that r is the standard double covering of A. o Let (G, W) be a primitive permutation group and Iet A be its 2-orbit (orbit of Gon

W x W) which is antisymmetric, i.e. if (x, y) e A then (y, x) e A. For (x, y) e A, Iet H 1

and H 2 be the stabilizers in Gof the elements x and y, respectively.

Lemma 2.5. If the closure (G, H 1, Hz) is maximal in its class then exactly one of the following two possibilities holds:

(i) r(G, H 1, Hz) is abi-primitive graph;

(ii) there exists a permutation group (D, W) in which (G, W) is a subgroup of index 2 suchthat (xd,yd) = (y, x) for some (x,y) e A and d e D.

Proof It is easy to see using Lemma 2.1 that r(G, H I> H 2 ) is the standard double covering of some graph A. Let us show that for vertex-transtivity of r the existence of a group D possessing the above properties is necessary and sufficient. Since the closure (G, H 1, H 2 ) is maximal in its class, and H 1 , H 2 are conjugate maximal subgroups of G, the stabilizer of an arbitrary vertex x e V (r(G, H 1> H 2 )) stabilizes exactly one additional vertex, say 4>(x). Moreover, x and (jl(x) lie in different parts of the graph r. The consideration of the action of Aut (r(G, H 1> H 2)) on the set of pairs {x, <)>(x)} Ieads to a homomorphism of this group into the symmetric group of the set W. Now the existence of an element 4> with the prescribed properties follows directly from Lemma 2.3. o

Lemma 2.5 gives a method for the construction of bi-primitive graphs from primitive permutation groups with certain properties. Starting with some primitive


representations of the nonabelian groups of order less then 106 [13] (for example with the representations of the group J 1 of degree 1463, 1540, 1596) it turnsouttobe possible to construct bi-primitive graphs. In these cases the maximality of (G, H 1, H 2) follows from the fact that these permutation groups are maximal uniprimitive permutation groups. The following lemma is a partial converse of Lemma 2.5.

Lemma 2.6. Let the closure (G, H1,H2) be maximal, H1 and H2 be conjugate in G and suppose the graph r(G, H 1, H 2) is bi-primitive. Then r(G, H t. H 2) is the standard double covering of some directed graph !l..

Proof. Again let tl>(x) denote the vertex of r( G, H 1 , H 2) such that tl>(x) "# x and the stabilizer of x coincides with that of tl>(x). Let us define a graph !l. by the following rule. The vertex set of !l. is one of the parts of r(G, H 1, H 2) and (x, y) e E (!l.) if and only if {x, tl>(y)} e E (1(G, H 1> H 2)). It is easy to see that r(G, H 1> H 2) is standard double covering of A and it follows from the bi-primitivity of r(G, H 1, H 2), and from Lemma 2.4, that if (x, y) e E (A) then (y, x) e: E (A), i.e. Ais directed. 0

It follows from above that it is reasonable to carry out the classification of biprimitive graphs of valency k separately in the parabolic and nonparabolic cases. In the latter case the graphs should be constructed as the standard double coverings of certain directed graphs (Lemma 2.6). In the case of parabolic type we'll use the following scheme of classification.

1. Describe all closures (G, H 1, H 2) where Hi is maximal in G, [Hi: Hin H 3_j] = k for i = 1, 2, and H 1 is not conjugate to H 2 in G.

2. Extract, using Lemma 2.2, the closures which are maximal in their classes.

3. For each maximal closure (G, H b H 2 ) determine the existence of an element 't e Aut (G) for which (lfi, H~) = (H 2• H 1 ).

By Lemma 2.3 the graph r(G, H b H 2) is bi-primitive if and only if such an element 't is nonexistent

3. Preliminary results

It is assumed below that r is a regular bipartite graph with parts V 1 and V 2 ;

G:;;; Aut(r) acts bi-primitively on r; H 1 and H 2 are the stabilizers in Gof two adjacent vertices V 1 and v2, Vj E vi, i = 1, 2. By Lemma 2.1, r is isomorphic to r(G, H}. H 2).

Let N be a nontrivial normal subgroup of G.

Lemma 3.1. If N fixes a vertex of r then r is a complete bipartite graph; in particular, r is vertex-transitive.

Proof. Suppose that N fixes a vertex in the part V 1. Since G is transitive on V 1 and N is normal in G, N fixes each vertex in V 1. Since N is nontrivial and G acts primitively on V 2 , N is transitive on V 2 and the claim follows. 0

It should be noted that if N S Hi , i = 1, 2, then G = H 1 N = H 2 N and hence

G IN :: H 1 I (H 1 n N) :: H 2 I (H 2 n N)

Lemma 3.2. If N is abelian then r is vertex-transitive.

(*)

Proof. We can assume that N n Hi = 1 , i = 1 , 2, since otherwise by (*) N n Hi is

a nontrivial normal subgroup of G and Lemma 3.1 can be applied. Thus

N n Hi = 1 , i = 1 , 2, and each element g e G can be written uniquely in the form

g = hn, where h e H 1 , n e N. By (*) the map h ~ hnh is an isomorphism of H 1 onto

H 2 . This map can be extended to the whole group G by the rule :

't: hn ~ hnhn-1 •

It can be checked directly that 't is an involutory automorphism of G and that H1 = H 2·

By Lemma 2.3 this implies that r is vertex-transitive. 0

It is assumed below that the valency of r is p, where p is an odd prime number.

The following Iemma is well known (see for example [2]).

Lemma 3.3. IHi I= pr' · p~ • · · · • pr:; • p where Pi, 1 ~ i < m, are prime

numbers less thenp. 0

Lemma 3.4. Suppose that NH, (H 1 n H 2) < NG (H 1 n H 2) and that r is not a

complete bipartite graph. Then p is a subdegree of G in its action on V 1•

Proof It is easy to see that the set of vertices fixed by the subgroup H 1 forms an

imprimitivity block of the group G. lf H 1 fixes more than one vertex in V 1, then H 1 acts

trivially on V 1 and r is a complete bipartite graph. Let us show that H 1 n H 2 fixes at

least one vertex of V 1 other than v 1 • In fact, N G (H 1 n H 2) preserves the set of vertices

fixed by H 1 n H 2• and if H 1 n H 2 fixes only v 1 then NG(H 1 n H 2) ~ H 1, a contradic

tion. Let w ::~; v 1 , w e V 1 be a vertex which is invariant under H 1 n H 2· Since wH' ::~; w

and H 1 n H 2• as a subgroup of index p, is maximal in H 1 the orbit of H 1 containing w

has length p. 0

Lemma 3.5. Let r be bi-primitive with the number of vertices in each of its parts

odd. Then K <1 G ~ Aut (K) for some nonabelian simple group K which acts edge

transitively on r. Proof Let us consider a minimal normal subgroup N in G. It is well known that

N = K 1 X K 2 X • · • X Kr ,

where the Ki are simple groups conjugate in G. By Lemmas 3.1 and 3.2,

N S Hi , i = 1 , 2, and N is nonabelian. Hence each Ki is also nonabelian. It follows from

the hypothesis of the Iemma that H 1 n H 2 is a Sylow 2-subgroup of G. Since I Ki I is

even and all subgroups Ki are subnormal in G, each subgroup Ki has a nontrivial inter

section with H 1 n H 2·

It is clear _that N ~ H 1 ~ H 1 ~ H 2 , since otherwise it follows from (*) that N ~ H 2 = N ~ H 1 ~ H 2 and hence N ~ H 1 ~ H 2 is a nontrivial normal subgroup in G = <H 1 , H 2 > which stabilizes an edge. The latter implies that N ~ H 1 ~ H 2 = 1. Since the valency of r is a prime number, Hi acts primitively on the set of vertices which are adjacent to Vj. At the sametimeN fixes no edges of r, so N ~ Hi acts transitively on these vertices. Thus N acts edge-transitively on r. Suppose that Ki ~ H 1 =Ki ~ H2 =Ki ~ H1 ~ H2 for some i. Then Ki ~ H1 ~ H2 fixes an edge and it is normal in N which is edge-transitive. This implies that Ki ~ H 1 = 1. But it was proved above that Ki ~ H 1 ~ H 2 '*- 1, a contradiction. So Ki ~ H 1 > K 2 ~ H 1 ~ H 2· Since the valency p of r is a prime number and the action of N ~ H 1 on the vertices adjacent to v 1 is transitive, this action is primitive. Hence Ki ~ H 1 also acts transitively on these vertices and I Ki ~ H 1 I divides p. Now since p 2 does not divide I H 1 I we have that r = 1.

So N = K and K is a nonabelian simple group. Suppose that Co(K) is non trivial. Then Co(K) <I G and Co(K) ~ K = 1. By arguments analogaus to those carried out above, it is easy to show that I Co(K) ~ H 1 I divides p and so I Hi I divides p 2, a contradiction to Lemma 3.3. Hence K <I G ~ Aut(K). 0

Remark 3.6. It follows directly from the proof of Lemma 3.5 that the index [G : K] is not divisible by p.

4. The nonparabolic case

In what follows (G, H 1, H 2) is the maximal closure in its class, [Hi: H 1 ~ H 2l = 3 for i = 1 , 2, and both H 1 and H 2 are maximal subgroups of G.

In this section we consider the situation when the pennutation group (G, Vi) has a subdegree 3 for each of i = 1, 2. This situation covers the parabolic case and, by Lemma 3.4, also the case when the cardinality of Vi is odd for i = 1 , 2. The principle role in the present setting is played by the following theorem of W. Wong which contains a complete description of the primitive permutation groups which have a subdegree of 3.

Theorem 4.1. [24] Let (G, W) be a primitive permutation group having a subdegree 3. Then eilher G has a normal elementary abelian subgroup acting regularly on W, or G is isomorphic to a group from the following list :

As, S 5, PGL2(7), PSL2(ll), PSL2(B), PSL2(q) for q = ± 1 (mod 16),

0

It was shown in Lemma 3.2 that if G has a normal elementary abelian subgroup then r cannot be bi-primitive. Let us consider the situation when G is a group from the above Iist.

Let r be a graph of nonparabolic type. lf Gis one of the groups As, S s. PGL2(1), PSL2(q), for q = ± 1 (mod 16), PSL3(3), Aut (PSL3(3)), then the cubic graph, with vertex set W which is invariant under G, is undirected. So Lemmas 2.4 and 2.6 can be applied. If G is PSL2(11) or PSL2(13), then Lemma 2.5 works where for D we take PGL2(11) and PGL2(13), respectively. So no bi-primitive cubic graphs of nonparabolic type exist

Let r(G, H 1o H 2 ) be ofparabolic type.

For all graphs in the Iist except the family PSL2(q), q = ±1 (mod 16), the maximal subgroups which Iead to representations with subdegree 3 are conjugate. If G = PSL2(q), q = ±1 (mod 16), these subgroups form two conjugacy classes in G. (All of these subgroups are conjugate in PGL2(q)). But subgroups from different classes cannot have intersection of index 3 in eachother. So the following Iemma holds.

Lemma 4.2. There is no bi-primitive cubic graph for which the automorphism group has a subdegree 3 in its action on each part. D

In partiewar the nonparabolic case does not Iead to a bi-primitive graph.

5. Tbe parabolic case

In this section we consider the situation when (H 1, H 2) is the closure which is maximal in its class, [H; :H1 f1H2l = 3, i = 1, 2; [G :H] is odd and H 1 , H2 aremaximal subgroups of G which are not conjugate in G. By Lemma 3.6 and Remark 3.6, in this case we have K <l G S Aut(K) for some nonabelian simple group K which acts edgetransitively on r(G, H1, H2) andforwhich [G :K] is apowerof2. UsingLemma 2.1 we conclude that the graphs r( G, H 1o H 2) and r(K, K 1o K 2 ) are isomorphic for K; = K f1 H; , i = 1 , 2. It is clear that K 1 f1 K 2 is a Sylow 2-subgroup of K.

By definition, K 1 +-- K 1 f1 K 2 ~ K 2 is a simple amalgam of index (3, 3). All such amalgams were classified by D. Goldschmidt in [10]; see Table 1 below. It follows from this description that in particular I K 1 f1 K 2 I divides 2 7. It was shown in [ 1] that all simple groups K whose Sylow 2-subgroup is of order at most 210 are known. So to obtain a complete list of the closures (K, K 1, K 2) it is sufficient to consider only the known simple groups K, and for the reconstruction of the closures (G, H 1, H 2) it is sufficient to consider the automorphisms of K whose orders are powers of 2.

N (K t. K2) K 1 nK2 n m

1 (Z3, Z3) 1 1 0 2 (S3, S3) z2 2 1

3 (S3, Z6) z2 2 1

4 (D12, D12) E4 4 2

5 (D12· A4) E4 4 2

6 (D24, S4) Ds 8 2

7 (D 8 xZ3, S4) Ds 8 2

8 (D 12 xZ2, A4 xZ2) Es 8 3 9 (Ds>.S3, Z2 xS4) z 2 xDs 16 3

10 (S4, S4) Ds 8 2

11 (S4 X Z2, S4 X Z2) Z2 xDs 16 3 12 ((Qs * Z4)· S3, (Z4 xZ4)· S3) (Z4 xZ4) x Z2 32 2 13 ((Qs * Qd · S3, (Z4 xZ4)· D12) (Z4 xZ4) >.E4 64 3 14 ((Qs * Qs)2 • S3, (Z4 X Z4) • D 12) (Z4 xZ4) x E4 64 3 15 ((Q 8 * Q 8)2 • D 12, (Z4 xZ4)· Z 3 xD 8) (Z4 xZ4) x D8 128 4

Table L

In Table 1 n denotes the cardinality of the intersection K 1 n K 2 and m is the 2-rank: of the intersection.

Remark. In the original table of arnalgarns of index (3, 3) in [10] there a few misprints which have been corrected in the table above.

Thus Iet K be a nonabelian simple group, containing an arnalgam K 1 ~ K 1 n K 2 ~ K 2 of index (3, 3). The following Iemma is a direct consequence of Table L

Lemma 5.1. The Sylow 2-subgroup K 1 n K 2 of K has order less than or equal to 27 .Moreover,eitherm <4or IK1 nK2 1 =27 •

Lemma 5.2. Let I K 1 n K 2 I > 2_ Then one of the following holds:

(i) K 1 nK2 =Ds;

(ii) K 1 = CK(t) where 't is a central involution of K_

Proof It follows from Table 1 that if I K 1 n K 2 I > 2 then either we have the amalgarn nurober 10 or K 1 contains a characteristic subgroup of order 2_ Let 't2 = 1 , <'t> char K 1- Since K 1 = H 1 n K ~ H 1, we have <'t> ~ H 1_ But H 1 is maximal in G, so H 1 = Cc('t) and K 1 = H 1 n K = CK('t)- At the sarne time K 1 contains a Sylow 2-subgroup of K, whence 't is a central involution of K. 0

Lemma 5.3. The group K has only one conjugacy class of central involutions.

Proof. Let K 1 ~ K 1 n K 2 --+ K 2 be the amalgam nurober i in Table 1. lf i = 1 , 2, 3 then I K 1 n K 2 I < 4, and it is well known that there is no such corresponding simple group K. lf i = 4, 5, 8 then K 1 n K 2 is an elementary abelian group of order either 4 or 8. Each corresponding simple group K is isomorphic either to PSL2(q) for suitable q or to the group 1 1 [16]. In either case K has a unique class of involutions. lf i = 9' 11 then K 1 (") K 2 = D 8 X z2. It is known (see for example [16]) that this Situation cannot lead to a simple group. For i = 12, 13, 14, 15 the center of the group K 1 n K 2 is cyclic so all central involutions are conjugate. Using the survey [16] we can list all simple groups K satisfying Lemma 5.1. The list is as follows:

PSL2(q), PSL3(q), PSL4(q), PSU3(q), PSU4(q)

G 2(q), 3D 4 (q), 2G 2(q) for suitable odd q, (1)

Lemma 5.4 Let K be a group from the list (1) and for an arbitrary central involution 't suppose the equality I CK('t) I = 2a3 holds for some a < 8. Then K is one of the following groups:

The proof follows from the properties of centralizers of central involutions in the known groups.

For the linear and unitary groups all calculations can be carried out directly.

For the groups G 2 (q) and 3D 4 (q) we can use a result from the paper [ 11], where it is shown that the centralizer of an involution in such groups is isomorphic to the central

product SL2(qt) * SL2(q2) for some Q1 and Q2·

The centralizer of an involution in the group 2G (q) is isomorphic to Z2 x PSL 2(q), q = 3n, n is odd (see for example [16]).

The order of the centralizer of an involution in the group Sz(8) is not divisible by 3, while for the group A 11 this order is divisible by 32 .

Finally, the centralizers of involutions in the sporadic groups M 22 , M 23 , 11 , 12 , 13 , M cL are nonsolvable [21 ]. D

Lemma 5.5. Let K be isomorphic to one of the following groups : PSL3(3), A7, A 10, M 11· and suppose K <l G ~ Aut (K). Then G cannot act biprimitively on a cubic graph.

Proof. Let us consider all cases, separately. Let K = PSL 3 (3). Since the order of a Sylow 2-subgroup of K is 16, by Lemma 5.2 K = CK('t) for some central involution 't. In

the situation considered CK('t) = GL2(3), but it is easy to see that there are no subgroups K 1 in Table 1 which are isomorphic to GL2(3), a contradiction.

K = A 7 . In this case G is isomorphic to either A 7 or S 7. But neither of these groups has a maximal subgroup of order 2a3 (see for example [13]). The case K = A 10 can be excluded by the same arguments.

K = M 11 • Since M 11 has no outer automorphisms, in this case G = M 11· The group M 11 has a unique conjugacy class of maximal subgroups of order 2a3. Hence H 1 and H 2 are conjugate in G so this is not the parabolic case. D

Let us consider the other groups in Table 1.

Let K = PSL 2(q) and let K 1 f- K 1 n K 2 --+ K 2 be the amalgam number 10 in Table 1. Then q = ± 7 (mod 16) and K 1 = K 2 = S 4. Now we are considering the parabolic case, so K 1 and K 2 are not conjugate in K. But then the graph r in question is vertex-transitive by Lemma 2.3 where the role of Dis played by the group PGL2(q).

Now let K 1 f- K 1 n K 2 --+ K 2 be nonisomorphic to the amalgam number 10. Then K 1 = CK(t). It is well known that in the case considered CK('t) = Dq±l· This and Table 1 imply that the considered amalgam has number 4, 5 or 6 and that q e { 11 , 13, 23, 25}. If we have amalgam number 4 then the graph r(K, K 1o K 2) is vertex-transitive by Lemma 2.3 where the role of Dis played by the group PGL2(q). For the amalgam number 5 we have q e { 11 , 13}. U sing Lemma 2.2 we see that the closures are embedded in (PGL2(11),D24,S4) and (PGL2(13),D24•S4), and the groups PGL 2(11) and PGL2(13) act bi-primitively on the corresponding graphs. Now let us consider the amalgam 6. In this case q e {23, 25}. Again, using Lemma 2.2, we conclude that the closures are contained in (PSL2(23), D24, S4) and (PrL2(25), D 8 xS3, S4 xS 2), where PrL 2(25) is the extension of the group PSL2(25) by a field automorphism. In the former case the action is bi-primitive while in the latter case it is not bi-primitive since s4 X z2 < PGL2(5) X z2 < PLL2(25). It should be mentioned that, for the value of q in question, the group PSL2(q) contains exactly two conjugacy classes of subgroups S4 which are conjugate in PGL2(q). Foreachgraph we take one such class. This finishes the consideration of the groups P SL 2 ( q ).

K = PSU3(3). This group contains two classes of maximal subgroups of order 25 ·3 (see for example [13]). Since PSU3(3)=G2(2)', it is easy to see that the graph r(K, K 1o K 2) is the graph of the parabolic geometry of the group G 2(2) = Aut(PSU 3(3)) and corresponds to the amalgam number 13.

K = M 12. This group contains exactly two conjugacy classes of subgroups of order 26 • 3 and these subgroups are maximal. By Lemma 2.2 the closure is contained in (Aut(M 12 ), H 1, H 2) and the latter corresponds to the amalgam 15.

Thus we have obtained five closures whose groups act bi-primitively on the corresponding graphs. All these closures are maximal. In fact the number of vertices in the graphs are all different so no one of the closures can contain another. For the same


reason embeddings in the closures with an elementary abelian normal subgroup and corresponding to the groups PSLz(q), q = ±7 (mod 16) are impossible.

Thus the five closures are maximal in their classes. This, and the fact that H 1 and H 2 are not isomorphic, implies that all the graphs are bi-primitive and Theorem 1.1 is proved.

Let us say a few words about the combinatorial description of the constructed graphs. The graph with automorphism group PGLz(l1) was constructed in [12] in terms of the Paley design P (11). As mentioned above the graph related to G 2 (2) is the first graph in the infinite family of bi-primitive graphs. In [2], page 164, a description of this graph in terms of the projective plane of order 9 equipped with a unitary form is given. The graphs with groups PGL 2(13) and PGL 2(23) can be described in terms of the projective line over GF (q) for q = 13 and 23, respectively. The graph with group Aut(M 1z) was described in [7], [19] in terms of the Steiner system S (5, 6, 12).

Notice that four of the five closures reconstructed in this paper were mentioned in [10].

It should be remarked that in fact the proof of Theorem 1.1 contains the classification of all pairs (r, G) where r is a cubic graph and G act bi-primitively on r. Namely the following proposition holds.

Theorem 5.6. Let r be a cubic graph and let G = Aut-(r) act bi-primitively on r. Then one of the following holds.

(i) r is the comp1ete bipartite graph K 3,3;

(ii) r is the standard double covering of a cubic graph ~ and G acts transitively on E (~)

and primitively on V (M;

(iii) r is the graph r(PSLz(q),HI>H2) where q = ±7 (mod 16), H1 =Hz =-S4, H 1 n H2 =Ds and H 1, H2 are not conjugate in PSLz(q). Moreover either q =p, p prime, and G = PSL 2(p); or q = p 2 , p prime, p = 3, 5, 11, 13 (mod 16), and G =PU2 (p 2 );

(iv) r is one of the five bi-primitive cubic graphs described in Theorem 1.1.

Let r satisfy the conditions of Theorem 5.6 and not be bi-primitive. Then Aut(r) acts vertex-transitively on r and the stabilizer of each part acts primitively on that part. In [9] these graphs are called semiprimitive. Thus, Theorem 5.6 contains, in particular, the classification of semiprimitive cubic graphs.

The graphs defined in (iii) of Theorem 5.6 were studied in [4]. In this paper these graphs are called sextet graphs. It was conjectured in this paper that each 5-transitive semiprimitive cubic graph is either a sextet graph or the standard double covering of the primitive cubic graph on 234 vertices related to the group PSL 3(3). Recall that a graph is 5-transitive if its automorphism group act transitively on the set of paths of length 5.


Theorem 5.6 proves in particular the aforementioned conjecture.

References

1. B. Beisiegel, Über einfache endliche Gruppen wit Sylow 2-Gruppen der Ordnung höchsterns 210 , Comm. Algebra, 5 ( 1977), 113-170.

2. N.L. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.

3. N.L. Biggs, D.H. Srnith, On trivalent graphs, Bull. London Math. Soc., 3 (1971), 155-158.

4. N.L. Biggs, M.J. Hoare, The sextet constructionfor cubic graphs, Combinatorica 3,

(1983), 153-165.

5. F. Buekenhout, Diagrams for geometrics and groups, J. Combin. Theory (A), 27

(1979), 121-151.

6. F. Buekenhout, Geometriesfor Mathieu group M 12 , Lect. Notes Math, 969 (1982),

74-85.

7. LV. Chuvaeva, On some combinatorial objects which admit the Mathieu group

M 12, In "Methods for Complex System Studies, pp. 47-52, Moscow, VNIISI, 1983

(In Russian).

LA. Faradzev, M.H. Klin, M.E. Muzichuk, Cellular rings and groups of automorphisms of graphs. ( see Ref 18)

8. J. Folkman, Regular line symmetric graphs, J. Comb. Theory, 3 (1967), 215-232.

9. A. Gardiner, Symmetry conditions in graphs, London Math. Soc. Lect. Notes Ser., 38 (1978), 22-43.

10. D. Goldschrnidt, Automorphisms of trivalent graphs, Annals of Math., 111 (1980), 377-406.

11. D. Gorenstein, K. Harada, Finite groups of low 2-rank and the families G2(q), 3D 4 (q), q odd, Bull. Amer. Nath. Soc., 77 (1971), 829-862.

12. A.A. lvanov, Computation of lengths of orbits of a subgroup in a transitive per

muiation group, In "Methods for Complex Systems Studies", pp. 3-7, Moscow, VNIISI, 1983 [see English translation in this volume].

13. A.A. Ivanov, M.H. Klin, LA. FaradZev, Primitive representations of the nonabelian simple groups of order less than 106 , Parts 1, 2. Moscow, VNIISI, 1982, 1984 [In Russian].

14. A.A. Ivanov, LV. Chuvaeva, The action of the group M 12 on Hadamard matrices [In this volume].

472 M. E. IOFINOV A AND A. A IV ANOV

15. A.V. Ivanov, On regular edge-symmetric but not vertex-symmetric graphs, In "Permutation groups and combinatorial objects". pp. 38-51, Preprint, 82.14, Kiev, 1982 [In Russian].

16. V.V. Kabanov, A.S. Kondratev, Sylow 2-subgroups of Finite Groups, Sverdlovsk, 1979 [In Russian].

17. M.H. Klin, On edge- but not vertex-transitive graphs, In "Algebraic Methods in Graph Theory", pp. 399-403, Budapest, 1981.

18. LA. FaradZev, M.H. Klin, M.E. Muzichuk, Cellular rings and groups of automorphisms of graphs [In this volume].

19. M.A. Ronan, G. Stroth, Minimal parabolic geometries of the sporadic groups, Europ. J. Comb., 5 (1984), 59-91.

20. J.P. Serre, Arbres, amalgams, SL 2 , Soc. Math. France, Asterisque, 46, 1977.

21. S.A. Syskin, Abstract properties of the sporadic simple groups, Uspehi. Mat. Nauk, 35 (1980), 43-55 [In Russian].

22. J. Tits, Buildings of Spherical Type and Finite (B,N)-pairs, Lect. Notes Math. 386, 1984.

23. V.G. Vising, Same unsolved problems in graph theory, Uspehi Mat. Nauk, 23 (1968), 117-134 [In Russian].

24. W. Wong, Determination of a class of primitive permutation groups, Math. Z., 99 (1967), 235-246.


ON SOME PROPERTIES OF GEOMETRIES OF CHEV ALLEY GROUPS AND THEIR GENERALIZATIONS

V.A. U stimenko

The Tits geometries turn out to be useful in the study of Lie groups and Lie algebras as well as arithmetic groups. A more general class of objects, the Buekenhout-Tits

geometries are closely related to the Tits geometries. The recently constructed geometries for sporadic groups are members ofthis class (see [17], [24], [25]).

It was mentioned in [ 17] that, on one hand, it is interesting to study the known

objects of this type and to find new interpretations for these objects (see, for instance, the interpretation of the exceptional Chevalley groups proposed by H. Freudental in [10]).

On the other hand, it is interesting to construct new geometries.

In the present paper, we study an operation over binary relations defined in [26], [27]. This Operation is applied here to the geometries of the Coxeter groups. In this way we obtain new interpretations of the geometries of the Chevalley groups, of the split sem

isimple algebraic groups and of the geometries related to the Kac-Moody algebras and some other Lie algebras. The considered operation also enables the construction of some new geometries. For instance, in Section 4, a family of geometries having finite rank is described. This family contains the geometries of some known BN-pairs in addition to some other ftag-transitive Tits geometries. As a generalization of the classical group cases some ftag-transitive geometries of infinite rank are defined.

The aforementioned interpretations of the geometries of the Chevalley groups turn out to be useful in the consideration of certain problems (see Sections 5,6). One of these

problems is the computation problern for the structure constants of the Hecke algebra of a

Chevalley group corresponding to a parabolic subgroup of this group (see [28], [33]). In Section 7, as a consequence of the formulas appearing in [33], we prove a theorem about

the simplicity of the Hecke algebras of the classical groups acting on certain homogeneaus spaces. Analogous results are obtained for subalgebras of the generating Tits algebras.

1. lncidence systems and geometries

An incidence system over a set ß of types is a triple (r,I,t) where r is the set of objects of the system, I is a symmetric reflexive incidence relation over r and t is a mapping of r onto ß. Sometimes we will write r instead of (r,I,t).

A ftag of an incidence system is a set of objects from the system which are pairwise incident. The residue Res F of a ftag F in[' is the set of objects from t-1(ß-t(F)) which

are incident to all elements of the ftag, along with the restrictions of I and t to this set.

473

474 V. A. USTIMENKO

A morphism of incidence systems r and r' over the same set of types is a mapping of r onto r' which preserves incidence and types of objects.

An automorphism group G of an incidence system r is said to act flag-transitively on r if its action on the flags of a fixed type is transitive.

The rank of an incidence system (resp., of a flag F) is the cardinality of .1 (resp., the cardinality of t(F)). The corank of a flag Fis defined as the cardinality of .1-t(F). The incidence system (r,/,t) is said to be connected if the graph with vertex set r and edge set I is connected.

An incidence system (r,I,t) over .1 is called a geometry if the restriction oft to each maximal flag F of r is a bijection of F onto A.

It is easy to see that if F is a flag of a geometry r then the residue Res F is a geometry over the set of types .1- t(F).

A diagram over .1 is a mapping D defined on the totality of 2-element subsets of .1

such that D (A) is a class of geometries (of rank 2) over the set of types A. A geometry r is said to be a geometry over the diagram D if, for each ftag F of corank 2, Res F is contained in D(.1- t(F)).

The most important class of rank 2 geometries is the class of so called generalized m-gons (see [24]).

The connected geometries over diagrams D such that D (A) is the class of generalized m-gons for some m, are called the Tits geometries.

Let G be a group and G;, i e J, be a collection of distinct subgroups of G. The group incidence system r(G,G;), i e J, is the set r of left (right) cosets of G over the subgroups G; with the incidence relation I = { a, 13 e r I the intersection of a and 13 as subsets of G is nonempty} and the type function t which assigns to g G; the element i of J.

LetS be a set of generators of G. Let r(G,S) denote the incidence system r(G,G5 ),

s e S, where Gs = < S- {s} >. It will be assumed that the set s-1 = {s-1 I s e S} coincides with S. With an element g e G we associate the length of the shortest word over the alphabet S which represents g. This is the length l (g) of the reduced representation of g in the sense of [3]. For an element a of r(G,S) we put /(a) = min l(g ). Let 1: be the set

ge a

of elements which are conjugate to elements from S. For a e r, put .1+(a) = {w e 1: I /(wa) < /(a)} and .1-(a) = {w e 1: I i(wa) > /(a)}.

Let (W,S) be a Coxeter system [3], i.e. W is the group with generators s; e S and

relations (s; Sj )m'i = 1, m;; = 1. Put W1 = <1 > for J ~ S. These subgroups W1 are the parabolic subgroups of W. The geometry r(W,S) is a Tits geometry. The flags of this geometry can be identified with the cosets over the parabolic subgroups. Then the maximal flags correspond to the elements of W. Notice that in this case the rank 2 residues are the ordinary m-gons, and there is a natural correspondence between the diagram of the

ON SOME PROPERTIES OF GEOMETRIES OF CHEV ALLEY GROUPS AND THEIR GENERALIZA TIONS 475

Coxeter group W in the sense of [3] and the diagram of the geometry r(W,S).

Let G be a group with a BN-pair (a Tits system (G,B,N,S) in the sense of [3]) and Iet (W,S) be the corresponding Coxeter system. The maximal subgroups of G which contain B (the maximal standard parabolic subgroups) have the form Ps=BWs-(sJB,se S. The incidence system r(G,Gs) is a Tits geometry. This geometry is called the geometry of the BN-pair or the geometry of G.

2. The main construction

Let (A, +) be an abelian group and I: be an arbitrary set. Let u denote a certain subgroup of the abelian group consisting of all functions from I: to A. For l:o ~I:, Iet u(l:o)

denote the subgroup of the functions from u which vanish outside l:o, and, for f e u, Iet f l:r.., be the function which coincides with f on l:o and vanishes outside I.o. We will

assume that on u a distributive operation * is defined:

(f+g)*h=f*h+g*h;

h* (f+g)=h*f+h*g.

Definition. Let ci> be a binary relation on a set N and Tl be a mapping of N into 21:. Let = (I:,u,*,TJ) be the relation on N = {(a,f) I ae N, f e u(TJ(a))} defined as follows: ((a,f), (ß,g)) e if and only if (a,J3) e ci> and

f- g IT)(a)n T!(ß) = f * g ~(a)n T!(ß)·

Then is called a blow-up of the relation cl>.

If u coincides with the set of all functions from I: to A (resp., with all functions having finite support) then a blow-up is said to be cartesian (resp., direct).

If Ais a module over a ring R, u is a submodule of Ar and * is abilinear operation on u, then the blow-up is said tobe linear over R.

If the relation ci> is symmetric and the operation * is alternating (i.e. x * y = -y * x) then the relation is symmetric as weiL In particular, if (r,/,t) is an incidence system over a set of types A then, for * an alternating operation, the covering (f,i,i), where t(a,f) = t(a), is an incidence system over A. Moreover the mapping (a,g) ~ a is a morphism. Such a rnorphism will be called a retract. The system (f,i,i) will be referred to as a blow-up of the system (r,/,t).

Example 2.1. Let N be the set of all subsets of a finite set n = { 1 , ... , n} and ci> be the symmetric inclusion relation on N. Put I:= { (i,j) E n2 ' i > j}. For a E N, Iet us consider A( a) = { (i,j) e I:, i e a, j ::F- a}. Let K be an arbitrary skew field. In the vector space V= Kr= {/:I:~ K}, we fix the basis elements vij suchthat

476

{ 1 , X = (i,j) ,

Vij(X) = 0, X:!: (i,j)

and we define an alternating bilinear multiplication o by the following rule:

{1 ,j=k,

Vij o Vfd = -1 , / = i , 0 otherwise.

V. A. USTIMENKO

We consider the relation = (E,K l:, o , L1). Let r(N,K) be the incidence system (IV, ,i)

where i(a,f) = I a I = t(a). It is clear that is a linear blow-up of <l>.

Let us show that the incidence system r(N,K) is isomorphic to the projective geometry PGn_1 (K). The latter is the set of allproper subspaces of the space Vn(K) with the incidence relation

(u,w)e/~[(u>w) v (w>u)]

and with the type function dim.

Let us fix a basis in Vn(K). To any arbitrary collection h 1, h2, ... , h1 of vectors from V n (K), we associate the number I ( { h 1 , ..• , hr}) = max { i I hsi :;: 0 for some

hs = (hs1• hs2• ... , hsn)}, i.e. the maximum position in which some vector hs has nonzero coordinate in that position with respect to the chosen basis.

- -Let W be a subspace of Vn (K), dim W = m. Then W = for some vec-- -

tors bi = (bi 1 , ••. , bin). Let us transform the basis bi into another basis by the following algorithm.

Step 1. Let Im= I({b 1, b2, ... , bm}) and bs be a vector such that bsr .. :!: 0. Let us

fix a vector f~~) = (bsr..)-1 x bs and consider the new basis -(1) -(1) -(1) -(1) -(1) -(1) - - -(1) . b1 ... , bs-1> bs+1 ' ... ' bm 'rr.. where bi - bi- bir,. X rr.. . Suppose that, With

respect to the new basis, b~l) = (b~~>, b~!> , ... , b}!> ), f~~> = (ge~, ge1 , ... , g~~~). Step 2. Put lm_1 =I( {bp> }). Let us consider the vector b}P such that bs'l,._, :;: 0. Put

rl:~l = (bs'l,._, )-1 X bs'• We consider also rl:) = rl:)- (gEL )-1 X fl:~l and arrive at

the system of vectors b~2) = !Jp>- (b~E-1 )-1 X re ' i :;: s'. We will use the notation

b-<2> _ (b<2> b<2> b<2>) . r<2> _ (g<2l <2>) r-(2) _ < <2> <2> ) l - l1 ' lz '· • ·' ln. 'l '# S' lm - lml '• • ·' glmn ' 1".,_1 - glm-11 '· · ·' gl"._1n ·

We continue this process and consider the m-th step. The system {b~m-1 ) , ... } contains a unique vector b~m-1 ) with all of its Coordinates having position > / 1 equal to 0, where !1 = /( {b~m-1 ) , ... }) = l(bi-1 ). Put fr 1 = f~~) = (b~f:- 1 ) )-1 x b~m-1 ) and exchange the vectors f<m-1) f<m-1) f<m-1) by f = f<m) = f<m-1) - (g ) • f<m-1)

12 • 13 • • • • • 1.. 11 11 11 1111 / 1 •

It can be shown that the outcome of the presented variation of the Gauss method does not depend on the choice of the basis b 1, b 2 , ... , bn and on the choice of the

ON SOME PROPERTIES OF GEOMETRIES OF CHEV ALLEY GROUPS AND THEIR GENERALIZATIONS 477

vector b~k-1 ) with the property that b81H '# 0 at step k of the algorithm.

Lemma 2.2. The system i\ , ... , I\. of vectors is uniquely determined by the sub

space W. - -

It is convenient to represent the vectors r1, , ••• , r 1 .. as rows labeled by numbers

/ 1 , ..• , Im in an n x n matrix r(W). In this case the coordinate gl,i , i < 18 , of the vector

r1, will be in the (/8 ,i)-position of the matrix. If we represent the vector ej, j '# ls, as the

j-th row of the matrix, we obtain a triangular matrix of a special type (the Gaussian matrix in the sense of [28]; see Fig. 1.1).

• • * * 1

• • • • * * 0 * 1

• • • • • •

Fig. 1.1.

Let us consider the algorithm which associates to a subspace W the subset {1 1 , /z, ... , Im} of Q and the triangular matrix r(W) = (gij) with gij = 0 if (i,j) e A( { 11, 12 , ... , Im}). It is clear that the matrix r(W) is uniquely determined by the function g(W) from I. to K suchthat g(W) ((i,j)) = gij· So the algorithm associates to W the element p(W) = ({1 1 , ... , Im}, gij) of the set N of the incidence system r(N,K), and so determines a mapping p : PGn_1 (K)--+ N. Notice that it is convenient to identify I. either with the set of transpositions of the group Sn (the Weyl group An_1 of the generallinear group PGLn(K)) or with the positive roots of the system An_1 (i.e. with the vectors ei -ej). Here ei denotes the i-th standard orthonormal basis vector of ndimensional Euclidean space. D

Proposition 2.3. The mapping p is an isomorphism of the geometries PGn_1 (K) and r(N,K).

Proof If U and W are distinct subspaces then p(U) and p(W) are distinct elements of N. In fact, the equality p(U) = p(W) implies equality of the Gaussian bases I\, rl2 , ... , rl., and rl, ', r/2', ... , rl .. ' of the subspaces u and w, respectively. In this

case U = <i\, f1 2 , ••• , f 1,. > = <f1, ', f1/, ... , f1 .. '> = W. On the other hand, the

image of an element (B,f) is nonempty. It contains the subspace <f1,, f1 2 , ••• , f1 .. > - - -

where {11, lz, ... , Im} = B and r1,, r12 , ••• , r1 .. is the Gaussian basis determined by

the function f Thus the mapping p is a bijection.

478 V. A. USTIMENKO

Let U be a subspace of Wandlet p(U) = (B,f), p(W) = ~.g). Let us show that B c A. Suppose, to the contrary that i e B and i e: A. The vector ri of the Gaussian basis of U is a linear combination of the vectors I'/ of the 9aussian basis of W. Moreover, in

this linear combination, the coefficients of the vectors rj' are all zero for j > i:

fi = L Ajf/ (2.1) j<i

But the i-th coordinate of the vector on the right is zero while this coordinate of the vector on the left is 1, a contradiction. Notice that if i E B then

f. -f·'= ~ J...,.f.' I I "-'1J" (2.2)

j <i

Finally let us show that

g- f 1,.\(A)nl\(B) = g 0 f.

Notice that the vectors from K 8(A) = (f I j(x) = 0 for x e: ö(A)} form an abelian subalgebra (since x o y = 0 for all x,y e K8(Al). This is due to the fact that if the elements of ö(A) are considered as the roots ei- ej of the system An-l, then the sum of two roots from ö(A) is not a root. So

g 0 f = g IL\(A)-L\(A)n L\(B) 0 f IL\(B)-L\(A)n L\(B)·

An element ei- ej from ö(A) 11 ö(B) has i- j- 1 presentations as the sum of two roots: ei- ej = (ei -ek) + (ek-ej), i < k < j. Hence

g o f(ei-ej)= L g(ei-ek)f(ei-ej). kEA&i>k>j

The right side of the above equality is just the j-th coordinate of the right side of (2.2). 0

Definition. Let (f,i,i) be a blow-up of an incidence system (r,I,t) with set of types ö. For an element y E U let us define a transformation y of the set fi by the following rule:

y: (a,x) ~ (a,x + (y +X* y) 111(a))

where (a,x) E fi. The incidence system (f,i,i) will be called a smooth blow-up of (r,I,t) if there is an element i e ö (the distinguished type) suchthat

(a) an element a E r is uniquely determined by the set Oa of elements from ti which are incident to a;

(b) for arbitrary y and a the transformation y maps 0 a into the set Ob for some b;

(c) the action of y on f defined by the rule: aY = b ~ (Oa)Y =Ob is an automorphism off.

The incidence system (IV,c'P,i) considered in Example 2.1 is a smooth blow-up of the system (N, <D,t). For the distinguished type one can take 1 or n - 1 (recall that

ON SOME PROPER TJES OF GEOMETRIES OF CHEV ALLEY GROUPS AND THEIR GENERALIZA TIONS 479

t(A)= lA I forA e N).

Example 2.4. (Geometry of the Weyl groups of classical groups.)

The relation cD on the set N considered in the previous example can be identified

with the incidence relation of the geometry of the Coxeter group An-1 (i.e. of the symmetric group S(Q) with the set of generating transpositions (1,2), ... , (n -1, n)).

Let us consider the geometry of the Coxeter group Bn. lts elements can be inter

preted as the partially defined functions on the set Q taking values in the field F 2· So

r(Bn) = {(B,f) I B c Q and f e F~ }. Put (B,f)-< (C,g) if and only if C cB and the

restrictions of the functions fand g to the set B coincide. The incidence relation cD' of the geometry is the symmetrization of the relation -<. The type of a partial function (A ,f) is

by definition I A I.

The incidence relation of the geometry r(Bn) can be considered as a direct blow-up

of r(An_1). In fact, Iet us consider the direct blow-up q, = (O,F~,o ,e) where e is the

identity mapping of N = 2n onto itself, x o y = 0 for all x,y e F~. Then it is easy to see that cD' = q, and that is a smooth blow-up of cD.

The geometry r(Dn) is a Subset r' of the partially defined functions from r(Bn)

with incidence relation and type function restricted to r'. The set r' consists of all func

tions which either are defined everywhere or have an even number of values 1.

Example 2.5. Let us consider the segment [0,1] of the realline. Let D be the set of

intervals [a,b ], where 0 :S: a y} and

T)([a,b]) = { (x,y) e l: I a :S: x :S: b, O:S: y :S: a }. The subgroup U in Rr. and the bilinear multiplication will be defined as follows.

Let us consider a partially continuous bounded function f which is defined on [0, 1] x [0, 1]. This means that [0, 1] x [0, 1] is divided into a finite number of domains

with continuous boundaries and the restriction of f to any of these domains is a continuous function. The space L of all such functions will be considered as a linear algebra with

multiplication

1 1

f(x,y) * g(x,y) = J f(x,z) g(z,y) dz- J g(x,z) f(z,y) dz. 0 0

Let us consider the subalgebra U in L consisting of all functions which are zero outside

l:. A direct check shows that (l:, U, *, T)) is a blow-up of the incidence system (D, cD, t ).

480 V. A. USTIMENKO

3. Blow-ups and Lie algebras

Let V be a vector space over a field K of characteristic zero. A reduced root system

is a finite subset R of nonzero vectors from V such that

(a) R spans V as a vector space over K;

(b) for each a e R there exists a* e V*, where V* is the space dual to V, suchthat

a* ( a) = 2 and the mapping s a (x) = x - a* (x) a preserves R;

( c) for each a e R the inclusion a* (R) c Z holds;

(d) if a E R then the elements from R which aremultiples of a are a and --a.

The set of forms a* for a E R is a reduced root system in V*.

A basis (or a set of simple roots) is a subset S of R which is a basis of V suchthat all

coefficients in the decomposition of an arbitrary element a with respect to this basis have

the same sign.

We will also consider infinite sets R (in spaces V of infinite dimension) which

satisfy conditions (a)-(d) and contain a basis. These sets will also be called root systems.

The group W of automorphisms of V generated by the reflections s a is called the

Weyl group of the system R.

If S is a basis in R then (W, {s a; a E S}) is a Coxeter system [3].

Now let (W,S) be a Coxeter system. For g E W let l(g) denote the length of the

reduced representation of g. Fora coset ß E r(W) put /(ß) = min l(g). Let k be the set of gE j3

reflections from the group W (i.e. elements which are conjugate with elements of S). The

elements s a and s -a coincide and the correspondence s a ---+ a E R + is a bijection of k ontoR+.

Let L be a Lie algebra over the field K admitting the root decomposition

L =Lo $ I, Lw$ I, Lw (3.1) WER+ WER"

where R+ and R- are the sets of positive and negative roots of R. Examples of such Lie

algebras are the semisimple Lie algebras over C, the split reductive Lie algebras over the

field K and some of their generalizations (see [22]). In the mentioned cases, Lo is the

Cartan subalgebra of the considered algebra. Notice that the subspaces Lw are not neces

sarily one-dimensional and that an algebra can admit more than one root decomposition.

To the decomposition (3.1), we associate the algebras L + = I, Lw and L- = I, Lw. wER+ WER-

Starting with the algebra L + (or L -) Iet us construct a blow-up of the geometry r(W) by

the following rule.

Let us identify the set of reflections k of the group W (R) with the set R +. Notice

that an element I = I, Iw E R + Uw E Lw) can be considered as a function f from k to wEl:

L +, where f(w) = Iw, S(f) = { w E k I f(w) ;t 0}. Let us consider the set r(L) = f(W) of

pairs (a,f) where o.e r(W), S(f)cß(o.)={we 1: 1/(wo.)</(o.)}. Let r(L) be

equipped with the following incidence relation i:

(a,f) i(ß,g) ~ o./ ß andf- g 1.:\(a)n A(ß) = [f,g] IA(a) n A(ß)· (3.2)

It is clear that f(W) is a blow-up of r(W). Notice that the set ß(o.), o. e r(W), is a closed set of roots in 1: = R + and that [f ,g] in the relation (3.2) vanishes outside ß( o.) n ß(ß)

(see [4]).

Remark. The blow-up r(N,K) of (N,ct>,t) defined in Example 2.1 (which is isomorphic to the projective geometry PGn(K) by Proposition 2.3), is the geometry r(L) for the simple Lie algebra L of type An-l over K. The latter is the algebra of n x n matrices over the field K with zero trace and multiplication [A,B] = A x B - B x A. In fact, the Weyl group for An-l is the symmetric group S(Q), I Q I = n, with the reflections (i,j)' i e n' j e n (or with the positive roots ei- ej' 1 < j < i < n of the root system An_1). The set S of transpositions (1,2), ... , (n -1,n) corresponds to the set of simple roots. An element A from N is identified with the set of cosets of Sn over the subgroup <S- {s} > for some s e S (see Example 2.4). The set ß(A) can be considered as the set of transpositions which decrease the length of A as a coset If we consider the Lie algebra of type An-l then L + and L- aretriangular rnatrices with zerodiagonal and nonzero elements below and above the diagonal, respectively. This means that an element (A,U)

from r(L) , U e L +, is identified with the element of PG n-l (K) for which A is a Gaussian set and U x E is a Gaussian matrix.

From consideration of the standard decompositions (3.1) for the algebras (for the p

Lie algebras) of the algebraic sernisimple groups, one can obtain a geometric interpretation for these groups. Let us consider in some detail the Chevalley groups of normal type.

Let L be a simple Lie algebra over the field q; with the Chevalley basis hr , r e ll, er, r e R, where R = R+ uR- is a root system. Let us consider the abelian group Lz of all integer linear combinations of the basis elements. The Chevalley basis is organized in such a way that Lz is closed under Lie multiplication. Let K be an arbitrary field. If we consider the tensor product of the additive group of K and Lz then we obtain

LK = K ® Lz. Let 1K be the identity element of K. Then each element of LK can be written in the form

I, Ar(lk®hr)+ I, llr(lk®er) re n reR

where Ar, llr e K and ll is the set of simple roots of R. Put hr = 1K ® hro er= 1K ®er. If we define on LK the Lie multiplication by the rule [lK ®x, 1K®y] = eK ® [x,y], then LK becomes a Lie algebra.

Let us define on LK a K-analog Xr(t) of the automorphism Xr(t) of the algebra L

which maps z onto exp[er A., z ]. It can be checked that Xr(t) is an automorphism of LK.

482 V. A. USTIMENKO

The Chevalley group corresponding to LK is defined by G = <XrO•) I A. E K, r E R >. This group has a BN-pair with B = Nc(U) where U = <Xr(A.) I A. E K, r E R+> and N = <wa(t) =Xa(t)X_a(t-1)xa(t) I CXE R+, t E K>. The subgroup W = <w a(l), a E R+> of the group N is isomorphic to the Weyl group of the root system R. The mapping which associates to w a(t) the element w a(l) is called the canonical homomorphism of N onto W. The group G has a split BN-pair and B can be presented as a semidirect product UTwhere T = <ha(t) =wa(t) Wa(t)-1 I CXE R, t E K> is the subgroup known as the maximal torus of G. The subgroup Xr generated by the element Xr(t)

is isomorphic to the additive group of the field K.

A subgroup 'I' of R is called a closed set of roots if, whenever r,s E 'I' with ir + js E R for integers i,j, we have ir + js e 'I'· Let us consider the case when R + is

presented as a disjoint union of two closed subsets: R+ = '1'1 u '1'2 where '1'1 n '1'2 = 0

and '1'1> '1'2 are closed. Put U1 = TI Xn and U2 = TI Xr. r E 'l't r E '1'2

Lemma 3.1 ([4], p. 114). There is a factorization U = U 1 U2 , U 1 n U2 = 1. Moreover an element u e U has a unique representation u = u 1 x u 2 , where ui e Ui, i = 1,2P

For w e W put '1'1 (w) = {r e R+ I w(r) e R+}, 'l'2(w) = {r e R+ I w(r) e R- }.

Lemma 3.2. ([4], p. 114). Let T be the set of ordered decompositions of R+ into two closed subsets. The mapping ll which associates with g e W the decomposition

('1'1 (w), 'lf5 (w)) is a bijection of W onto T. 0

The subgroups <Xr IrE 'l'1(w)> and <Xr IrE 'l'2(w)> are denoted by u:;, and u;., respectively.

Lemma 3.3. For w e W, choose a representative nw of the preimage of w under the canonical homomorphism of N onto W. Then each element g e G has a unique represen

tation of the form

g = b nw u where b e B and u e u;.,.

0

This Iemma can be considered as a specification of the Bruhat decomposition (see [4], p. 115).

Corollary. An arbitrary element g e G possesses a unique representation of the form g = u 1 h nw u, where u 1 E U, h E T ' w E T ' u E u-;;.,. 0

Let a 1, a 2 , ... , CXJ be a basis of the root system R (an ordered set of simple roots). Then an arbitrary positive root can be written as a linear combination of ai with nonnegative coefficients. On the set of positive roots :E, let us introduce an ordering by the rule: r 1 > r 2 if the first nonzero coefficient of r 1 - r 2 is positive.

Lemma 3.4. (see Theorem 5.33 (ii) in [4]). Each element of the group U has a unique representation of the form rr ri(t ), ri E R + where the roots in the product increase with respect to the aforementioned ordering. 0

Theorem 3.5. Let G be the Chevalley group corresponding to the Lie algebra LK. Then the geometries r(G) and r(LK) are isomorphic.

Proof. Let (W,S) be a Coxeter system. To the geometry r(W) let us associate the incidence system rF(W). The latter consists of sets of cosets of W over subgroups <1>, 1 c S, as its objects, has type function t : rF --+ 25 defined by t(g <1 >) = 1, and has incidence relation I' defined by the rule: a/' ß c:=:=> a n ß '*- 0. It is clear that r(W) c rF(W) and that the restriction of I' to r(W) coincides with I. The elements of rF(W) are identified with the flags of r(W). Recall that the incidence relation /p on the flags is defined tobe the Subset {(a,ß) I Cl.::> ß V ß ::::> a} ofthe set rF(W) X rF(W). It is clear that al' ß implies alp ß.

Let I. be the set ofreflections of (W,S) (i.e. elements conjugate to elements of S). To each element a e rF(W) we associate the triple T(a) = (A+(a), O(a), A-(a)), where O(a) =I.- A+(a) u A-(a). Let n be the set of ordered partitions T(a) of the set I. for all Cl. E rF(W), nJ = {T(a) I a E rF(W) and t(a) = J} for allJ c s.

The following Iemma is analogaus to Lemma 3.2.

Lemma 3.6. The mapping J.1.: rF(W) --+0 which associates to a the triple (A+(a), O(a), A-(a)) is a bijection. D

Now we assume that W is the Weyl group of the root system R and that S is the set of reflections in the hyperplanes orthogonal to the simple roots. We further identify I. with the set R + of positive roots. The proof of the following Iemma is analogaus to that of Lemma 3.2.

Lemma 3.7. Let Cl. E rF(W) and t(a) = 1. Then

(a) O(a) is a root system with Weyl group isomorphic to W1 = <1>;

(b) the sets A±(a) and A±(a) u O(a) are closed systems ofroots. D

Notice that if 1 =S then the sets A+(1x) and A-(a) are identified with 'I'I (a) and 'l'2(a). Fora e rF(W), let U(a)± denote the groups <Xr I r e A±(a)>. The following Iemma is analogaus to Lemma 3.3.

Lemma 3.8. Let G be a Chevalley group and P J = B W1 B be its standard parabolic subgroup. Then an arbitrary element of the coset g P 1 can be presented in the form u- w h u where w is an element of W such that B g P 1 = B w P 1 , h e H , u e U, and uis a uniquely determined element from u-(a). D

Corollary. Let r be the set of pairs (a,u-) where Cl. E rF(W)' u- E u-(a). The mapping ywhich assigns to an element g PJ the pair (w WJ, u-), where wand u- are the elements defined in Lemma 3.8, is a bijection of r(G) onto r. o

Let rF(G) be the set of left cosets of G over the standardparaballe subgroups (the set of flags of r(G)) with type function tc: rF(G)--+ 25 defined by t(g B w1 B) = 1 and incidence relation defined by alc P c:=:=> a n ß = 0. It is known that the set F B of elements ofrF(G) which are incident toB, along with the restrictions of lc and t to this set,

484 V. A. USTIMENKO

is isomorphic to the system rF(W). Each orbit of the group B acting on rF(G) contains a unique element from FB (this fact follows from Lemma 3.3). The mapping p of rF(G)

onto rF(W) (constructed from an arbitrary orbit <1> of the group) which is identical on <1> 11 FB is said tobe a retract with center B. It is known that p is a morphism of rF(G) onto (FB,l,t). It is clear that 'Y(p(A)) = (o.,e) where e is the unit of u~. If we identify the elements (a.,e) and a., then y o p can be considered as a morphism of rF(G) into rF (W) (rF (W) c r).

Let o. and 13 be two elements of rF(W) with o.;:::) 13 (in particular o./p l3). Then ~(o.) c ~(13) (see [3]) and Ua. and the element A from rF(G) suchthat 'Y(p(A)) = o.. Let us define r 13 (A) as the set of allX e rF(G) suchthat )'(p(X)) = 13 andX Ic A.

Lemma 3.9. The group Ua.-ll acts regularly on each set r 11 (A). 0

Let A 1 and A 2 be incident elements of the geometry r( G ). Let us consider the element A = A 1 11 A2 of the geometry of flags of G. Let )'(Ai)= (o.i, Vi), i = 1,2, and 'Y(A) = (o.,h). It is clear that o.i = )'(p(Ai)). Since y o p is a morphism we have

(3.3)

It follows from Lemma 3.8 that the elements A 1 and A 2 satisfying (3.3) are incident if and only if there existx and y from ua.,-a., i = 1,2, which satisfy

(3.4)

Let us equip the subset r' = { ( o., u) e r I o. e r(W)} of r with the structure of a geometry. Two elements (o.1 ,V 1) and (o.z, V 2) will be incident in this geometry if the relations (3.3) and (3.4) are satisfied and t(o.1 ,V 1) = t(o.1 ). Then the restriction y0 of the mapping y to r(G) is an isomorphism of r(G) onto r'.

Let us identify the subset (o.,O) of r(Lk) with r(W). Let v : r' ~ r(Lk) be the mapping which assigns to (o.,xr, (tt), ... , XrN(tN)) the element (o.,f) where f(ri) = t 1. (Here

{r1, rz, .. . , rN} = ~(o.) and the product is taken over an increasing sequence of roots.) By Lemma 3.4 v is a bijection. To prove the theorem it is sufficient to checkthat (3.4) is equivalent to the linear relation (3.1) for v (V 1) and v (V z). Indeed, if so, y0 o v will be an isomorphism of r(G) and r(L +).

We will prove the equivalence of (3.1) and (3.4) for the case of the root system B 2· The root system B 2 contains the following positive roots: E2- E1 = ~Z-1> ~1+ 1 = 2E1, Ez + E1 = ~2+1, 2Ez = ~2+2• where E1, E2 is an orthonormal basis of the space IR 2. The roots ~2-1 and ~1+1 are simple. Let us fix the basis ~ 1+ 1 , ~2-1 and define the order relation > on the set of simple roots .Then ~2+2 > ~2+ 1 > ~2- 1 > ~1+ 1 . The Weyl group B 2 is isomorphic to the dihedral group D 8 • As above, we make a distinction between reflections and the corresponding positive roots. The set { 1,2,3,4} of points is the set of cosets of B 2 over the subgroup <1, ~1+ 1 >, while the set of edges of the quadrangle (see Fig. 3.1) is

ON SOME PROPER TIES OF GEOMETRIES OF CHEV ALLEY GROUPS AND THEIR GENERALIZA TIONS 485

identified with the cosets of B 2 over the subgroup <1, ~2-1 >. It follows that Ll( 1) = 0, L1(2) = {~2-d. A(3) = {~2-1> ~2+1}, A(4) = {~2-1> ~1+d· Analogously, for the lines (edges), we have L1(12)=0, L1(23)={~2-1>~2+2}, il(34)={~2+2·~2+3•~1+1},

Ll(14) = {~1+1 }.

1 2 o-----o

o-----o

4 3

Fig. 3.1.

The elements :X~ (t) where ~ e B 2 will be interpreted as the following matrices of order 4.

1 0 0 0 1 0 0 0 t 1 0 0 0 1 0 0

:x~2-l (t) = 0 0 1 -t ; :x~l+l (t) = 0 0 1 0 0 0 0 1 0 t 0 1

1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0

:x~2+1 (t) = 0 t 1 0 ; :x~M (r) = t 0 1 0 t 0 0 1 0 0 0 1

An arbitrary element of the group u can be written in the form

X~l+l (a) X~2-l (ß) X~u1 (y) X1;2+2 (S).

Let us study the incidence condition on parabolies of "general position" (representations of the orbits of the Borel subgroup having the largest size). An element ä = (3,c ~2-1 +b ~2+ 1 +a ~2+2) from r(L) (i.e. a function f e Kr. written as the formal sum L f( w) w) corresponds to a coset a from r( G) whose canonical representation in

wE!.

U(3) is equal to

1 0 0 0 c 1 0 0 ab 1 -c =x~2-l(c)x~2+1(b)x~;2•2(a).

b 0 0 1

A line ß=([34],a~2+2+ß~2+1 +y~1+1) corresponds to a parabolic ß' with unipotent

486

representative

1 0 0 0 0 1 0 0

(X ~ 1 0

~ 'Y 0 1

V. A. USTIMENKO

Let <i and ~ be incident. Then the intersection of parabolies a' and ~, contains a par

abolic V which corresponds to flag '9 = { <i, ~}.

Since y/ <i in the flag incidence system, the unipotent representative x of the para

bolic y can be written in the form

1 0 0 0 1 0 0 0 1 0 0 0 c 1 0 0 0 1 0 0 c 1 0 0 ab 1 -c X 0 0 1 0 = ab -ct 1 -c

b 0 0 1 0 t 0 1 b t 0 1

for some t. It follows from V I ~ that

1 0 0 0 0 1 0 0

x= (X ~ 1 0

~ 'Y 0 1

1 0 0 0

11 1 0 0

a+~11 ~ 1 11 ~+y 11 0 1

From the matrix equality

1 0 0 0 c 1 0 0 ab-ct1-c

b t 0 1

X

1 0 0 0 11 1 0 0 0 0 1 -11 = 0 0 0 1

1 0 0 0 11 1 0 0

= a+ßl1 ~ 1 -11 ~+i'J.I. 'Y 0 1

we find the values of the free variables: 11 = c and t = y. The equality b - ct = ~ is

equivalent to

(3.5)

In addition we have


a = a+ ßc. (3.6)

The relations (3.5) and (3.6) are equivalent to the equality

[a~2+2 +b ~2+1 + c ~2-1> a~2+2 + ß~2+1 +Y~l+d IA(3)nA(34) = (b-ß)~2+1 + (a -a)~2+2

in the Lie algebraT of formal combinations of the form a ~2+2 + b ~2+1 + c ~1+1 + d ~2-1 with the bilinear multiplication ® defined as follows: ~2-1 ® ~ = ± ~2+ I>

~2-1 ® ~2-1 = ± ~2+2; the products of all other generators equal to zero. It is clear that T is isomorphic to L! for the case B 2• D

Remark. The geometries of the simple Chevalley groups 'tK of twisted type over K are blow-up of the geometry r(L) of the root decompositions of a certain Lie algebra. This blow-up is linear over some subfield k over K.

Let us describe a blow-up f of the geometry r(Bm) which is isomorphic to r(A~(K)).

Example 3.10. Let K be a quadratic ex:tension of a field k and w be the involutory automorphism of K which fixes each element from k. Let us define a mirror blow-up of the geometry of the group B2m. The set l: of reflections of the Weyl group B2m will be identified with the set of positive roots of the system B 2m· This set consists of the vectors ei- ej, i > j, ei + ej, i > j, and ei where e 1, ... , e2m is a Standardbasis in IR 2m (see [3]). Let us consider in the space Kr. the abelian subgroup U ={je Kr. I f(ei) e k}. Let W be the set of pairs (A,B) of matrices of order m x m over K, where A is a triangular matrix with nonzero elements under the diagonal and B is a Hermitian matrix (i.e.

B = (bij) where BT =Bw = (bij)). Such a pair will be denoted by [~]·Let us transform

W into an abelian group:

[Al [A '] [ A + A '] B + B' = B +B' ·

We define multiplication o in W by the following:

[~]0[~]=[~]. where Y=A·C-C·A, X=BC-(cwr1 B+A-1 D-DA. Notice that [ ~] can be

identified with the matrix: [ ; (A ~ )w] of size 2m x 2m. In this case o can be considered

as the commutator of the corresponding matrices. Let us consider the isomorphism

4> : U ~ W which associates to a function f from U the element [ ~] where

aij = f(ei -ej), bij = f(ei +ej) for i > j and bkk = f(ek). It is clear that * = 4> o 4>-1

488 V. A. USTIMENKO

determines a distributive multiplication in U. Let a = (B,f) where A c n = {1,2, ... ,m}, be an element ofthe group B2m,A = {x e B I f(x) =0} (see Example 2.2). Then the subset n(a)!:: I: which consists of the mappings w such that /(wo.) < /(a) is the set of roots ei- ei where i e A, je: A, ei+ej, ei where i e B-A, je A. Since * is abilinear mapping of the vector space U over k, the blow-up ci» = ci»(I:, U, *, 11) of the incidence relation <!» of the geometry r(B 2m) is linear over IR.

Notice that in the case k = fC, the algebra (U, *) is isomorphic to L + for the Lie algebra of the complex Hermitian matrices (Ä =AT) over IR. The relation <!» can be obtained by twice applying the blow-up Operation to the incidence system r(Am-1) (see Example 2.2).

4. Other possibilities

Let us consider a generalization of the algebra r(L) where it is assumed that a decomposition (3.1) of L exists.

Let (W,S) be an arbitrary Coxeter system. A blow-up of r(W), which is linear over a field D, will be called a mirror blow-up if

(a) I: is the set of refl.ections of W (the set of elements conjugate to generators from S);

(b) n(a)iseithertheseta+(a)={we I: 1/(wa)<l(a)whereae r(W)}ortheset a-(a)={we I: ll(wa)>l(a)};

(c) u = l: Mw(D) where Mw(D) is a vector space over the skew field D. The elewel:

ment u is considered as the function from I: to l: Mw(D) which assigns to w e I: the wel:

element 1 from M w (D) c U. Thus u is contained in the additive group {/ : I: --+ n } .

The incidence system defined above is denoted by r± = r± (W, * ,D ). Here * is the considered bilinear multiplication and the sign of r± is determined by that of a±(a). Notice that the geometries r+ and r are not isomorphic if the group W is infinite. Indeed, in this case a + ( a) is finite while a-( a) is infinite. In the case of the geometry L defined by a Lie algebra L of a Chevalley group, the exchange of a + ( a) by K ( a) does not lead to a new incidence system. Notice that some interesting geometries and other incidence systems can be obtained as nonlinear blow-up of geometries of the Weyl groups satisfying conditions (a)-(d). In [32] a blow-up r(A 2) (a projective plane) is associated to an arbitrary cartesian group (D, o ,+,*)in this manner. In that paper, a blow-up r(An) is obtained whose points and hyperplanes form a symmetric block design with the

tn+1_1 tn-1 tn-1_1 parameters v = 1 , k = --, Ä = , where t is the order of the cartesian

t- t-1 t-1 group D. Some COnnections between blow-ups of r(B 2) and generalized quadrangles are also studied there.


It is natural to generalize the notion of a mirror blow-up to an arbitrary group G with a set S of generators. In this case reflections are the elements of G which are conjugate with elements from S.

1. The Lie algebras of Kac-Moody type possess a decomposition (3.1) (cf. [13]). The Weyl group W is infinite in this case and some reflections arenot conjugate to generators. Forthis reason the roots are divided into real roots (of the form rK where r is a simple root and g e W) and imaginary roots (all the others). In this case, for the algebra L +, Iet us consider the subspace L' = :I: Lw where Q is the totality of real roots. It is

wen clear that Q can be identified with the set :E of the group W. Let us define the mirror blow-ups P(W) of the geometry r(W) as the set of pairs (a,f) where a e r(W), f e L' and s(f) c ~±(a), where the incidence relation is determined by the equality (3.2).

It can be shown that the geometry r+(L) is isomorphic to the geometry of the RN

pair which is associated with Las in [22].

2. Let M be a set of arbitrary size. Let us consider the "integer lattice", i.e. the set of linear functions from M to :z with finite Support. The elements of r will be written as

formallinear combinations :I: Ai Si• where Si is the function which is 1 at the point i ieM

and 0 elsewhere. Let r+ = { a = :E ki Si I ki ~ 0 , :E ki > 0}. Let us consider the mapping f fromM x M to Z such that

(1) 'V i E M f(i,i) = 2;

(2) f(i,j) ~ 0 for i "# j;

(3) f(i,J) = o ~ !U.i) = o. This function f of two variables will be called the Cartan function. We shall also

consider the Dynkin function, i.e. the mapping a from M x M to Z such that d(i,j) = f(i,j) xfU,i).

Let US consider the linear functions (with finite Support) $i from r to :Z suchthat Ms1) = f(i,J), and the transformations

ri(a)=a-clli(a)si, i e M.

The group Wt generated by 'ti, i e M, will be called the generalized Weyl group. Let us consider the set :E = {ri I i e M, w e W} of "positive roots". Let~ be the set of elements of w1 which are conjugate to elements from S = {ri I i e M}. Then the mapping J.L: ~--+ :E which maps g-1 ri g to sf is a bijection. Forthis reason the sets ~ and :E may be identified.

Starting with an arbitrary field lF Iet us construct a blow-up of the incidence system r(Wt,S). The space V= IFr. of functions from :E to lF having finite support will be identified with the expressions of the form :I: ki ai, ki e Z. Let us equip V with the

a;er.

490

altemating bilinear multiplication o defined as follows:

·o ·-{±A.ij(ai+aj), ifai+ajei. a, a,- 0 if a· + a· ~ I.

' I J

V. A. USTIMENKO

(4.1)

where Aij "# 0 and Aij = -Aji. Let us associate with the coset a = g < S - { s} > the subspace \1 = /Ft:..±(a.) = {f e V I f(x) = 0 for x ~ d(a)}. Put

f± = { (a,f) I a e r(W1,S), f e V a.}, i(a,f) = t(a)

and define the incidence relation i by

(a,f)i(ß,g) ~ (a/ ß) andf- g lt:..(a.)n t:..(~) = f o g lt:..(a.)nt:..(~)· (5)

Remark. If M is finite then Wr is defined by the relations rr = e' (ri X rj)nij = e,

where nij can be calculated from the Dynkin function

Uij 0 1 2 3 ~4

n·· 1) 2 3 4 6 00

The set I. is an infinite system of positive roots in the sense of [14]. The matrix A = (j(i,j)) is the Cartan matrix of this system. If K is a field of characteristic zero then A

determines a Kac-Moody algebra L and, for a suitable choice of Aij in ( 4.1 ), A determines the geometry P(L) introduced in the beginning of the section. So the system r+(L),

defined by means of the relation (4.1) over an arbitrary field, is an analog of the BN-pair

geometry introduced in [22].

Example 4.1. The root system Ä 1, A =~ : 2 ~l In this case d contains the vectors

k a 1 + (k -1) a2 and (k -1) a 1 + az. The sum of two elements from d is not a root. The imaginary roots for the Kac-Moody algebra of type A 1 are the elements (a1 + a 2) k. The dihedral group Doo with the generators a,b is the Weyl group for Ä1• The geometry r(D oo) is the totality of points (which are the cosets over the subgroup ) and the totality of lines (which are the cosets over subgroup <a > ). So r(D 00

) is the ordinary polygon with an infinite number of venices. The set I.(ß) of reftections contains all words of odd length over the alphabet a,b. The sets d+(a1) and d+(a2) for the points a 1 = {(ab) 1 , (ab)1 a} and a 2 = {b(abt, b(ab)s a} are equal to {a, aba, ... , (ab)s-1 aba(ba)s-1 } and {b, bab, ... , b(ab)s-1 aba(bay-1 b}, repectively.

Let IF be an arbitrary field. In order to construct the blow-up P(A, /F) in this case, let us consider the vector space V= /Fr. with the multiplication f o g = 0 for f,g e V. This means that a point (a,f) and a line (ß,g) (where f,g e V, S(f) c d±(a), S(g) c d±(ß)) are incident if and only if a and ß are incident and the functions fand g

coincide on d(a) n d(ß).

Theorem 4.2. Let A =~ : 2 ~~~ and let IF be a field. Then the geometry r(A, IF) is

flag-transitive.

Proof. A direct check shows that the covering r(A, IF) of the geometry r(D oo) is smooth (as the distinguished type we can take either points or lines). The elements of the group V* = {/ E V} acting by the rule (a,g I= (a.,g + fiA(a)) preserve the incidence relation. It is clear that, by suitable transformations from V*, the elements (a,f), (ß,g) can be carried to (a.,O), (ß,O). 0

Let ri, i = 1,2, be the sets ofpoints and lines ofthe geometry r(Doo), respectively, and let fi be the sets of points and lines of the geometry P(A, IF). Let us consider the space L = {f: r 1 ~ IF I S(j) < oo} and the set P(L) of lines of L. To an element (a.,f) from the set f 1, let us associate the function g from L, where

{1 ,ifx=a.,

g(x) = f(w) , if x = a.w

0 , otherwise.

and w E ~±(a.) ,

Let p a,f be the line from P (L) which contains the vector g. The mapping 'I': (a.,f) ~Pa.! is a bijection of f 1 onto P(L). In fact, all elements from r have different length /(a.); if a.1 and a.2 sarisfies /(a.1) > /(a.2) then there is a unique reflection w

suchthat w a.1 = a.2 • To each element g of Doo, we associate a linear transformation g of the vector space L. This transformation g maps f(x) to f(gx) and induces a transformation g of the space P(L). The rule g ~ g ='I' g '1'-1 determines an action of the group D oo on r 1. It can be checked that an element of (D oo,r I) maps the set of points from f 1

incident to a fixed line onto the set of points incident to some other line. This enables one to define the action of the Weyl group Doo on r2 and on f2 u f1o which preserves the incidence relation. The subset T = { (a.,O) I a. E r(D oo)} is invariant under the action of the group (Doo,f2 u ft) and the group (Doo,T) is similar to (Doo,r(Doo)). This means that an arbitrary pair of incident elements (a,O) (a point) and (ß,O) (a line) can be

transformed to the pair ((a),O), ((b),O). So the group <V,Doo> acts flag transitively on

fl u f2.

Let us consider a generalization ofExample 4.1.

Let Wn be the Coxeter group with ge'lerators a 1 , ••• , On and defining relations o'f = e , i = 1 , ... , n (i.e. a free product of groups 7k 2). Let us consider the set 1: of reflections consisting of the elements g ai g-1 , i = 1 , ... , n, g E Wn, and the totality r(W) of left cosets of W n over the subgroups <S - { oi } > , S = { o 1 , ••. , On } .

Let N be an arbitrary set with the distinguished element 0. Let us define a blow-up P(WmN) of the geometry r(Wn). This blow-up consists of the pairs (a.,f), where jis a function with finite support s(j) ~~±(/).Elements (a.,f) and (ß,g) of different type will

be incident if a./ ß and f IA±(a)n A•(ß) = g IA(a)n A(ß)· It can be shown by induction on n

492 V. A. USTIMENKO

that r(Wn,N) sarisfies the axioms of a Tits geometry. In this case the rank 2 residues are generalized polygons with an infinite number of vertices. Notice that if the structure of an abelian group is defined on N and the distinguished element is the zero element of the group, then P(Wn, /F) can be considered as a smooth blow-up (it is assumed that f(x) o g(x) = 0 everywhere). Now suppose that the structure of a field is defined on N. Then, by arguments similar to that in the proof of Theorem 4.2, it can be shown that P(W, /F) is ftag-transitive.

The simplest example of a geometry r(A, /F), for the case of an infinite Weyl group and nonzero multiplication in IF l:, can be obtained from the root system Ä 2 (i.e. from the affine Weyl group for the diagram Az). The set of positive roots of Az consists of the vectors k a.i 1 + (k + 1) ai 2 + (k + 1) ai 3 , k "2:. 0, {i 1oi z,i 3} = { 1,2,3 }, where a.l> a.z, a.3 are the simple roots. The irnaginary roots are the roots which are multiples of the root a.1 + a.z + a.3 .

Let us consider the blow-up of an incidence system of infinite rank. Let n be a set and let (~2 • 0) be the symmetric inclusion relation on the subsets of n. Let us fix a strong linear order < on n. Put 1: = { (i,j) E n X n I i > j} and let 11 : 2n ~ 21: be such that 11(M)={(i,j)el:lieM,je:M}. Let K be a field and put U = {/ e K l: I supp f < oo}. Let us consider the functions eij from U which are equal to 1 on (i,j) e A and zero elsewhere. The bilinear altemating multiplication * on the left vector space U will be defined by the relation

{±eij, if j=l;

eij * elk = 0 , if j * l , j "2:. I.

lt is easy to check that the covering ci>(l:, U, 11, *) is smooth. The set {j I f e U} = U is closed under Superposition. Let us define the action of the group S(Q) of bijections of n with finite support on the set 20..

The group S(O) acts on the vectors of the space V= {g e Kn I supp f < oo} by the rule g 11 (x) = g (x 11 ) w here 1t e S (Q) , g e V, and permutes the elements of the set P (V) of

lines of the space V. The set 1 1 of elements ({i}, f) (the "lines" of in) will be identified with P (V). Namely ( {i}, f) will correspond to the line containing the vector g such that g(i) = 1 and

. { f(i,j) ' if j I.

The group (S(Q), 1 1) maps the set 1 1 of lines incident to (A,f) onto the set of lines

which are incident to some other element. Thus S(O) permutes the elements of in and preserves the relation ci>.

Let us consider the group G(O) generated by the elements of U and S(Q). It is clear that G(O) is contained in Aut(~). Let F(Q) be the set of nonempty subsets in 2n and ~'

ON SOME PROPERTJES OF GEOMETRIES OF CHEVALLEY GROUPS AND THEIR GENERALIZATIONS 493

be the restriction of to F(Q). Then F(Q) is an invariant block in S (Q) and G (Q) preserves the relation ci>'=ci>'(I:,U,tl,*) on the set F(Q). Let us turn F(Q) into an incidence system by putting t((o.,f)) = I a I where a e F(Q), (o.,f) e F(Q).

It can be seen from the construction of G(Q) and of the incidence system r = (ci>, F(fl), t) that

(a) G(Q) acts transitively on the set of finite flags of a fixed type;

(b) if I Q I = n is finite, then G(Q) = GLn(K) and r is the ordinary projective geometry (see Example 2.1 );

(c) if n is countable then G(Q) is the general linear group GL(K) (the natural inductive Iimit of the groups GLn (K) [2]).

In an analogaus way, generalizations of the geometries of other classical groups can be obtained.

Another way to obtain analogs of the geometries of classical groups is considered in Example 2.5.

Notice that if Q is countable then the incidence system r coincides with the geometry P(j,K), where fis the Cartan function on n satisfying f(i,i) = 2,f(i,j) = -1 if I i- j I = 1, andf(i,j) = 0 otherwise.

5. Some linear algebra problems for RN-pairs

With an incidence system (r,I,t) over the set of types ~. we associate the incidence structures (rs u rt,J)' t,s E A. In studying these systems the following problems arise naturally:

(I) Let a and b be elements of r such that t(a) = s, t(b) = t. Decide whether or not they are incident.

(Ln) Let the elements a ~oa2, ... , an of type s be given. Does there exist an element x of type t which is incidence to all of them?

The complexity of these problems (in the ordinary sense, see [11]) depends on the way in which the system (r,I,t) is described. Let us assume that the geometry of a Chevalley group is described as a linear mirrar blow-up of the geometry of its Weyl group as defined in Section 4. It can be deduced from the proof of the theorem that this description means the following. The cosets over the standard parabolic subgroups are described in terms of the canonical representatives from u-. In this case it can be proved that the complexity of the problems I and Ln is realized for elements of "general position", i.e. if a,b,ai,x are in the maximal orbit of the standard Borel subgroup containing u-. This means that /(r(a)), /(r(b)), /(r(ai)), l(r(x)) have the maximal value (Condition 4.1), and r (a) Ir (b) and r (x) Ir (ai) (Condition 4.2).

494 V. A. USTIMENKO

Theorem 3.5 enables one to solve the problems (/) and (Ln) by means of the methods of linear algebra if 4.2 is satisfied. In fact, let (r,i) be a linear blow-up of a graph (i.e. of a binary relation) (r,J) and let r be a retract from f to r. Then the condition for the elements a = (a,X) and b = (ß,Y) tobe incident when a/ ß can be written in the

form x-y ITJ(cx) n TJ(ß) = x * y ITJ(cx) n TJ(ß). Here r(a) = a, r (b) = ß and * is the bil~ear multiplication which determines the blow-up. The condition for an element y = (a,* ), with the given retract a, to be incident to elements 'ä; = (ß;,a;) can be written as a system of k linear equations:

X- ~i ITJ(cx)n TJ(ß) =X X ~i ITJ(cx)n TJ(ß)· (5.1)

Here k = l11(a) r1 'll(ß) I x n.

The known results (see [11]) conceming the estimation of the complexity of certain problems in linear algebra enables one to estimate the complexity of the problems I and Ln for the geometries of the Chevalley groups.

Example 5.1. Let us consider the geometry r of the Chevalley group An-1 (q) and the incidence structure rs,t where s and t correspond to the terminal nodes of the Dynkin diagram. The points and the hyperplanes of "general position" are the elements a and b with r(a) = n, r(b) = {2, ... ,n}, whose Gaussian bases are

1 1

0 1 0

1 and ß2.1 1 0

ßn,1 0 Xn, 1 ''' Xn,n-1

respectively. These elements are incident if

Xn,n-1 - ßn-1,n = Xn, 2 ß2. 1 + Xn, 3 ß3,1 + · · ' + Xn,n-1 ßn-1,1 ·

The problern Ln_1 for the elements x and b; = (r(b), ß~ 1 ~21 + · · · + ß~-11 ~n-n) is equivalent to the solvability problern of the following system of linear equations.

- (ß1 ßn-1 ) - n-1,n' · · · ' n-l.n ·

1 1 1

ß~-1.1 ßL1.1

ßn-1 2,1

ßn-1 n-1,1

=

Example 5.2. The blow-up of the symmetric inclusion relation on the intervals of the segment [0, 1], considered in Example 2.5, is linear over the field IR. The problern Ln for the elements x = ([1,a], u(x,y)) and ai = ([l,b], gi(x,y)) is equivalent to the

solvability with respect to u(x,y) of the following system of equations:

1

J (u(x,z) gi(z,y)- gi(x,z) u(z,y))dz = 0

= u(x,y)- gi(x,y) I[O,b]x[O,a]• i = 1,2, .•• ,n.

Let a = (a,i') be an element of (f,I) and ß be an element of r incident to r(a). Since the set V ß (a) = { (ß,x) E f I (ß,x) i a} is determined by a system of equations of the form 4.1 for n = 1, a vector space structure is defined on V ß (a ), and we can speak about linearly independent elements of V ß (a ).

Lemma 5.3. Let r(G) be the geometry of a Chevalley group Gof normal type over the field K with Weyl group W. Let a E r(G), ß E r(W), r(a)/ß. Then the space V ß(a) is isomorphic to K.1.(ß)--.1.(ß)n.1.(a).

To prove this lemma it is sufficient to check that, in the considered situation, the system 5.1 is in canonical form and that the values of the function x E Kr. (corresponding to (ß,x)) restricted to the set ~(ß)- ~(ß) n ~( a) can be considered as free parameters. In this case all other coordinates of x (i.e. the values of x at the others points of I:) are uniquely determined. 0

To solve problems (/) and (Ln) in the general case, for the geometries of the Chevalley groups under the present interpretation, it is necessary to solve these problems for the geometries of the corresponding Weyl groups. Let us consider an interpretation of the geometry of the Coxeter group (W,S) which is useful for the study of the mirror blow-ups ofr(W).

With an element a from r(W) (i.e. with a coset of W with respect to the subgroup <S - { s } >, s E W), let us associate the sets ~ + ( a) and ~- ( a). It is clear that the elements ~+(a), O(a) =I:- ~+(a) u ~-(a), ~-(a) determine a partition of I: of the form t = (t',t",t'") = (~+(a), O(a), ~-(a)) for all a E r(W). Let 'V denote the mapping from r(W) to r which maps a to (~+(a), O(a), A-(a)). We will write t 1 V t 2 if t 1, t 2 E T and

p (t 1• t2) = I t 1' n t 2"' I + I t 1 "' n t 2' I = 0. Let us consider an equivalence relation ~ on r(W) x r(W) defined as following:

Lemma 5.4. Let (W,S) be a Coxeter system and r(W), T, V, 'P be the objects defined above. Then

(a) the mapping 'V is a bijective morphism of (r,/) onto (T, V');

(b) the group W acting on r(W) x r(W) preserves the relation ~- 0

Corollary 5.5. To solve the problern (/) for the incidence system r(W) ((T, V')) at most I I: I operations of comparing elements are required. 0

496 V. A. USTIMENKO

Remark 5.6. Lemma 5.4. is still valid if, instead of r(W), we consider the set of flags r F(W) with the following incidence relation: o./ ß if and only if the cosets o. and ß with respect to the subgroups and <1> have a nonernpty intersection.

Remark 5.7. Lemma 5.4 and its generalization rnentioned above arevalid for the generalized Weyl groups considered in Section 4.

Remark 5.8. It is known that the binary relations of the group W over <S- {s} > = p, s e S, are in a one-to-one correspondence with the double cosets of the group W over P. The graph corresponding to the class P s p is called the Coxeter graph.

Corollary 5.9. Two elernents o. and ß frorn W I <S- {s} > are joined by an edge in the Coxeter graph if and only if p(ljl(o.), 'ljf(ß)) = 1.

Let G be a group and S be a system of generators of G. Let us consider a blow-up f of the incidence system r(G,S). With the incidence structure rss' = {x E r I t(x) E {s,s'}} let us associate the function llss'(t) which determines the cornplexity of the problern (/) under the restriction max(/(r(a)), /(r(b ))) < t (the problern Ir).

Lemma 5.10. For the incidence structures of the system P(j,K) defined by a Cartan function f over the field K, the problern Ir is solvable.

To prove the lernma, it is sufficient to notice that the problern (fr) is solvable for r(Wr) and that the space Kll(a)ull(b) has a finite dimension. D

Let us consider the problern Ir for the incidence structures rss' of a system r = P(A,K) suchthat the subgroups <S- {s }> of W(A) with systern of generators S are finite. We assume also that an elernents from W(A) is deterrnined by the pair ß+(o.), O(o.). The problern can be solved by an algorithrn which checks the condition p('ljf(r(a)), 'ljf(r(b ))) = 0. The latter is equivalent to the relo~:~ms

ß+(r(a)) n O(r(a)) :::> ß+(r(b)),

ß+(r(b)) n O(r(b)) :::> ß+(r(a)).

After that, we check condition 4.1 for n = 1. Observe, for the geometry r(WA), that the value llss' (t) can be bounded by the function 2(t + ~) where ~ is maximurn for the nurober of reflections of the Coxeter system.

For a Coxeter systern (W,S), let us consider the value ~(W) which is the rnaximum over s of the numbers of reftections in the Coxeter systems ( <S - { s } , { S} - { s } > ).

Lemma 5.11. Let (W,S) be a Coxeter system for which ~ = ~(W) < oo. Then

(a) the function llss'(t) for the geometry r(W) does not exceed 2t + ~;

(b) the function llss'(t) for an arbitrary mirror cover P(W) does not exceed 12t3 + 4t + 2~.

Proof. Claim (a) follows directly frorn relations 4.2. Let (o.,f) and (ß,g) be ele

ments of r+ (W). Verification of the condition f- g lll(a) r. ll(ß) = f * g lll(a) r. ll(ß) requires

at most 3(2t)2 x 2t + 2t operations. In fact, the dimension of the space K.d(a)uA(ß) is at

most 2t. So to calculate the value of the bilinear form f * g(w) at a point w, it suffices to perform 2 x (2t)2 multiplications and (2t)2 additions. Since IA(a) n A(ß) I < t, to calculate the value on the right side of the relation, it is sufficient to do at most 12t3 operations. 0

Foreach concrete system r(A + ,K) with ~(W) < oo, one can obtain a stricter bound. It is clear that J.l.ss•(t) does not depend on Kandis determined by the Cartan matrix (i.e. by the additive structure of the positive roots). The set of functions Yss•(t) which determine the multiplicative complexity (i.e. number of multiplications) of problern Ir for f ss',

will be called the diagram of complexity for P(W,K). If W is finite, we put

Yss' = max Yss•(t). The values of y for generalized m-gons are 2,4, 7,0 form= 3,4,6,oo, respectively.

6. On the Hecke algebras of finite BN-pairs and the Tits generating algebras

Let G be a finite group with a BN-pair with Weyl group W. Let us consider the Coxeter system (W,S), S = { w b ... , Wn}, for W. Put W1 = <S- J > where J c S (it is assumed that Ws = <e >) and Iet P = B W1 B be the parabolic subgroups of G. The Hecke algebra H(G, B W1 B) of the group G is the V-ring (of Schur) of the permutation group G1 corresponding to the action of G on the left cosets over the subgroup P. The Bruhat

decomposition establishes a one-to-one correspondence between the sets G1 and w1 .

Here G1 and W 1 are the sets of cosets of the groups G and W over the subgroups B W1 B and Wb respectively (see [3]). It is known that the binary orbits of the permutation group G1 (i.e. the generators of H(G,B W1 B)) are in a one-to-one correspondence with the elements of G1 and hence with the elements of W1 (see [6]). In particular the generators ag

of the algebra H(G,B) are indexed by the elements g of the Weyl group W. The following proposition is well known.

Theorem. (Iwahori-Matsumoto; see [6]):

Let ag, g e W, be a basis of H(G,B). Then

{ aw;g , if /(wig)";;?: l(g);

a · a = . W; g Qi" aw;g , tf l(Wjg) < l(g) (*)

for all g e W and wi e S, where ai = indB wi = IBn B w; I. Moreover, H(G,B) can be considered to be the algebra having generators aw; , 1 S: i S: n, and relations

a~. = Qi 1 + (qi -1) aw. (the quadratic relation), aw aw. aw . ... = ... aw. aw. aw (the I I I J I ) 1 J

homogenity relation). 0

Example. Let W = S3. Then H(G,B) has two generators aw 1 and aw2 and is deter

mined by the relations (aw.)2 = Qi 1 + (qi -1) aw., i = 1,2, and aw n aw = n n n 0 I I 1 -w2 } -wz -w} -w2•

498 V. A. USTIMENKO

The parameters Qi and Qj are equal if wi and Wj are conjugate in W. If we put all Qi

equal to 1 then we obtain the group algebra 7l [W] of the group W. In order to study the relations between the Hecke algebra and the group algebra 7l [W], J. Tits introduced the notion of the generating algebra Aw of a finite Coxeter group W defined as follows.

Let R = Q [x 1 , ... , Xn] be the ring of the polynornials over the rational number field Q in the variables Xi which correspond to the elements Wi. Suppose that Xi = Xj if Wi

and Wj are conjugate in W. This implies that for an irreducible Coxeter group the ring R can have only one or two indeterminates. The generaring algebra Aw is the algebra over R with basis { aw) , w E W, and with multiplication determined by the relations

{

aw.w , if /(wi w) ~ /(w);

aw,w= Xi .. aw,w+(Xi-1)aw, if /(wiw):5:/(w).

The notion of generaring algebra can be generalized if the field Q is replaced by a commutative ring with a unity. The K'-algebra over the ring K' = K[x 1 , ... , Xn] with generators x 1 , ... , Xn and relations Xi =Xj for conjugate wi and Wj, will be denoted by Aw(K).

The structure constants of the generaring algebra Aw(K) are polynomials in x 1 , ... , Xn. Let f: K' ~ K be the homomorphism defined by means of "specialization", i.e. by means of substitution of the elements Qi E K for the variables Xi· This homomorphism maps P(x1, ... , Xn) to the element P(q1, ... , Qn) from K. The K-algebra with the same generators as Aw and structure constants obtained by application of f, is called the special algebra and is denoted by Aw.r(K). Notice that the algebra obtained by means

of the specialization map g : Xi -t Qi, i = 1, ... , n, where the Qi are as in Theorem 5.1, is isomorphic to the adjacency algebra H(G,B,K) of the homogeneous coherent configuration of G over K acting on the cosets of B. In particular Aw• (K) is isomorphic to the Heckealgebra H(G,B).

Theorem 6.2. (On multiplicities [6]). Let { G(q)) be the system of finite Lie groups with irreducible Weyl group W, where W is not isomorphic to the dihedral group of order

16. Then the indices { qd in G (q) are of the form q c, for some integers { c 1 , ... , Cn }. 0

Let W be the Weyl group for G(q) and ci be the constants as in the above theorem. Let us consider the homomorphism p' : K' -t K [x] defined by means of the parametriza

tion Xi =xc', i = 1, ... , n. The K[x]-algebra with generators aw, w E W, whose structure constants can be obtained by means of application of p to the structure constants of Aw(K), is denoted by Aw(K) and is called the monogen generating algebra. The image of Aw(K) under the specialization x = a, a E K, will be denoted by At\r(K). It is clear that At\r(K) =A~(K) wheref(xi) = ac'.

Definition. Let 't = {T;), i E /, be a partition of Wand bi = L ag be an element ge T1

of D, where Dis one of the algebras Aw(K), Aw(K), A"(K). If the module B spanned by the vectors bi, i e /, is a subalgebra of D, then B, with the basis bi, i e /,will be called a block subalgebra and 't will be called a block partition for D.

Proposition 6.3. Let (W,S) be a Coxeter system, W1 = <S -1>. A decomposition of W into double cosets with respect to W1 determines a block subalgebra Aw(l,K) in

Aw(K).

This proposition follows directly from the definition. o The algebra Aw(J, I<) will be c:alled the standard subalgebra of Aw(I<). Let us c:on

sider as well the universal standard algehra Uw(I<), whose basis is the union of the

bases of Aw(J, I<) for all JE 28 , and with multiplic:ation as in Aw(T<).

It is easy to see that the parametrization and specialization mappings map the block subalgebras of Aw(K) into block subalgebras of Aw(K) and Awt(K). The property of being a block subalgebra is inductive (i.e, if B > B' > B", and B' and B" are block subalgebras of B and B", respectively, then B" is a block subalgebra of B). For an arbi

trary specialization g, let us define Awa (J,K) and Uwa (K) analogously to Aw, (K).

The following proposition is a direct consequence of the definitions and the theorem of Iwahori-Matsumoto presented above.

Proposition 6.4. Let W be the Weyl group of a finite BN-pair with the indices Qi, i = 1, ... , n, and let f: K' ~ K be the homomorphism such that f(xi) = Qi· Then Awt(J,K) (resp., Uwa(K)) coincides with the adjacency ring H(G,BW1 B, K) (resp., H(G,K)) of the coherent configuration of the group Ci] (resp., of the group G) acting on the cosets of all parabolic subgroups B W1 B. 0

It is clear that for an arbitrary Coxeter group W and specialization 1 : K' ~ K with l(xi) = 1, i = 1, ... , n, the K-algebras Aw'(K), (resp., Uw'(K)) are the adjacency rings H(W, W1 , K) (resp., H(W,K)) over K of the coherent configurations of the group W acting on the cosets of W1 (resp., on the set of all cosets of W1 for I e 2s).

Notice that the block subalgebras of H(G,B W1 B, K), H(G,K), H(W, W1 ), H(W)

are cellular subrings in the sense of [21].

It follows from the defining relations that the structural constants of Aw(~) and Aw(K), where K is a field of characteristic 0, coincide. For this reason the symbols ~ and K will be omitted below.

Let us consider some problems.

a) The computation problern for the structure constants for the algebras Uw, Aw(l), Uw, Aw(l), H(G, B W1B), H(G), H(W,WJ), H(W) where W is the Weyl group of the BN-pair G.

Notice that a solution to this problern for Uw gives the solution for each of the aforementioned algebras. Let G = G(q) satisfies the hypothesis of the theorem on multiplicities. Then the cases Aw(l) and H(G,B W1 B) (Uw and H(G)) are equivalent since Awa = H(G, B WB) (and Uwa = H(G)).

500 V. A. USTIMENKO

b) The computation problern for the characters of the algebras Aw( {s}), Aw({s }),

H(G,B W (s) B), H(W, W (s)) for s e S.

The algebras mentioned in this section are commutative and semisimple. The characters are just the structure constants with respect to the basis of orthogonal idempotents. Notice that the characters of the adjacency algebra over Q of a coherent configuration which is a metric association scheme can be effectively calculated from the structure constants. For the case G = PSLn(q), G (s} is the classical group acting on the maximal isotropic subspaces. The configurations G (s J and W (s J = (W, W I W (s J) are metric association schemes.

References on the known results conceming problems a) and b), as well as some motivations for these problems can be found in [28].

c) The characterization of block subalgebras of the algebras considered in this sec-tion.

A block subalgebra will be called simple if it does not contain block subalgebras. For the algebras H = H(G, B W (s) B), it is natural to conjecture that the lattice of cellular subalgebras for H corresponds to the lattice of overgroups of G {s) in the symmetric group S(M) of the set M =GIB W {sl B. The latter lattice was described in [31]. Notice that the V-ring of a group (X,M), where G (s) <X < S(M), is a cellular subalgebra of the algebra H. The cellular subalgebras of H (W, W {s 1 ), for the case W = Bn and s corresponding to the terminal node of the Coxeter diagram, are described in [20].

In the paper [28] of the present collection, the structure constants of the generating algebra for An are determined using the construction of a mirror blow-up (i.e. the Gaussian basis from Example 2.1). In an analogous way in [35], formulas for the structure constants of the Hecke algebra of the unitary group (in the sense of [7]) acting on the maximal isotropic subspaces are obtained. In the next section, as a corollary of these results, theorems about simplicity of certain algebras H(G, B W (sJ B), Aw({s}) will be proved. Recall that an algebra is simple if it contains no proper block subalgebras.

7. Simplicity theorems for block algebras

Let K and K' be rings. A homomorphism 4> of K onto K' can be extended to the canonical homomorphism K[x 1 , ... , Xn] -t K'[x 1 , ... , Xn]. Let us define a mapping cj;· of the ring Aw(K) onto the ring Aw(K') by

~ - ~ cj>(I,P w(X 1 , ... , Xn) aw)- I,(P w(X 1 , ... , Xn)) aw. w w

It is clear that $ is a homomorphism of the rings which maps a block ubalgebra of Aw(K)

onto a block subalgebra of Aw(K').

If f is a specialization of Aw(K), then r = (cp(j(xi))), i = 1, ... , n, is a specialization of Aw(K,f) -t Aw(K,r), where

~f(1:Pwlf(x1), ... , f(xn))Ow) = "f.Pwlf'(x1), ... , r<xn))aw. w w

Let fx be a parametrization of Aw(K). Then the following diagrams are commutative.

Aw(K) ~ Aw(K')

.l.j J.r Aw(K,f) ~

Aw(K'.r>

Aw(K) ~ Aw(K')

.l.fx

Aw(K)

J,fx

Aw(K')

It follows from the defining relations that the structure constants of the algebras Aw(Q) and Aw(.Z) coincide. So the symbols Q and .Z will be omitted below.

Lemma 7.1. (cf. [30]). Let C(q) be a finite group of Lie type with an irreducible Weyl group W, where W is not isomorphic to the dihedral group of order 16. Let B W1 B be a parabolic subgroup of W where J is a subset of the set of generators of the group W. Let Z q-1 be the ring of integers modulo q -1. Then the Hecke algebra H'B:.,_, (C(q), B W1 B) coincides with Hz,_, (W, W1 ).

Proof. The algebra Hz,_,(C(q),B) is the image of Hz,(C(q),B) under the homomorphism ~. where eil is the canonical homomorphism of the rings .Z and .Z q-1 .

This means that the defining relations of the algebra H '8.,_1 (C(q), B) can be obtained from consideration of the relations (*) (see Section 6) modulo q -1. By the theorem on multiplicities, the values qi in the considered relations are powers of q. So we obtain the relations Ow1 xaw=Ow1w which determine Hz,_,(W, W0 )=Zq_1[W], i.e. the group algebra of W over the commutative ring Z q-1. The algebra H '8.,_1 (W, W1 ) is the image oftheblock subalgebra Hz (C(q), B W1 B) of H '8. (C(q), B) under the homomorphism ~p

Lemma 7.2. Let eil be the canonical homomorphism of .Z onto Zn, (W,S) be a finite Coxeter system, and J c S. Then the homomorphism ~ : H '8. (W, w,) -+ Hz. (W, w,) establishes a bijective correspondence between the block subalgebras of Hz (W, w,) and Hz. (W, w, ), for n ~ I W: W1 I.

Proof. Let the sets T 1 , ••• , Ts form a partition 't of the set ( a 1 , ••• , a1} of the generators of the algebra Hz (W, W 1 ) (resp., H '8.. (W, W1 ) ). Let us consider the elements ti = "f. ai, 1 ~ i ~ s. The partition 't determines a block subalgebra of H z.(W, W1 ) if

jeT1

and only if the relations

ti • tj = "f. J.}j • ts (7.1) s

hold. Since no ')...~ exceeds the degree of the group (W, W I W1 ) for n ~ I W: W1 I, the validity ofrelation (7.1) implies that 't determines a block subalgebra in Hz (W, W1 ). o

Remark. The nurober I W: w, I in the formulation ofLemma 7.2 can be exchanged by the maximal coefficient of a generator in the decomposition of the square of an

502 V. A. USTIMENKO

element from H (W, W1 ), where the maximum is tak:en over generators of all subalgebras of H(W, W1 ). It is clear that an analogous Iemma can be formulated for an arbitrary Kalgebra with a fixed basis.

It is shown in [15] that the Hecke algebra H(Sn,Sm X Sn-m) (Sn =An-1) is simple for n > a(m), where a(m) depends only on m. It follows from [18] that a(m) can be taken as m 4 + 4m + 1. This result, in view of Lemmas 7.1 and 7 .2, implies the following

Theorem 7.3. For q > C': and n > a(m), the Hecke algebra H(An_1(q),BWs-(sJB), where S={(1,2), ... ,(n-l,n)} and s=(m,m+1)e S, is simple. o

Corollary. The algebra As.+, ({s}) is simple for n > a(m). o

In what follows a. and ß will denote the generators of the Weyl group Bn and Dn+l corresponding to the right terminal nodes of the Dynkin diagram in accordance with the numbering ofvertices from [3].

It is known that for an arbitrary field K, the permutation group corresponding to the action of Bn(K) on the set of left cosets over the parabolic subgroup P =B Ws-(a) Bis contained in the group Dn+l (K) acting on the cosets of the subgroup B Ws:!:}~J B [8]. Moreover a direct check shows that the following proposition holds.

Proposition 7.4. Let AB. and Av.+, be the monogen generating algebras corresponding to the groupsBn(q) andDn+t(q). ThenAv.+, (ß) is a block subalgebra ofAB.(a.). 0

Remark. It is known that the Weyl groups of the root systems Bn and Cn coincide. This implies that AB. =Ac •. Hence the specializations corresponding to the substitution x = q arealso the same for these Weyl groups. Nevertheless, if q is odd, the Hecke algebras H(Bn(q), B Ws-{a} B) and H(Cn(Q), B Ws-(a} B) having the same structure constants are not isomorphic as cellular rings in the sense of [9]. The metric association schemes SB. (q) and Sc. (q) corresponding to these Hecke algebras are also nonisomorphic [1]. The subschemes of SB. (q) and Sc. (q) corresponding to the specialization Av.+, (ß) will be denoted by SB.'(q) and Sc.'(q), respectively. It is clear that the automorphism group of SB.'(q) contains Dn+l (q) and acts distance-transitively on this scheme. By a result from [35] on the maximality of the normalizer of a symplectic group acting on the maximal isotropic subspaces, the group Aut Sc. '(q) does not act distancetransitively. This implies that SB. '(q) and Sc. '(q) are not isomorphic if q is odd. The scheme Sc. '(q) should be added tothelist of known metric association schemes given in [1].

Theorem 7.5. For q > 2n we have the following:

(a) the Heckealgebra H(Dn+l (q),B Ws-(s) B) is simple;

(b) the algebras H(Bn(q), B Ws-(a} B) and H(Cn(q), B Ws-(a} B) have no subalgebras except that corresponding to Aoo+, (ß);


(c) the Hecke algebras of the classical groups, which are Chevalley groups of twisted type, acting on the maximal totally isotropic subspaces are simple.

Proof. Let G be a classical group over F q with Weyl group W coinciding with Bn or Dn+l· The permutation group W = (W, W I Ws-(a)) is either the exponentiation S2 i Sn+t or the subgroup of S2 i Sn+l which stabilizes the hyperplane

n H = { 1 e V I L Xi = 0} and acts on the vectors of this hyperplane. In the latter case the

i=l

group (W, W !Ws-(a)) is the semidirect product Vn(2)A.Sn+l and contains S2 i Sn. So the group W contains the exponentiation of the symmetric groups S 2 and Sn and its V

ring Vz (W) isasubring of Wz (S2 i Sn). In view ofLemma 7.1, H-z,_1 (G, B Ws-(s) B)

coincides with V -z,_1 (W). For q > 2'+1 the lattice of cellular subalgebras of V z,_1 (W)

coincides with that of V z (W) (cf. Lemma 7.2) and is contained in the lattice of subalgebras of V z (S 2 i Sn). The description of the latter lattice is given in [20]. It follows from this description that the number of cellular subalgebras of V(S 2 i Sn) does not exceed 16. Since there is a natural correspondence between the generators of V(G, B W1 B) and of V(W), to determine the cellular subalgebras of V(G, B W1 B) it is sufficient to check which partitions corresponding to subalgebras in V(W) give subalgebras in V(G, B W1 B). Let 't be a partition of the set {\Pd, i = 0, l, ... ,n -1, into orbits of the group Vn(2) A.Sn+l· Using the formulas for the structure constants of V(G, B w, B) presented in [33], [34], one can check that a partition of the set {'Pd, i = 0, l, ... ,n -1, into generat0rs of a cellular subalgebra of V(S 2 i Sn) other than 't fails to determine a block subalgebra only for V -z,_1 (G, B W; B). The partition 't determines a subalgebra

only in the exceptional situation given in the theorem. 0

Corollary.

(a) The subalgebra of AB. (a) described in Proposition 7.4 is the only block subalge-

bra.

(b) The algebras AB. (a), Av.+1 (ß), as well as their parametrizations corresponding

to the classical groups different from B n (q ), are simple. O

Notice that it follows from Theorem 7.2 and from the Chow lemma that the group P1Ln(q) acting on the rn-dimensional subspaces is maximal for n > a(m). Analogously, from Theorem 7.5 one can obtain a description of the lattice of unitransitive overgroups of the classical group over Fq, with the diagrams Bn and Dn+l• acting on the maximal isotropic subspaces for q > 2n. More detailed results in this direction are obtained in [31 ], [35] by means of another technique.

Recently M.E. Muzichuk has made an improvement ofTheorems 7.3-7.5 (cf. [19]). He has shown that the conditions q > C'J:, n > a(m) and q > 2n can be omitted.

504 V. A. USTIMENKO

References

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6. C.W. Curtis, Representation of finite groups of Lie type, Bull. Amer. Math. Soc. (New Series), 1 (1979), 721-757.

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1971.

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11. M.R. Garey, D.S. Johnson, Computersand Intractability, A Guide to the Theory

of NP-Completeness, San Francisco, 1979.

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13. V.G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Acad. Nauk

SSSR 32 (1968), 1323-1367 [In Russian].

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1983.

15. L.A. Kaluznin, M.H. Klin, On certain maximal subgroups of symmetric and alternaring groups, Mat. USSR Sb., 16 (1972), 95-123.

16. L.A. Kall1Znin, V.I. Suschanskii, V.A. Ustimenko, Application of the Computer to the theory of permutation groups and their applications, Kibernetika, 6 (1982), 83-94 [In Russian].

17. W. Kantor, Furtherproblems concerning finite geometries and finite groups, Proc. ofPure Math. 37 (1980), 479-486.

18. M.H. Klin, Investigations of algebras of invariant relations for certain classes of

permutation groups, Ph. D. Thesis, Nikolaev, 1974 [In Russian].

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20. M.E. Muzichuk, The subschemes ofthe Hamming scheme [In this volume).

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Springer, 1976.

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L-functions, Proc. Symp. Pure Math. 33, pp. 3-27, AMS, Providence, 1979.

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1980.

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27 (1985) 110-112 [In Ukrainian].

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Combinatorics, pp. 137-148, Kiev, 1978 [In Russian].

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130, 1982 [In Russian].

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The preliminary version of this paperwas originally published in IATC0-85, pp. 134-148.

SUBJECT INDEX

absolute point (of a polarity) 414 adjacency algebra of a scheme 10

matrix 10, 286 of a co1oured graph 79

affine group 343 matehing 413 nine 411 plane 171,409 ring 172

almost simple group 343 alternating forms graph 357 amalgam 417, 453, 461

method 419 of index (3,3) 466

amorphic cellular ring 169, 406 antipodal block 296

covering 296 folding 296

antireflexive relation 12 antisymmetric association scheme 75 apartment 332, 428 asymptotic simplicity 19 association scheme 10, 187, 286

metric 286 of 2-orbits 286 P-polynomial 286 Q-polynomial 287 Schurian 286 symmetric 286

automorphic graph 321 automorphism group 154, 182

of a cellular ring 16, 221 geometry 444 multigraph 266 relation 13

Balaban-Randicgraph 68 base of a permutation group 88

wreath product 161 basis graph (relation) 4 Burnside mark 29, 97,276 Bigg' s Iist of intersection arrays 322, 389 Biggs-Smith graph 320 bilinear forms graph 357 bi-primitive graph 279, 459

of parabolic type 460 nonadmissible graph 122

507

block subalgebra 499 b1ow-up of a relation 475 BM-algebra (Bose-Mesner algebra) 10, 286, 293 BN-pair 475 Bochert-Manning bound 100

theorem 155 Bruhat decomposition 482

Cartan function 489 subalgebra 480

cartesian blow-up 475 Cauchy-Frobenius-Burnside Iemma 94 cell6, 168

isomorphism 86 cellular algebra 4

computable 17 equivalent 17 idempotent 52 ring 4, 19, 168, 226 subring 6, 84, 168, 226

centralizer ring (V -ring) 7 characteristic matrix 315

vector of a vertex 313 Chevalley basis 481

group 111, 251 classical generalized polygon 423 Clebsch graph 181 clique 302, 402 closure of an amalgam 461 2-closure 61 coefficient of projection 39 coherent configuration 9, 253 collineation group 411 coloured clique 388 coloured graph of a cellular ring 404 commutative cellu1ar ring 43 comp1ete affine ring 171

coloured digraph 4 connected component of a graph 269 convo1ution of functions 15 Conway group 35 Coxeter graph 320

group 435, 491 system 435, 474

cyclic (circulant) graph 78, 123, 129

decomposition (of a graph) 411 deleting procedure 410

508

Desargues graph 320 diagram of a geometry 443

incidence system 474 diameter of a graph 420 direct blow-up 475

sum of permutation groups !55 discrete topology 216 distance-regular

graph (d.r.g.) 286 antipoda1 295 bipartite 295 halved 296 primitive 295

distance-transitive graph (d.t.g.) 34, 286, 379,414 representation 285

Doro graph 351 double coset 7 Dynkin function 490

eigenmatrix (first and second) II eigenvalue of a graph 173 elation 333 element of an amalgam 417 embeddable amalgam 453 embedding of a subgraph 380 endomorphism of cellular rings 212 enumeration of graphs 91 exchange condition 427 exact (complete) invariant 26 exponentiation ofpermutation groups 65, 103,

160,188,206,226 even graph 344

factorization of a graph 169 feasibility condition 293 feasible partition 255 first (second) partition 4

standard basis 12 Fischer's graph 307 fixed points method 397 flag 26,252

of geometry 443 flag-transitive action 453 formula-expressible relation 14 formula-equivalent relations 14 Postergraph 320 Frobenius group 69

Galois correspondence 13, 16 Gaussian basis 477 G-distance-transitive graph 285

SUBJECT INDEX

generalized polygon (n-gon) 303, 422, 474 Weyl group 489

generating algebra 254, 498 function 94

generously transitive permutation group 133 geometric cycle 312

girth 312 geometry 443 Gewirtz graph 350 girth of a graph 303,419 Goldschmidt's Iemma 423 good subset 83 Gosset graph 351 Grassmangraph 347

manifold 261 Grassmanian 108 graph of

Latin square 172 parabolic geometry 304

Hadamard matrix 40 I Hamming distance 160

scheme 104, 188 Heawood graph 320 Hecke algebra 254, 497

ring 8 Hermitian forms graph 357

matrix 487 Higman-Sims graph 339 Hoffman-Singleton graph 304, 431 homogeneaus coherent configuration 9

imprimitive cell 51 incidence graph 302

matrix 410 system 474

induced symmetric group 99, 156 inducing 7 intersection array (of d.r.g.) 286,293,413

graph 268 index 8 number 10, 254

invariant relation 13, !54 isomorphism of multigraphs 266 iterative classification 86

Janko's group 35, 279, 463 Johnson graph 344

association scheme 4 7, I 0 I Jordan-Dedekind condition 62

SUBJECT INDEX

Kac-Moody algebra 490 k-closed permutation group 14 k-closure 14 k-compact p-subgroup 159 k-equivalent permutation groups 14 k-funtion 15 k-mu1tire1ation 15 k-orbit 13, 154, 253 k-point 12 Krasner a1gebra 13 Krein condition 294

parameter 12 k-relation 12

Latin sq uare 182 lattice of order n 181

subrings 229, 236 Levi graph 413 Lie algebra 481

multiplication 483 line graph 157, 305 locally s-transitive action 301, 420 logarithmical convexness 298

mark of a subgroup 276 Marggraf theorem 154 Mathieu group 27, 28, 31, 395 maximal torus 482 maximality of a permutation group 104, 158 member of an amalgam 417 merging of classes (basis sets) 46 method of

Gauss 260 subsequent splitting 27

minimal degree of a permutation group 100, 155 model (of a projective plane) 410 Moore graph 303 multigraph 266 multiplicity freeness 270, 300 multiset 228

negative Latin square graph 180 non-abelian simple group 112, 344 non-admissible graph 122 non-degenerate cycle 390 non-Schurian (non-Schur-type) cellular ring 16 nonstandard flag 453 normalizer of a group 96

odd graph 336 O'Nan-Scott theorem 123, 225, 343

O'Nan-Sims group 438 orbit 154 2-orbit 7 3-orbit 39

(P and Q)-polynomial scheme 107 Paley graph 130, 245

design P(l1) 279, 398 partially ordered set 62 partition of a number 182

set 231 subgraph 381

pentagraph 340 Perkel graph 351 permutation character 32, 95

group 154 Petersen subgraph 320, 351, 381,445 p-distinguished partition 158 polynomial computable 17

equivalent 17 position in a graph 381 primitive cell 51

graph 123 group 414 representation 112

product of permutation groups 155 projective geometry 157, 252, 476

plane 39,410 pseudo-affine cellular ring 172, 406 pseudo-lattice 181

q-analogue of Hamming scheme 111 Johnson graph 357

scheme 108 quasi-order 214

rank function of a semi-Iattice 62 of a cellular ring 4

geometry 443 permutation group 7, 265, 279

rectagraph 341 recurrence for the intersection numbers 109 reduced root system 480 regular graph 420 residual geometry 443 residue of a flag 4 73 retract 475 root 480

system 480

Schreier method 79 Schurian (Schur-type) cellular ring 16, 245 semifield 368

509

510

semi-lattice 62 separating of points 66 simple amalgam 417

quantity 8 ring 19

Sims conjecture 313 smooth blow-up 478 split BN-pair 482 S-ring 8, 56, 130, 188 S-subring 188 S-system of an S-ring 58 stabilizer 154 standard basis 4

doubling of a graph 462 flag 453 scheme of investigation 19

step basis of a chain 253 s-transitive graph 301,417 strong conjugacy condition 397

generating system 88 strongly doubly primitive group 431

regular graph 71, 73, 168 structure constants 6, 81, 228 subconstitient 420 subdegree 7

of permutation group 265, 279 primitive group 34

suborbit of a permutation group 66 subscheme of a scheme 187 symmetric association scheme 10

power of a graph 225 cellular ring 225

k-relation 13 relation 12

symmetrized k-point 78

t-apartment 428

SUBJECT INDEX

tensor multip1ication of cellular rings 210 product of relations 227

Tits geometry 474 group 34

topological space 213 trace of an S-ring 209 transitive extension 132, 402 transitivity module 8 transposition !55 triangle in a semi-lattice 62 tri valent graph 417 Tutte graph 320 t-vertex condition 71 twisted field 369 two-graph 131 type of

a flag 253 geometry 443

uniqueness condition 427 universal group of an amalgam 418

(v,k,A.)-graph 406 valency of a graph 420 V-ring of a permutation group 209, 286

Weyl group 254, 480 wreath product I 02, 160, 266

over a partially ordered set 222

investigations in algebraic theory of combinatorial objects

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