combinatorics 1984: finite geometries and combinatorial structures: colloquium proceedings: finite...

COMBINATORICS '84

NORTH-HOLLAND MATHEMATICS STUDIES 123 Annals of Discrete Mathematics (30)

General Editor: Peter L. HAMMER Rutgers' University, New Brunswick, NJ, U. S.A.

Advisory Editors C. BERGE, Universite de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.-H. VAN LINT California Institute of Technology, Pasadena, CA, U.S.A. G .4 . ROTA, Massachusetts Institute of Technology, Cambridge, MA, U. S.A.

NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD .TOKYO

COM BI NATORICS '84 Proceedings of the International Conference on Finite Geometries and Combinatorial Structures Barl; ItalK 24-29 September, 1984

edited by

A. BARLOlTI Universita di Firenze, Firenze, Italy

M. BILIOITI Universita di Lecce, Lecce, Italy

A. COSSU Universita di Bart Bari, Italy

G. KORCHMAROS Universita delta Basilicata, Potenza, Italy

G.TALLINI Universita 'La Sapienza: Rome, Italy

1986

NORTH-HOLLAND -AMSTERDAM 0 NEW YOAK QXFORD .TOKYO

@ ELSEVIER SCIENCE PUBLISHERS B.V., 1986

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ISBN: 0 444 87962 5

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Library of Congress Catalogingin-Publication Data

International Conference on Finite Geometries and Combinatorial Structures (1984 : Bari, Italy) Combinatorics '84 : proceedings of the International

Conferenco on Finite Geometries and Combinatorial Structures, Bari, Italy, 24-29 September 1984.

(Annals of discrete mathematics ; 30) (North-Holland mathematics studies ; 123) Includes bibliographies. 1. Combinatorial geometry--Congresses. I. Barlotti,

A. (Adriano), 1923- . 11. Title. 111. Title: Combinatorics eighty-four. IV. Series. V. Series: North-Holland mathematics studies ; 1 2 3 . QA167.158 1984 511l.6 85-3 1 12 1 ISBN 0-444-87962-5

PRINTED IN THE NETHERLANDS

V

PREFACE

Every year, since 1980, an International Combinatoric Conference has been held in Italy: Trento, October '80; Rome, June '81 ; La Mendola, July '82; Rome, at the Istituto Nazionale di Aka Matematica, May '83.

The International Conference Combinatorics '84, held in Giovanazm (Bari) in September '84 is part of the well established tradition of annual conferences of Combi- natorics in Italy. Like the previous ones, this Conference was really successful owing to the number of participants and the level of results.

The present volume contains a large part of these scientific contributions. We are indebted to the University of Bari and to the Consiglio Nazionule delle Ricerche for fmancial support. We are pmfoundly grateful to the referees for their assistance.

A. BARLOTTI M. BILIOTTI

A. COSSU G. KORCHMAROS

G. TALLINI

This Page Intentionally Left Blank

vii

Intervento di Apertura del Prof. G. Tallini al Convegno “COMBINATORICS 84”

Vorrei dare, a nome del Comitato scientific0 e mio, ilbenvenuto aimoltipartecipanti che dall’Italia e dall‘estero sono qui convenuti per prendere parte a questo convegno. Esso si ricollega e fa seguito ai congressi internazionali di combinatoria tenuti a Roma nel giugno del 1981, a La Mendola nel luglio del 1982, a Roma presso I’Istituto Nazionale di Alta Matematica nel maggio del 1983. Questi incontri, ormai annuali in Italia e che spero possanc continuare, s’inquadrano nell’ampio sviluppo che la combinatoria va acquistando a livello internazionale.

Come 8 noto il mondo modern0 si va indirizzando ed evolvendo sempre di pib verso la programmazione e l’informatica, al punto che un paese oggi B tanto pih progredito, im- portante e all’avanguardia quanto pib B avanzato n e b scienza dei computers. I1 ram0 della Matematica che B pi^ vicino a questi indirizzi e che ne I! la base teorica B proprio la combinatoria. Essa a1 gusto astratto del ricercatore, del matematico, associa appunto le applicazioni pib concrete. Cib spiega il prepotente affermarsi di questa scienza nel mondo e ne prova il fervore di studi e di ricerca che si effettuano in quest’ambito, le pubblica- zione dei molti periodici specializzati, i numerosi convegni internazionali a1 riguardo.

Vorrei ringraziare gli Enti che hanno permesso la realizzazione di questo convegno, tutti i partecipanti, in particolare gli ospiti stranieri che numerosi hanno accolto il nostro invito e tra i quali sono presenti insigni scienziati.

Concludo con I’augurio che questo convegno segni una tappa da ricordare nello sviluppo della nostra scienza.

ix

CONTENTS

Preface

G. TALLINI, Intervento di apertura a1 Convegno

V. ABATANGELO and B. LARATO, Translation planes with an autornorphism group isomorphic to SL(2,5)

L. BENETEAU, Symplectic geometry, quasigroups, and Steiner systems

W. BENZ, On a test of dominance, a strategic decomposition and structures T(t,q,r,n)

A. BEUTELSPACHER and F. EUGENI, On n-fold blocking sets

A. BEUTELSPACHER and K. METSCH, Embedding finite linear spaces in projective planes

A. BICHARA, Veronese quadruples

M. BILIOTTI, S-partitions of groups and Steiner systems

M. BILIOTTI and G. KORCHMAROS, Collineation groups strongly irreducible on an oval

B. BIONDI and N. MELONE, On sets of Plucker class two in PG(3,q)

F. BONETTI and N. CIVOLANI, A free extension process yielding a projective geometry

F. BONETTI, G . 4 . ROTA, D. SENATO and A.M. VENEZIA, Symmetric functions and symmetric species

R. CAPODAGLIO DI COCCO, On thick (Qi2)sets

P.V. CECCHERINI and N. VENANZANGELI, On a generalization of injection geometries

P.V. CECCHERINI and A. SAF'PA, A new characterization of hypercubes

V

V i i

1

9

15

31

39

57

69

85

99

105

107

115

125

137

X Contents

P.V. CECCHERINI and A. SAPPA, F-binomial coefficients and related oombinatorial topics: perfect matroid designs posets of full binomial type and F-geodetic graphs

L. CERLIENCO, G. NICOLETTI and F. PIRAS, Polynomial sequences associated with a class of incidence coalgebras

M. DE SOETE and J.A. THAS, R-regularity and characterizations of the generalized quadrangle P(W(s), (-))

M. DEZA and T. IHRINGER, On permutation arrays, transversal seminets and related structures

G. FAINA, Pascalian configurations in projective planes

P. FILIP and W. HEISE, Monomial code-isomorphisms

S. FIORINI, On the crossing number of generalized Petersen graphs

J.C. FISHER, J.W.P. HIRSCHFELD and J.A. THAS, Complete arcs in planes of square order

M. CIONFRIDDO, A. LIZZIO and M.C. MARINO, On the maximum number of SQS(v) having a prescribed PQS in common

A. HERZER, On finite translation structures with proper dilatations

M. HILLE and H. WEFELSCHEID, Sharply 3-transitive groups generated by involutions

F. KRAMER and H. KRAMER, On the generalized chromatic number

P. LANCELLOTTI and C. PELLECRINO, A construction of sets of pairwise orthogonal F-squares of composite order

D. LENZI, Right S-n-partitions for a group and representation of geometrical spaces of type “n-Steiner”

G. LO FARO, On block sharing Steiner quadruple systems

G. MENICHETTI, Roots of affine polynomials

S. MILICI, On the parameter D(v,tv. 13) for Steiner triple systems

C. PELLECRINO and P. LANCELLOTTI, A new construction of doubly diagonal orthogonal latin squares

143

159

171

185

203

217

225

243

25 1

263

269

275

285

29 I

297

303

311

33 1

Contents xi

C. PELLEGRINO and N.A. MALARA, On the maximal number of mutually orthogonal F-squares

C. PICA, T. PISANSKI and A.G.S. VENTRE, Cartesian products of graphs and their crossing numbers

G. TALLINI, Ovoids and caps in planar spaces

B.J. WILSON, (k,n#arcs and caps in finite projective spaces

N. ZAGAGLIA SALVI, Combinatorial structures corresponding to reflective circulant (0,l)- matrices

H. ZEITLER, Ovals in Steiner triple systems

PARTICIPANTS

335

339

347

355

363

373

383

Annals of Discrete Mathematics 30(1986) 1-8 0 Elsevier Science Publishers B.V. (NorthHolland) 1

TRANSLATION PLANES WITH AN AUTOMORPHISM GROUP ISOMORPHIC TO SL (2,5)

Vito Abatangelo Bambina Larato Universith di Bari

Italia

In this paper translation planes of odd order q , 51q -1 , are constructed. Their main interest consists in the fact that their translation complement contains a group isomorphic to SL(2,5) . At first these planes were obtained in other ways by 0. Prohaska in the case

51q+l ( [lo] ,1977) and by G. Pellegrino and G. Korchm5ros in the case 51q-1 ( [9] ,1982), but in both papers the Authors did not establish the previous group property. Moreover we show that Pellegrino and Korchmhros plane is not a near-field plane of order 11 .

2

2

1. - AN AUTOMORPHISM GROUP OF THE AFFINE DESARGUESIAN PLANE OF ORDER q , 51q -1 , ISOMORPHIC TO SL(2,5)

Set K = GF(q J , q odd. We may assume that the elements of K can be written in the form g + t T with 5 , V € F = GF(q) and t = s , where s is a non-square element of F . Let a be the affine Desarguesian plane coordinatized by K : points are pairs (x,y) of elements of K and lines are sets of points satisfying equations of the form y = mx+b or x = c with m,b,c elements of K . The affine subplane no coordinatiz@d by F is an affine Baer subplane; the image of no under a composition of a linear transformation with a translation of 3c is also taken to be an affine Baer subplane. The lines at infinity of Baer subplanes are called Baer sublines at infinity. By standard arguments (similar to those of [7]. p. 80-91) one can show the following facts: Baer sublines at infinity are sets of elements of

(1.1)

2 2

2 2

K u { oo} of the form: [ ap+b I a,b€K , p runs over F u{m} j

o r (1.2)

Let v and r be any two elements of K . For any d E K and g EK , such that gq+l = 1 , the set of all points (x,y) for which

(1.3) y = xv + rxqg + d

Research partially supported by M.P.I. (Research project "Strutture Geometriche Combinatorie e l o r o Applicazioni'').

2 V. Abatangelo and B. Larato

is t h e po in t - se t o f an a f f i n e Baer subplane. Its Baer s u b l i n e a t i n f i n i t y has equa t ion (1 .2 ) as t h e l i n e y = xmtd i n t e r s e c t s (1.3) e i t h e r i n q p o i n t s o r i n t h e on ly p o i n t ( 0 , d ) accord ing as (1 .2) ho lds o r does n o t hold. The a f f i n e Baer subplanes with equa t ions of t h e form ( 1 . 3 ) c o n s t i t u t e a n e t on t h e a f f i n e p o i n t s o f n . The l i n e a t i n f i n i t y becomes t h e Miquelian i n v e r s i v e p l ane M(q) i f we t a k e a s c i r c l e s t h e sets (1.1) and ( 1 . 2 ) of elements o f

I n o r d e r t o cons ide r an automorphism group o f $C isomorphic t o SL(2 ,5) we

d i s t i n g u i s h two c a s e s , accord ing a s 51q+l o r 51q-1 .

K ~ { c o } .

CASE 5 ( q + l

By assumption 5 1 q t l t h e r e e x i s t s an element a o f K such t h a t a5 = 1

(cf . [4]). We p u t b = (a-aq)-' and c such t h a t c q + l t bq+l = 1. After t h i s l e t u s cons ide r t h e fo l lowing a f f i n e mappings o f a

a : x ' = ax , yt = aqy f l : x ' = bx t cy , y ' = -cqx t bqy

and t h e i d e n t i t y & : x ' = x , y ' = y . Note t h a t < a , f l> 2 S L ( 2 , 5 ) , ( c f . 141, p. 199) .

Now our purpose is t o s tudy t h e a c t i o n o f < a , p > i . e . t h e a c t i o n o f I' = <a@/<-& >ZPSL(2,5) on r . l(q).

PROPOSITION 1. - I n M(q) t h e group I' maps t h e c i r c l e C on to i t s e l f and l eaves t h e s e t V = { C1 , C 2 , ... , C 6 } C : mqtl = 1 and

on t h e l i n e a t i n f i n i t y ,

q+ 1 0 :. = -l i n v a r i a n t , where

1 i ( 1 - q ) q 2 -1 q t l 2 -2

C i + 2 : (m-2a b c ( 2 b +1) ) = (2b +1) , ( i = 0, ... , 4 ) .

PROOF - Some long and easy c a l c u l a t i o n s prove t h a t a and f l send C o n t o i t s e l f and a c t on t h e set $f as fo l lows 0

a : (C1)(C2C3C4C5C6) , f l : ( C 1 C 2 ) ( C 3 C 6 ) ( C 4 ) ( C 5 ) . PROPOSITION 2 . - The group I' a c t s on as PSL(2 ,5) a c t s i n i ts usua l two- t r ans i t i ve r e p r e s e n t a t i o n on s i x o b j e c t s .

x' =

c1 We i d e n t i f y

C with 4 and A5 as a and get ou r assert.

PROOF - Assuming B. Hupper t ' s t e rminology, [41 , ( 1 . 4 ) l e t d = GF(5) u { m ) be t h e s e t c o n s i s t i n g of s i x o b j e c t s ; and (1.5) l e t PSL(2,5) be t h e permuta t ion group on d o f t h e form

(ax+b) (cx+d) - ' , ad-bc €{1 ,4 ) , a , b , c , d E G F ( 5 ) . with t h e symbol w , C2 wi th 0 , Cg wi th 3 , with 1 ,

C wi th 2 ; then a': x' = x+3 and 8 ' : x ' = x a c t on 6 f l , r e s p e c t i v e l y , a c t on V . AS < a ' , p ' > N P S L ( ~ , ~ ) , we

c41

Translation Planes 3

PROPOSITION 3. - Any two circles of have two o r zero common points according as q E 3 (mod 4 ) o r q 5 1 (mod 4).

PROOF - By Prop. 2 it is sufficient to prove that the circles C and C 1 2

satisfy our assert.

STEP 1 - If C 1 and C 2

satisfy both their equations the following

(1.6)

As { m = ch I h EGF(q) } is (1.61, the whole system (1.6)

have any common point, its coordinates would which form a system which is equivalent to

1 =: 1

1 c q m -c mq = o ,

the set of solutions of the second equation of has any solution if and only if the equation

cq+l h2 = 1 is solvable, i.e. if and only if cq+l is a square in F . STEP 2 - cq+l is a square of F or not according as q 5 3 (mod 4 ) o r q 1 (mod 4) . In order to prove this, let x and x be elements of F such that a = x + tx * note that

(1.7)

and (a ) = 1 , so

1 2

2 2 1 2

1 2 '

aq+l = x - sx E F q+l 5

(1.8) a'+' = 1

(because in F there is no element of order 5 ) and therefore 5x + lox x s +

+ x s = 0 , By means of (1.7) and (1.81, we get

+.r/;; = 8x - 3 : this proves that 5 is a square in F . Now cq+l = c(a-aq)2 + 13la-a' = p(l+sx ) - 3 7 ( 4 x 2 s ) = (4x - 3 ) ( 4 x s) =

4 2 2

1 1 2 4 2 4 2

2 1 16x1 - 12x + 1 = 0 , i.e.

2 1

2 2 -1 2 2 -1 2 1 2

-

-1 -2 = -1 s (1 2 m 2 ( 4 x 2 )

q+l. and therefore c is a square in F if and only if -1 is not a square in F , i.e. if and only if q 3 (mod 4).

PROPOSITION 4 . - Co is disjoint from any circle of % . PROOF - It is easy to check that C and C are disjoint. Our assertion follows by means of Prop. 2.

0 1

CASE 5)q-1

By assumption 51q-1 there exists an element a of F such that a5 = J. . Let b,c,d be elements of K such that b = (a-a ) , cd = -1-b . Now let us consider the following affine mappings of 7C

-1 -1

-1 -1 y : x ' z a x , y' = (2+t)b x + ay

6 : X' = [-b+(Z+t)d]x + dy , y' = [2(2+t)b+c-(2+t) dJx + [b-(2+t)ay 2

and note that<y,d >--SL(2,5)

We shall now look at the action of< 7,6> action of the group A = < Y , ~ > / < - E >on

(cf.[4], p . 198).

on the line at infinity, i.e. the

M(q).

4 V. Abatangelo and B. Larato

PROPOSITION 5. - In M(q) the group A maps the circle Do : 2tmq+l-m+mq =O

,D6 } invariant where D :4mq+1+m+mq=0

2 2i 2 -1 a-icd-ll .q+1 + C(2tt)-a 2i (1+2b2)(2bd)-3m+ 1 onto itself and leaves the set { D1e - 0 .

and D : C(4-t ) - 4a (1+2b )(2bd) - i+2

2i 2 + [(2-t) - a (1+2b )(2bd)-Y mq + 1 = 0 , (i = 0.1, ... ,4). PROOF - Long and easy calculations prove that y and 6 map D onto itself and act on the set { D1 ,D2, . . . , D ~ ] as follows

y : (D1)(D2D3D4D5Ds) and 6 : (D1D2)(D3D6)(D4)(D5) . 0

PROPOSITION 6. - The group d acts on { Dl,D2, ... ,D6 } in the same way as PSL(2,5) acts in its usual two-transitive representation on six objects . PROOF - The proof is similar to that one of Prop. 2 , provided we identify D with the symbol 00 , D2 with 0 , D3 with 1 , D4 with 2 , D5 with 3

and D6 with 4 and moreover y and 6 with 7' and 6 ' , respectively, where

1

y ' : x' = x + 1 and 6' : x ' = 4x . Assume now 9 = { Do ,D1 , .. . ,D6 } ; by direct calculations and by Prop.6 we get the following

PROPOSITION 7. - (i) Any two circles of 9 have two common points; (ii) no three circles of 9 have a common point.

2. - DERIVED AND R-DERIVED PLANES

In this section we discuss briefly the processes of derivation and multiple derivation due to T.G. Ostrom and of R-derivation and multiple &derivation due to A.A. Bruen. The reader can find some details in Hughes-Piper c37,

Ostrom c7,8] and Bruen Ll].

Let us assume q > 3 ; every circle C of M(q) is a derivation set, i.e. every pair of affine points lies in a unique Baer affine subplane, whose subline at infinity is C . From now on we will consider circles C with equation (1.2). An affine plane can be obtained by replacing lines whose equation is y = mx+b , for each m satisfying ( l . Z ) , with the q+l Baer affine subplanes, called also components, with equation y = xv + rx g

where g runs over A = { g E K 1 gq+' = 1 } , together with their translates. This construction is said derivation by C and the obtained plane is a translation plane of order q . Derivation can be repeated many times, when there is a set of circles C l , C a , C 3 , . . . ,C each disjoint from the other; in this case the process is said multiple derivation with respect to the circles

About R-derivation, suppose that M(q) admits a chain of circles, i.e. a family % of circles satisfying the following properties:

(i) any two circles of W have two common points;

(ii) no three circles of % have a common point; (iii) % consists of (q+3)/2 circles.

q

2

k. C 1' C 2' * . * 'Ck

Let I denote the point set covered by the circles of V . It follows that I contains exactly (q+l)(q+3)/4 points and each point of I belongs ex-

actly to two circles of % . Hence any two points on a line with its ideal point on I belong exactly to two affine Baer subplanes whose sublines at infinity represent circles of V

The affine Baer subplanes with Baer sublines at infinity representing circles of %' do not form a net on the affine points of 7c . However, for small q ( 5 1 qS 13), several Authors have constructed a net N on the affine points of by taking a suitable half of such affine Baer subplanes; namely, for each circle C of %? , (q+l)q /2 affine Baer subplanes with the same Baer subline representing C . If such a net N exists, we can obtain a translation plane on the affine points of JZ by replacing the affine lines of R whose ideal pointslie on I with N . The resulting translation plane is said to be R-derived from 7c . Of course if there is a set of disjoint chains of circles on M(q) and the corresponding nets exist, then the above method can be repeated: the resulting translation plane is said to be multiple R-derived from 7c .

.

2

2 3. - A CLASS OF TRANSLATION PLANES OF ORDER q , 51q+l , CONTAINING IN THEIR TRANSLATION COMPLEMENT AN AUTOMORPHISM GROUP ISOMORPHIC TO SL(2,5)

In this section we assume that q - 1 (mod 4 ) . By Propositions 3 and 4 we get three planes n . (j = 1,2,3): 7c by derivation from 7c with respect to the circle C ( s o 7c is the well-known Hall plane of order 81, cf. c23,

J 1

1 p. 225), a2 by derivation with respect to the circles C1'C2' 'C6 t

R3 by derivation with respect to the circles C o'cl* -.- ,C6 ; each of

them contains in its translation complement the group<u,p >?SL(2,5) . Now we prove the following propositions

PROPOSITION 8. - The group < a,P > leaves invariant each of the q + l

components corresponding with the derivation set co *

PROOF - The q + l components are the Baer subplanes B with equations

B : y = rx g where g

rqtl = -1 and g runs over A . A straigh 9

g culation shows that U as well as p leaves each B invariant.

PROPOSITION 9. - The group splits the set of the 6(q+l corresponding with the multiple derivation set C I U C U ... (q+1)/2 orbits each of length 12.

PROOF - Let H, (i = 1,2, ... ,6) be the set consisting of the

g

2

forward cal-

components u c6 into

q+l compo-

nents which corrispond with the derivation set . Then <a,p > acts on the set { H1, ... ,H6 } in the same way as on the set { C 1, . . . ,C6 } . By Prcp. 2,

in < a$ > < a,p>acts transitively on 3 2

is< a , l >with A=papap , i.e.

Ci

{ H1, . . . ,H6} . The stabilizer of H 1

2 A : x ' = c(2b -acq+l-aqb2)y , Y' = - ~ ~ ( 2 b ~ - a ~ c ~ + ~ - a b ~ ) x .

6 V, Abatangelo and B. Larato

The q+l components belonging to H are the Baer subplanes E with 1

equations y = xqg where g runs over A . A straightforward calsulation shows that a leaves each E invariant and 1 maps E onto E .

g g -g PROPOSITION 10.- The line-orbit of < a.8 > containing the line joining the origin to Ym has length 12.

PROOF - The vertical line through 0 is left invariant by a and - & but not by 3, . Since each subgroup of < a , f l > containing properly <a,-&> contains also 1 , it follows that the stabilizer of the vertical line through 0 has order 5. This proves our assert.

We point out that for q = 9 , which is the first non trivial case, Proposi- tions 8,9 and 10 yield:

PROPOSITION 11.- (i) The group r =<a, >/<-&> splits the line at infinity of fC into one orbit of length ten and six orbits of length twelve; (ii) tie group r =<a,p > / < - E > splits the line at infinity of f~ into six

orbits of length twelve and ten orbits of length one.

Now we state the following

PROPOSITION 12.- Let (I be a plane obtained by derivation from a. (j = 2,3). If SL(2,5) is an automorphism group of (I , then (I coincided with

8

3

R or f C 3 . 2

PROOF - As it is well known, the number of disjoint circles of M(q) is q-1 and when it occurs they form a linear flock by a theorem due to W.J. Orr (cf. 161). In our situation belong to no linear flock. So if C

is any circle which determines a derivation of J C , , C must coincide with some circle C (i = 0,1, ... ,6) or C must not intersect each of them. If

C E{Co ,C1, ... ,C6 } , then necessarily C = C ; on the other hand C cannot

stay on Hg because, by Prop. 8, no orbit of SL(2,5) is long 10 or less than 10 in H

C1 ,C2, . . , ,C6

i 0

3 '

2 4. - A TRANSLATION PLANE OF ORDER q , 5(q-1 , CONTAINING IN ITS

TRANSLATION COMPLEMENT AN AUTOMORPHISM GROUP ISOMORPHIC TO SL(2,5)

In the previous section 1 we determine the set of circles 9 which is a family satisfying properties (i) and (ii) of chains of circles. Moreover, when q = 11 , 9 satisfies property (iii) and, therefore, is a chain of circles.

By means of the automorphism w : X' = (2tt)x , y ' = (8+7t)x + (2tt)y

of M(11) , we can check that 9 is equivalent to the following chain: c ~ : m - m l 1 = 0 , C' : m t m l 1 = 0 ,

C; : (m - 2 ) - (-2) , i = 1,2, ... ,5 ,

which was studied by G. Pellegrino and G. KorchmBros, So 9 determines a translation plane (cf. C9-J 1.

P -2i 11 - 1+1

Pellegrino and Korchmiros used a geometrical construction and so they cannot notice that the translation plane associated to the chain 9 admits SL(2,5) as automorphism group.


Finally we want to remark that Pellegrino and Korchmlros plane surely is not a near-field plane of order 11 (cf. [q, p. 88), though it satisfies the same group property. The near-field planes have only tdo orbits on the line at infinity: the first has length 2 and the other consists of all the remaining points. In the present case the orbit length are 2 and 120, while the Pelle- grino and Korchmlros plane has an orbit of length 42 = (H 1 on its line at infinity .

2

2

REFERENCES

A.A. Bruen, Inversive geometry and some new translation planes I, Geom. Dedic., 7 (19771, 81-98.

P. Dembowski, Finite geometries (Springer-Verlag, Berlin-Heidelberg-New York. 1968).

D.R. Hughes-F. Piper, Projective planes (Springer-Verlag, Berlin-Heidel- berg-New York, 1973).

B. Huppert, Endliche gruppen I (Springer-Verlag, Berlin-Heidelberg-New

York, 1967).

H. Luneburg, Translation planes (Springer-Verlag, Berlin-Heidelberg-New York, 1980).

W.J. O r r , A characterization of subregular spreads in finite 3-space, Geom. Dedic., 5 (19761, 43-50.

T.G. Ostrom, Finite translation planes (Springer-Verlag, Berlin-Heidel- berg-New York, 1970).

T.G. Ostrom, Lectures on finite translation planes, Conf. Sem. Mat.

112, Annals

Univ. Bari, n. 191, 1983.

G. Pellegrino-G. KorchmSros, Translation planes of order

of Discrete Math., 14 (19821, 249-264.

0. Prohaska, Konfigurationen einander meidender kreise in Miquelschen Mobiusebenen ungerader ordnung, Arch. Math. (Basel), 28 (1977), n. 5, 550-556.

V. Abatangelo-B. Larato Dipartimento di Matematica Via Giustino Fortunato Universitl degli Studi 70125 - B A R I

Annals of Discrete Mathematics 30 (1986) 9-14 0 Elsevier Science Publishers B.V. (North-Holland) 9

SYMPLECTIC GEOMETRY, Q U A S I G R O U P S , A N D STEINER SYSTEMS

L u c i e n B e n e t e a u

U n i v e r s i t e P a u l S a b a t i e r

U E R - M . I . G .

3 1 0 6 2 TOULOUSE - C E D E X FRANCE

Zassenhaus's process o f c o n s t r u c t i o n o f H a l l T r i p l e Systems can be genera l i zed . I t t u r n s o u t t h a t t h e r e i s a canon ica l correspondence between equ iva lence c lasses o f non ze ro a l t e r n a t e t r i l i n e a r forms o f V(n,3) anf isomorphism c lasses o f rank ( n t l ) HTSs whose o r d e r i s 3(" 1, Thus t h e problem o f c l a s s i f y i n g these designs and t h e r e l a t e d S t e i n e r quas i - groups may be presented as a s p e c i a l case o f a more genera l c l a s s i f i c a t i o n problem o f e x t e r i o r a lgebra . As an i l l u s t r a - t i o n o f these ideas we s h a l l dea l comp le te l y w i t h t h e case ns6. Fo r n=6 one o b t a i n s e x a c t l y 5 isomorphism c lasses o f HTSs .

1-INTRODUCTION - Sec t ion 2 g i v e s a b r i e f i n t r o d u c t i o n t o t h e H a l l T r i p l e Systems (HTSs) and t o t h e r e l a t e d groups and quasigroups. There a r e two statements g i v i n g p r e c i s i o n s about t h e correspondence between t h e HTSs on one s ide , and t h e c u b i c hypersur face quasigroups and t h e F i s c h e r groups on t h e o t h e r s ide . We r e f e r t h e reader t o t h e l i t e r a t u r e f o r t h e connect ions w i t h o t h e r p a r t s o f a lgeb ra and des ign t h e o r y (C7,10,11).

F u r t h e r on a process o f e x p l i c i t c o n s t r u c t i o n o f HTSs i s r e c a l l e d ( s e c t i o n 3 ) . Th i s process i s n o t canon ica l . Bu t i t a l l ows t o g e t a l l t h e non a f f i n e HTSs whose 3-order s equa ls t h e rank p. As usua l t h e rank i s t o be understood as t h e minimum p o s s i b l e c a r d i n a l number o f a genera tor subset. The e q u a l i t y s = p corresponds t o an ex t remal s i t u a t i o n , t h e non a f f i n e HTSs obey ing sdp, w h i l e t h e a f f i n e ones obey s=p-1 (see [ll). It i s t h e c l a s s i f i c a t i o n o f non a f f i n e HTSs o f g i ven rank whose o r d e r i s minimal t h a t l e d us t o a problem o f symp lec t i c geometry.

Given some v e c t o r space V, t h e r e i s a n a t u r a l a c t i o n o f GL(V) on t h e s e t o f sym- p l e c t i c t r i l i n e a r forms o f V. We s h a l l be coun t ing o r b i t s i n some s p e c i a l cases. For f u r t h e r i n v e s t i g a t i o n s t h e most i m p o r t a n t r e s u l t i s some process o f t r a n s l a - t i o n i n case t h e f i e l d i s GF(3) : t h e r e i s then a one-to-one correspondence between t h e o r b i t s o f t h e non-zero forms and t h e isomorphism c lasses o f some HTSs. Th is w i l l be used here t o o b t a i n an exhaus t i ve l i s t o f t h e HTSs o f o rde r g2187 whose ranks a r e #6. We s h a l l a l s o c l a s s i f y t h e non a f f i n e HTSs a d m i t t i n g a c o d i - mension 1 a f f i n e subsystem.

2-HALL TRIPLE SYSTEMS, MANIN QUASIGROUPS AND FISCHER GROUPS - A S t e i n e r T r i p l e System i s a 2-(v,3,1) design, namely i t i s a p a i r (E,L) where E i s a s e t o f " p o i n t s " and L a c o l l e c t i o n o f 3-subsets o f E, c a l l e d " l i n e s " , such t h a t ony two d i s t i n c t p o i n t s l i e i n e x a c t l y one l i n e 11 c L. The co r res ond ing S t e i n e r quas ig roup c o n s i s t s o f t h e same s e t E under t h e b i n a r y law : Ef +E ; x,y-xoy d e f i n e d by xox=x and, whenever x#y, xoy=z, t h e t h i r d p o i n t o f t h e l i n e

10 L. Be'ne'reau

through x and y. The Ste iner quasigroups can be a lgeb ra i ca l l y characterized by the f a c t t h a t the law i s idempotent and symmetric. Recall t h a t a law i s sa id t o be symmetric when any e q u a l i t y o f the form xoy=z i s i n v a r i a n t under any permu- t a t i o n of x,y,z ; t h i s i s equiva lent t o the conjunct ion o f the commutativi ty and the i d e n t i t y xo(xoy)=y. For a f i x e d set E, t o endow E w i t h a f am i l y o f l i n e s L such t h a t (E,L) be a Ste iner T r i p l e System i s equiva lent t o provide E w i t h a s t ruc tu re o f Ste iner quasigroup. So i n what fo l l ows we s h a l l i d e n t i f y (E,L) w i t h

A Ha l l T r i p l e System (HTS) i s a Ste iner T r i p l e System i n which any subsystem t h a t i s generated by three non c o l l i n e a r po in ts i s an a f f i n e plane =AG(2,3). This add i t i ona l assumption i s equiva lent t o the f a c t t h a t the corresponding S te ine r quasigroup i s d i s t r i b u t i v e (a o (xoy)=(aox)o(aoy) i d e n t i c a l l y ; see Marshall H a l l J r . [ 6 ] ) . Therefore the HTSs are i d e n t i f i e d w i t h the d i s t r i b u t i v e Ste iner quasi groups.

Let K be a commutative f i e l d . Consider an absolute ly i r r e d u c i b l e cubic hypersur- face V o f the p r o j e c t i v e space Pn(K). Let E be the set o f i t s non-singular K-points. Three po in ts x,y,z o f V w i l l be sa id t o be c o l l i n e a r (no ta t i on : L(x,y,z)) i f there e x i s t s a l i n e L conta in ing x,y,z such t h a t e i t h e r l l c V o r xtytz=R.V ( i n t e r s e c t i o n cyc le ) .

The best known case i s when dim V = l , and n=2 : V i s then a plane curve, i t does not conta in any l i n e and o v e r a l l f o r any x,y i n E, there i s exac t l y one p o i n t z i n E such t h a t L(x,y,z). The corresponding law x , y ~ xoy=z i s obviously symme- t r i c . The set of the idempotent po in ts o f (E,o) i s the set o f f l exes ; i t i s isomorphic t o AG(t,3) w i t h t<2 (endowed w i t h the mid-point law). L a s t l y f o r any f i x e d u i n E, x,y- x*y=uo(xoy) makes E i n t o an abel ian group.

Let us now consider the case dim V>1. Assume t h a t K i s i n f i n i t e . We have the fo l l ow ing f a c t t h a t we mention here wi thout a l l t he requi red d e f i n i t i o n s ( f o r a more complete account see Manin [9] pp. 46-57, espec ia l l y theorems 13.1 and 13.2):

Theorem o f Manin : I f V admits a p o i n t o f "general type", then i n a su i tab le f a c t o r s e t E o f E, the three-place r e l a t i o n o f c l l i n e r i t y gives r i s e t o a symmetric law obeying (aox)o(aoy)=a2o( xoy) and xjox2=xq i d e n t i c a l l y .

As a r e l a t i v e l y easy consequence we have :

Coro l l a ry : The square mapping xcf x =p(x) i s an endomorphism. The set o f t h e idempotent elements o f (E,o) i s I= Im p ; i t i s a d i s t r i b u t i v e Ste iner quasigroup. A l l the f i b r e s A=p-l(e) o f p are isomorphic elementary abel ian 2-groupsY and

Let us say t h a t a F ischer qroup i s a group o f the form G=<S> where S i s a conju- gacy c lass o f i nvo lu t i ons o f G such t h a t O(xy)h3 f o r any two elements x and y from S ( i n o ther terms the dihedral group generated by any two elements o f S has order ~ 6 ) . I n case we have O(xy)=3 f o r any x,y S, x#y, G i s , say a specia l Fischer qroup. I n any special F ischer group G t he re i s j u s t one c lass o f i nvo lu - t i o n s S (namely, the se t o f a l l the i nvo lu t i ons from G ) , and S may be provided w i t h a g t ruc tu re o f HTS by s e t t i n g xoy=xY=yxy(=xyx). We c a l l ( 5 ,o) the HTS corresponding t o 6. This group-theoretic const ruct ion o f HTSs i 4 canonical. More p rec i se l y :

Theorem : Given any HTS E, the (non-empty) family 7 o f special Fischer groups whose corresponding HTS i s E admits : (i) a universa l ob jec t U ; any G i n ( i i ) a smallest ob ject I = U / Z ( U ) , which i s a l so the unique centerless element o f F .

(E,o).

2

(E,o) = . I x A ( d i r e c t product).

i s o f the form G=U/C where CcZ(U).

Symplectic Geometry, Quasigroups, and Steiner Systems 1 1

3-A PROCESS OF EXPLICIT CONSTRUCTION : Le t E be a vec to r space over GF(3) w i t h dim E=*+ l . P i k u some basis e ,e )... en:entl. Besides choose a non-zero sequence o f (i) eyements from G$(3f, say .

u=(A ) j = n + l . i j k lGi< j<k$n i = n t l

Define a b inary law i n E by s e t t i n g that , i f x= C

xi,yJ6 GF(3), then :

x ei and y= C yJe. where . . i = l j = 1 J

i i j k - k j x 0 Y = -x-Y+( 1 A i j k ( x -Y ) ( x Y x Y 1) en+l

l $ i< j<k$n

Proposi t ion : (E,ou) i s a HTS of rank n t l . Any rank n+l HTS o f order 3"' ar ises i n t h i s manner.

For n=3, u i s j u s t a non-zero sca la r : ?1 ; up t o isomorphism, one gets on l y one HTS which i s the order 81 d i s t r i b u t i v e Ste iner quasigroup discovered by Zassenhaus (see Bol [53) . The e q u a l i t i e s : t(e.,e.,ek) = A . .

t r i l i n e a r form t : V3 * GF(3) w i t h V=<e ,e2,. . .e >. Since several d i f f e r e n t sequences u'=(Aijk) y i e l d isomorphic H h s , ther@ are several such forms r e l a t e d

t o the same HTS (E, o ) . I n f a c t a l l these t r i l i n e a r forms are "equiva lent" i n a sense t h a t we are goigg t o precise.

4-THE PROBLEM OF CLASSIFICATION OF ALTERNATE TRILINEAR FORMS : Consider a n-dimensional vect r space V=V(n,K) over a commutative f i e l d K. Recall t h a t a

e t r i c ) i f t(a(x),n(ty),a(z))=sign(a) . t(x,y,z) f o r every permutation IT o f -:* ny two t r i l i n e a r forms t and u are equiva lent i f there e x i s t s L i n GL (V) such t h a t u(x,y,z)=t(!L(x),a(y),!L(z)) i d e n t i c a l l y . Desig a te as A(n,K) the f a h i l y

se t of equivalence classes. We s h a l l be concerned w i t h the c l a s s i f i c a t i o n o f the t r i l i n e a r forms up t o equivalence i n two specia l cases : f i r s t when the re i s a t o t a l l y i s o t r o p i c vector space, second when ns6. I n the f i r s t s i t u a t i o n , n can be any i n t e g e r >3 b u t we r e s t r i c t a t t e n t i o n t o the forms t h a t vanish i d e n t i c a l l y on M3 where M i s a (n-1)-dimensional subspace. When ns6 we ob ta in an almost complete c l a s s i f i c a t i o n , and i n f a c t a complete one i n case K=GF(3). Now any r e s u l t concer- n ing the special case K=GF(3) can be i n t e r p r e t e d i n terms o f HTSs i n view o f the fo l l ow ing r e s u l t :

Theorem 4.1 - ( t r a n s l a t i o n theorem) : There i s a one-to-one correspondence b@yeen X(n, GF(3)) and the isomorphism classes o f HTSs o f rank n+ l whose order i s 3

This t rans la tes the problem o f the c l a s s i f i c a t i o n o f non a f f i n e HTSs o f maximal rank i n t o an e x t e r i o r a lgebra ic c l a s s i f i c a t i o n problem. We s h a l l study two specia l cases where the number o f classes can be spec i f i ed . As examples l e t us g i ve a couple o f statements cons is t i ng f i r s t o f c l a s s i f i c a t i o n theorems o f symplectic geometry and second o f an app l i ca t i ons t o the case K=GF(3) through the t r a n s l a t i o n theorem.

Theorem 4.2 - For any commutative f i e l d K the elements o f A(n,K), n23, admi t t i ng a t o t a l l y i s o t r o p i c codimension 1 subspace form ["l /2] complete equivalence classes. I f n i s odd there i s on l y one such c lass f o r which V i s non s ingu la r ; i f n i s even the re i s none.

Coro l l a ry - The rank n t l HTSs o f order 3"', 1123, admi t t i ng an a f f i n e subsystem o f order 3n form exac t l y ["l/2] complete isomorphism classes. I f n i s odd there i s e x a c t l y one such system wi thout non t r i v i a l a f f i n e d i r e c t f a c t o r ; i f n i s even there i s none.

l s i < j < k s n determine a unique a l t e r n a t e 1 J 1 J k '

t r i l i n e a r form t : V 9 +K, (x.y,z)t-+ t(x,y,z) i s a l t e r n a t e ( o r : symplecfic, skew-

o f the non-vanishing a l t e r n a t e t r i l i n e a r forms o f V and # (n,K) the corresponding

.

12 L. Bkniteau

For instance if n=9, the re are 4 rank 10 HTSs o f order 31° admi t t i ng an a f f i n e order 39 subspace ; they can be described i n the process o f sect ion 3 by s e t t i n g a

X123=1y X145=ay X167=B, X189=y and, f o r (i , j . k l 4 {{1,2,31, ~1,4,51, {1,6,7},{1,8,9)f

Xijk=Oy where ct,B,y

a=B=y=l.

are 0 or 1. The non s ingu la r case a r i ses when one takes

Theorem 4.3 - For any commutative f i e l d K, 11(3,K)I=l=llk(4,K)I and 1%(5,K)I=2.

Besides 1%(6, GF(3))1=5.

Coro l l a ry - Up t isomorphism there are exac t l y (i) one rank 4 (resp. 5) HTS o f

o f order 3 . The l i n k between theorem 4.3 and t h e c o r o l l a r y fo l l ows from t h e t r a n s l a t i o n theorem. It ives a qu ick proof o f (i) and (ii) hat had been prev ious ly

co l l abo ra t i on w i t h J. Lacaze.

The f o l l o w i n g t a b l e g ives the number o f isomorphism classes o f HTSs corresponding t o a g iven 3-order ~ $ 1 3 and a given rank ~ $ 8 .

m r g s p . 3 5l ) , ( i i ) two rank 6 HTSs o f order 36, and ( i i i ) f i v e rank 7 HTSs

establ ished q6,8,1,2,3,4]. The determination o f i (6, GF(3)) was obtained i n

x 3 4 5 6 7 8 9 10 11 12

16-55 4 1 lJ1///////

5 1 1 1 1 4 ? ? ? 1 - 1 2J2 ? ? ? ? ?

1 J3 ?

/ The l a s t c o r o l l a r y s e t t l e s the case (s,p)=(7,7). I t turns out t h a t there are as many HTSs corresponding t o t h i s case as the t o t a l number o f non a f f i n e HTSs w i t h ss6. I n the c e l l s corresponding t o s=p>7 the complete c l a s s i f i c a t i o n seems d i f f i c u l t . But the HTSs w i t h an a f f i n e maximal sub-system are now determined, and f o r s=2nt2=p the re i s j u s t one such system Jn indecomposable.

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C21 L. Beneteau ; Une classe p a r t i c u l i e r e de matro'ides p a r f a i t s , i n "Combinatorics

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[41 L. Beneteau ; Ha l l T r i p l e systems and r e l a t e d top i cs . Proc. In terna. Conf. on

C51 G. Bol ; Gewebe und Gruppen, Math. Ann. 114 (1937) 414-431 ; Zbl. 16, 226.

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Stev in 54 (1980) 107-124.

79", Annals o f Discrete Math. 8 (1980) 229-232.

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Combinatorial Geometries and t h e i r appl icat ions, 1981 ; Annals o f Discrete Math. 18 (1983) 55-60.

Develop. (1960), 406-472. MR 23AX1282 ; Zbl. 100, p. 18.

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C7 Marshall H a l l J r . ; Group theory and b lock designs, i n Proc. I n t e r n . Conf. on t h e Theory o f Groups, Camberra 1965, Gordon and Breach, New-York. MR 36/2514 ; Zbl. 323.2011.

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Annals of Discrete Mathematics 30(1986) 15-30 0 Elsevier Science Publishers B.V. (North-Holland) I S

ON A TEST OF D O M I N A N C E ,

A STRATEGIC DECOMPOSITION AND STRUCTURES T ( t , q , r , n )

Walter Benz

Mathemtisches S e m i n a r der Universitat Hamburg

1. A t a s k i n psychology or economics i s t o compare d i f f e r e n t group- s g s G1, ..., Gr of o b j e c t s ( p e o p l e , goods) conce rn ing a p r o p e r t y E ( o r p r o p e r t i e s E l , ..., En) and t o o r d e r them conce rn ing E s t a r t i n g w i t h t h e dominant g roup ing . T h i s can be done by c o n s i d e r i n g i n t e r a c - tions ( i . e . r-sets i n t e r s e c t i n g e v e r y g roup ing i n one e l e m e n t ) and by judg ing a l l t h e p o s s i b l e i n t e r a c t i o n s : I f I = [ g l , . . . , g r ) , giEGi, i s an i n t e r a c t i o n one l o o k s f o r an o r d e r i n g g i 1 2 . . . t g i r , where 'I >_

s t a n d s for "is E - b e t t e r than" o r " is E-equal t o " . Hence e v e r y i n t e r = - t i o n l e a d s t o a p o s i t i o n number f o r a c e r t a i n g roup ing and one has t o add up t h e s e numbers i n o r d e r t o g e t t h e E-ordering f o r a l l t h e g roup ings .

Usua l ly there are t o o many i n t e r a c t i o n s , and t h e problem is t o f i n d a ba lanced s u b c l a s s o f i n t e r a c t i o n s on which t h e judgement shou ld be based .

When I became c o n f r o n t e d wi th t h i s problem i n a s p e c i a l c a s e , my pro- p o s a l w a s t o use g e n e r a l i z e d Laguerre g e o m e t r i e s i n c a s e o f one s i n - ple n r o p e r t y E (and g e n e r a l i z e d Minkowski geomet r i e s for s e v e r a l p r q - e r t i e s E l , . .., En) : T h e p a r a l l e l classes o f p o i n t s o f a p l a n e La-guer- re geometry may f o r i n s t a n c e r e p r e s e n t the g r o u p i n g s , and these l ec t - ed i n t e r a c t i o n s c o u l d be g i v e n by t h e b l o c k s o f the geometry. The f a c t t h a t through t h r e e Fairwise non p a r a l l e l p o i n t s t h e r e i s e x a c t - l y one b l o c k c o u l d s e r v e as p r o p e r t y o f b a l a n c e conce rn ing t h e cho- s e n s u b c l a s s o f i n t e r a c t i o n s . A s a m a t t e r o f f a c t t h o s e g e n e r a l i z e d Laguerre g e o m e t r i e s are s t u d i e d i n t h e l i t e r a t u r e under d i f f e r e n t names l i k e o r t h o g o n a l a r r a y s (Bush [ 2 ] ) , op t ima l geometr ies(%lckr ,&h [4]) and t h e y p l a y an impor t an t r81e i n c o n n e c t i o n w i t h optir , ial codes i n cod ing t h e o r y (Halder , Heise [ 4 1 ) .

I n s e c t i o n 2 w e l i k e t o s o l v e a p r a c t i c a l problem which comes up i n c a r r y i n g o u t a t e s t under c o n s i d e r a t i o r , : T o f i n d a d i s j o i n t decompo- s i t i o n of t h e s e t o f b l o c k s o f an op t ima l geometry such t h a t a l l t h e components o f t h e decomposi t ion are p a r t i t i o n s o f t h e s e t o f p o i n t s . By u s i n g such a decomposi t ion i n a p r a c t i c a l c a s e one can d i v i d e t h e whole t e s t i n a number o f s u b t e s t s such that a l l o b j e c t s are i n v o l v e d i n a s u b t e s t .

I n s e c t i o n 3 w e l i k e t o d e e l w i t h a simultaneous t e s t of a se t o f ok- j e c t s conce rn ing p r o p e r t i e s El ,..., E n . The c o m b i n a t o r i a l s t r u c t u r e s T ( t , q , r , n ) which we o f f e r i n t h i s c o n n e c t i o n a r e g e n e r a l i z a t i o n s o f

-

16 W. Benz

c e r t a i n cha in geometr ies ( [ l ] ) . The c l a s s of opt imal geometr ies can be i d e n t i f i e d w i t h the c l a s s o f T ( t , q , r , l ) . The Minkowski-m-struc- m s a r e t h e s t r u c t u r e s T ( t , q , q , 2 ) w i t h t = m+2. - I n Theorem 3 we show tha t a necessary c o n d i t i o n f o r t h e e x i s t e n c e of T ( t , q , r , n ) is t h a t rn-1 i s a d i v i s o r of q . I n Theorem 4 we determine t h e number of blocks of a T ( t , q , r , n ) and a l s o t h e number of t h e s o c a l l e d g l o b a l i n t e r a c t i o n s . I n Theorem 5 , 6 we c h a r a c t e r i z e t h e T ( t , X r n - l , r , n ) ( c a s e s X = l , X >1) by apply ing permutat ion s e t s and s p e c i a l c l a s s e s of f u n c t i o n s .

2 . Let q , r , t be i n t e g e r s such that q > 1 and 2 I t 5 r. Consider t he mat r ix -

where t h e ordered p a i r s ( i , j ) a r e c a l l e d p o i n t s ( o r o b j e c t s ) , and where we put ( i , j ) = ( i l , j l ) i f f i= i l and j = j r . The columns of M a r e a l s o c a l l e d groupings. An i n t e r a c t i o n of M is an r - s e t c o n t a i n i n g one element of every column. There a r e qr i n t e r a c t i o n s of M . By I ( M ) we denote t h e s e t o f a l l i n t e r a c t i o n s of M . Consider now a s u b s e t B ( t ) of I ( M ) and c a l l t h e i n t e r a c t i o n s of B ( t ) b l o c k s . We a r e then i n t e r e s t e d i n the fo l lowing p r o p e r t y of ba lance

( * ) T o every t - s e t S having a non-empty i n t e r s e c t i o n w i t h t d i s t i n c t groupings of M t h e r e is e x a c t l y one b lock c o n t a i n i n g S.

By T ( t , q , r ) ( o r T ( t , q , r , l ) wi th r e s p e c t t o s e c t i o n 3 ) we denote a s t r u c t u r e ( M , B ( t ) ) s a t i s f y i n g ( * ) . Many examples of s t r u c t u r e s T ( t , q , r ) f o r c e r t a i n t , q , r and a l s o non e x i s t e n c e s t a t e m e n t s f o r cer - t a i n t , q , r a r e known ( s . f o r i n s t a n c e Halder , Heise [ 4 1 , Heise 151, Heise , Karzel [ 6 ] ) . Two s t r u c t u r e s T ( t , q , r ) , T ( t l , q l , r l ) a r e c a l l e d isomorphic i f f t h e r e is a b i j e c t i o n ( c a l l e d isomorphism) of t h e s e t o f p o i n t s of T ( t , q , r ) on to t h e se t of p o i n t s of T ( t l , q B , r t ) such t h a t the b locks of t h e first s t r u c t u r e are mapped onto b l o c k s of t h e second s t r u c t u r e . Since two d i s t i n c t p o i n t s of T ( t , q , r ) a r e i n t h e samegmup-

map columns onto columns. Obviously, isomorphic T ( t , q , r ) , T ( t l , q t , r l ) co inc ide i n t h e parameters , 1 . e . t = t l , q = q ' , r=r ' . I n [ l ] we have s t u d i e d cha in geometr ies , The f i n i t e c h a i n geometr ies of Laguerre type ( [ l ] , p . 144) C ( K , L ) are s t r u c t u r e s T ( t , q , r ) . Here K is a Galo is f i e l d GF(Y) and L > K is a f i n i t e l o c a l r i n g wi th L/N 2 K , where N denotes t h e maximal i d e a l of L. The parameters a r e given by t = 3 4 = !=i N , r = ~+1. The c l a s s of c h a i n geometr ies of Laguerre type Z ( K , L ) , L = K [ E ] / < ~ ~ , , c o i n c i d e s w i t h t h e c l a s s of miquel ian Laguerre p l a n e s . Two Laguerre geometr ies C ( K , L ) , C ' ( K ' , L ' ) wi th c h a r K $: 2 4 char K ' a r e isomorphic i f f t h e r e is an isomorphism

i n g i f f t h e r e i s no b lock j o i n i n g them ( n o t e t L 2 ) isomorphisms

0 : L -t L ' such t h a t o l K is an isomorphism o f K on to K 1 ([l], p . 176,

On a Test of Dominance 17

Sa t2 3 . 1 ) . Consider a Galois f i e l d G F ( y ) w i t h 2 ,/ yand put K = G F ( y ) = K , . Let n 2 3 be an i n t e g e r and V be the v e c t o r space of dimension n-1 over K. Define l o c a l r i n g s L : = K[ ' ] /< ,n , and L' : = I ( k , v ) ( k E ~ ,vrv1 w i t h

( k l v l ) + ( k 2 , v 2 ) :=

(kl V 1 ) . ( k 2 , v 2 ) :=

Hence L + L , because of N n - l + 0, ( N 1 ) 2 a r e t he maximal i d e a l s of L , L , . We thus t u r e s ~ ( 3 , ~ - 1 , y + l ) , T I (3 ,yn-1, y+l).

= 0 , where r e s p e c t i v e l y N , N ' g e t two non isomorphic s t r u c -

The s t r a t e g i c decomposition we have announced i n s e c t i o n 1 concerns the fol lowing c l a s s of s t r u c t u r e s T ( t , q , r ) which is a subc la s s of those s t r u c t u r e s def ined i n Halder , Heise [ 4 ] on pages 268, 269 by us ing l i n e a r forms. L e t K be a Galo is f i e l d GF(Y) and l e t V be a vector space over K wi th 1 < dim V < -. For an i n t e g e r t such t h a t 3 2 t < y + 1 now de f ine T ( t , # V , u + l ) as follows: The s e t o f p o i n t s i s given by K ' x V w i t h K ' : = K u : - l and the blocks a r e given by

By us ing Vandermonde's determinant i t is easy t o check t h a t t he pro- p e r t y of balance ( * ) is s a t i s f i e d f o r t .

Theorem 1: L e t A be the Galo is f i e l d G F ( y and l e t f E K[x] be the minimal polynomial of a p r imi t ive element 6 of A over K . Assume V ~ , . . . , V ~ - ~ V . Then the s e t B(vl , ..., v t -1 ) of blocks

t-1)

t-1 t - 1 - v r ( a , v f ( a ) + E ) l a E K ) U I(m,v)l , v E V , v = l v v a

i s a p a r t i t i o n of the s e t of p o i n t s and

is a d i s j o i n t decomposition of B ( t ) . (Not ice that the degree of f is t-1).

Proof . Let v,w be two the two blocks

t ( a , v f ( a ) +

t ( r : , w f ( 6 ) +

d i s t i n c t elements of V. We l i k e t o show t h a t

of B(v l s . . . , v t - l ) have no po in t i n common. Assume t o the con t r a ry tha t (C,x),C E K ' , x E V , is a po in t i n both blocks. This impl ies

5 $ - because of v w. Hence

18 W. Benz

i . e . f ( 6 ) = 0 which i s n o t t r u e s i n c e f is i r r e d u c i b l e ove r K.- c o n t a i n s as many b l o c k s as t h e r e a r e e l emen t s i n

V. The number of p o i n t s on a b l o c k is k K’ = ~ + 1 . Hence B(v1, ..., vt-3 c o n t a i n s (y+l) .# V many p o i n t s and is t h u s a p a r t i t i o r . o f the s e t o f p o i n t s .

Now we l i k e t o show

‘Ihe set E(vl,. . . ,vtdl)

B ( v l I . . . , V ) fl B(wl,. . . , W t-1 t-1 = 8

i n case t h a t t h e two o rde red ( t - 1 ) - p l e t s ( V ~ , . . . , V ~ - ~ ) , ( ~ ~ , . . . , w ~ - ~ ) a r e d i s t i n c t . Assume t o t h e c o n t r a r y t ha t the b l o c k s

a r e e q u a l . T h i s i m p l i e s v=w and hence

t,l t - 1 - u v = l ( w u - v u ) a = o

for a l l a E K.Because of t < Y + 1 there e x i s t p a i r w i s e d i s t i n c t e lements a l , . . . , a s - l i n K . We hence have i n m a t r i x n o t a t i o n

t:l} = The Vandermonde m a t r i x P he re is r e g u l a r because of # ( a l , ..., a = t-1 and by m u l t i p l y i n g t h e m a t r i x e q u a t i o n w i t h P - 1 from t h e r i g h t w e g e t (wl-vl.. .wt-l-vt-l) = 0 which is n o t t r u e . - There are 6cfV) t -1 many sets B ( v l , ..., v t - 1 ) . Every B(v l , . . . , v t - l ) c o n t a i n s #V many b locks . Since t h e r e are qt many b l o c k s i n a s t r u c t u r e T ( t , q , r ) these t

B(v l , ..., v 1 U t-1

c o n t a i n s a l l t h e b l o c k s .

Example. Consider K = GF(3). V = K and t = 3. A r e q u i r e d decomposi- t i o n h e r e is g iven by


aApP bBqQ cCrR

aArR bBpP cCqQ

aAqQ

cCpP bBrR

aBrP bCpQ cAqR

aBqR bCrP CAPQ

aBp Q bC qR cArP

aCqP bArQ cBpR

aCpR bAqP cBrQ

aCrQ bApR cBqP

3. Let q,r,t,n be integers greater than 1 and such that 2 5 t r. Eet P be a set of cardinality qr. The elements of P are called points ( o r objects). Consider moreover n matrices

such that

P = t(l,l,i), ..., (r.l,i), ..., (l,q,i), ..., (r,q,i)] f o r all i=l,.,.,n. A global interaction of MI ,..., Mn is a r-set which is an interaction for all the matrices MI, ... Mn. Two points are called competitors if they are not in a common column for all the matrices M1. ..., Mn. By G(M1, ..., Mn) we denote the set of all global interactions of Ml, ..., M,. We are now interested in a set B(t)c G(M1, ..., Mn) such that the following two conditions are satisfied (the elements of B(t) are called blocks)

(i) Through t distinct points which are parwise competitors there is exactly one block .

(ii) For every integer j with 1 5 j 5 n the following holds true: If D1 is the point intersection of j distinct columns (of MI, ..., I+,) such that no two of them belong to the same M, and if D2 is another such intersection of j columns then # D1 = fc D2.

We denote a structure (MI, ..., Mn; B(t)) by T(t,q,r,n). Conditions (i), (ii) serve as properties of balance.

Example. Assume that four firms are offering each a comparable collection of four wines and that four other firms are offering each a comparable collection of four bottles manufactured to be filled up with wine. The question is to test sinultaneously the quality of the wine collections and that of the bottle collections. We like to do this with a T(3,4,4,2):

20 W. Benz

M1 = M2 = (2J. The columns o f M I r e p r e s e n t t h e f o u r wine c o l l e c t i o n s and the columns of M 2 t h e b o t t l e c o l l e c t i o n s . Now t h e b o t t l e c o l l e c t i o n A,q,c ,R f o r i n s t a n c e is f i l l e d up i n A,q,c ,R with wine of t h e 2 . , 3 . , 1 . , 4 . wine producer r e s p e c t i v e l y . Now check b o t h q u a l i t i e s a long t h e fo l lowing s e t of b locks :

bBqQ rAPd bqDS bBRs Ap aQ BqPd cpSD C a R s rcCQ brRD cBPs CqSd brAQ p a R D aAP s ApSd qCaQ bAsS pBdR aqDP BcpQ c c s s rCdR rcDP.

(This example is t h e miquel ian Minkowski p lane of o r d e r 3 . )

Theorem 2 : Let K be t h e Galo is f i e l d G F ( r ) and l e t n > 1 be an i n t e - g e r . Denote by Ln t h e r i n g K x . . . x K wi th n f a c t o r s . The c h a i n geome- t r y Z ( K , L n ) i s then a s t r u c t u r e T ( 3 , ( u + l ) n - l I ~ + l , n ) .

Proof . a ) Consider t h e fo l lowing maximal i d e a l s

J. := {(kl, ..., kn) E Ln 1 ki = 01

of Ln f o r i = l , . . . , n . A p o i n t of E(K,Ln) is given by

R(pl , P,) := { ( r p l , r p 2 ) I r E R},

where R denotes t h e group of u n i t s of Ln and where p l r p 2 a r e e lements of Ln such t h a t t h e i d e a l genera ted by p l , p 2 is t h e whole r i n g Ln. For t w o p o i n t s P = R(p1,p2) Q = R(q1,q2) w e d e f i n e

b ) The p o i n t s R ( a , b ) with a = ( a l , . . . , a n ) , b = ( b l l a . . , b n ) can be i d e n t i f i e d wi th t h e ordered n - p l e t s ( K ( a l l b l ) , . . . , K ( a n , b n ) ) of p o i n t s K(ai ,bi) of t h e p r o j e c t i v e l i n e n over K . Moreover: R(a ,b) I l iR(c,d) i f f K(ai ,bi)= K(ci ,di) . Hence11 i is an equiva lence r e l a t i o n on t h e s e t of p o i n t s and t h e r e are (y+l)"- l many ordered n-p le t s (PI , ... , P n ) , P l , . . , , P n e n , such t h a t P i = cons t . Thus t h e number of p o i n t s i n an equiva lence c l a s s concerning 1 1 i s ( y + l ) n - I . The number of equiva lence c l a s s e s concerning I l i is u + l s i n c e t h e r e a r e r+l p o i n t s K(a i ,b i ) i n n ,

rows and y t 1 c01um.s. T k matrix Mi can be chosen a r b i t r a r i l y up t o t h e f a c t t h a t t h e columns a r e supposed t o be t h e 1 1 -equivalence c l a s s e s .

n-1 c l We now d e f i n e t h e m a t r i x M i ( i { l l . . . l n l ) wi th ( y + l )

i


d ) Two p o i n t s R(a1, a 2 ) , R ( b l r b 2 ) a r e obviously competi tors

i f f lal a 2 1 c R . - I n cha in geometry [ l ] i t is proved t h a t through three b1 b2 p o i n t s A , B , C such that

the re is exac t ly one cha in and t h a t f o r two d i s t i n c t p o i n t s P .Q the

i n t e r a c t ions (note t h a t any cha in con ta ins u + l points)ofM,, ...&. element of R . Chains a r e hence global

Define the s e t of cha ins t o be the s e t B(3 ) . Then ( i ) holds t r u e f o r t=3. In order t o v e r i f y ( i i ) l e t j be an i n t e g e r w i t h 1 2 j 5 n and l e t i l , . . . , i j ~ t l , . . . ,ni be j d i s t i n c t i n t e g e r s . Consider equivalence c l a s s e s E(il) , ..., E ( i j ) of the relations ( ( i l , ..., IIij r e spec t ive ly . Then

f o r j = l , ..., n. For i f we f i x the components i l , ..., i j i n ( P i , ..., P n ) , Pi 6 n , then the number of the remaining n-p le t s i s (Y +l)n- j .

Remark: A b i j e c t i o n of t he s e t of p o i n t s of a s t r u c t u r e T ( t , q , r , n ) is c a l l e d an automorphism i f f images and inverse images of blocks a re blocks. A s a s p e c i a l case of a theorem of Schaeffer [ l o ] the automorphism group of Z ( K , L n ) is known for Ln semi loca l , #K > 3 and char K#2(inmse t h a t K is f i n i t e , obviously, Ln must be semi loca l ) : This i s the group p r L ~ ( 2 . L ~ ) . - TheZ(K,L2) are the miquelian Minkowski p lanes , K an a r b i t r a r y f i e l d .

Theorem 3: Consider a s t r u c t u r e T ( t , q , r , n ) . Then rn-' must be a d i - v i s o r of q. Moreover: If D is the po in t i n t e r s e c t i o n of j ( l 5 j 5 n ) d i s t i n c t columns (o f M1, ..., M n ) such that no two of them belong t o the same MV then

Proof: The formula is t r u e f o r j=1. Assume now 2 2 j 2 n. Let E, be a column of M, M j : Observe t h a t the C 1 , ..., C r a r e pairwise d i s j o i n t and t h a t t h e i r union i s the whole se t of po in t s . Hence

f o r v=l, . . . , j - 1 and l e t C1, ..., Cr be the columns of

and thus a = r a.. This proves the theorem.

A t the beginning of s ec t ion 3 we r equ i r e n > l f o r a s t r u c t u r e T( t ,q , r ,n ) . But obviously the s t r u c t u r e s T ( t , q , r ) of s e c t i o n 2 can be considered a s s t r u c t u r e s T ( t , q , r , l ) , s ince ( i i ) p lays no r61e i n case n=l.

j-1 J

The s t r u c t u r e s T ( t , q , r , Z ) have been s tud ied ex tens ive ly i n the l i ter- a t u r e i n case q = r . See f o r ins tance the r e s u l t s i n Ceccherini [31 ,

22 W. Benz

Heise, Karzel [ 61, Heise, Quattrocchi [ 71 , Quattrocchi [ 81 , where Minkowski-m-structures are considered. Concerning the real case z ( I R , IR x IR x IR) compare Samaga [ 9 ] .

Theorem 4: Let (M1, ..., Mn; B(t)) be a structure T(t,q,r,n). Then

and the cardinality of the set of all global interactions of MI, ..., Mn is given by

Proof. If b is a point, denote by [b]i the column of Mi through b. Consider t distinct columns C1, ..., Ct of M1.We now like to define sets D1, ..., Dt.Those sets (but not their cardinalities) will depend on certain points al,a2,. .. . Put D1 := C1. In case Du(l <- v ft-1) is defined, take a E D and put v v

n .- \ u ([allp u ... u [a 1 1. D v + l * - %+1 p = 2 V P

We then have

t ff B(t) = ,,il ff Dv)

since the number of blocks must be equal to the number of (bl, . . . ,bt), bi E Ci, such that the b's are pairwise competitors. Put

for V E ( ~ , . . .,ti and I . I E { ~ , ..., nl . For 4 I J ~ we get

and hence by Theorem 3

since the sumands of the right hand side of (0) are pairwise disjoint. Sirnilary we have

for pairwise distinct p 1' - * - 9 Ps in 12, ..., n}, Because of n

cs p:2 B v P Dv =


U-1 n-1 = q . ( l - -) f o r V = 2 , . .. , t . r

( T h i s e s p e c i a l l y i m p l i e s t ha t D f 0.) Hence

1 n-1 t-i n-1 ff B ( t ) = q . q ( l - 7 ) - . . q ( l - -) r

I n o r d e r t o g e t t he c a r d i n a l i t y of t h e s e t of a l l g l o b a l i n t e r a c t i o n s w e determine

Remarks: 1) I n case z(K,L,) w i t h K = GF(Y) and. n > 1 w e g e t f o r t h e number of b l o c k s by Theorem 4

ff B(3) = (6*('i1)

The number o f g l o b a l i n t e r a c t i o n s

[ ( y +1) !

3 n-1 = ( Y -Y) . i n t h i s c a s e i s g i v e n by

2 ) If al,a2 a r e c o m p e t i t o r s o f a s t r u c t u r e T ( t , q , r , n ) t hen acco rd ing t o t h e c o n s t r u c t i o n i n t h e proof o f Theorem 4 t h e r e is a b lock ( n o t i c e t 1_ 2 ) c o n t a i n i n g a l , a 2 . On t h e o t h e r hand: Two d i s t i n c t p o i n t s of a b l o c k must be c o m p e t i t o r s . We t h u s have: Two d i s t i n c t p o i n t s a r e c o m p e t i t o r s i f f t h e r e i s a b l o c k jo inb ig them.

I n Theorem 2 we have p r e s e n t e d s t r u c t u r e s T ( 3 , ( y + l ) ,~+l,n), where n > 1 is an i n t e g e r and where Y is a prime power. We now l i k e t o de- f i n e f u r t h e r s t r u c t u r e s T ( t , q , r , n ) .

n- 1

A . Consider n m a t r i c e s MI, ..., Mn a l l w i t h q > 1 rows and r > 1 columns and a l l f i l l e d up w i t h t h e e l emen t s of P,# P = qr, such that p r o p e r t y

24 W. Benz

(ii) is satisfied. Taking then the set B(r) of all global interac- tims VE get a T(r,q,r,n),

Let H = [hl,.,.,hrl be a set containing r 1. 2 elements. Define P to be the Cartesian product

with n factors for a given integer n > 1. In order to define matrices Mi, i E 11,. . . ,nl , put

P = H x ... x H

:= {(xl, ".,X 1 E p IXi = hv) 'iv n

for the columns Cil,...,Cir of Mi. Then property (ii) is satisfied for the matrices Ml, ..., Mn.- Let t be an integer with 2 5 t 5 r and let r2, ..., rn be sets of permutations of H such that every ri is sharply t-transitive on H. We then like to define a set B(t) of glcb- a1 interactions for MI, ..., Mn : For given permutations n E r , p = 2,. ..,n, call

P P

a block, i.e. an element of B(t). Since hl + h2 implies TI (h ) =TI (hd ( W = 2 ,..., n) we are sure that two distinct elements of b(v2 , . . . , nn) are competitors. Consider now t distinct points (hWrx2V,...,xr,y),

V = l,...,t, such that no two of them are competitors. There is exactly one permutation TI,,E~,, with for v = l , ..., t. There is thus exactly one block, b(n2, . . . ,vn), joining the t points. Hence we get a T(t,rn-l, r,n). The Mathieu groups M ,M12 for instance lead to structures T(4,11n-l , 11,n) (put r2 = ... T(5,12"-l, 12,n) for every n > 1.- In case that H is the projective line over K = GF(r) and r2 = . .. = r n is its projective group we get z(K,Ln) up to an isomorphism.

Theorem 5: Every structures T(t,rn-',r,n), n > 1, can be described as was done before by an r-set H and permutation sets r2,,..,rn on H which are sharply t-transitive on H.

v 1 u

r , , (h,) = x,,,

= Mll) and

Proof. Let H : = {hl ,..., bribe a block of T(t,rn-l,r,n) and let x be an arbitrary point. Put xi := hV iff x,h are in the same column of Mi. We like to show that x is completely'determined by (XI, ..., xn) and that every (yl, ...,yn\yi E H , occurs as a point: This is a consequence of the fact that n columns, no two of them in the same matrix,

intersect in a = - = 1 points. Consider now a block +- 1

n .n-l

(xpl ,... ,x 1 , P=l,...,r, of ~(t,r~-l,r,n). pn

Then for a given j c i 2 , ,..,nl

) E X ( P = I , . ..,r) p j

TI (x j p 1

is a permutation of H. Define r j t o be the set of all 'j stemming from

blocks , j = 2 , . . . , n . I t i s rj3 f o r every joc{2,..,,rd sha rp ly t - t r a n s i - t i v e on H: Consider t d i s t i n c t elements y11, of H and moreover t d i s t i n c t e lementsyl jo , . . . , y t j o of H. L e t

(X pl,...'X ) , P = l , ..., t , pn

be t d i s t i n c t p o i n t s such tha t no two of them a r e compet i tors and such t h a t

x - = Y f o r ~ = l , , . . , t ,

Since there is e x a c t l y one jo in ing b lock t h e r e must be e x a c t l y one n j o fi0 such t h a t

Pl - Y P 1 ' p j o p j 0

f o r p = l , ..., t .

B. Let u s have mat r ices M 1 , . . . , M n as w a s descr ibed a t the beginning of A. Suppose n > 1. Then a l s o Mi,.. .,Mn-1 s a t i s f y (ii) and a s t ruc- t u r e T ( r , q , r , n ) thus l eads t o a s t r u c t u r e T ( r , q , r , n - l ) .

C . Der iva t ion process . Consider a s t r u c t u r e T ( t , q , r , n ) such t h a t t, 3 and r 2 3. Let MI, ..., Mn be the underlying ma t r i ces , P be the po in t s e t and B ( t ) be the s e t o f blocks. Take a po in t p E P . In every m a - t r i x M u cance l t he column [ p l V through p g e t t i n g the mat r ix M: Denote

.

by 4 and cancel i n 4 . Define

PI1 ... [ P I n

every remaining column of M1, ..., Mn t he p o i n t s o f

Then M i , ..., MA t oge the r w i t h

B ' ( t - I ) := E \ [ P I I B E Bp(t)I

n- 1 i s a s t r u c t u r e T ( t - l , q ' , r - l , n ) w i t h q ' = 9 * (r-1) .

.n- 1

Remark. S t a r t i n g w i t h the s t r u c t u r e T ( 3 , ( ~ + 1 ) " - ~ , Theorem 2 , we ge t a s t r u c t u r e T ( 2 , v n Y 1 , Y , n ) by the d e r i v a t i o n proc- e s s . T h i s s t r u c t u r e can be descr ibed as follows: Take the K n , K =

= GF(Y), and de f ine the columns of Mi t o be the hyperplanes X i = = const ( i=l , ..., n ) . The blocks a r e given by the l i n e s

r + l , n ) n > 1, of

D . For i n t e g e r s t ,X,r such t h a t 2 5 t <_ r and X 2 2 we denote by

26 W. Benz

o(t,x,r) a set of functiom

’$’: tl, . . . , rl * t l , . . . , A }

satisfying the following condition:

( * ) For given distinct il, . . . , i t € (1, . . . , rl and f o r given J1, ..., Jt E {l, ..., A 1 there is exactly one YC @ such that %i,) = j, for v = l , ..., t.

Examples:

1) o ( 2 , 2 , 3 ) :

1 2 3 4

Remark. Consider a class of functions @(t,h,r) satisfying ( * ) . Then obviously

with the blocks

b(9) := {(i,q(i))li E 11 , . . . , 1 - 1 1 , Y E o(t,A,r),

is a T(t,A,r,l).- On the other hand consider a T(t,h,r,l). Define to every block


the function : {l,,..,r) + U, ..., X I byVb(i) = ji (i=1 , . . . , r ) . Then it is clear that the class of functions vb satisfies ( * ) .

Let H = {hl,...,hr1 be a set containing r 2 2 elements. Consider integers t,A,n greater than 1 such that 2 5 t 5 r . Let r2, ...,rn be permutation sets which are sharply t-transitive on H and let @(t,X,r) be a class of functions as defined above. We then like to define a structure T(t,Arn-I,r,n). The points are the ordered (n+l)-plets (XI, . . . , xn,w) such that XI,.. .,xnE H and u E A :={l,...,XI . In order to define matrices Mi, i c {1,2, ..., n), put

for the columns Cil, ..., Cir of Mi. Then property (ii) is satisfied for the matrices MI,...,Mn. The set B(t) of blocks is defined as follows: Far given permutations n p c T p , ~ = 2 , ..., n, and f o r a given Q E @ call

a block, i.e. an element of B(t). This construction leads to a T(t,Xn-l,r,n).

n- 1 Theorem 6. Every structure T(t,Xr ,r,n) n > 1, A > 1, can be described as was done before by an r-set H, by permutation sets r2,...,Tn on H which are sharply t-transitive on H and by a function set @(t,A. r ) satisfying ( * ) .

Proof. For given points a,b write a - b iff they are in the same columns # (C1n ... n C,) = X,Ci a column of Mi (i=l,...,n) the equivalence classes contain exactly X points. Let H := {hi, ..., hrl be a block of T(t,AP-l,r,n) and let x be an arbitrary point. Put Xi :=hw iff x,hV are in the same column of Mi. The n-plet (XI, ...,xn) does not determine the point x. But there are X points equivalent to x. We call them (XI, ..., xn,u), u=1, ..., A . We construct the permutation sets r 2 , ..., rn the same way it was done in the proof of Theorem 5. To every block

of M1, ..., Mp. This is an equivalence relation and since

of T(tlAp-l,rtn) we like to associate a function Cp : {l,,..,X}: Put Cp(i) = v p in case xpl = hi. Call 4(t,X,r) the set of all such functions stemming from blocks. Given now distinct il, ..., it E {l, ..., 1-1 and elements jl, ...,jt E {I, ... XI. Let then

(1, ..., d *

be t distinct points such that no two of them are competitors. Since there is exactly one joining block there must be hence exactly one function qin @(t,X,r) such that 9(il) = jl,...,T(it) = jt.

28 W. Benz

Remarks.1) With t h e examples @ ( 2 2 , 3 ) , 4 ( 3 , 2 , 4 ) one can c o n s t r u c t

T(3,2.4"- l ,4 ,n) ( p u t r 2 = ... = r n =: S4) i n case n > 1. s t r u c t u r e s T ( 2 , 2 ~ 3 " - ~ , 3 , n ) ( p u t r2 = ... = rn =: S3) ,

2) Because of the comection o f f u n c t i o n c l a s s e s O ( t , X , r ) and geomet r i e s T ( t , X , r , l ) and because o f t h e non-exis tence o f T ( 3 , X , A + 2 ) ,

f o r A odd and n E N accord ing t o Theorem 6 . X odd, ( s . Heise [ 5 ] ) , t h e r e do n o t e x i s t T(3,A.(X+2)"- l ,A+2,n)

3) By app ly ing t h e d e r i v a t i o n p r o c e s s i t i s e a s y t o v e r i f y t h a t t h e r e do n o t e x i s t T ( t , A , r ) i n c a s e A 2 < ( A - l ) ( r - t + 2 ) . ( F o r i n s t a n c e t h e r e does n o t e x i s t a T ( 3 , 1 0 , 1 3 ) ) . Th i s i m p l i e s by Theorem 6 t h a t t h e r e do n o t e x i s t T ( t , A r n - I , r , n ) i n c a s e nE N and A 2 < ( A - l ) ( r - t + 2 ) .

n- 1 4 ) A s f a r as t h e number of b l o c k s o f a T ( t , A r , r , n ) i s

concerned, Theorem 5 , 6 l e a d t o a new p roof o f Theorem 4: A s h a r p l y t - t r a n s i t i v e pe rmuta t ion s e t on a r - s e t c o n t a i n s r . ( r - I ) . . . ( r - t+r) =

= t! ( r ) many e l emen t s . Hence f

because of # O ( t , X,r) = A t . - T h i s remark does n o t concern t h e number of g l o b a l i n t e r a c t i o n s o f a r b i t r a r y M1,,..,Mn s a t i s f y i n g ( i i ) which is determined by t h e proof of Theorem 4.

Re fe rences

t8l

W . Benz, Vorlesungen iiber Geometrie d e r Algebren. Springer-Verlag, Berlin-New York 1973.

K . A . Bush, Orthogonal a r r a y s o f i ndex u n i t y . Ann. Math. S t a t .

P.V. C e c c h e r i n i , Alcune o s s e r v a z i o n i s u l l a t e o r i a d e l l e r e t i . Rend. Acc. Naz. L i n c e i , 40 (1966) , 218-221.

H.R. Ha lde r , W. H e i s e , Kombinatorik. H a n s e r Ver l ag , Munchen - Wien 1976.

W . He i se , E s g i b t ke inen op t ima len (n+2,3)-Code e i n e r ungeraden Ordnung n . Math. Z . 164 ( 1 9 7 8 ) , 67-68.

W. Heise, H. K a r z e l , Laguerre und Minkowski-m-Strukturen. Rend. 1st. Mat. Univ. Tr ies te I V (1972) .

W . He i se , P . Q u a t t r o c c h i , Survey on Sha rp ly k - T r a n s i t i v e S e t s o f Pe rmuta t ions and Minkowski-m-Structures. A t t i Sem. Mat. F i s . Univ. Modena 27 ( 1 9 7 8 ) , 51-57.

P . Q u a t t r o c c h i , On a theorem of P e d r i n i conce rn ing t h e non-exi- s t e n c e o f c e r t a i n f i n i t e Minkowski-m-structures. Journ. Geom.

23 ( 1 9 5 2 ) , 426-434.

13 (19791, 108-112.

On a Test of Dominance

[ 91 H. -J. Smaga, Dreidimensionale reelle Kettengeometrien. Journ.

[lo] H. Schaeffer, Das von Staudtsche Theorem in der Geometrie der

Geom. 8 (1976), 61-73.

Algebren. J. reine angew. Math. 267 (1974), 133-142.

29

Annals of Discrete Mathematics 30 (1986) 3 1-38 0 Elsevier Science Publishers B.V. (North-Holland) 31

ON n-FOLD B L O C K I N G SETS

A l b r e c h t B e u t e l s p a c h e r a n d F r a n c o E u g e n i

F a c h b e r e i c h M a t h e m a t i k d e r U n i v e r s i t a t M a i n z F e d e r a l R e p u b l i c o f G e r m a n y

I s t i t u t o M a t e m a t i c a A p p l i c a t a F a c o l t a ' I n g e g n e r i a L ' A q u i l a , I t a l i a

An n - f o l d b l o c k i n g s e t i s a s e t o f n - d i s j o i n t b l o c k i n g s e t s . We s h a l l p r o v e u p p e r a n d l o w e r b o u n d s f o r t h e n u m b e r o f c o m p o n e n t s i n a n n- f o l d b l o c k i n g s e t i n p r o j e c t i v e a n d a f f i n e s p a c e s .

INTRODUCTION

A b l o c k i n g s e t o f a n i n c i d e n c e s t r u c t u r e ,a=( P , . y , I ) i s a s e t B o f p o i n t s s u c h t h a t a n y e l e m e n t o f 9 ' ( a n y " l i n e " o r " b l o c k " ) c o n t a i n s a p o i n t o f B a n d a p o i n t o f f B . An n - f d d b l o c k i n g s e t o f .a i s a s e t B = { B , ,Bz ,..., B, 1 P . Any s e t B , i s s a i d t o ~e a c o m p o n e n t o f B . W h i l e b l o c k i n g s e t s h a v e b e e n s t u d i e d f o r a l o n g t i m e ( c f . f o r i s t a n c e [ l ] , 161 , [ 1 2 ] , 1151, [ 1 7 ] , [ 1 8 ] ) , t h e r e a r e n o t many p a p e r s d e a l i n g w i t h n - f o l d b l o c k i n g s e t s .

G e n e r a l i z i n g a t h e o r e m o f H a r a r y [9 ] ( w h i c h w a s a l r e a d y k n o w n t o Von Newmann a n d M o r g e n s t e r n [ 1 9 ] ) , K a b e l l [ 1 4 ] r e c e n t l y , p r o v e d t h e f o l - l o w i n g a s s e r t i o n . (We s h a l l u s e o u r a b o v e t e r m i n o l o g y ) .

RESULT. I f a p r o j e c t i v e p l a n e o f o r d e r q h a s a n n - f o l d b l o c k i n g s e t , t h e n n s q - I . Any n - f o l d b l o c k i n g s e t o f a n a f f i n e p l a n e o f o r d e r q s a t i s f i e s n 5 q-2.

I n S e c t i o n 2 we s h a l l p r o v e a t h e o r e m w h i c h u n i f i e s , g e n e r a l i z e s a n d i m p r o v e s t h i s r e s u l t . L a t e r o n , w e s h a l l c o n s i d e r b l o c k i n g s e t s i n p r o j e c t i v e a n d a f f i n e s p a c e s . L e t B b e a p r o j e c t i v e o r a f f i n e s p a c e . A s e t B o f 2 i s s a i d a t - b l o c k i n g s e t i f a n y t - d i m e n s i o n a l s u b s p a c e o f H c o n t a i n s a t l e a s t a p o i n t o f B a n d a p o i n t o f I-B. A s e t B = { B, ,..., B , } o f n m u t u a l l y d i s j o i n t t - b l o c k i n g s e t s i s s a i d a n n - r x d t - b l o c k i n g s e t o f P . I n S e c t i o n 3 w e s h a l l d e a l w i t h t h e m a x i m a l n u m b e r n o f c o m p o n e n t s o f a n n - f o l d t - b l o c k i n g s e t i n P G ( r , q ) o r A G ( r , q ) . We s h a l l p r o v e u p p e r a n d lower b o u n d s f o r t h i s m a x i m a l n u m b e r .

I n S e c t i o n 4 we s h a l l c o n s t r u c t e x a m p l e s o f n - f o l d b l o c k i n g s e t s . I n p a r t i c u l a r , w e s h a l l p r o v e t h e f o l l o w i n g f a c t : G i v e n a p o s i t i v e i n t e g e r n , t h e r e i s a n i n t e g e r q,, s u c h t h a t a n y p r o j e c t i v e o r a f f i n e p l a n e o f o r d e r q zqo h a s a n n - f o l d b l o c k i n g s e t .

We w a n t t o r e m a r k t h a t we u s e t h e w o r d " n - f o l d b l o c k i n g s e t " i n a l i g h t l y d i f f e r e n t m e a n i n g a s H i l l a n d M a s o n [ l o ] ,

o f n m u t u a l l y d i s j o i n t b l o c k i n g s e t s o f

32 A . Beutelspacher and F. Eugeni

2. B L O C K I N G SETS IN STEINER SYSTEMS.

We b e g i n w i t h t h e f o l l o w i n g

2 . 1 THEOREM, Let S b e a n S ( 2 , k , v ) S t e i n e r s y s t e m . I f S a d m i t s a n n - f o l d b l o c k i n g s e t , t h e n n s k - 2 .

PROOF. D e n o t e b y a n n - f o l d b l o c k i n g s e t o f S . C o n s i d e r a p o i n t x o u t s i d e a c l a s s e Bi , S i n c e a n y l i n e t h r o u g h x i s i n c i d e n t w i t h a t l e a s t o n e p o i n t o f Bi , i t r e s u l t s IB, I 5 r , w h e r e r = ( v - l ) / ( k - 1 ) i s t h e n u m b e r o f l i n e s t h r o u g h x . S u p p o s e we h a v e IB i l - = r . T h e n a n y l i n e t h r o u g h a p o i n t x o u t s i d e Bi meets B, i n j u s t o n e p o i n t . I n o t h e r w o r d s , a n y l i n e j o i n i n g two p o i n t s o f Bi i s t o t a l l y c o n t a i n e d i n B i . S i n c e B i {B , , , . , , B n } c a n n o t b e a n n - f o l d b l o c k i n g s e t , S o , I B i I 2 r + l f o r a n y 1 E 1 , 2 , .. . , n I . ( T h i s i s a l s o a c o n s e q u e n c e o f T h e o r e m 1 i n 1 7 1 ) . T h e r e f o r e ,

{ B, ,..., B,}

h a s a t l e a s t t w o p o i n t s ,

n

i - 1 v = EIB, I z ( r + l ) n .

On t h e o t h e r h a n d , we h a v e v-1 = r ( k - 1 ) . T o g h e t h e r w e g e t

n ( r + l ) s v = r ( k - l ) + l

H e n c e n s k - l - ( k - Z ) / ( r + l ) < k - 1 a n d so n s k - 2 .

EXAMPLES. T h e r e e x i s t : n - f o l d b l o c k i n g s e t s i n some S ( 2 , k , v ) w i t h n = k - 2 : T h e p r o j e c t i v e p l a n e o f o r d e r 3 a n d t h e a f f i n e p l a n e o f o r d e r 4 h a v e b l o c k i n g s e t s ; t h e s e b l o c k i n g s e t s f o r m , t o g e t h e r w i t h t h e i r r e s p e c t i v e c o m p l e m e n t s , 2 - f o l d b l o c k i n g s e t s . A l s o , t h e p r o j e c t i v e p l a n e o f o r d e r 4 a d m i t s a p a r t i t i o n i n 3 Baer s u b p l a n e s . T h i s i s a 3 - f o l d b l o c k i n g s e t .

We r e m a r k a l s o t h a t n o S ( 2 , 4 , v ) S t e i n e r s y s t e m h a s a 3 - f o l d b l o c k i n g s e t . ( A s s u m e t o t h e c o n t r a r y t h a t a n S ( 2 , 4 , v ) h a s a 3 - f o l d b l o c k i n g s e t ( B , ,B, , B 3 1 . S i n c e a n y Bi i s a b l o c k i n g s e t a n d s i n c e b y [18, ( 2 . 1 3 ) ] a n y b l o c k i n g s e t i n S ( 2 , 4 , v ) h e s a t l e a s t ( v - h ) / 2 p o i n t s , we h a v e 3 ( v - f i ) / 2 r v , i . e . v 5 9 , a c o n t r a d i c t i o n . ) A s i m i l a r a r g u - m e n t a t i o n h o l d s i n S ( 3 , 4 , v ) s i n c e ( c f . 118)) i f a b l o c k i n g s e t t h e r e e x i s t s t h e n i t c o n t a i n s e x a c t l y v / 2 p o i n t s .

Now, w e c o n s i d e r p r o j e c t i v e p l a n e s . T h e f o l l o w i n g r e s u l t i s a s u b - s t a n t i a l i m p r o v e m e n t of K a b e l l ' s r e s u l t .

2 . 2 THEOREM. D e n o t e b y p a p r o j e c t i v e p l a n e o f o r d e r q . a n d l e t B = { B, , . . . , B , } b e a n n - f o l d b l o c k i n g s e t o f P . T h e n n s q - S p + l w i t h e q u a l i t y i f a n d o n l y i f a n y B i i s a B a e r s u b p l a n e .

PROOF. Any c l a s s Bi E B i s a b l o c k i n g s e t i n t h e u s u a l s e n s e . H e n c e , t h e t h e o r e m o f B r u e n [ 6 ] i m p l i e s l B i l t q + S q + l . H e n c e

n ( q t f i t 1 ) s q * + q + l = ( q + G + l ) ( q - s q + l )

i m p l y i n g t h e i n e q u a l i t y o f a s s e r t i o n . If n = q - @ + l , t h e n a n y Bi h a s p r e c i s e l y q + \ r q + l p o i n t s . A g a i n u s i n g B r u e n ' s r e s u l t , B i i s a B a e r s u b p l a n e . T h e o t h e r d i r e c t i o n i s t r i v i a l ,

We r e m a r k t h a t a n y c y c l i c (so, i n p a r t i c u l a r , a n y d e s a r g u e s i a n ) p r o - j e c t i v e p l a n e of s q u a r e o r d e r h a s a p a r t i t i o n i n B a e r s u b p l a n e s ( s e e H i r s c h f e l d 1111 4 . 3 . 6 ) . S o , t h e b o u n d i n T h e o r e m 2 . 2 i s s h a r p ,

On n-Fold Blocking Sets 33

3 . THE PROJECTIVE A N D AFFINE CASE

L e t 2 b e a n r - d i m e n s i o n a l p r o j e c t i v e s p a c e o f o r d e r q . I n t h i s S e c t i o n , we a r e i n t e r e s t e d i n t h e q u e s t i o n , how many c o m p o n e n t s a n n - f o l d t - b l o c k i n g s e t o f H c a n h a v e . C l e a r l y , t h e e x i s t e n c e o f a n n - f o l d t - b l o c k i n g s e t i m p l i e s t h e e x i s t e n c e o f a n m - f o l d b l o c k i n g s e t f o r a n y m s n . T h e r e f o r e , we may d e f i n e n p = n p ( t , r . I ) a s t h e g r e a t e s t i n t e g e r w i t h t h e p r o p e r t y t h a t L h a s a n n - f o l d t - b l o c k i n g s e t f o r a n y n s n p ( t , r , 2 ) . I f H i s d e s a r g u e s i a n o f o r d e r q , t h e n w e w r i t e a l s o 5 ( t , r , q ) i n s t e a d o f n p ( t , r , 2). S i m i l a r l y , t h e f u n c t i o n s na ( t , r , A ) a n d n , ( t , r , q ) f o r a n a f f i n e s p a c e A o f d i m e n s i o n r a n d o r d e r q a r e d e f i n e d .

I n 2 . 2 . T h e o r e m w e h a v e a l r e a d y s h o w n t h a t n p ( 1 , 2 , q ) < q - q + I . Now we s h a l l d e a l w i t h t h e h i g h e r d i m e n s i o n a l case . F i r s t , w e s h a l l s t a t e s o m e e a s y - t o - p r o v e u p p e r b o u n d s f o r n p . By ? ( q ) = q ' + ...+ 1 w e d e n o t e t h e n u m b e r o f p o i n t s i n P G ( r , q ) .

o n i y i f r-

PROOF. Let ( a ) B Y ("2

B = (B, ,..., B n } be a n n - f o l d t - b l o c k i n g s e t i n P G ( r , q ) . , 2 . 1 1 ) , a n y t - b l o c k i n g s e t B i i n P G ( r , q ) s a t i s f i e s

T h e r e f o r e , n

f i r ( q ) 2 ,r IBi I 2 n [ i + r - t ~ ) + ~ i+r - t - l (q) ] * I - I

( b ) S i n c e r < 2 t , by [41 , a n y t - b l o c k i n g s e t B , i n P G ( r , q ) h a s a t l e a s t a , - , ( q ) . p o i n t s , e q u a l i t y h o l d s i f a n d o n l y i f B, i s t h e p o i n t s e t o f a n ( r - t ) - d i m e n s i o n a l s u b s p a c e . T h e r e f o r e :

n

I f e q u a l i t y h o l d s , t h e n .9. ( 9 ) d i v i d e s a r ( q ) , w h i c h i m p l i e s t h a t r - t - l l r + l , a n d s o r - t + l l t . r - t

REMARK. S u p p o s e r < 2 t a n d r - t + l l t . T h e n i n P G ( r , q ) a t o t a l ( r - t ) - s p r e a d ( s e e [8] ) i s a n n - f o l d t - b l o c k i n g s e t w i t h n = # ( q ) / 6 ( q ) . r r-t

S i m i l a r l y , t h e f i r s t a s s e r t i o n o f t h e f o l l o w i n g t h e o r e m f o l l o w s .

THEOREM. ( a ) I f H ( t , r , q ) d e n o t e s t h e m a x i m a l c a r d i n a l i t y of a

P u t r = a ( r - t t l ) t b , w h e r e a a n d b a r e i n t e g e r s y i t h a > O a n d t i a l t - s p r e a d i n P G ( r . q ) , t h e n n p ( t , r , q ) z M ( r - t , r , q ) .

b 5 r - t - 1. T h e n

i = l PROOF. By (131, T h e o r . 4 . 2 ) , t h e r e e x i s t s a p a r t i a l ( r - t ) - s p r e a d w i t h t h e c a r d i n a l i t y i n q u e s t i o n .

Now, we c o n s i d e r t h e c a s e r = 2 t , .

3 . 3 T h e o r e m . D e n o t e by P = P G ( P t , q ) t h e p r o j e c t i v e s p a c e o f d i m e n - s i o n 2 t 2 4 a n d o r d e r q . S u p p o s e t h a t q i s a s q u a r e . D e f i n e t h e p o s i - t i v e i n t e g e r s b y s = [ t + 2 / 2 ] . T h e n

n ( t * 2 t s q ) 2 # * 2 ( t - - s + l ) -h qt-s.

P PROOF. L e t B b e a s u b s p a c e o f d i m e n s i o n 2 ( t - s t l ) o f P. ( N o t e t h a t i n v i e w o f t L 2 , t h e d e f i n i t i o n o f s i m p l i e s t - s & O . ) D e n o t e by R a c o m p l e m e n t - o f B i n €'.Then R i s a s u b s p a c e o f d i m e n s i o n 2 t - 2 ( t - s t l ) - l = 2 s - 3 . L e t B,, ..., B,, b e a p a r t i t i o n o f B i n B a e r s u b s p a c e s o f d i - m e n s i o n 2 ( t - s + l ) . ( I t i s w e l l k n o w n t h a t s u c h a p a r t i t i o n e x i s t s ; f o r a p r o o f see f o r e x a m p l e [ l l] 4 . 3 . 6 T h e o r . ) T h e n

S i n c e a n ( s - 2 ) - s p r e a d o f R h a s e x a c t l y q s - ' e l e m e n t s , b y o u r h y p o t h e - s i s s ; : ( t + 2 ) / 2 , t h e r e e x i s t s u b s p a c e s R , , ..., R n o f R of d i m e n s i o n s - 2 w h i c h a r e m u t u a l l y s k e w , C o n s i d e r t h e " B a e r c o n e s "

( i = 1 , . . . , n ) . %i = % ( R , , B i ) = X E ~ i ( X , R i ) t U

B y [13] i t f o l l o w s i n p a r t i c u l a r t h a t t h e s e B a e r c o n e s a r e t - b l o c k i n g s e t s . S i n c e B a n d R a r e s k e w , t h e s e t s $f, a r e m u t u a l l y d i s j o i n t . H e n c e n ( t , 2 t , q ) > n .

EXAMPLE. T h e o r e m s 3 . 1 a n d 3 . 3 i m p l y f o r i n s t a n c e P

q- G t l I np ( 2 , 4 , q ) 5 q2 -q S q t q - S q + l ,

i f t h e p r i m e p o w e r q i s a s q u a r e .

Now we c o n s i d e r t h e a f f i n e _c_a_s~.

3 . 4 T H E O R E M . n , ( t , r , q ) s q r [ ( t + i ) q r - t - t I-'. PROOF. Any c o m p o n e n t B, o f a n n - f o l d t - b l o c k i n g s e t o f A = A G ( r , q ) ( w i t h n t 2 ) i s a t - b l o c k i n g s e t o f A. So , by [ 2 1 , Cor 2 . 2 3 , w e h a v e

I B i I 2 ( t t l ) q r - t - t ,

H e n c e , t h e a s s e r t i o n f o l l o w s .

As a c o n s e q u e n c e w e h a v e

3 . 5 C O R O L L A R Y . n a ( l , r , q ) s q / 2 .

I t i s wel l k n o w n [ 1 5 ] t h a t t h e r e e x i s t s a f u n c t i o n b p = b ( t , q ) ( a n d a f u n c t i o n b , = b , ( t , q ) ) s u c h t h a t t h e r e e x i s t s a t - b l o c e t i n g s e t i n P G ( r , q ) ( o r A G ( r , q ) ) if a n d o n l y i f rib, ( o r r l b , , r e s p e c t i v e l y ) . T h e s e f u n c t i o n s h a v e b e e n c a l l e d t h e Mazzocca-~Ta.llln_i_f.u~nI_nctio_ns_. By [18] we h a v e b , ( t , q ) < _ b p ( t , q ) . I f a p r o j e c t i v e o r a f f i n e s p a c e c o n t a i n s a n n - f o l d t - b l o c k i n g s e t ( w i t h n > 2 ) , t h e n i t h a s a l s o a t - b l o c k i n g s e t . C o n s e q u e n t l y , t h e r e e x i s t f u n c t i o n s bp ( n , t , q ) a n d b a ( n , t , q ) s u c h t h a t P G ( r , q ) ( o r A G ( r , q ) ) c o n t a i n s a n n - f o l d t - b l o c - k i n g s e t i f a n d o n l y if r n . N o w w e p r o v e a g e n e r a l i z a t i o n o f t h e m e n t i o n e d t h e o r e m o f T a l l i n i .

3.6 THEOREM. b a ( n , t , q ) < _ b p ( n , t , q ) .

PROOF. A s s u m e t o t h e c o n t r a r y t h a t b, > _ b p t l . F i x i n P = P G ( b p t 1 , q ) a h y p e r p l a n e H , a n d d e n o t e by B a n n - f o l d t - b l o c k i n g s e t of H . By our a s s u m p t i o n , t h e r e e x i s t s a n n - f o l d t - b l o c k i n g s e t B' i n t h e a f f i n e s p a c e P-H. C o n s e q u e n t l y BuB' w o u l d b e a n n - f o l d t - b l o c k i n g s e t

-

On n-Fold Blocking Sets 35

o f P =PG ( bp + 1 , q ) , a c o n t r a d i c t i o n .

We f i n i s h t h i s S e c t i o n b y d e t e r m i n i n g a p a r t i c u l a r c l a s s of v a l u e s b p ( n , t , q ) , w h e n q i s a s q u a r e .

3 . 7 PROPOSITION. I f t h e p r i m e - p o w e r q i s a s q u a r e , t h e n

bp ( q- Sq+ 1 , 1 , 9 ) = 2.

PROOF. I t i s known t h a t i n t h e d e s a r g u e s i a n p r o j e c t i v e p l a n e P G ( 2 , q ) t h e r e e x i s t s a n ( q - S q + l ) - f o l d b l o c k i n g s e t . A s s u m e t h a t P G ( 3 , q ) c o n - t a i n s a ( q - S q + l ) - f o l d 1 - b l o c k i n g s e t . T h e n , by 3 . l ( a ) ,

q - q + l = n s (q3 + q 2 + q + l ) . ( q 2 + q f i + q + ~ + l ) - l < q - f i + l , a c o n t r a d i c t i o n .

4. EXAMPLES

I n t h i s S e c t i o n , w e s h a l l c o n s t r u c t some e x a m p l e s o f n - f o l d t - b l o c - k i n g s e t s i n p r o j e c t i v e a n d a f f i n e s p a c e s . F i r s t , we s h a l l d e a l w i t h p r o j e c t i v e s p a c e s o f s m a l l o r d e r ,

4 . 1 . THEOREM, L e t r b e a p o s i t i v e i n t e g e r w i t h r 2 3. ( a ) n p ( l , r , 3 ) = 0 ( b ) n p ( l , r , 4 ) 2 2 ( c ) n p ( l , r , 5 ) 5 3 ( d ) n p ( l , r , 7 ) 2 4 ( e l n p ( l , r , 8 ) 5 5

PROOF. ( a ) B y [ 1 7 ] , t h e r e d o e s n o t e x i s t a 1 - b l o c k i n g s e t i n P G ( r . 3 ) f o r r 5 3 ; ( b ) a n d ( c ) f o l l o w s i m m e d i a t e l y f r o m 3.1 ( a ) . N o w we p r o v e ( d ) . A b l o c k i n g s e t o f P G ( 2 , 7 ) h a s a t l e a s t I 2 p o i n t s ( c f . [ l l ] 1 3 . 4 . 8 T h . ) . S u p p o s e t h a t S i s a n i r r e d u c i b l e b l o c k i n g s e t i n P G ( r , 7 ) . H e n c e , V x E S t h e r e e x i s t s a t l e a s t a l i n e L w i t h I L n S l = l E a c h o f t h e ( 7 r - 1 - 1 ) / 6 p l a n e s t h r o u g h L c o n t a i n s a t l e a s t 11 p o i n t s o f S - i x l . S o , IS1 2 1 + 1 1 ( 7 r - ' - 1 ) / 6 , T h e n

n p ( l , r , 7 ) 5 ( 7 ' + ' - 1 ) / ( 1 1 - 7 r - 1 - 5 ) < 5 .

N o w w e p r o v e ( e ) . A b l o c k i n g se t of PG(2,8) h a s a t l e a s t 13 p o i n t s ( c f . 1111 1 3 . 4 . 9 T h . ) . By a s i m i l a r a r g u m e n t a t i o n a s a b o v e , i t f o l - l o w s IS1 2 1 + 1 2 ( 8 ' - ' - 1 ) / 7 . So

r t l n p ( l , r , 8 ) a ( 8 - 1 ) / ( 1 2 8 r - ' - 5 ) < 6 .

4 . 2 THEOREM. Any p r o j e c t i v e p l a n e nq o f o r d e r q B 7 h a s a 3 - f o l d b l o c k i n g s e t .

PROOF. C o n s i d e r t h r e e l i n e s a , b , c t h r o u g h a p o i n t C , . F i x t h e p o i n t s A , , A, E a - {C,) a n d B, , B , E b- I C , t . D e f i n e

S:=A, B, f l c , C, : = A , B z fl c 9 Q, : = A , B, fl AZB, 1

Q, : = A , B, fl A, B, , T, : = C , Q, fl C, B, . F i n a l l y , f i x a p o i n t T, o n B,C,- I B,, C,) w h i c h is n o t o n a . T h e n

J3, := a U b U I Q , ,Qz) - ( A l , A, , B l , €3, t

a r e d i s j o i n t b l o c k i n g s e t s . S i n c e A , , B , # _B, U _B2 a n d q 27, a l s o _B, U _Bp f o r m s a b l o c k i n g s e t . H e n c e &,B2 a n d t h e c o m p l e m e n t o f f o r m a 3 - f o l d b l o c k i n g s e t of nq.

N o w w e s h a l l d e a l w i t h t h e f o l l o w i n g q u e s t i o n . G i v e n a p o s i t i v e i n t e g e r n , i s t h e r e a n i n t e g e r qo s u c h t h a t a n y p r o j e c t i v e p l a n e o f o r d e r q z q , h a s a n n - f o l d b l o c k i n g s e t ? T h e f o l l o w i n g t h e o r e m a n s w e r s t h i s q u e s t i o n i n t h e a f f i r m a t i v e .

4 . 3 THEOREM, I f 3tq d e n o t e s a p r o j e c t i v e p l a n e o f o r d e r q , t h e n n q h a s a n n - f o l d b l o c k i n g s e t w i t h I n o t h e r w o r d s , n p ( l , 2 , q ) 2 ' d ( q + 2 4 ) 4 . PROOF. L e t n b e a p o s i t i v e i n t e g e r w i t h n s31/ i ;24) /4 ' . F i x a l i n e 2 i n r$ , a n d l e t PI ,? !..., Pn b e n p o i n t s o n S t e p 1. F o r a n y i ( 1 - 1s n ) t h e r e a r e t w o a , , b , 4 1 t h r o u g h P, s u c h t h a t t h r o u g h n o p o i n t o f n, p a s s t h r e e of t h e l i n e s a , , a , ,..., a n ,

n s a . d ; q 2 4 ) / Z .

I .

. . b, sbz t - . . t b n i-1 I n o r d e r t o c h o o s e a i a n d bi , a t most 4 ( j - 1 ) = 2 ( i - l ) ( i - Z ) of i -1 t h e a l i n e s d i f f e r e n t f r o m 1 t h r o u a h P; a r e f o r b i d d e n . B v t h e h y p o i h e s i s 2 ( n - l ) ( n - 2 ) 5 q - 2 , h e n c e f o r e v e r y i s n o n e c a n f i n d t w o l i n e s a i , bi s a t i s f y i n g t h e a s s u m p t i o n .

Now, we d e f i n e f o r a n y i w i t h 1 s is n

B ; . . * b i n b , , a i fl b,-l ,..., a i n b, , b i n a i - , , . . . , b i n a,

= \ a , U b i - a i fl a i + , , . . . , a i n a , , bi fl b i + , , . , . I

S t e p 2. Any l i n e ai o r b i c o n t a i n s a p o i n t o f e a c h o f t h e s e t s

N a m e l y : L e t j b e a n i n t e g e r w i t h 11 j c n . I n o r d e r t o s h o w t h a t a i a n d bi I f j > i , t h e n a i c o n t a i n s t h e p o i n t a i fl a , 5 Bit , a n d bi i s i n c i d e n t w i t h bi fl bi EE,' , On t h e . o t h e r h a n d , i f j < i , t h e n bj fl a i l i e s i n g / n a i , a n d b i p a s s e s t h r o u g h a i n bi E B_! . Now, we e m b e d d _B: i n a b l o c k i n g s e t _ B i . I n v i e w o f S t e p 2 , we m u s t a d j o i n p o i n t s i n o r d e r t o b l o c k e x a c t l y 4 ( n - 1 ) ' l i n e s c o n n e c t i n g a p o i n t o f bi w i t h a p o i n t o f b , , Any s u c h l i n e s h a l l c o n t a i n j u s t o n e p o i n t o f g , . M o r e o v e r , we w a n t t o d o t h i s i n s u c h a w a y t h a t t h e r e s u l t i n g b l o c k i n g s e t s B 1 , B 2 , ...,En a r e m u t u a l l y d i s j o i n t . C l e a r l y , El; c a n b e e n l a r g e d t o a b l o c k i n g s e t _B i i n t h e way d e s c r i b e d a b o v e . N o w , l e t i b e a n i n t e g e r w i t h l < i i n , a n d s u p p o s e t h a t t h e b l o c k i n g s e t s B, ,_Bz,. . . , E n h a v e b e e n c o n s t r u c t e d . On a n y l i n e x w h i c h i s n o t b l o c k e d b y Flf , t h e r e a r e a t most

2 2 2 i + ( i - l ) 4 ( n - l ) 5 2 n + ( n - 1 ) 4 ( n - 1 )

a ; 2; 1 . 0 0 9 % *

h a v e a p o i n t of B i , we may a s s u m e t h a t j + i.

I

= 4 n 3 - 1 2 n 2 + 4 n - 4

p o i n t s w h i c h a r e c o n t a i n e d i n G , , g t ,..., Bi-, . By o u r h y p o t h e s i s t h i s n u m b e r i s a t m o s t

q + 2 4 - 1 2 n 2 + 1 4 n - 4 = q-( 1 2 n 2 - 1 4 n - 2 0 ) < - q .

T h e r e f o r e , i t i s p o s s i b l e t o a d j o i n a p o i n t X t o i n o r d e r t o b l o c k t h e l i n e x .

T h e r e m a i n d e r o f t h i s S e c t i o n i s d e v o t e d t o t h e a f f i n e p l a n e - c a s e .

4 . 4 THEOREM. n , ( 1 , 2 , a q ) < [ n p ( 1 , 2 , n q ) - 1 ] / 2 , w h e r e aq i s a n a f f i n e

On n-Fold Blocking Sets 3 7

p l a n e o f o r d e r q w i t h p r o j e c t i v e c l o s u r e n,.

PROOF. L e t B = {g, ,B,,...B,j b e a n n - f o l d b l o c k i n g s e t i n j e c t i v e p l a n e nq. D e f i n e

t h e p r o -

B' = { Ei u Bz ,B,u -B, * . . . s _ B n - l u -B,} i f n i s e v e n , a n d

A *

i f n i s o d d . T h e n B' i s a A s e t o f m ~ ( n - 1 ) / 2 s e t s El , g , ,.,.,En w i t h t h e p r o p e r t y t h a t a n y s e t Bj h a s a t l e a s t 2 p o i n t s o n a n y l i n e o f JCs .

C o n s e q u e n t l y , B' i n d u c e s a n n - f o l d b l o c k i n g s e t i n a, ,

A s c o r o l l a r i e s w e h a v e t h e f o l l o w i n g t w o t h e o r e m s ,

4 . 5 THEOREM. L e t a , b e t h e d e s a r g u e s i a n a f f i n e p l a n e o f o r d e r q . I f q i s a s q u a r e , t h e n n , ( 1 , 2 , a q ) 2 ( 9 - S q ) / 2 .

PROOF. By t h e r e m a r k a f t e r 2 . 2 T h e o r e m , n p ( 1 , 2 , n,) = ( q - G + l ) / Z .

4 . 6 THEOREM. Any a f f i n e p l a n e o f o r d e r q h a s a n n - f o l d b l o c k i n g s e t w i t h n z ,f( q / 2 4 + 1 )' -1 / 2 .

T h e p r o o f f o l l o w s b y 4 . 4 a n d 4 . 3 .

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[ 2 ] Berardi,L.,Beutelspacher, A . , a n d E u g e n i , F . , O n ( s , t ; h ) - b l o c k i n g s e t s i n f i n i t e p r o j e c t i v e a n d a f f i n e s p a c e s , A t t i Sem. M a t . F i s . M o d e n a 31 ( 1 9 8 3 ) 1 3 0 - 1 5 7 .

[3 ] B e u t e l s p a c h e r , A . , P a r t i a l s p r e a d s i n f i n i t e p r o j e c t i v e s p a c e s a n d p a r t i a l d e s i g n s , M a t h . 2. 145 ( 1 9 7 5 ) 2 1 1 - 2 3 0 .

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[ l l ] H i r s c h f e l d , J.W.P., P r o j e c t i v e g e o m e t r i e s o v e r f i n i t e f i e l d s ( C l a r e n d o n P r e s s , O x f o r d 1 9 7 9 ) .

[12] H o f f m a n n , A . J . , a n d R i c h a r d s o n , M . , B l o c k d e s i g n g a m e s , C a n a d . J . M a t h . 13 (1961) 1 1 0 - 1 2 8 .

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[ 1 7 ] T a l l i n i , G . , k - i n s i e m i e b l o c k i n g s e t s i n P G ( r , q ) e i n A G ( r , q ) , Q u a d e r n o n . 1 1 s t . M a t . A p p l i c a t a U n i v . L' A q u i l a ( 1 9 8 2 ) .

[I81 T a l l i n i , G., B l o c k i n g se t s n e i s istemi d i S t e i n e r e d - b l o c k i n g se t s i n P G ( r , q ) e A G ( r . q ) , Q u a d e r n o n . 3 1 s t . M a t . A p p l i c a t a U n i v . L' A q u i l a (1983).

[19] Von Newmann, J . , a n d M o r g e s t e r n , O . , T h e o r y o f g a m e s a n d e c o n o - m i c b e h a v i o u r ( U n i v e r s i t y P r e s s , P r i n c e t o n 1 9 7 2 ) .


EMBEDDING FINITE LINEAR SPACES IN PROJECTIVE PLANES

Albrecht Beutelspacher and Klaus Metsch

Fachbereich Mathematik der Universitat Saarstr. 21 D-6500 Mainz

Federal Republic of Germany

It is shown that a finite linear space in which all points have degree n+l can be embedded in a projective plane of order n, provided that the line sizes are big enough.

INTRODUCTION

A linear space is an incidence structure S = (p,L,I) of points, lines and incidences such that any two distinct points are on a unique line and any line contains at least two points. A great variety of linear spaces can be obtained as follows. Let P be a projective plane and denote by x a set of points of P with the property that outside x there are some non-collinear points. Then the incidence structure S = P-x whose points are the points of P outside x and whose lines are the lines of P which have at least two points outside x is a linear space. Any linex space isomorphic to such an P-x is called embeddable in the projective plane P.

An old result of M. HALL [ 4 ] asserts that any linear space can be embedded in a projective plane, which is usually infinite. There is a conjecture dating back to the time of M. HALL’S paper which says that any finite linear space can be embedded in a finite projective plane. This seems to be an extremely difficult problem.

We want to deal, however, with an even more difficult question. For a finite linear space S denote by n+l the maximal number of lines which pass through a common point; the so-defined integer n is called the ordar of S. Clearly, the order of any projective plane in which S is embedded is at least n. The question we are interested in is the following: When can a finite linear space of order n be embedded in a projective plane of the same order n ? One of our answers can be formulated as follows (cf. corollaries 3 . 3 and 3 . 4 below).

Let S be a finite linear space in which any point has degree n+l. Denote by n+l-a the minimal number of points on a line. If

1 n > s( a’ -1) ( 5aZ -3a+20),

then S is embeddable in a projective plane of order n.

In other words: For a given a, there is at most a finite number of such linear spaces, which cannot be embedded in a projective plane of order n.

As corollaries we obtain the theorem of THAS and DE CLERCK [6] on the embedding of the pseudo-complement of a maximal arc and the theorem of MULLIN and VANSTOPJE [5] on the embedding of the pseudo-complement of a pencil of lines.

This paper contains the main results of the second author’s Diplom- arbeit.

40 A . Beutelspacher and K. Metsch

1. THE MAIN IDEA

Throughout this section, S = ( p r L , I ) denotes a finite incidence structure consisting of points, lines and incidences such that any two distinct points are incident with at most one common line. For a point p (or a line L ) , its degree is the number r (or kL) of lines through p (or points on L, respectively.) P

Two lines are called paraZle1, if they have no point in common. For two distinct lines L1 and h(L ,L2) denotes the number of lines parallel to both, ani2'L2. hhroughout, we denote the number of points of S by v,L&he number of lines of S by b. Also, S has maximal point degree n+l, and z is the integer defined by z = b - (n'+n+l). The theorems which we are going to prove in this section will play a crucial role in the embedding of linear spaces in projective planes. Our method generalizes the method introduced by BRUCK [ 3 ] and BOSE

THEOREM 1.1. Fix a line H of S , and let a, c, d, e and x be integers with the following properties: (1) The degree of H is n+l-d < n+l. (2) The number of lines parallel to H is nd+x > 0. ( 3 ) For every parallel L of H we have n+a 5 h(L,H) 5 n+c. ( 4 ) For any two intersecting lines Llf L2 parallel to H we have

[21.

h(L1,L2) 5 e. ( 5 ) 2n > (d+l)(de-d-2a-2) + 2x. ( 6 ) n > (2d-l)(c+l) + e - 1 - 2x. (7) At least one of the following assertions is true:

(i) On every parallel of H there is a point of degree n+l, (ii) n(d-s) > s(c+l) - x for every integer s with 0 5 s 5

Then we have:

(a)There is an integer m with the following property: If M1,.

Mt are all maximal sets of mutually parallel lines with

H € hi and lmil m,

or d-1.

. I

then t = d and every line parallel to H is contained in exactly one Mi. (If d 2, we can take m = n - (d-l)(c+l) + x + 1.) (b) If S is a linear space such that any point outside H has degree n+l, then the sets M . are parallel classes (i.e., every M . induces a partition of the point set of

The proof of this theorem will be prepared by several Lemmas. From now on, we suppose that S fulfilles the hypotheses of Theorem 1.1.

By a claw we mean a set S of lines with the following properties:

S ) .

(a) H { s f and H is parallel to every line of 5, (b) any two lines of S intersect.

A claw S is called normal, if ( s l = d. The order of a claw is the number of its elements.

A clique is a set of mutually parallel lines of S. The clique h is called maximal , if

(a) H € M , (b) h is a maximal set of mutually parallel lines, and (c) Ihl > n - (d-l)(c+l) + x.

Now we can state our first Lemma.

Embedding Finite Linear Spaces 41

LEMMA 1. Let S be a claw of order s. Denote by the set of lines parallel to H which do not lie in 5. For 0 2 y 5 s let f(y) be the number of lines in 7 which have exactly y parallels in 5 . Then s 5 d. Moreover,

f(0) - f f(y)(y-l) 2 n(d-s) + x - s(l+c), y=l

5 f(y) = nd + x - s. y= 0

PROOF. From our assumption ( 2 ) we get

(i)

By double couiting we have

Using hypothesis ( 3 ) we obtain therefore

(ii) 2 f(y)y 5 s(n+c) and (ii)' y=O

Now let L1 and L be two distinct lines parallel to L an& L there are at most which are parailel to 2; and L2. Hence

y=l

Together with (i) and

(iii) 0 5 f(O)

- < n(d-s)

(ii)' we conclude

If we assume s = d+l, then by (iii) we would obtain the contradiction to ( 5 ) :

1 n = n(s-d) 5 x - s(l+a) + ~s(s-l)(e-l)

= Z(d+l)(de-d-2a-2) + x. 1

and

s(n+a).

H is lines in i

f ol lowing

Hence there exists no claw of order d+l, therefore any claw has at most d elements.0

LEMMA 2. For every line L parallel to H, there exists a normal claw containing L.

PROOF. Consider first the case that there is a point p of degree n+l on L. Let 3 be the set of lines through p parallel to H. Since H h a s degree n+l-d, we have 1st = d. Obviously, 5 is a claw containing L. Thus, we may suppose that hypothesis (7)(ii) is true. Let 5 be a claw of maximal order with L E 5 . With the nota-

tion of Lemma 1 we get

f(0) 2 n(d-s) + x - s(l+c).

Assume now s < d. Then (7)(ii) implies f(0) > 0. Therefore, there

is a line L' in which intersects every line of 5. But then

5 ' = .S [L'I is a claw with L € 5' and ( 5 ' 1 > 1x1. This contradiction proves Lemma 2.0

LEMMA 3 . Let L be a line parallel to HI and denote by 5 a nor-

mal claw containing L. Moreover, let m be the set of all lines

L' 5-tLI which are parallel to H and intersect every line of

5-{LI. Then Mu (HI is contained in a maximal clique through L.

PROOF. Clearly, 5' = 5-ILl is a claw of order d-1. If L1, L2 E f i ,

then 1 5 ' ~ [L1,L211 = d+l, and therefore 5'u {L1,L21 is not a claw.

This shows.that L1 and L2 are parallel and that f l u {HI is a

clique. Lemma 1 applied to 5 ' gives

l f l l = f(0) n - (d-l)(c+l) + x.

Therefore, m u [HI is contained in a maximal clique. Since L €

kl u {HI, Lemma 3 is proved.0

LEMMA 4 . If fll and f i2 are different maximal cliques, then

f11nf12 = (Hl.

PROOF. Because /)Il and f12 are maximal cliques, there exist inter-

secting lines L1 € M1 and L2 € m 2 , From our hypothesis ( 4 ) we get

IM1n f121 5 h(L1,L2) 5 e.

Assume now Ih1nh21 2 2 . Then there is a line L H in h1nfi2.

From hypothesis ( 3 ) we obtain therefore

I f 1 1 C J f i 2 1 5 h(L,H) + ItL,HII 5 n+c+2, and

Ifill -k

I flll +

On the other hand

together we get a

/112 1

I M 2

we

con

= I"1uf12

have

- > 2(n -

radiction

+ IA1n M 2 1 Q n+c+2 + e.

d-l)(c+l) + x + 1);

to our hypothesis ( 6 ) . 0

Now we are ready for the proof of theorem 1.1.

(a) If d = 1, then the statement is obvious. Therefore we may sup-

pose d ? 2 . Since there are lines parallel to H, there exists a

normal claw 5 (cf. Lemma 2 ) . We shall use the notation of Lemma 1

for 3 . Because 0. From Lemma 1

there exists no claw of order d+l, we have we set

f(0) =

y=l i f so I

d y ) = nd + x - d, and - f(y)(y-1) x - d

y=2 l+c).

d d f(1) = nd + x - d - f(y) 2 nd + x - d - f(y

- > nd + x - d + x - d(l+c) = nd - d(c+2) + 2x. y=2 y=2

Put S = {L1, ..., Ldf, and define I?,’ as the set of all lines S-{L. 1 which are parallel to H ahd intersect every line of ILill Obviously, the sets I?; are distinct; therefore,

- -

l f i l - [ H I I +...+ (fid-IHII

= Ihl-IL1ll +...+ Ihd-ILdll = f(1) + d.

1f i ; I +...+ IfiiI

Assume that there is another maximal clique fld+l. Then

- < f(1) + d + Ihd+l-IH)I 5 Ihl-IHII +...+ [Ad+ f(1) + d + n - (d-l)(c+l) + x

On the other hand, Lemma 4 yields

[hl-{H)l +...+ Ihd+l-{HII = I(h1 ... hd+l)-

- < number of parallels of H 5 nd + x. Together we get

nd + x 2 f(1) + d + n - (d-l)(c+l) + x.

(y-1

L E S-

Since f(1) > nd-d(c+2)+2x, we conclude n 2 (2d-l)(c+l)-2x. Hence condition (6) implies e 5 0. But in view of d 2 2 and h(L1,L2) - > I { H I 1 = 1, this contradicts condition ( 4 ) .

Therefore, there are exactly d maximal cliques. N o w Lemma 3 and 4 show that every parallel of H is contained in exactly one of the sets Mi. This proves (a); (b) is obvious.0

In the following corollary, we handle an important particular case.

COROLLARY 1.2. Let S be a finite linear space of order n, and let H be a line with kH 5 n such that every point outside H has degree n+l. Let the integers d, x, z be defined in the following way: The number of lines of S is b = n2+n+l+z, kH = n+l-d, and H has exactly nd+x+z parallels in S. Suppose that there exists positive integers c and 5 with the following properties: 1) n+l-d 5 kL 5 n+l-i for every parallel L of H.

2) 2n > (d+l)(da’ + d - 2dG + 2: - 2) - 2dx + d(d-l)z. 3 ) n > (2d-l)(d-l)(d-l) + a* - 1 + (2d-3)x + 2(d-l)z. Then assertion (a) of Theorem 1.1 is true. Furthermore, the sets h; are parallel classes of S.

PROOF. Define a = dd-d-d+x+z, c = dd-d-a+x+z and e = d’ Using 2) and 3 ) we see t h t (5) and (6) of Theorem 1.1 are (note that x‘ replaces x and x is replaced by x+z). the conditions (l), (21, (7) of 1.1 are fulfilled. Now, let L be a parallel of H with k = ntl-d’. Since sects kL(d-l) parallels of H and H kas nd+x+z para we get

I

x ’ = x+z. satisfied Obviously,

L inter- lel lines,


h(L,H) = nd + x + z - kL(d-l) - 1 = n + (dd'-d-d') + x + z. So, condition ( 3 ) of Theorem 1.1 is true (note that 2 5 d' 5 d ) . Let L1 and be two different intersecting lines parallel to H, and put kLL2= n+l-di (i = 1 , 2 ) . Then

1

h(L1,L2) = dldZ + z 5 el which shows that also condition (4) of Theorem 1.1 is fulfilled. Therefore, the corollary follows from Theorem 1.1.0

REMARK. If pl, ...'p are the points on H and if we denote the degree of p. b9'AFb-d. then we have x = d +. . . 1 +dn+l-d in the corollary. IA particula;, x = 0, if every point of S has degree n+l.

2. CONSTRUCTION OF THE PROJECTIVE PLANE

In this section, S = (p,L,I) denotes a finite linear space of order n with b = n2+n+1+z lines. First, we show the following theorem.

THEOREM 2 . 1 . Suppose that S satisfies the following conditions: (a) b 2 n'. ( b ) For every line L of S there is an integer t(L) with the following property: If kL = n+l-d, then there are exactly d maximal sets M of mutually parallel lines with L € f i and ( M I 2 t(L) Furthermore, every line parallel to L appears in exactly one of these d sets M .

Then S is embeddable in a projective plane of order n.

For the proof of this theorem we shall use the following notation. A cZique is a maximal set f i of mutually parallel lines with ] M I

if I f i l = n. By we denote the set of all cliques of S.

For p € p and 18 € j we define

For h E we put

t(L) for at least one line L E f i . A clique M is called norrnaZ,

p - M , if p 1 L for at least one line L of M .

51(h) = I F ) ' I f i ' € j r mnh ' = @ I ,

Z,cm = G1(M) u { M I , 4 p I ) = Ip I p E p , p 1. M I . For every normal clique M we define

Now we can dFfine the incidence structure S ' = ( P d , L the following way:

p I ' L * P I G for all p € p r L €

p I' 5 ( f i ) Q p E 4 ( M ) for all p 6 p , G ( f l M I ' L - L E A for all M E j, L E

M I' ~(4') * f i E G ( M ' ) for all ill E /3, 5 ( f i

L ; E 2;

L ;

) E 2. As in section 1, we shall prepare the proof by several lemmas. From now on we suppose that S satisfies the hypotheses of Theorem 2.1.

LEMMA 1. (a) A line of degree n+l-d is contained in exactly d cliques. ( b ) If L and are parallel lines, then there is a unique clique f i l with E A .

PROOF. Let L be a line of degree n+l-d. Then there are exactly d cliques Mi with L E Mi and I/tlil 2 t(L). Furthermore, every line parallel to L appears in exactly one of theses cliques. Assume that there is another clique f l with L E h. Then I f l l 5 t(L). By definition, h contains a line L' with Ihl 2 t(L'). In L' 4 L, and L' is parallel to L. Let j be the index with L' E M . . Then we have L,L' E f l . , f l , and ~ h . ~ , ~ f l ~ 1. t(L'). Now condition ( a ) of Theorem 2 . 1 gives M 3 = M . contraaicting IhI < t(L) 5 I f l j I .O

LEMMA 2. (a) Let L be a line, and denote by p a point off L. If kL = n+l-d and r = n+l-y, then there are exactly d-y cliques f l with L E h and 'p y f l . (b) If p is a point of degree n+l, then p % /11 for every clique m . (c) We have 1/31 + v = n2+n+l.

PROOF. (a) There are exactly r -kL = d-y parallels of L through p. Therefore, the assertion folaows from Lemma 1 (b). (b) Let L be a line with p 1 L. From (a) and Lemma l(a) we infer p % /rl for every clique M with L E A . (c) Let p be a point of degree n+l, and let L1f...,Ln+l be the lines through p . If the degree of Li is n+l-di, then we have

1

n+l n+l

i=l 1 i=l v-1 = (kL.-l) = (n+l)n - 2 di.

In view of (b) and Lemma 1 we conclude

n+ 1 n+l

i=l i=l = I { f i € 13, Li E M I 1 = 1 di.

Together, our assertion fol1ows.U

LEMMA 3. Let f l be a normal clique, and denote by L a line with L 4 h. Put = n+l-d, and l e t t denote the number of points p on L with kb % h. Then there are exactly 1-t cliques which are disjoint to f l and contain L. In particular, we have 0 5 t 5 1. PROOF. L has kL-t points p with p Q M . Therefore, there are exactly

lines LA! ..., L in h which are parallel to L. Let hi be the clique w ich coatains L and L.. Since L { f l , we have h f hi for all i. Therefore, by Lemma l(b), hi m . for i f j . Obviously,

Now, Lemma l(a) shows

m := I f l l - (kL-t) = d + t - 1

I I { M ' I h ' E /3, M n f l ' $. @, L E f l ' l l = m.

1 { h ' I h ' E 13, h n h ' = @, L E h ' l l = d - m = 1 - t.0

LEMMA 4 . Denote by f l a normal clique. Then (a) l G 2 ( M ) l 5 1. In other words: There is at most one point p with

(b) lG(h) I = n+l. P 1. f i .

PROOF. (a) Assume that there are two distinct points p and q with p,q + h. Let L be the line which passes through p and g , and define t as in the preceding lemma. Then t 2 2 , contradicting our Lemma 3.


(b) If M = iL1, ..., Ln) and kL, = n+l-d. then 1 n

I G 2 ( M ) I = 1 Ip € p I p Q M } I = v - 1 i=l

(n+l-di).

From Lemma 1 we get

and the assertion follows in view of + v = n'+n+l.O

LEMMA 5. Let hl be a normal clique. (a) If L is a line, then L and G ( M ) intersect in S ' in a unique point; i.e. one of the following cases occurs:

G 2 ( h ) f 0, then L is not incident with the point of 4 ( M I .

cident with the point 02 (b) Any two cliques of Gl(3) are disjoint. PROOF. (a) We may suppose L E M . Using the notation of Lemma 3 we get t € I0,l) by Lemma 3 . Moreover, Lemma 3 implies that (1) (or (ii)) occurs if and only if t = 0 (or t = 1, respectively). (b) is a consequence of (a1.0

LEMMA 6. Let M1 and M 2 be two distinct normal cliques. Then

(i) There is a unique clique in G ; , ( A ) containing L. If

(ii) NO clique of G ( M ) contains L, I G , ( M ) I = 1 an& L is in- 4 ( & ) .

(a) 4 ( A 1 ) = 5 ( A 2 ) 0 A 1 n M2 = 0. (b) 14(M1) 5 ( M 2 ) I = 1 0 Mln M 2 9 0. PROOF. (a) One direction is obvious. Suppose therefore Then M € 51(fll); hence, by Lemma 5(b), cl(hl) c Gl(M2). SirnilarZy, we have In view of Lemma 4 we may assume without loss of generality that 5 ( 4 . ) = {p.) with points p1 and p . Lemma 5(a) sa s that a line L2 it incidknt with p. if and only ig no clique of contains G1(Mlt = T1(fi2), this shows that a line is incident with p1 if and only if it is incident with p2. Since r 1. 2, we have 5,(fll) = 5 , ( M 2 ) . Pi

(b) In view of (a), one direction is obvious, Let us suppose A1nM2 f 0. Then fil and f12 intersect in a line L.Define

Since M1, M 2 [ 5(M1) G ( M 2 ) , we have

M1n M 2 = 0.

G1(h2) c G l ( f l l ) , hence equality.

L. Since

T = r M I ki € ,E,

I5(A1) n 5 ( A 2 ) I

= I M E F I M n M l = ia = M n M 2 ) 1 + 152(M1)n52(M2)1

= r M E i~ I M n M 1 = $ 1 1 - li-I + 142(fi l)n52(M2)1

= IG1(f l1)l - li-l + 152(M1)n52(M2)1.

M n M l = 0 + f l n M 2 } .

Now we distinguish three cases.

C a s e 1. 52(/111) = 0. Then by Lemma 5(a) every line is contained in a unique clique of T ( 4 ) . Because no clique of G1(hl) contains two lines of M 2 , we have' 171 = IA2-ILfI = n-1, and so

l5 ( f l1)n5( f i2) l = 151(f11)l - ( T I = n - (n-1) = 1, since IG1(M1)I + 1 = IT1(M1)I = I G ( M 1 ) I = n+l.


152(fi1)I = 1 and G2(M1) = 52(fi2).

1 G 2 ( f i , ) = S 2 ( f i 2 ) , no line of f i2 f i 1. As in case 1, this implies I T ( = I f i -ILlI = n-1. Since I = 1 we now have

is incident with the point

I G , ( A , ) I = n-1, and t6is imp1 es

15(f i1)nG(f12)l = lGl(fil)l - I T 1 + l G 2 ( f i 1 ) n G 2 ( f i 2 ) l = 1.

C a s e 3 . I G 2 ( f i 1 ) I = 1 and G2(fi1) c G 2 ( M 2 ) .

Since 4 ( f i ) G 2 ( M Z ) , there is a unique line L' in f i 2 which is incident2wi&h the point of G ( f i l l . In view of L € f i l we have L $. L', and so 171 = ( M 2 -{L,L')y = n-2. This shows again

IG(PIl)n G(M2)l = I G 1 ( M 1 ) I - I T 1 = 1.

Since proved. 0

LEMMA 7. Any two distinct lines of S ' intersect in a unique point of S ' .

PROOF. If one of the two lines is an element of l, we already proved the assertion in Lemma 6(a) and Lemma 7. If both lines are elements of L , the assertion follows from Lemma l(b1.O

Now we are ready for the proof of theorem 2.1.

Let S* = L L u L , p u p , I ) be the dual incidence structure of S ' . By Lemma 7, S is a linear space with n2+n+l lines (Lemma 2(c)) and at least n2 points (hypothesis (a)).Furthermore, in view of Lemmas l(a) and 4(b), any poipt of S has degree n+l. Now, by the theorem of VANSTONE [ 7 ] , S is embeddable in a projective plane of order n. But then also S ' is embeddable in a projective plane P of order n.

This completes the proof of Theorem 2.1.0

We remark that S' = P if b > n2. (Assume to the contrary that S ' P. Since I p I + 1/71 = n2+n+1, there is a line L of P which is not a line of S ' ; so, L ( Lu 2. Because S is a linear space, at most one of the points P1t---rPn+l incident with L is a point of p . Every line of S is in P incident with exactly one of the points pi. Since points among the p.'s which are incident with n lines of S. But one of these two po$nts, say pl, is an element of /3. Hence p1 ,is a normal clique of G(p,) = tpl,...,pn+ll = L, contradicting

IG2(fii)I 5 1, we have handled all cases. Thus. Lemma 6 is

*

b > n', this shows that there are at least two

S , and L t Z.) The following theorem is probably the main result of this paper.

THEOREM 2 . 2 . Suppose that the hypotheses of part (a) of Theorem 1.1 or its corollary are satisfied for every line of S which has not degree n+l. If b 2 n', then S is embeddable in a projective plane of order n.

PROOF. We show that for every line L of S there exists an integer t(L) such that the hypothesis (b) of Theorem 2.1 is fulfilled. If kL = n+l, we put t(L) = 2 . If kL n+l, then Theorem 1.1 (or its corollary) show that such an t(L) exists. Therefore Theorem 2.2 follows from Theorem 2.1.0

48 A . Beutelspacher and K . Mehch

3 . LINEAR SPACES WITH CONSTANT POINT DEGREE

Let A be a finite set of nonnegative integers. We say that the linear space S is A-semiaffine, if r -k < A for every non-incident point-line pair (p,L) of S . TRe kinear space S is called A-affins, if it is A-semiaffine, but not A‘-semiaffine for every proper subset A’ of A.

Throughout this section, S will denote an A-affine linear space in which every point has degree n+l. Because the lO)-affine linear spaces are the projective planes, we will assume A I 0 1 throughout.

Denote by a the maximal and by a the minimal element in A-fO}. The integer z is defined by b =-n’+n+l + z. The following two facts shall be used frequently.

For any line L whose degree is not n+l we have n+l-a 5 k n+l-g. If L kL n+l.

LEMMA 3.1. (a)- We have z 2 - g z . - (b) If n > - aa(a-a), then z 5 (a-a-l)a. PROOF. (a) Since a € A and since every point has degree n+l, there is a line L- of degree n+l-a. Let L ’ be a line intersecting L at a point q. Then every point of L’ other than q is on precisely 5 lines parallel to L. Thus, L has at least

parallels. On the other hand, L intersects exactly kLmn = (n+l-a)n lines. Hence,

-

is a line which has at least one parallel line,LtEen

(kL,-l)g 1. (n-a)a -

n2+n+1+z = b 2 (n-a)g + (n+l-a)n + 1 = n’+n+l - 22, -

i-e. z -aa.

(b) If S has a line of degree n+l, then b = n2+n+l, and so z=O. Therefore, we may assume n+l-a 6 k n+l-a for every line X . Let L be a line of degree n+l-a, and 8ekte by 4 the set of all lines parallel to L. Then

[ M I (n+l-a) 5 x k X = (rp-kL) = (v-k )a L -- X€A PkL

Because every line has at most n+l-g points, we have

v 5 kL + n(n-g). Together it follows

(v-k )a n(n-g)g (Z-l)a(Z-a-l) I M I 2 L - , - = na- + (2-g-1)~ +

n+l-Z n+l-Z n+l-Z Our hypothesis yields

therefore I M I 2 ng + (2-2-1)s. It follows

i.e. z 5 (2-g-1)a.m

- - n+l-Z > gZ(Z-5) + 1 - a 1. (a-1)3(Z-g-l),

b = 1 + kL*n + I M l 5 n2+n+l + (a-g-l)~,

THEOREM 3.2. Suppose b 5 n’+n+l and assume that S satisfies the following conditions:

(1) n > i(~2-1) ( ~ ’ + ~ - 2 g + 2 ) + T z ( f i - 1 ) z I 1


( 2 ) n > 2(Z-1)(Z2-Z+l) + 2(?-1)z, (3) b n2 or n 2 g; - 1.

Then S is embeddable in a projective plane of order n.

(b) If b > n2+n+l, then one of the following inequalities holds:

n 5 q(52-1)(~2+~-2g+2) 1 + k ( ~ - l ) z , or 2

n 5 2(Z-l)(Z2-Z+1) + 2(Z-l)z. PROOF. (a) If n 2 a;-1, then b = n2+n+l + z 2 n2 by Lemma 3.1. Hence we have b 2:’ in any case. In view of Theorem 2.2 it suffices to show that for every line L with kL n+l the hypotheses of Corollary 1.2 are fulfilled.

- Consider therefore a line L- of degree n+l-a 5 n. Put d = a, a = a, 4 = g and x = 0. Then d 2 d 2 4, and our hypothesis (1) shows

- > (2d-l)(d-l)(a-l) + 8’ - 1 + Z(d-l)z. Therefore, the hypotheses of part (a) of Corollary 1.2 are fulfilled. Hence the assertion follows in view of Theorem 2.2.

(b) Assume that our statement is false. Then, as in part (a), we would be able to embed S dicting b > n’ +n+l .O

COROLLARY 3 . 3 . If b 5 n2+n+l and n > +(a2-l)(Z‘+Z-2g+2) is embeddable in a projective plane of order n.0

n 5 7(a2-l)(a2+z-2a+2 - Ta(a l)(a-g-l)g.

PROOF. Since b > n2+n+l, there is no line of degree n+l. first a = a. Then every line has degree n+l-a, we have

v = 1 + (n+l)(n-g) and v(n+l) = b(n+l-a).

in a projective plane of brder

1 - OROLLARX 3.4. If b > n’+n+l, then f- -_

n, contra-

then S

+

Assume

We obtain b 5 n’+n+l, a contradiction. Hence we may suppose 1 5 g C a. Assume that our statement is false. Then

n > $ ( ~ ’ - 1 ) (~‘+~-2a+2) 1. L ( ~ 2 - 1 ) 2 (%‘-a) 1. ~ ( ~ - 1 1 2

- - > aZ(Z-a),

- and from Lemma 3.1 we get z 5 (Z-a-l)g. In view of z > 0 we have a 23+2. Now we get

n > $ ( ~ 2 - 1 ) (~’+~-2g+2) + $(i-l) (~-5-l)C - > 2(Z-l)(2-Z+l) + 2(Z-l)z.

This is a contradiction to Theorem 3.2(b).0


In the remainder of this last section we shall study the case IAl = 2. Let a and c be non-negative integers with a c, and denote by S an ta,c)-affine linear space in which every point has degree n+l. Then every line has either n+l-a or n+l-c points. We call a line of degree n+l-a l o n g : the other lines are said to be s h o r t t . The number of long lines (or short lines) of S is denoted by ba (or b , respectively). Let t be the (constant) number of long lines Fhrough a point. Then we have

v-1 = t(n-a) + (n+l - t)(n-c) = t(c-a) + (n+l)(n-c). The proof of the following assertion is straightforward and will be omitted here.

LEMMA 3.5. (a) We have

((c-a)(t-a)+l-a)t n+l-a ba = (n-c+a)t +

and

(b) If b = n2+n+l, then

bc = n2+n+l - c - (n-c+a)t - (t(c-a)+l-c) (t-c) n+l-c

(c-a)’t2 - (c-a)[(n+l)(c-a)+n]t + cn(n+l-a) = 0, or

J D t = t t- o 2(c-a) ‘

where n+l

2(c-a) and

D = [ (~-a+1)~-4c]n~ + 2(c2+a2-c-a)n + (c-a)’ . U

COROLLARY 3.6. Let c be a positive integer, and denote by S a finite tO,c)-affine linear space in which every point has degree n+l. If

1 n > ?(c2-1) (c2 -c+2),

then S is the complement of a maximal c-arc in a projective plane of order n. In particular, c divides n.

PROOF. Since there is a line of degree n+l, we have b = n2+n+1. By corollary 3.3, S is embeddable in a projective plane P of order n. Hence there is a set c of points of P such that S = P-c. It follows that c is a set of class (0,cI of P, so c is a maximal c-arc of P . 0

REMARK. Corollary 3.6 is a slight generalisation of a theorem of THAS and DE CLERCK [6].

COROLLARY 3.1. Let c 1 be a positive integer, and denote by S a finite tl,cl-affine linear space in which every point has degree n+l. If

n > zc(c-1)(c2+3c-1), 1

then S is embeddable in a projective plane P of order n. More- over, one of the following cases occurs:

( a ) S is the complement of c concurrent lines of P. (b) There is a maximal (c-1)-arc c and a line L in P such

that L does not contain a point of c . S is obtained by removing the line L and the points of c from P.

(cl b = n2+n+l, c-1 divides n, and c(c-4)n’ + Zc(c-l)n + ( ~ - 1 ) ~

is a perfect square.

PROOF. First we show that S is embeddable in a projective plane of order n. From our hypothesis we get

( * )

Therefore, in view of Corollary 3 . 3 , we may assume b > n’+n+l, i.e. z > 0. Hence, in view of Theorem 3 . 2 , it suffices to show that

n > 7(c2-l)(c2+c) 1 + zc(c-l)(c-2). 1

(2) n > 2(c-l)(c2-c+l) + 2(c-l)z. Let L be a long line. Then through every point outside L there is precisely one line which is parallel to L. Hence L together with its parallel lines forms a parallel class n of S. Since b = kL*n + I I I 1 , we have I I I l = n+l+z. Since t 5 n we have

(n+l+z)(n+l-c= = I~~l(n+l-c) 5 v = 1 + t(c-1) + (n+l)(n-c) - < n2+l-c.

Now we claim z 5 c-2. (Otherwise, we would have (n+c) (n+l-c) 5 n’+l-c,.

s o n 5 c2-2c+l =

contradicting ( * ) . I NOW

Hence S is embeddable particular, b 5 n2 +n+l. two cases.

(c-l)2 I

(1) and (2) follow immediately from ( * ) .

in a projective plane P of order n. In Let L and n be as above. We distinguish

C a s e 1 . All long lines are contained in II. Then t 5 1 and s o t = 1. From Lemma 3 . 5 we get

b = b + b = n’+n+l - c. a c Hence there are exactly c lines L1, ..., L in P which are not lines of S. Since b = n2 + 1 I I I , we have Y I I l = n+l-c. Now it is easy to see that S is the complement of the c concurrent lines

C a s e 2 . There is a long line L’ outside II. Since every point of L’ is on a unique line of n , we have I n 1 2 n, and so b 2 n2+n. This means b € tn2+n,n2+n+11.

Consider first the possibility b = n‘+n. Then there is exactly one line X in P which is not a line of S. Because all the points of S have degree n+l, none of the points of X is a point of S. Ad- ding X to S we get a Il,c-ll-affine linear space s ‘ , in which every point has degree n+l. Corollary 3 . 6 shows that S ’ is the complement of a maximal (c-1)-arc.

Suppose finally b = n2+n+l. Then I n 1 = n+l. Let s be the number of lomg lines in n . Then

v = sn + (Inl-s)(n+l-c) = s(c-1) + (n+l)(n+l-c). On the other hand, we have v = 1 + t(c-1) + (n+l)(n-c). Together we get n = (t-s)(c-1). Therefore, c-1 divides n. From Lemma 3.5(b) we obtain furthermore that c(c-4)n2+2c(c-1)n+(c-1)’ is a perfect square. Thus, Corollary 3 . 7 is proved completely.0

REMARKS. 1. Corollary 3 . 7 has already been proved by BEUTELSPACHER and KERSTEN [l] under the additional hypothesis b 5 n’+n+l.

C‘ L1,. . . ,L

52 A . Beutelspacher and K . Metsch

2. Case ( a ) of Corollary 3.7 can be obtained from the theorem of MULLIN and VANSTONE [ 5 ] . 3. If P is a projective plane of order (c-l)’, and c is a Baer- subplane of P, then P-c meets the conditions of 3.7(c).

COROLLARY 3 . 8 . Let a and c be positive integers with 2 5 a c, and let S be a finite Ia,cl-affine linear space in which every point has degree n+l. Suppose that S satisfies the following conditions:

(1)

and (2)

Then S is embeddable in a projective plane P of order n and one of the following cases occurs:

(a) There is a positive integer x with a = x2+1 and c = x’+x+l; S is the complement of a subplane of order x in a projective plane of order n.

(b) n’+n+l-a 5 b 5 n2+n, c-a divides n and c-1. In particular, c 5 2a-1. (c) b ; n 2 + n + l , c-a divides n, and [ ( ~ - a + 1 ) ~ - 4 c ] n ~ + 2 ( c ~ + a ‘ - ~ - a ) n +(c-a) is a perfect square.

PROOF. From Lemma 3.5 we get b = n’+n+l + z = n2+n+l - c + f(t), where

n > T ( C Z - ~ ) ( C ‘ + C - ~ ~ + ~ ) 1 + c3z’+4(c-a+1)2 - 2

1 n > z(c2 -1) (c2+c-2a+2).

f(t) = [(t-a)(c-a)+l-a]t - [t(c-a)+l-cl(t-c) . n+l-a n+l-c (3)

Obviously, f(t) is a polynomial of second degree with negative coefficient in t‘, which has its maximum in

2 (c-a) - (n+l)(c-a) + n t =

From (2) we get

( 4 ) f(0) = - n f(-) = c-a

First we show that n. If z 5 0, this may assume z > 0.

c(c-l) > -1, and n+l-c

f(n+l) = c-a - - a(a-1) > c-a-1. n+l-a

S is embeddable in a projective plane of order follows from ( 2 ) and Corollary 3.3. Therefore, we Then

n+ 1 < n+l < - < (n+l)’ (n+l-a)(n+l-c) n+l-a-c n+l-2c -

Z(C*-C3+3C2+C-4) 1 + 1 < c3-c2+4

c3 -c2 - 5 1 - -(c‘-c3 +3c2 +c-4)+1-2c 2

Thus I

and

(5)

In view of ( l ) , this implies 1 1 2

(c-n+l)’ - c. c3 -c2 +4 2 = f(t) - c 5 f ( t ) - c < 4(c,-cd)

n > -(c2-1)(c2+c-2a+?) + Zc(c-1)z.

n 5 2(c-l)(c2-c+l) + z(c-1)~. On the other hand, Theorem 3.2(b) yields

Together, we have a = 2, c = 3 and z 2 2 0 , which contradicts ( 5 ) . Therefore, b 5 n’+n+l, and S is enbeddable in a projective plane P of order n.

Denote by x the set of points of P which are not points of S. Consider a point p of x . Since the lines of S through p con- stitute a parallel class of s , every line L of degree n+l-d is contained in exactly d parallel classes nl(L),...,IId(L). Further- more, every line parallel to L lies in exactly one parallel class ni(L). It follows in particular

b = 1 + (n+l-d)n + I nl(L) I + . . .+- I nd(L) 1 . Now we distinguish two cases.

C a s e 1 . There is a parallel class n of S having exaczly n+l lines. If s denotes the number of long lines in TI, then

Inl(n+l-c) + s(c-a) = v = 1 + (n+l-t)(n-c) + t(n-a). Hence s(c-a) = t(c-a) - n, so c-a divides n. Furthermore,

t = s + - 2 L . n c-a c-a ‘

hence ( 4 ) implies b 2 n2+n+1-a. If b = n2+n+1, then Lemma 3.5(b) shows that we are in case (c) of Corollary 3.8. Therefore we may assume that

n‘+n+l-a 5 b 5 n2+n.

Let X be a line of P which is not a line of S . Since every point of S has degree n+l, each point p. of X lies in x (i E t 1,. . . ,n+l}). Let hi be the number o* lines of S through pi. Then h. 5 n and h + ... + hn+l = b. Since b n2+n+l-a, there is a j l € {l ,...,n+l{ with h. = n. Therefore the lines of S through p. form a parallel ciass n ’ with exactly n elements. If s ’ deAotes the number of long lines in n ‘ , then

n(n+l-c) + s’(c-a) = v = 1 + (n+l-t)(n-c) + t(n-a). Hence s’(c-a) = t(c-a) - (c-1). Consequently, c-a is a divisor of c-1, and now we are in case ( b ) of Corollary 3.8.

C a s e 2 . Every parallel class of S has at most n elements. Consider a short line L of S . Then

C n’+n+l-c 5 b = 1 + k n + C ( I I I ~ ( L , ) ~ - ~ ) 5 n2+n+l-c.

i=l

Hence b = n’+n+l-c and

( 6 ) Ini(L)I = n (i E [l, ... ,cl) for every short line L.

In particular we have Consider now o long line L of S. We get

f(t) = 0. By (4) we obtain t < A. . - c-a a

i=l n2+n+l-c = b = 1 + k n + ( Ini(~)I-l) 5 n’+n+l-a.

Hence there exists a parallel class n which contains L and has fewer than n elements. Since n is a parallel class corresponding to a point of x, it follows in view of ( 6 ) t.hat every line of n is long. Hence, n+l-a divides v ( = l+(n+l)(n-c)+t(c-a)); so,

( 7 ) n+l-a I (t-a)(c-a) + 1 - a. Since t 5 =, we have (t-a)(c-a)+l-a < n+l-a

we get from (2) that

(t-a)(c-a)+l-a 2 -a(c-a)+l-a > -c(c-1

n

so, I(t-a)(c-a)+l-al < n+l-a. NOW, ( 7 ) implies particular

a(c-a)-1 c-a v = (n+l-a)(n-c+a), and t =

Toaether with

On the other ha.nd,

+l-c = -c2+1> -(n+l-a),

t-a)(c-a)+l-a = 0, in

a -

t2 (c-al2 - t.(c--a) [ (c-a) (n+l)+n] + c(c-1) (n+l-a) (n+l-a)(n+l-c) 0 = f(t) =

we get

~i[a!c-a)~ +2!a-1) (c-a)-(c-I)* ] = (a-1) [a(~-a)~+Z(a-l) (c-a)t(c-1)2 1. Since n f a-1,

a(c-a

and so c = a + satisfying a = (x2+x+l), ba =

Moreover. if n

we obtain

+ 2(a-l)(c-a) - ( ~ - 1 ) ~ = 0 ,

Ja-1. Therefore, there is a positive integer x x2+l, c = x’+x+l, v = (n-x2)(n-x), b = n’+n+l - x’+x+l)(n-x), and bc = n2+n+l - (x2+x+1) - ba. is a parallel class corresponding to a point of P

outside S , then one of the following possibilities occurs:

(I) 1x1 = n, and n contains x2 long and n-x2 short lines;

( 1 1 ) 11!1 = n+3.-a , and n consists of lomg lines only.

Using these properties it is now easy to see that we are in case (a) of our corol1ary.n

REMARKS. 1. Suppose a = 2 and c = 3 . If we are in case (a) of the above corollary and if n > 42, then S is the complement of a triangle in a projective plane of order n. (This result has been proved in case n > 7 by DE WITTE 18.1.) 2. The existence of strectures in case (b) and ( c ) satisfying (1) and (2) is not known to the authors.

COROLLARY 3.9. If S is a finite (2,4l-semiaffine linear space in which every point has degree n+l, then n E I5,7,131.

PROOF. Since a short line has n-3 points, we have n 2 5 . By 3.5 we get

b2 = (n-2)t + (2t-5)t ( 8 1 n-l , b4 = n2+n-3 - and

V

(2t-3jft-4) n-3

b = n2+n-3 + f(t) with f(t) = (2t-5)t - (2t-3)(t-4). n-1 n-3

Emhediiiug Finite Linear Spaces 5 5

Obviously, f is a polynomial of second degree with negative leading coefficient, which takes its maximum at t = (3n+2)/4. Since n 2 5, we have

therefore f(t) 5 3. From n-1 1 (2t-5)t and n 2 5 we get t 1. 4; consequently,

12 1 f(t) 2 f(4) = n-l > 0 or f(t) 2 f(n+l) = 2 - n-l 1.

Hence f(t) = s with s E (1,2,3}. By (l), we obtain

3n+2r JD 4 with D = (9-4s)n2 + (16s-36)n + 52-12s. t =

Now we distinguish three cases.

C a s e 1 . s = 3. Then D = -3n’ + 12n + 16 2 0, and therefore n = 5.

C a s e 2 . s = 2. Then D = n’ - 4n + 28 = (n-2)’ + 24. Since C is a perfect square, we get n = 7.

Case 3 . s = 1. Then D = 5(n2-4n+8) and b = n‘+n-2. Assume n > 135. Then, by Corollary 3.3, S is embeddable in a projective plane of order n. Because b < n’+n+l, there is a parallel class n of S with n elements. If s denotes the number of lonq lines in n , then

v = lnl(n-3) + 2s = n2 - 3n + 2 s .

On the other hand, we have

v = 1 + (n+l)(n-4) + 2t = n2 - 3n - 3 + 2t.

Together we get 2(t-s) = 3, a contradiction.

Consequently n 2 135. Since D is a perfect square, it follows n E 16,13,31,78}. In view of t = (3n+2tJD)/4 and t 5 n we get n 6,31,78. So, n = 13.0

REMARK. The authors do not know, whether the structures considered in 3.9 exist in the cases n = 7 or n = 13. For n = 5 we give the following example. Let A be an affine plane of order 4, and let S ‘ be the linear space which is obtained by removing one of the points of A. Then there are five lines of degree 3 in S ’ . Replacing each of these lines by three lines of degree 2, we get a 12,4l-affine linear space S of order 5 with 15 lines of degree 4 and 15 lines of degree 2.

REFERENCES

[l] Beutelspacher, A. and Kersten, A., Finite semiaffine linear

[2] Bose, R.C.: Strongly regular graphs, partial geometries and

[3] Bruck, R.H.: Finite nets 11. Uniqueness and imbedding, Pacific

[ 4 ] Hall, M.: Projective planes, Trans. Amer. Math. SOC. 54 (19431,

spaces, Arch. Math. 44 (1985), 557-568.

partially balanced designs, Pacific J. Math. 13 (1963), 389-419.

J. Math. 13 (19631, 421-457.

229-211.


[5] Mullin, R.C. and Vanstone, S.A.: A generalization of a theorem of Totten, J. Austral. Math. SOC. A 2 2 (1976), 494-500.

[ 6 ] Thas, J.A. and De Clerck, F.: Some applications of the fundamental characterization theorem of R.C. Bose on partial geometries, Lincei - Rend. Sc. fis. mat. e nat. 59 (1975), 86-90.

third Manitoba conference on numerical math. 1973, 409-418.

plane, to appear.

[7] Vanstone, S . A . , The extendability of (r,l)-designs, in: Proc.

[ 8 ] De Witte, P., On the complement of a triangle in a projective


VERONESE QUADRUPLES

Alessandro B ichara

D ipa r t imen to d i Matemat ica I s t i t u t o " G . Castelnuovo"

U n i v e r s i t a d i Roma "La Sapienza" 1-00185 - Rome, I t a l y

ABSTRACT. The c l a s s i c a l Veronese v a r i e t y r e p r e s e n t i n g t h e con ics i n a p r o j e c t i v e p lane i s genera l i zed s t a r t i n g f rom Buekenhout ova ls . Th is leads t o t h e d e f i n i t i o n o f a Veronese quadruple which i s comple te ly c h a r a c t e r i z e d as a p roper i r r e d u c i b l e p a r t i a l l i n e a r space c o n t a i n i n g two d i s j o i n t f a m i l i e s o f suspaces s a t i s f y i n g s u i t a b l e axioms.

1. INTRODUCTION

The p a i r - S = ( P , L / i s s a i d t o be a p roper i r r e d u c i b l e p a r t i a l l i n e a r space

( P L S ) i f P i s a non-empty se t , whose elements a re c a l l e d p o i n t s , L i s a p roper

f a m i l y o f subsets o f P , l i n e s , and t h e f o l l o w i n g h o l d [3]:

( i ) Through any p o i n t of - S t h e r e i s a t l e a s t one l i n e .

( i i ) Any two l i n e s have a t most one p o i n t i n common.

(iii) Any l i n e o f - S i s on a t l e a s t t h r e e p o i n t s .

( i v ) There e x i s t two d i s t i n c t p o i n t s such t h a t no l i n e con ta ins bo th o f them.

Through t h i s paper ,? = ( P , L J denotes a p a r t i a l l i n e a r space.

Two d i s t i n c t p o i n t s p and q i n S a r e s a i d t o be c o l l i n e a r , if t h e y l i e on

a common l i n e ; i n t h i s case we w r i t e p s q .

A subset H o f 7' i s s a i d t o be a p roper subspace o f 2, i f H c o n s i s t o f c o l -

l i n e a r p o i n t s , a t l e a s t t h r e e o f which a re n o t on t h e same l i n e .

Now we c o n s t r u c t an i r r e d u c i b l e p roper PLS 5 c o n t a i n i n g proper subspaces.

L e t P = ( 7 , B ) be an i r r e d u c i b l e p r o j e c t i v e p lane o f o rde r g r e a t e r t han t h r e e and

denote by ? t h e s e t o f a l l unordered p a i r s [ l , s ] o f l i n e s i n P. For a l i n e 1 i n

P, we d e f i n e

al = t ! I , s 1 : s E i31.

Such a nl i s n a t u r a l l y endowed w i t h t h e s t r u c t u r e o f a p r o j e c t i v e p lane

58 A . Biclzara

( t h r e e p a i r s [ l , s l 1, [ l , s , ] , [ l , s , ] are s a i d t o be c o l l i n e a r i f t h e t h r e e l i n e s

s , , s,, s , a re concur ren t i n P ) . Such a p lane i s isomorphic t o t h e dual p lane

o f P.

Def i ne

: 1 E D I , P I = I n 1

Next, f o r p E 3 d e f i n e

a t [ l , s l , p E 1 , P E S , 1, S E 01 P

If a Buekenhout ova l [ 2 ] B (p ) i s d e f i n e d on t h e p e n c i l F (p1 o f l i n e s i n P

th rough p, then t h e s t r u c t u r e o f l i n e a r space ( B ( p ) ) can be g iven t o a as f o l -

lows: t h e t h r e e p a i r s [ l , ,sl 1 , [ l p , s z 1, [ l 3 , s 3 1 are c o l l i n e a r i f e i t h e r an i n -

v o l u t i o n o f B(p) in te rchanges li ans s ( i = 1,2,3), o r 1 , = l2 = 1 ). Therefore,

t h e l i n e a r spaces a ( B ( p ) l i s isomorphic t o t h e dual space o f t h e one c o n t a i n i n g

B (p ) .

P

i

I n what fo l l ows , i t w i l l be assumed t h a t f o r any p o i n t p i n 3 a Buekenhout

ova l B ( p ) i s g iven . THen we d e f i n e P , = ( a ( B ( p ) ) : p € 9 1 .

Denote by L t h e f a m i l y o f l i n e s be long ing t o some elment i n P , u P 2 . Hence

t h e p a i r - S = ( P , L ) i s a proper i r r e d u c i b l e PLS c o n t a i n i n g t h e c o l l e c t i o n s P I and

P, o f p roper subspaces.

The quadruple ( P , L , PI, P I ) w i l l be s a i d t o be t h e Veronese space o f P as-

soc ia ted w i t h t h e f a m i l y i B(p ) : p~ 71 o f Buekenhout ova ls .

I n o rde r t o c h a r a c t e r i z e t h e Veronese space o f a p lane assoc ia ted w i t h a

c o l l e c t i o n o f Buekenhout ova ls , we d e f i n e a Veronese quadruple as a quadruple

( P I L, P , , P 2 ) s a t i s f y i n g t h e f o l l ow ing cond i t i ons ,

(i) The p a i r - S = ( P , L ) i s a p roper i r r e d u c i b l e PLS.

( i i ) P and P, are two d i s j o i n t f a m i l i e s o f p roper subspaces such th rough

any l i n e o f - S t h e r e i s a t l e a s t one subspace o f $ , U p , and

(1.1)

(1.2) Any two d i s t i n c t subspaces i n P (i = 1 , 2 ) have p r e c i s e l y one p o i n t i n

Fo r any n E P , , and any a E P , e i t h e r a n n = 0 o r any l i n e i n a meets n .

i common.

Veronese Quadruples 59

(1.3)

(1.4)

( 1 . 5 )

(1 .6 )

( 1 . 7 )

( 1 . a ) (1 .9 )

Through any p o i n t t h e r e a re a t l e a s t t h r e e elements o f P I u P , and a t

most two elements o f P I . I f a p o i n t p i s on two elements o f P,, t hen p i s on e x a c t l y one element

o f P I .

Any t h r e e elements i n Y', meet ing t h e same element i n P , have a common

p o i n t .

Any t h r e e elements o f P I meeting t h e same element i n P, meet every e l e -

ment i n 7 , i n c o l l i n e a r p o i n t s .

I f t h r e e elements i n 7 , meet a subspace i n P , i n c o l l i n e a r p o i n t s then

t h e r e e x i s t s an element i n P, hav ing a n o n - t r i v i a l i n t e r s e c t i o n w i t h

each o f them.

An isomorphism between two Veronese quadruples ( P . L , P l $P,) and ( ? ' , L ' ,

?', , P I 2 ) i s a b i j e c t i o n f : P + p ' such t h a t

Both f and f - ' map l i n e s on to l i n e s .

Both f and f - ' p rese rv . t h e two c o l l e c t i o n s o f p roper subspaces.

It i s easy t o check t h a t t h e Veronese space ( P , L , P , .P, ) o f a p r o j e c t i v e

p lane P ( 7 , f l ) assoc ia ted w i t h a f a m i l y i B ( p ) : p E S ) o f Buekenhout ova ls i s

a Veronese quadruple. Furthermore, i f P can be c o o r d i n a t i s e d by a (commutat ive)

f i e l d K and each B(p ) i s assoc ia ted w i t h a con ic i n P , t hen !pzL P I $, 1 i s i s o -

morphic t o t h a t p a r t o f t h e cub ic su r face Mt i n PG(5,K), r e p r e s e n t i n g t h e c o n i c s

i n P which s p l i t i n t o two l i n e s i n P [ l ] .

I n t h i s paper t h e f o l l o w i n g r e s u l t s w i l l be proved.

I . - I f Q = ( P , L , P , , P , i s a Veronese quadruple, t hen t h e r e e x i s t s a p ro -

j e c t i v e p lane o f o rde r g r e a t e r t hen t h r e e such t h a t f o r each p o i n t p i n P a

Buekenhout ova l B ( p ) i s d e f i n e d on t h e p e n c i l o f l i n e s th rough p. Furthermore,

Q i s i somorph ic t o t h e Veronese space o f P assoc ia ted w i t h t h e f a m i l y ( B ( p ) : p ~ P l

When p i s f i n i t e , axiom ( 1 . 7 ) w i l l be shown t o be a consequence of t h e r e -

mai n i ng ones.

11. - L e t Q = ( ? , / . , P I ,?,) be a quadruple i n which ( P , L ) i s an i r r e d u c

PLS and ? 1 and f', are two f a m i l i e s o f p roper subspaces such t h a t th rough

l i n e o f - S t h e r e i s a t l e a s t one subspace o f P I U y',, Suppose moreover t h a t Q

f i l s axioms (1 .1 ) ,..., ( 1 . 6 ) . I f - S i s f i n i t e , then a l s o (1 .7 ) ho lds .

b l e

any

u l -

60 A . Bichara

2. SOME PROPERTIES OF VERONESE QUADRUPLES

L e t Q = (P , .L , P I , ? ' , ) be a Veronese quadruple.

111. Denote by V t h e s e t o f a l l p o i n t s i n 2 through which e x a c t l y one e l e -

ment i n P , passesand by A t h e se t o f a l l p o i n t s i n 2 through which p r e c i s e l y two

elements pass o f P I . Then b o t h A and V a re non-empty and P = A uV.

__ Proof . By (1.3), th rough any p o i n t p i n 2 a t most two elements pass of

P I . I f p IE A, i .e. th rough p a t most one element passes o f P I , then two elements

e x i s t i n 8 , c o n t a i n i n g it; the re fo re , th rough p p r e c i s e l y one subspace passes

o f P, (see (1 .4 ) ) and p e V . Consequently, P = A uV.

Next, A # 0 w i l l be proved. Take q , E P and v , a subspace i n P , th rough q,

(by t h e p rev ious argument such a subspace e x i s t s i n P I ) . Since ( P , L ) i s a prcjp-

e r e r PLS and V, i s a subspace a p o i n t q, e x i s t s i n P\ v l . L e t n 2 be an element

o f P , th rough q 2 . Obviously, ~ l , f IT, and by (1 .2 ) t h e p o i n t n , n a , belongs t o

A; thus , A # 0.

F i n a l l y , V # 0 w i l l be proved. Through bo th q and q 2 a t l e a s t one element

ai E P2, i = 1,2, passes (see ( 1 . 3 ) ) . I f a , # a 2 , then C I , ~ CI, i s a p o i n t i n V.

Assume a , = a z and l e t CI be an element i n P2 th rough a p o i n t q o f f a , ; thus

a na,E V and t h e statement i s proved.

The nex t p r o p o s i t i o n i s a s t r i g h t f o r w a r d consequence o f axioms (1 .3 ) and

(1 .4 ) .

I V . Through any p o i n t i n V a t l e a s t two d i s t i n c t subspaces o f P , pass and

th rough any p o i n t i n A p r e c i s e l y two subspaces o f P I and one o f P , pass.

V. I f n € P , ; then I n n V I ~ l .

- Proof . Assume TI con ta ins two d i s t i n c t p o i n t s i n V, say p , and p 2 . By prop.

I V , th rough p , two d i s t i n c t subspaces a , and a : o f P2 pass which share j u s t t h e

p o i n t p , (see (1.2)) ; hence, t h e y have a non-empty i n t e r s e c t i o n w i t h TI. S i m i l a r -

ly, two subspaces a,and a; e x i s t s i n P,meeting a t p, and n o t skew w i t h IT. The

subspaces a., a ' . , i = 1,2, o f P , share a p o i n t by (1.5). Therefore, p I =

belongs t o a, na; so t h a t a , n a ~ > t p , , p , ] ; a c o n t r a d i c t i o n , s i n c e p 1 # p2 . The

statement fo l l ows .

1 1

V I . I f T E P , , a E P , , t hen e i t h e r n n a = 0 o r l n n a l 2 3.

Proof . Assume a n n # 0; thus , a p o i n t p e x i s t s i n a n v . L e t 1 be a l i n e i n

a th rough p and q a p o i n t i n a n o t on 1 ( s i n c e a i s a p roper subspace i t i s n o t

__


c o i n c i d e n t w i t h 1; hence, q e x i s t s ) and q ' a p o i n t on 1 . By ( 1 , l ) t h e l i n e

th rough q and q ' meets n a t a p o i n t p ' E - S and - o b v i o u s l y - p ' # p. The sub-

spaces TI and o f ( P , L ) meet a t a subspace which con ta ins t h e l i n e th rough i t s

d i s t i n c t p o i n t s p and p ' . Since ( F ' , L ) i s i r r e d u c i b l e , 111 2 3.

V I I . I f TI E P , , a € $ , , then e i t h e r n n a = 0 o r n n a E L .

- Proof . By pr0p.s V and V I , i f T n a # 0 then a t l e a s t two d i s t i n c t p o i n t s

p 1 and p , o f A l i e on n n a . The l i n e 1 i n L th rough them i s con ta ined i n n n a ,

as two subspaces meet a t a subspace. To prove t h a t TI n o = 1 , t a k e a p o i n t p, i n

TI n a n A . By prop. I V , th rough p i = 1,2,3, p r e c i s e l y one subspace TI o f $,pas-

ses o t h e r than n. Moreover, T, nn = p , . The t h r e e subspaces T., i = 1,2,3 a l l

meet a ( a t l e a s t a t pi, resp . and by axiom (1.6) meet T i n c o l l i n e a r po in ts ;

hence, p l , p,, p , are c o l l i n e a r and n n a n A c l . I f n n a = T I ~ G ~ A , then t h e

statement i s ?roved. On t h e o t h e r hand, i f t h e r e e x i s t s a p o i n t q o f V i n n n a ,

then, by prop. V, n n a nV = q, whence n n a = 1 U L q ) . The subspace n n a o f ( $ , L )

i s an i r r e d u c i b l e l i n e a r space, t h e r e f o r e , q €1 o the rw ise any l i n e i n n n (I j o i n -

i n g q w i t h a p o i n t p on 1 would c o n s i s t s of j u s t p and q, a c o n t r a d i c t i o n . The

statement f o l l o w s .

i' i

1

3. THE PLANE (P2,B)

L e t TI be an element i n P1. By prop. V I I , every subspace a o f Y', v e e t i n g

TI a t a p o i n t meets i t a t a l i n e i n L. Thus, t h e f o l l o w i n g subset o f P 2 i s d e f i n -

ed

B(n) = { a € $ , : ( 1 n n ~ L 1 .

V I I I . If Ti E P I , then B ( T ) i s d i s t i n c t f rom P , and con ta ins a t l e a s t two

elements i n P,; hence, B ( n ) i s a p roper subset of P2.

Proof . By prop. I V , th rough any p o i n t p , i n TI a t l e a s t one element o f

P, passes which meets n a t a l i n e 1 i n L ( s e e prop. V I I ) ; thus , (I, B ( n ) . Through

a p o i n t p, i n T I \ 1 t h e r e i s a subspace a 2 o t h e r than a 1 o f P, ( p , € a , and p,BaI).

O f course, ~ , E B ( T I ) and s ince a,€ B ( n ) , IB(TI) I ,Z.

-

To prove t h a t B ( T T ) # P 2 , assume, on t h e c o n t r a r y , B ( T I ) = P 2 . Thus, any sub-

space o f P , meets TI i n a l i n e i n L , so t h a t , by axiom (1.5); a p o i n t q e x i s t s

i n V th rough which a l l t h e elements i n P , pass. Therefore, s i n c e I B ( T I ) ~ ~ ~ , by

axiom ( 1 , 4 ) , th rough q p r e c i s e l y one element n ' E P I passes. T I ' con ta ins t h e

62 A . Bichara

p o i n t q which i s on a l l elements i n P 2 ; hence, i s met by every subspace o f

P , a t a l i n e i n L th rough q (see prop. V I I ) . Next, l e t q , and q z be two p o i n t s

i n 9 ' o t h e r than q and n o n - c o l l i n e a r w i t h it. By prop.V, q , and q , be long t o A

and th rough then t h e subspaces TI, and n , , resp. of P1 pass; moreover, TI, , TI, #

TIi and IT, # TI, . Through t h e p o i n t q ' = n,nn, p r e c i s e l y one element a EP? pas-

ses, The subspace u meets TI, a t a l i n e 1 , i n L and i s n o t skew w i t h ~ l ' ( s i n c e

no e leme i t i n 8 , i s d i s j o i n t f rom TI ' 1 . Thus, by axiom (1.11, t h e l i n e 1 on

a meets 11' a t a p o i n t . Since 1, i s con ta ined i n n, i t passes th rough t h e p o i n t

q 1 = IT'^ IT,. Consequently, q , i s on 1 , and s ince I , C U , q , l i e s on D. By a s i m i -

l a r argument, q, E D . The subspace a E P , passes th rough q, t h e p o i n t on a l l t h e

elements i n ?,, thus i t con ta ins q, q l , and q 2 . These t h r e e p o i n t s a re a l s o on

T I ' ; hence, I q,q , , q , ~ c u n TI I , a c o n t r a d i c t i o n s ince these p o i n t s a re non-co l -

l i n e a r (see prop. V I I ) . The statement fo l l ows .

I X . I f n , , n 2 ~ P I y IT, # TI^ , then I B ( n , ) n B ( n 2 1 I = 1.

~ P roo f . Through t h e p o i n t p = n , n n , a un ique element E P, passes (see

prop. I V ) . Obv ious ly , U E B ( T , 1 n B ( n , 1. Take D ' ~ B ( T I , ) ~ B ( T I , ) ; D' meets TI, a t a

l i n e 1, E L and IT, a t a l i n e 1 , ~ L. 1 i s conta ined i n a ' and by axiom ( 1 . 1 )

meets n2 a t a p o i n t p i E n 2 . Since p ' E l , and 1 ,C n,, p ' E n, whence p ' E TI^ Tin,.

Since TI^ and TI^ meet j u s t a t p, p ' = p. Therefore, a' passes th rough p and i s

c o i n c i d e n t w i t h u t h e unique element th rough p i n P 2 . Thus, D = B ( n , ) n B ( n , )

and t h e statement i s proved.

By prop. I X , t h e f a m i l y B = { B ( n ) : n c P , } o f p roper subsets o f P 2 i s d e f i n -

ed.

X. The p a i r (F '2yB) i s a p r o j e c t i v e plane.

~ P roo f . By pr0p.s V I I I and I X , 0 i s a p roper c o l l e c t i o n o f proper subsets

o f P , and any two d i s t i n c t elements i n 13 share p r e c i s e l y one element i n P2.

Next, we prove t h a t two d i s t i n c t elements a, and a 2 i n P z are conta ined i n

a un ique element o f B. Set p = a , n a , ; through p e x a c t l y one element TI i r i

P I passes (see prop. I V ) . Obviously, u , , a , E B ( T I ) and t h e statement f o l l o w s .

X I . TI € P I i m p l i e s I n n V l 1 . Furthermore, t h e l i n e s i n TI which are t h e i n -

t e r s e c t i o n s o f n by elements i n B ( T I ( G P ~ ) are p r e c i s e l y t h e l i n e s i n t h e pen-

c i l ( i n TI) w i t h c e n t r e a t t h e p o i n t v = n n V . Thus, p a i r w i s e d i s t i n c t p o i n t s i n

n \ { v l a r e c o l l i n e a r w i t h v i f they belong t o t h e same element i n B(TI).

__ Proo f . L e t a , and a 2 be two d i s t i n c t elements o f B(n 1 ( t hey do e x i s t by

Verotiese Quadruples 63

prop. V I I I ) and se t v = a l n a 2 (see axiom ( 1 . 2 ) ) ; o f course, v belongs t o V . A s -

sume v a n; th rough v a un ique element IT' of P , passes o t h e r than IT. Thus, t h e

l i n e s 1 , = a, n n ' and 1, = a, n n ' a r e d i s t i n c t , as l a , n a , 1 = 1 and 1 1 . 1 2 2 ,

i = 1,2, and belong t o a, and a z , resp. Since a, and a, belong t o B ( n ) , t h e

l i n e s 1, and 1, meet n a t t h e p o i n t s q, and q,, r e s p e c t i v e l y , which a r e d i s t i n c t ,

o the rw ise v would be c o i n c i d e n t w i t h q, = q,, imposs ib le as v a n . Hence, ~ n n ' con ta ins t h e two p o i n t s q, and q 2 , a c o n t r a d i c t i o n (see axiom (1 .2 ) ) . There fore

v belongs t o IT and by prop . V ~i nV = [ v ) .

1

By axiom ( 1 . 5 ) any element i n B ( n ) con ta ins t h e p o i n t v = a , n a 2 ; thus , i t

meets n a t a l i n e i n L th rough v (see prop. VII: Since th rough eve ry p o i n t i n

n t h e r e pass a t l e a s t one element i n P,, hence i n B ( I I ) , t h e l i n e s i n n , which

a re t h e sec t i ons o f n by t h e elements i n B ( n ) , a r e p r e c i s e l y t h e l i n e s th rough

v. The statement fo l l ows .

X I I . (P2,5) i s a n i r r e d u c i b l e p r o j e c t i v e plane.

Proof. Take n E P , and v = i~ n V. A l i n e 1 does e x i s t i n n n o t th rough v,

as n i s a p roper subspace of ( a , L ) . Since (a , L ) i s i r r e d u c i b l e , 111 33. The

l i n e s j o i n i n g v w i t h p o i n t s on 1 a r e d i s t i n c t and con ta ined i n n . Thus, a t

l e a s t t h r e e l i n e s e x i s t i n IT th rough v. By prop. X I , on any l i n e i n ( P z , B ) a t

l e a s t t h r e e p o i n t s l i e and t h e statement i s proved.

-

4. THE PROOF OF PROPOSITION I

I n t h i s s e c t i o n prop. I w i l l be proved.

Take CZE P , and La be t h e s e t o f a l l l i n e s i n L on a; cons ide r t h e p e n c i l

F ( a ) o f t h e l i n e s i n t h e p r o j e c t i v e p lane ( p , ,o) th rough t h e p o i n t a E P 2 ; ob-

v i ous l y ,

F ( a ) = i B ( n ) €5 : a n ~i E L 1

I f l c L a , a correspondence i : F(a) + F(a) i s de f i ned as f o l l o w s 1

\ = B ( n ' ) w n n n ' €1, n # n ' ,

( 4 . 1 ) il ( B ( v 1 ) 1 = B ( n ) n VE I.

By axiom (l.l), any element i n P I meet ing a i n a l i n e shares a p o i n t w i t h

1. Moreover, t h e p o i n t s on 1 n o t i n V be long t o A and (see prop . 111) th rough

each of them e x a c t l y two elements o f P , pass. There fore , il i s a b i j e c t i o n and

64 A . Bichara

and i n v o l u t i o n o f F( a) whose f i x e d l i n e s a re a l l t h e l i n e s B ( n ) i n F ( a ) such

t h a t n nV € 1 ; fu r thermore , i in terchanges t h e l i n e s B(n) and B ( n ' ) , * # 11') o f 1

F (a ) i f n n s ' ~ 1. Thus, t h e nex t statement has been proved.

X I I I . I f a e P 2 and l E 6 , t h e b i j e c t i o n i :F(a ) + F ( a ) de f i ned by (4.1) i s 1 an i n v o l u t i o n .

I f a E P2 then t h e f a m i l y e ( a ) = 6 . 1 E L a } o f i n v o l u t i o n s o f F ( a ) i s de- 1 '

f i ned .

X I V . I f a ~ 7 ' ~ , then IF( a l l 24. Furthermore, t h e p a i r ( F ( a ) , e ( a ) ) i s a

Buekenhout ova l .

Proo f . By prop. X I I I , each b i j e c t i o n i : F ( a ) + F(a (1 EL,) i s an i n v o l u -

t i o n . Since 111 '3, I F ( a ) l '3 . Next, i t w i l l be shown t h a t ( F ( a ) , e ( a ) ) i s a

Buekenhout ova l , i .e. t h a t (21

- 1

( i l every element o f e ( o ) i s an i v o l u t i o n o f F ( a )

( i i ) for any two p a i r s (B(n, ) , B ( n 2 ) ) and (B(n:),B(n:)), mi # n' i,j = 1,2

o f l i n e s i n F ( a ) p r e c i s e l y one i n v o l u t i o n e x i s t s i n e ( a ) i n te rchang ing B( n,)

and B(n , ) and B ( n : ) w i t h B(n:) .

j '

From prop. X I 1 1 (i) f o l l o w s . Thus, ( i i ) w i l l be proved. I f B(n, ) = B(n, 1

and B ( n ; ) = B (n ; ) , then 11, = n 2 and n I l = . A unique l i n e 1 e x i s t s i n a th rough

t h e p o i n t s n l nV and n', n V , bo th on a s ince B ( n , ) , B(n : ;E F ( a ) . I f t h i s occurs

then i i s t h e unique element i n e ( a ) f i x i n g b o t h B(n,) and B(n:) . 1

On t h e o t h e r hand, i f B(n, 1 # B(n,) and B ( n , ' ) # B(n:), then n 1 # n 2 and

n; # nI2; t h e two p o i n t s TT, nn2 and n ' , nn', i n A belong t o a and are d i s t i n c t ;

hence, t h e r e i s a unique l i n e 1 i n L a th rough bo th of them. Again, i i s t h e

unique element i n e ( a ) i n te rchang ing B ( n , ) w i t h B(n,) and B ( n , ' ) w i t h B(n;). 1

A s i m i l a r argument proves ( i i ) i n t h e remain ing cases. The statement f o l -

1 ows. - - -. . Next, l e t ( F ' , L , P , , P be t h e Veronese space o f p2,8) assoc ia ted w i t h t h e

c o l l e c t i o n { ( F ( a ) , O ( a ) ) : aEp210 f Buekenhout ova ls .

Consider t h e mapping @ : P + P a s s o c i a t i n g w i t h eve ry unordered p a i r

[ B ( n ) , B ( n ' ) ] o f d i s t i n c t l i n e s i n L i ( i . e . n f n ' ) t h e p o i n t n n n ' E P and w i t h

t h e p a i r [ B ( n ) , B ( n ) ] o f c o i n c i d e n t l i n e s i n D t h e p o i n t TI n V i n P . - 5 -

X V . The mapping 0 i s an isomorphism between ( r J , P 1 , F 2 ) and ( P , L , P l , p 2 ) .

Proof . By pr0p.s 111, X I and axiom (1,2), - (4.2) @ i s one-to-one and on to .

Vrronrse Quadruples 65

From t h e d e f i n i t i o n o f (F(a),e(n)) t h e n e x t statement f o l l o w s . - - ( 4 . 3 ) Any l i n e i n L on an element i n , i s mapped by @ on to a l i n e i n

L o n an element i n P , ; fu rh te rmore , t h e i n v e r s e image under @ o f any l i n e i n

L on an element o f 8, i s a l i n e i n L on an element i n d , . By axiom (1.61, t a k i n g i n t o account prop. X I ;

( 4 . 4 ) Any t h r e e c o l l i n e a r p o i n t s o f an element i n P , a r e mapped by 0 on to

t h r e e c o l l i n e a r p o i n t s o f an element i n P1. From axiom (1 .7 ) and prop. X I t h e

nex t statement f o l l o w s .

(4.5) Three c o l l i n e a r p o i n t s on an element i n P , are mapped by @-' on to

t h r e e c o l l i n e a r p o i n t s on an element i n F1. The statement f o l l o w s f rom ( 4 . 2 ) -

( 4 . 5 ) .

From pr0p.s X, X I I , X I V , X V , prop. I f o l l o w s .

5. THE PROOF OF PROP. I 1

F i r s t l y , remark t h a t a l l p rev ious r e s u l t s b u t ( 4 . 5 ) were proved w i t h o u t

t h e he lp o f axiom (1 .7 ) . Thus, w i t h t h e same n o t a t i o n as be fo re , t h e nex t propo-

s i t i o n can be s ta ted .

X V I . Under t h e assumptions i n prop. I1 f o r t h e mapping @ : P * P ( 4 . 2 1 ,

( 4 . 3 ) and ( 4 . 4 ) ho ld .

Next, any fl E ? , determines bo th t h e l i n e B(. be long ing t o B and t h e sub- - E b,in ( P , L ) d e f i n e d by (see sec t . 1 ) :

B ( space 4 = n -

n = n = ( [ B ( n ) , B ( n ' ) ] : V ' E P I ] . B( n)

Consider t h e mapping 6 :13 -+ o f t h e dual p lane o f ( 2 , , /I) o n t o ? de f i ned

by

(5.1 1 6 ( B ( n ' ) ) = [ B ( n ) , B ( n ' ) 1 . C l e a r l y (see sec t . 11,

( 5 . 2 ) $ i s an isomorphism betwecn t h e dual p lane o f (P2 , D ) and t h e sub-

space % o f (P , L ) .

Def ine a mapping a ' : ii n by

(5.3) @ ' ( [ B ( n ) , B ( n ' ) 1) = @ ( [ B ( a ) , B ( n ' ) 1); obv ious l y , 0 ' i s one-to-one and on to . S ince @ and @ a r e b i j e c t i o n s ,

( 5 . 4 ) t h e mapping @'6: 13 +TI i s a b i j e c t i o n .

Next, a s s m e 7) i s f i n i t e . Then P , i s f i n i t e and so i s t h e p r o j e c t i v e .

66 A . Bichara

p lane ( ' P , , B ) ; i f i t i s o f o rde r q, t hen 101 = q z t q t 1; hence (see (5 .4 . ) ) ;

(5 .5 ) l f l l = q 2 t q t 1.

By (5.2), (5.31, and (5.4)

(5.6) 0' J, maps t h r e e concur ren t l i n e s o f B on to t h r e e c o l l i n e a r p o i n t s

i n TI.

Furthermore, t a k i n g i n t o account prop. X I , a l i n e i n passes th rough t h e

p o i n t v =

th rough a p o i n t on B (n ) . Consequently,

(5.7) There a re p r e c i s e l y q t 1 l i n e s i n th rough v = n V and on

each o f them q t 1 p o i n t s l i e .

n n V if i t i s t h e image under $ ' 6 o f a p e n c i l o f l i n e s i n h 'zJ

Next, l e t 1 be a l i n e i n 1~ n o t th rough v. S ince every p o i n t on 1 i s j o i n e d

t o v by a l i n e , by ( 5 . 7 ) 111 5 q + 1; hence

(5 .8 ) Any l i n e i n c o n s i s t s o f q t 1 p o i n t s a t most.

The q t q t 1 l i n e s i n (P z ,B) a l l have s i z e q t 1 and each o f them i s map-

ped by O ' J I on to a l i n e i n n (see (4.4) and (5.8)); t h e r e f o r e , on 11 a t l e a s t

q t q t 1 l i n e s l i e each o f them having s i z e q t 1 . Since * i s a subspace of

( B , L ) , by (5.51,

(5,9)

morphism between t h e dua l p lane o f (P2,B) and n.

1~ i s a p r o j e c t i v e p lane o f o rde r q and t h e mapping 0 'J, i s an i s o -

L e t pi, i = 1,2,3, be t h r e e p o i n t s i n n. C l e a r l y ,

Since bo th JI and $ I $ a re isomorphisms t h e t k r e e p o i n t s Q- l (p .1 a re c o l -

@-'(pi) = ( e ' l - ' ( p . ) 1 = ( J , d J - l ( O ' ) ' ) (P i ) = J,(O'J,)-'(pi)

1

l i n e a r iff t h e p o i n t s p . a re c o l l i n e a r whence (4 .5 ) f o l l o w s . 1

By t h e p rev ious argument and prop. X V I , under t h e assumptions i n prop. 11,

$: P + ? s a t i s f i e s (4 .2 ) t o (4 .5 ) so t h a t i t i s an isomorphism between Q = (d, L , ? , , F 2 ) and Q = (P ,L , P,, P J . Since f o r t h e Veronese space Q axiom (1 .7 ) ho lds ,

t h e same i s t r u e f o r Q and prop. I 1 i s proved.

REFERENCES

[ l 1 E. B e r t i n i , I n t roduz ione a l l a geometr ia p r o i e t t i v a d e g l i i pe rspaz i , Pisa,

E. Spoer r i (19071.

[ 2 ] F. Buekenhout, Etude i n t r i n s e q u e des ova les , Rend. d i Mat. V (1966) 333-393


[ 3 ] G. T a l l i n i , Spazi p a r z i a l i d i r e t t e , spaz i p o l a r i . Geometr ie subimmerse,

Sem. Gem. Comb. Univ. Roma 14 (1979) .


S-PARTITIONS OF GROUPS AN0 STEINER SYSTEMS

Mauro Biliotti

Dipartimento di Matematica Universit2 di Lecce

Lecce - ITALIA

In this paper we investigate a special class of S-partitions of finite groups. These 5-partitions are used for the construction of resolvable Steiner systems. Several classification theorems are also given.

The concept of S-partitions may be traced back to Lingenberg [ 131 although the

actual introduction was made by Zappa [ 2 4 ] in 1964. Zappa developed some ideas of

Lingenberg so as to provide a group-theoretical description o f linear spaces with

a group of automorphisms such that the stabilizer of a line acts transitively on

the points of that line.

Afterward Zappa [ 26 ] and Scarselli [17] mainly investigated the following ques-

tion: find conditions on a S-partition Z: o f a group G relating the existence o f C

to that of a partition - in the usual group-theoretical sense - o f a subgroup o f

G. In this case the linear space associated to C is simply the translation AndrC

structure associated to that partition [ 3 ] . From a geometrical point of view, the

work of Zappa [ 2 5 ] , Rosati [16] and Brenti [6] on the so-called Sylow S-partitions

seems to be more interesting as Sylow S-partitions are useful in constructing some

classes of Steiner systems. In this connection, another class of S-partitions is

noteworthy. These S-partitions are those considered by Lingenberg [ 131 and later

bv Zappa [ 2 4 ] , We shall call these S-partitions "Lingenberg S-partitions".

Lingenberg S-partitions were inspired by a reconstruction method of the affine

geometry A G ( n , K ) , K a field, by means of a special class of subgroups of SL(n,K).

In this paper, we study Lingenberg 5-partitions o f finite groups. We mainly inves-

tigate "trivial intersection" 5-partitions which we call type I S-partitions (see

section 2). For type I S-partitions, we give a "geometric" characterization and

somewhat determine the corresponding group structure and action. Also we obtain a

classification theorem for Lingenberg S-partitions o f doubly transitive permutation

groups. We note that for some simple groups, Lingenberg S-partitions are useful in

constructing resolvable Steiner systems. In these cases, the Steiner systems might

be regarded as a natural affine geometry for the groups.

70 M . Biliotti

1. PRELIMINARIES

Groups and incidence s t ruc tu res considered here are always assumed t o be f i n i t e .

I n general, we s h a l l use standard notat ion. I f G i s a group and H 2 G, K 9 G, then

O(G) i s the maximal normal subgroup o f odd order o f G, S (G) i s the set o f a l l P

Sylow p-subgroups of G and HK/K i s denoted by A. If H l l K = <1> then K X H denotes

the semid i rect product of K by H. I f G i s a permutation group on a set R and r G f i

then G denotes the g loba l s t a b i l i z e r o f i? i n G. A set R i s a G-set i f the re i s a

homomorphism cp from G i n t o the symmetric group on G. Usually we s h a l l w r i t e G

instead o f v ( G ) .

Le t G be a group and S a subgroup of G with SzG. A se t C o f n o n - t r i v i a l subgroups

o f G such t h a t ICl22 i s sa id t o be a (keguLatr) S-pwrLLtiion o f G i f the fo l l ow ing

condi t ions are s a t i s f i e d :

( i )

( i i )

( i i i )

The above d e f i n i t i o n i s due t o Zappa [ 2 4 ] . Here we are i n te res ted i n the fo l l ow ing

spec ia l c lass o f S -pa r t i t i ons :

a S - p a r t i t i o n C o f a group G i s sa id t o be a L i n g e n b a g S . - p a t , t i L i o ~ wl th respect

t o the subgroup T o f G i f the fo l l ow ing hold:

(j) (jj) <T' : x t ~ > = G , n s x = ;

(jjj) C = { T * : xeG-NG(T) }u {NG(T) } .

We p o i n t out t h a t C i s determined by the t r i p l e (G,S,T). Now we g ive some geome-

t r i c a l d e f i n i t i o n s .

As usual a Lineah npace i s a p a i r ( T I , R ) , where II i s the set o f p o i n a and R i s a

fami ly o f subsets o f Il whose elements are c a l l e d l ines , such t h a t two d i s t i n c t

po in ts l i e i n exact ly one l i n e . Here we assume a l so t h a t 11 contains a t l e a s t three

non-col l inear po ints .

An AndkL btkuctuhe ( A - n h c t u h e ) i s a t r i p l e (n,R,//), where ( n , R ) i s a l i n e a r

space and 1'//1' is an equivalence r e l a t i o n on R such tha t each equivalence c lass

gives a se t - theo re t i ca l p a r t i t i o n o f Il.

L e t 1' = ( I I , R , / / ) be an A-structure and l e t I I o = I I . Assume t h a t whenever P;Q,Rt n o wi th P#Q then the l i n e PQ and the p a r a l l e l l i n e through R t o PQ are wholly con-

ta ined i n n o . I f we s e t R o = { h : h b R , itcIIo} then (no,Ro,//) i s an A-st ructure

(or a l i n e ) and i t i s sa id t o be a dubspace o f Y . Isomorphisms and automorphisms

o f l i n e a r spaces and A-structures are def ined i n the usual manner with the r e q u i r

ement t h a t the p a r a l l e l i s m i s preserved f o r A-structures.

A h u o t w a b t e SXeinet oybtem w i t h pakametm ( v , f i ) i s an A-structure with v p o i n t s such t h a t each l i n e contains exac t l y 12 points .

R r

S H n s K = S f o r each H , K e Z with H#K;

f o r each g t G there e x i s t s H t C such t h a t g f SH;

H 4 C impl ies d- 'Hs t C f o r each o E S.

T 6 S < NG(T) < G;

xt G

Spartitions of Groups ond Steirrer Systems 7 1

Following Zappa [24] a quwi - . t na~h . t ion A-4tJLuctuhe is an A-structure $=(N,K, / I )

together with a set (or pencil) 8 of lines and a family 8 = I T ( & ) : a c 8) of

automorphism groups of 0 satisfying the following conditions:

(Q,) for each P c IT there exists exactly one line J L t 0 such that P c h;

( G I 2 ) for each n E R there exists exactly one line k E 0 such that c / / 4 ; ( Q 3 ) for each a E o the group T(K) fixes every point of k and every line parallel

to t, leaves 0 invariant and acts transitively on the points of each line 4

such that 4 / / @ and b # t .

In the following we shall say that (D is a quasi-translation A-structure with res-

pect to the pencil 0 and the family 8. It is very easy to see that, starting from

0 , a new A-structure 5 may be obtained by adding to 0 a new point 0 incident with precisely the lines of 0. 5 will be called the compLetion of 0. Any automorphism

a of @ extends to $ by setting a(0) = 0.

Lingenberg S-partitions are essentially the geometrical counterpart of quasi-tras-

lation A-structures. If Z is a Lingenberg S-partition of a group G with respect t o

the subgroup T, define [G,S,T] as the triple given by

- the set of right cosets o f S in G;

- the set S(C) of the complexes of the form SXg with X E 1, y 6 G ;

- the following relation "//" on S(Z) :

i f a = SN ( T ) q = NG(T)y, then a'//& if and only if either 4' = h or a' = STXz

with TX#T and Tg=Txz;

if J L = STXy with TX+T, then ~ ' / / J L if and only if either h ' = NG(T)z where TZ=TXY

or J L ' = STWz with Tw#T and Txg=Twz.

G

The definition of the parallelism seems to be very involved, but it will make clear

in the following.

PROPOSITION 1.1 (Zappa [24] ,2 .2) . The t G p L e [G,S,T] h a t h e 6 a t t o w i n g pkopek.tiu:

[ 1 1 t h e mappingh : Sx + S x y w i t h g E G o d t h e neR 0 6 t i g h t coht& 0 6 5 i n G i n t o i t 4 e t 6 60m u point-tJLaMnitiue gkaup i b 0 t n O t p h i C Ro G ;

( 2 ) [G,S,T] i n a quuni - t tamtat ion A-n,i3uctutuhe w i X h henpect t o t h e pen& 0 whobe

eLemen& a m the l inen NG(T)x w i t h x E G, and to the 6 a m i L y 9 = I T x : x t G j .

a6 autornohphi~mo 06 [G,S,T] w h i c h i b

Y

Obviously in [G,S,T], we have that the point Sx belongs to the line SXg if and only

if S x c S X y . Also the converse holds. Indeed, we have:

PROPOSITION 1.2 (Zappa [24],2,1). Let 0 be a quaoi- tkumLaLhn A-Athuctuhe w i t h rtenpect t o .the penciL 0 und t o t h e t ;amdg 9 = I T ( & ) : J L 6 El}. Fuhthemohe unnume 3

2, a cvmpte.te d a b 06 conjuga.te 4 u b g ~ ~ o u p ~ 0 5 G .- < T ( k ) : J L C 8>. 16 P 0 a poin t 06 @, p & t h e Pine 06 8 . t hough P and S = Gp, .then C = {T()L) : a E 8- {p } } LJ

u { N G ( T ( p ) ) }

0 2 I G , S , T ( p ) l .

u Linyenbety S - p a h R i L o n o d G d R h ~ ~ e n p e o t to T ( p ) and

I 2 M. Biliotti

Le t 5 be a quasi - t rans lat ion A-structure w i t h respect t o the p e n c i l 0 and f o r each

J L E O l e t H(h) denote the group o f a l l the automorphisms o f 5 f i x i n g every p o i n t o f

4. and every l i n e p a r a l l e l t o JI and which leave 0 i n v a r i a n t . Then @ i s a quasi-

- t r a n s l a t i o n A-structure with respect t o Q and t o the fami ly 8 = { H ( d : k c O } .

Moreover, 9 i s a complete c lass o f conjugate subgroups o f G = c H ( h ) : h e @ > so

t h a t 5 may always be represented i n the form [G,Gp,H(p)l, where P E ~ E O .

2. PROPERTIES AND EXAMPLES OF LINGENBERG S-PARTITIONS

LEMMA 2.1. LeR (G,S,T) d e t e m i n e u Lingenbetlg S-pakL i t ion , t h e n TXnTY = <l> 6vt l

each x,y t G w i t h T X # T Y .

Paood. Consider the A-structure [G,S,T] and l e t ? t T X n T Y . Each p o i n t P of [G,S,T]

i s on a p a r a l l e l l i n e t o NG(T)x and a l so on a p a r a l l e l l i n e t o NG(T)y. Since these

l i n e s are d i s t i n c t and both are f i x e d by ? then ?(P) = P and the re fo re ? = I.

Lingenberg S -pa r t i t i ons may be d i v ided i n t o two classes according t o the f o l l o w i n g

d e f i n i t i o n :

a Lingenbetlg S-pwc.tLtion C ad G w i t h kenpect t o t h e bubgtloup T LA 06 t y p e 1 4 6

S n T X = <1> d o t ench x t G w i t h T X # T . ld t h e S -pah t iL ion 0 not 06 type I, we 4haU

bay t h a t it i~ a6 t ype 11.

For Lingenberg S -pa r t i t i ons o f type I we have:

PROPOSITION 2.2. (G,S,T) de teminen a Lingenbekg S-pakLLtihion 0 6 .type 1 4 and o d y

id t h e automotlpkiom gkoup 7 0 6 [G,S,T] act6 berniheguLahey on t h e Linen 0 6 0 d i b -

. tinct 6kom NG(T). Fuhthemohe, 4 (G,S,T) de teminen a Lingenbekg S-pahtLtion 0 6 t y p e I t h e n t h e duUowing hold: ( I ) 7 uc . i~ t l e g U y on each . !he p a k a f i d t o NG(T) and dinLLnct @om .them, 121 N G ( T ) n T X = <1> 6ok each TK#T, and

W N

131 [NG(T):S] = I T I - 1 .

R o o d . Assume (G,S,T) determines a Lingenberg S - p a r t i t i o n o f type I and l e t

x t NG(T) n T y w i th TY#T. Then ? f i x e s both the l i n e NG(T) and each l i n e p a r a l l e l t o

N (T)q, so t h a t NG(T) i s pointwise f i x e d by x. Therefore, x c S and from SnTY=<1>,

i t fo l lows t h a t x = 1 and (2 ) holds. Now l e t z 6 T and NG(T)w,z = NG(T)w f o r some

w t G w i t h ?#T. Then Z E NG(Tw) and hence z t NG(Tw)nT. But, by (21, NG(TW)nT=<l>

and so z = 1 and 7 ac ts semiregularly on the l i n e s o f 0 d i s t i n c t from NG(T). The

argument may be reversed t o prove the converse. Le t k be a l i n e p a r a l l e l t o NG(T)

and d i s t i n c t from them and assume ?(R) = R f o r some z t. T, R 6 a. Then the l i n e o f 0

through R, which i s d i s t i n c t from NG(T), i s f i x e d by z . Since ? ac ts semiregular ly

on 0 - {NG(T)} , i t fo l l ows t h a t z = 1 and ( 1 ) i s proved. Now l e t lGl=g, ING(T)I=n,

IS(=b and ( T I = t . I n G, there e x i s t g/n - 1 d i s t i n c t complexes o f t he form STX with

Tx#T and each complex contains exac t l y n t elements since S n T X = c1>. I f we take

G

S-Partitions of Groups and Steiner Systems 73

account o f the f a c t tha: SNG(T) = NG(T) conta ins n elements o f G and (G,S,T) deter-

mines a S - p a r t i t i o n of G, then we must have ( n t - n ) ( g / n - 1) t n = g so t h a t ( 3 )

now fo l lows.

COROLLARY 2.3. 16 (G,S,T) d e t e h m i n u a Lingenbefig S-pahtition 0 6 t ype I t h e n

[G,S,T I h a huolwabde SReinek nyotem w i t h paharneteu [ w , k l whehe w = [G:S1 + 1

and k = I T I .

P J L U O ~ . This i s an immediate consequence o f (1) and (3 ) o f P ropos i t i on 2.2.

PROPOSITION 2.4. Let 0 be a quabi-Xhun&!ativn A-bthuCtuhe w i t h henpeCt t o t h e pencil 0 and t o t h e damily 8 = {T (k ) : h e 01. An auhomokpkinm a 0 6 @ c e n t ~ ~ a l i z e n T(t)

d o h each k e 0 id and o n d y 46 it 6 ixe4 &Why &ne 0 6 0. A non-idenLicCLe automohpkcnm

0 6 0 6 ix ing evehy f i n e 0 6 0 a c d 6 .p .6 . on Rhe n e t ofi p o i n d 0 6 0.

Phoo6. I f u f i x e s every l i n e o f 0 then, by [24],3.1, a c e n t r a l i z e s T(k) f o r each

~ € 0 . Conversely, assume n c e n t r a l i z e s T(h) f o r each J L C O and there e x i s t s 6 E 0

such t h a t a ( n ) # n . T(o) f i x e s every p o i n t o f n and every l i n e p a r a l l e l t o 4 . Like-

wise, T(n) = a - ’ T ( n ) a f i x e s eve rypo in t o f u ( n ) and every l i n e p a r a l l e l t o u ( n ) .

Since n#a(4 ) , i t i s easy t o see t h a t t h i s y i e l d s T(n) = <1>, which i s impossible.

Now l e t a. be an automorphism o f 0 f i x i n g every l i n e o f 0 and assume a(P) = P f o r

some p o i n t P. I f JL i s a l i n e through P and A / / & , 6 E 0 then u ( b ) = n and so a ( & ) / / &

which imp l i es ~ ( h ) = fi. Therefore, a f i x e s every l i n e through P. L e t Q be a p o i n t

d i s t i n c t from P and assume PQ=q { 0. I f w denotes the l i n e o f 0 through Q, we have

t h a t a ( Q ) = a ( q f \ w ) = a ( q ) n u ( w ) = q i ? w = Q. I f , on the contrary , q t 0 then the

r e l a t i o n a ( Q ) = Q can be obtained by us ing the same argument as above by s t a r t i n g

from a p o i n t P ’ { 4 . The t h e s i s a = I now fo l lows.

Now we s h a l l g i ve some examples o f Lingenberg S -pa r t i t i ons .

We assume the reader i s acquainted w i t h the s t r u c t u r e o f groups SL(2,q); PSU(3,qz), q=p h , p a prime; S Z ( ~ ~ ~ ’ ’ ) ; R(32nf ’ ) , n > l , and a l so w i t h the elementary

p roper t i es o f l i n e a r groups. General references are i n [ll] and [12]. I n p a r t i c u l a r ,

f o r Suzuki groups S Z ( ~ ‘ ~ ” ) , Ree groups R(3“”) and PSU(3,q’) see [20], [22] and

[23], [7] respec t i ve l y .

EXAMPLE I. G 2 SL(2,q), q = p , q>2. Le t P c S (G) and assume T = S = P; then i t i s

an easy exerc ise t o show t h a t (G,S,T) determines a Lingenberg S - p a r t i t i o n o f type

I and t h a t [G,S,T], the completion o f [G,S,T], i s the a f f i n e plane over GF(q).

This i s the c l a s s i c a l example which i n s p i r e d the work o f Zappa [24]. I t a l so ex-

p l a i n s the d e f i n i t i o n o f the p a r a l l e l i s m i n [G,S,T ] as given i n sec t i on 1.

EXAMPLE 11. G 7 S z ( q ) , q = Z Z n f ’ , ~ 2 1 . Le t P E S2(G) and l e t Z(P) be the centre o f P.

I f we assume T = Z(P) and S = P then (G,S,T) determines a Lingenberg S - p a r t i t i o n

o f type I. Indeed, as i t i s w e l l known, NG(T) = N (P) and i f x {NG(T) then

NG(T) nTX = <l> so t h a t NG(T) n S T X = S. Now l e t g E S T X n S T Y w i t h T # T X # T Y # T , then

h P

G

74 M . Biliotti

g = h,.t : = h2RY wi th b 1 , n 2 e S, R l , . t t 2 E T and hence n;'n, = d1.t; 1 x 1 . If .t,f l, t n f l

and G i s regarded as ac t i ng i n i t s usual doubly t r a n s i t i v e representat ion o f degree

q z + l then the element .t$(.t;'JX, being the product o f two i n v o l u t i o n s wi thout common

f i x e d po in ts , f i x e s an even number o f po ints . But h;'o1 l i e s i n a Sylow 2-subgroup

o f G and hence i t f i x e s exact ly one p o i n t which i s a con t rad i c t i on . As we have

prev ious ly shown, we cannot have R =1 f o r only one 4=1,2 and so R =1 for 4=1,2 and

~ E S . This y i e l d s S T x n S T Y = S. We S t i l l have I T 1 = q , IS1 = q ' , :NG(T)I =

= q 2 ( q - I ) , I G / = ( q z + l ) q 2 ( q - I ) and hence, i f T X ' , ..., TXQ2 are the q 2 subgroups o f

G which are conjugate t o T and d i s t i n c t f r o m them, i t i s e a s i l y seen t h a t the f o l - lowing r e l a t i o n holds:

This proves the asser t ion.

The completion o f the A-structure [G,S,T ] i s a resolvable Ste iner system w i t h

parameters (q(q2-q+I ) , q ) .

EYAMPLE 111. G 2 PSU(3,qz), q = 2 h , h > l . Le t Pe S2(G). I t i s w e l l known t h a t NG(P) =

= P X C, where C il c y c l i c o f order (q2-I)/d with d=(3,q+l). Denote by Cl t he sub-

group o f C of order (qti)/d and set T = Z(P), S = P X C,. Then (G,S,T) determines

a Lingenberg S - p a r t i t i o n o f type I. Indeed, we have again NG(T) = NG(P) and, i f

x f NG(T), NG(T)nTX = <1>, so t h a t NG(T)nSTX = 5. Now l e t TX, Ty be such t h a t

T#TX+TYgT. By w e l l known p roper t i es o f G, we have t h a t M = <Tx,TY> = SL(2,q) and

M'ING(P) = <1> or Z(P) X C,, wi th C, c y c l i c o f order q - I . Since q i s even, we have

also t h a t (q-I,qt]/d) = I and so, i f S n M # < l > then S n M = Z(P) = T. But, as we

have seen i n Example I. (M,T,T) determines a Lingenberg S - p a r t i t i o n o f type I and

hence T X T Y n T = c l > . I t fo l l ows t h a t T X T Y n S = <1> and therefore S T X n S T Y = S. The

thes i s can now be achieved by a c a l c u l a t i o n s i m i l a r t o t h a t c a r r i e d out i n Exam-

p l e 11.

The completion o f the A-structure [G,S,T] i s a resolvable Ste iner system w i t h

parameters ( q ( q 3 - q 2 + I ) , q ) . I n the case q=Zh, with h even, t h i s Ste iner system has

been already obtained by Schulz [la].

EXAMPLE I V . G 2 R ( q ) , q =

i n P G ( 6 , q ) due t o T i t s [22],§5. Let xI,x2,. ..,x7 be a coordinate system f o r P G ( 6 , q )

and l e t ocAu t (GF(q ) ) , a : x + x

P G ( 6 , q ) o f equation x , = O and denote by A the a f f i n e space obtained from P G ( 6 , q ) by

assuming I as the i d e a l hyperplane. Then

4

4' 1 ( I S T ' ~ ~ - IS^) t I N ~ ( T ) I = I G ~ .

i- I

, ~ 2 1 . We s h a l l make use o f the representat ion o f G 32n+ I

3n+ I . Furthermore, l e t I be the hyperplane o f

x = x , / x 7 , y=x2/x7, z = x 3 / x , , u=xs/x7, u=xs/x7, w = x b / x 7

i s a non-homogeneous coordinate system for A . F i n a l l y , se t (m) : ( I ,O,O,O,O,O,O) and denote by r - { ( m ) } the set o f po in ts o f A whose coordinates s a t i s f y the

equations

(1)

u = x z y - X Z t yo - p + 3

u = p y a - za f xy' t yz - X'CJ'3

1o = xzo - x o + i y x2y2 - - zz X ~ a f 4


Then G 2 PGL(6,q)r and G ac ts on r i n i t s usual doubly t r a n s i t i v e representat ion

o f degree q 3 + l . L e t P be the unique Sylow 3-subgroup o f G l y i n g i n G(,+. By us ing

the r e s u l t s o f T i t s [ 2 2 ] , § 5 , about the representat ion o f the elements o f P as w e l l

as the f a c t t h a t I Z ( P ) I = q , i t i s no t hard t o prove t h a t t he p r o j e c t i v i t i e s l y i n g

i n Z(P) are exac t l y those o f the form

tc : (XI,xZ,x3,X4,x-,xb,X-) +

+ ( X I ,XL ,X3+CX7 , - C X I +&+ ,cx~+X~-C'X~ ,C'X1-2CX3+X6-CZX7 , X 7 ) , c E GF(r().

According t o T i t s [ 2 2 ] , § 5 , we have a l so t h a t

i s an i n v o l u t o r i a l p r o j e c t i v i t y o f G which does no t l i e i n G Therefore,

uZ(P)W i s the centre o f a Sylow 3-subgroup Q o f G which i s d i s t i n c t from P. Now

i t i s our aim t o prove t h a t i f c,dc GF(q), c+O, d#O, then '""tCidd does no t belong

t o any Sylow 3-subgroup o f G. Since NG(P)nNG(Q) = E, where E i s c y c l i c o f order q - l , and Z(P)E i s a Frobenius group wi th Frobenius ke rne l Z(P) (see [23],111.4),

we can suppose, wi thout loss o f genera l i t y , d=l. We then have

fA : ,xZ,xJ,X4,x5,X6,x7) (X5,xb,X3,x2,xl ?-x7,-X6)

(a).

dCdI : (X,,X2,X3,X,,Xj,X6,XI) (X I + CX, + coxE, x 2 - cx j, ( J + ZC) x g - C'X - ( C+ cP ) x 6 + x , , - x + ( I - c)x, -c"xE, x 2 - 2cx3+ ( I -C+C~)X,+C'X 6 - X 7 ,

X,-(2*2C)X3+CX,,'CuXSf ( It2CtC"c')X6-X7 2CX3'CoXg-c2Xgfx7)

A s t ra igh t fo rward c a l c u l a t i o n shows t h a t dcdJ possesses the eigenvalue I whose

eigenspace i s generated by the vector ( 0 , I ,l/'Z,-cu~*,O,l/c,l). Now suppose dcdJ

l i e s i n a Sylow 3-subgroup o f G, then the f o l l o w i n g hold:

- ili.tcdI does no t have any eigenvalue d i f f e r e n t from I , f o r we are i n charac-

- LK wx must f i x a p o i n t o f r - { ( m ) } . C I

From t h a t which we have proved prev ious ly , we can i n f e r t h a t the f i x e d p o i n t o f

d 2 d l on r - { (a) ] must have non-homogeneous coordinates ( O , l , I /2 , - c O - ~ , 0, I /c) . But these coordinates do no t s a t i s f y (l), a con t rad i c t i on . Now we may argue as i n

the previous examples t o show t h a t i f we se t T = Z (P) and S = P then (G,S,T)

determines a Lingenberg 5 - p a r t i t i o n o f type I.

The completion o f the A-structure [G,S,T] i s a resolvable Ste iner system w i t h

parameters ( q ( q 3 - q '+ I ) , q 1.

EXAMPLE V. G = SL(n,q), q = p , n>3. L e t K = GF(q), V = K" and U a 1-dimensional

subspace o f V. Denote by T ( g , p ) the t ransvec t i on y -f l-u(l)a where g c V and

ptHomK(V,K) w i t h p(2) = 0, p#O. For a f i x e d non-zero vector b o f U, se t

Then T(U) i s a subgroup o f G (see [11],11, H i l f s s a t z 6.5). F i n a l l y , denote by S(U)

t he subgroup o f G f i x i n g U pointwise. Then (G,S(U),T(U)) determines a Lingenberg

S - p a r t i t i o n o f type 11. It i s indeed enough t o observe t h a t the A-st ructure

which i s obtained from the a f f i n e space A associated t o V by removing the o r i g i n

- 0 i s a quas i - t rans la t i on A-st ructure w i t h respect t o the p e n c i l 0 o f the l i n e s

t e r i s t i c 3 ;

h

T ( U ) = {I, T(b,u) : O # U E HomK(V,K), u(b)=gj.

16 M. Biliotti

through 0 (d isregard ing the p o i n t 0) and t o the fam i l y 3 o f t h e subgroups o f G

which a re conjugated t o T(U). Furthermore, a t ransvect ion o f T(U) with hyperplane

ff f i x e s a l l the l i n e s o f 0 l y i n g i n H , so t h a t T(U) i s no t semiregular on 8. The

asser t ion now fo l lows from Proposi t ions 1.2 and 2.2.

3. FURTHER RESULTS ON LINGENBERG S-PARTITIONS OF TYPE I

We w i l l r equ i re the f o l l o w i n g lemma.

LEMMA 3.1. Let G be one 06 t h e 6oCCowing ghoupn: h

SL(?,q), q = p , p p’Lime, 4 2 4 ;

i, S z ( q ) , q = p 2 n f 1 p=2, @ I ;

S U O , ~ ~ ) , q-p , p phime, 4 > 2 , 3 1 ~ 7 ; h

P S U ( ~ , ~ ’ ) , q = p , P phime, ~ 2 2 ; R(q), q=pZn*l, p=3, el;

and l e i P be a SgCow p-nubgmup 0 4 G. 16 T

condi t ioMn: ( I 1 / T I - I [NG(P):Tl and ( 2 1 TnZ(G) = <l>, t h e n T = Z (P) .

P m o d . We s h a l l i n v e s t i g a t e the var ious cases separately.

Let G = sL(z,q), q s 4 . Assume q i s odd. Then IZ (G) (= 2, ]PI = q , ING(P)I = q ( q - 1 )

and

by [ l l ] , V , Satz 8.16, we have t h a t e i t h e r 7 <

fo l lows t h a t T < PZ(G). But, T n Z ( G ) = <1> and hence T < P, which imp l i es T = <l>

since P i s a minimal normal subgroup o f NG(P). I n the l a t t e r case, cond i t i on (1)

y i e l d s T = P. Since P i s elementary abel ian, t he p roo f i s achieved. The case q

even i s s i m i l a r .

Le t G = S z ( q ) . N (P) i s a Frobenius group w i t h Frobenius ke rne l P, moreover

/NG(P)/ = q 2 ( q - l ) , lP1 = q z , IZ(P) I = q . We have t h a t e i t h e r T 2 P or T c P. Con-

d i t i o n (1 ) i s u n s a t i s f i e d when T 2 P. I f T < P, then e i t h e r T 2 Z(P) or Tn Z(P) =

= <1> s ince Z(P) i s a minimal normal subgroup o f NG(P). As T Q NG(P), we have t h a t

q - l 1 I T [ - I . So, i n the former case, i t fo l l ows T = Z(P) from cond i t i on (1). I n

the l a t t e r case we have P = T X Z(P). However, P/Z(P) i s abel ian and hence P must

be abel ian which is a contrad ic t ion.

Let G = S U ( 3 , q 2 ) , 3 1 q + 7 . We have /Z(G)I = 3 and N (P) = P X C , where /P I = 9’ and

C i s c y c l i c o f order q2-I and conta ins Z(G) . Moreover, I Z ( P ) ) = q .

N = NG(P)/Z(P)Z(G) i s a Frobenius group with Frobenius ke rne l

plements isomorphic t o c. Since i o N, we must have e i t h e r

c l e a r l y p i s a minimal normal subgroup o f

By cond i t i on ( 2 ) and since ( 3 , q ) = I , we then have t h a t T 5 Z(P) and thence T = Z(P)

a nomat dubgtoup 06 NG(P) b ~ ~ q 4 u 2 g

= N (P)/Z(G) i s a Frobenius group w i t h Frobenius ke rne l p. Since T Q N, then G or r h P. I n the f i r s t case, i t

G

G

- and Frobenius com-

 i n the f i r s t case.

because Z(P) i s a minimal normal subgroup o f NG(P). I n the l a t t e r case we cannot

have T 2 P by cond i t i on (l), whi le TnZ(P) = <1> forces P t o be (TnP)Z(P) , b u t

as we have seen before then P must be abe l i an which cannot be the case.

Le t G = PSU(3,q'). The p roo f i s s i m i l a r t o the previous one.

L e t G = R ( q ) . We have NG(P) = P X C, where ( P I = q 3 and C i s c y c l i c of order q-1 .

Moreover, Z(P) < P' = @(PI , IZ(P)I = q , I P ' I = q 2 . Since

i u s group w i t h Frobenius ke rne l p (see [23],111.11), as i n the prev ious cases, we

have e i t h e r 7 = <1> or 7 2 p. I n the f i r s t case T 5 PI. I f TnZ(P) # <l>, then

T 2 Z(P) since Z(P) i s a minimal normal subgroup o f NG(P), bu t by [ 2 3 ] , I I I . 2 ,

T > Z(P) imp l i es T = P ' and cond i t i on (1) i s no t s a t i s f i e d . So T = Z(P). We can-

n o t have TfiZ(P) = -1> s ince if X E PI - Z(P) then i t s c e n t r a l i z e r i n NG(P) has order 2qz (see [23],111.2) and hence I T ( > q , con t ra ry t o T S PI. I n the l a t t e r ca-

se, we cannot have T n P ' = <1>, since for each x E P - P' we have o ( x ) = 9 and x 3 6

t Z(P) < PI (see [23], Theorem). Nevertheless, T n P ' # <1> imp l i es I T 1 2 q' (see

[23],111.2) and again cond i t i on (1) i s no t s a t i s f i e d . This completes the proof .

The fo l l ow ing theorem i s concerned w i t h Lingenberg S - p a r t i t i o n s o f type I i n the

case o f T being o f even order.

THEOREM 3.2. L e L (G,S,T) d e t w i n e a ling en be^ S-pcmLiAon 06 type I. 1 6 T ha^

even o t d m t h e n one 0 6 t h e doLLowing h u t & :

( u )

( b . 2 ) G 2 SZ(~), q - 2 2ntJ

(6.3) G 2 PSU(3,qz), q = 2 , h22; T = Z(P) w i t h . P G S2(G) and S = P X CI w i t h I C 1 I =

Phood. I n the A-structure [G,S,T], the p e n c i l o f l i n e s 0 i s a t r a n s i t i v e k s e t

w i t h 10l>l. I f 4 = NG(T) t 0 then E = Ne(?) and, by P ropos i t i on 2.2, ? ac ts semi-

r e g u l a r l y on 0 - { a } . Then by [ l o ] , Theorem 2, e i t h e r the case (a ) occurs or

2 S L ( ~ , C ( ) , S Z ( C ( ) , PSU(3,qz), SU(3,q') w i t h q = Z h , h > l . I n the l a t t e r case by [ l o ] , Lemma 3, E ac ts on 0 i n i t s usual doubly t r a n s i t i v e representat ion o f degree q + l ,

q 2 + 1 , q 3 t 1 , q 3 + 1 respec t i ve l y . Then, i t is w e l l known t h a t e = Nc(F) with P"eS2(G)

and hence Nc(?) = N c ( B ) . By t a k i n g account o f P ropos i t i on 2.2, we see t h a t 7 s a t i s -

f i e s cond i t i ons (1) and (2 ) o f Lemma 3.1. Therefore ? = Z(P) . When E = SL(Z,q) ,

S z ( q ) or PSU(3,qz) we ob ta in (b.1) - (b.3) i n view o f P ropos i t i on 2.2,(3). I f

E = SU(3,q2) w i t h 31qtJ we have 171 = I Z ( p ) l = q and hence I ~ l ( = q - l . Since 1Z(G)I=3,

t h i s imp l i es t h a t 3 Iq- I by Propos i t i on 2.4, a con t rad i c t i on . Therefore, the case

G" = SU(3,q2), wi th 31q+l, cannot occur.

We p o i n t ou t t h a t Examples I, I1 and I11 o f sec t i on 2 show t h a t t he cases (b.11,

(b.2) and (b.3) a c t u a l l y occur. On the contrary , i t seems very d i f f i c u l t t o achieve

a complete c l a s s i f i c a t i o n o f Lingenberg S - p a r t i t i o n s o f type I i n the case (a) .

= NG(P)/P' i s a Froben-

G = O(G)T und T iA a Fhobeniun cornpLement; h

(6 .1) G 2 SL(2,q), 9.2 , h62; T = S = P w i t h P t S2(G);

a21; T = Z(P), S = P w i t h P6S2(G); i,

= ( q + l l / d , &eke d=13,4+11.

Jl =

h

78 M. Biliotti

I n succession, we g i ve some r e s u l t s and examples concerning t h i s case.

PROPOSITION 3.3. LeA (G,S,T) d&tehmine u Lingenbehg S - p a d t i o n ud type I und M-

dume [ T I 2 3. 16 E induced a F h o b e u p m u t a t i o n gmup on t h e pen& 0 i n %he

A-n&ucRuhe [G,S,T l , then Rhe ~oUoiu4ng h o l d : h (I) G = M X T, whehe M A a nonabe14un n p e c b l p-ghoup 0 4 m d u q 2 m t ’ w i t h q = p ,

( 2 1 IZ(G)I = lZ(M)I = 4, I T 1 = q + I , S = T, NG(T) = TZ(M).

m,hLJ;

P m a 6 . By Propos i t i on 2.4, Z(E) i s the ke rne l o f the representat ion o f E on 0.

Therefore, Go = E /Z (c ) and c’ ac ts on 0 as a Frobenius group by our assumptions.

Denote by G the Frobenius ke rne l o f

By Proposi t ions 2.2 and 2.4, we have t h a t IZ(G)I I / T I - 1 and hence ( l T l , l Z ( G ) l ) = l .

Moreover, IT], I i l ) = I s ince i s contained i n a Frobenius complement o f c. There-

fore, T n M = <1> and MT = M X T. I f x t G then, c l e a r l y , T X C M T and hence G = MT

and F = i@. We have r = 2 NG(T) 2 T, so t h a t T = NG(T) and NG(T) = TZ(G) .

Set (TI = R , then (Z(G)( = [NG(T):T] 2 [NG(T):S] = t-I by Propos i t i on 2.2. Since,

as we have prev ious ly seen, IZ(G)I I R - 1 , i t fo l lows t h a t /Z(G)I d-I and S = T .

Note t h a t since ii i s n i l p o t e n t so i s M (see [11], V.a.7). L e t P be a Sylow p-sub-

group of M with P $ Z(G) and l e t N = PT. Since (G,S,T) determines a Lingenberg

S-partitionof type I, the fo l l ow ing r e l a t i o n holds

where

hence c=X-I and Z(G) < P. This y i e l d s M = P. Consider the commutators o f the form

[x,g] with X E T and g t M-Z(G). We have [x,g] = X-’(g-’xg) and hence [ X , g ] E TT’.

Each complex TTg conta ins exac t l y t- J non- ident ica l d i s t i n c t commutators o f the

l y i n g i n TTg are d i s t i n c t from those l y i n g i n TT‘. Since I I T X : X E G } I = 121, then

by s e t t i n g (GI = 6 there e x i s t a t l e a s t ( X - I ) ( i - 7 ) t I d i s t i n c t commutators l y i n g i n

[ M , T ] . Since [ M , T ] 2 M and I M I = m(X- I ) , i t fo l l ows t h a t [ M , T l = M. Now suppose

there e x i s t s a c h a r a c t e r i s t i c abe l i an subgroup A o f M such t h a t A 6 Z(G). Then the

group AT contains exac t l y a = [ A : Z ( G ) n A ] d i s t i n c t conjugate elements of T. By

using the same argument as before, we have

( t - I ) a 2 / A \ 2 I [ A , T ] I 2 ( t - I ) ( a - I ) t J , where a>?. From t h i s i t fo l lows t h a t A = [ A , T ] and ( A ( = ( t - I ) a . The l a t t e r r e l a -

t i o n y i e l d s Z(G) < A , bu t t h i s con t rad i c t s a r e s u l t o f Zassenhaus [11],III, Sat2

13.4(b), since IZ(G)) > I. Therefore, a c h a r a c t e r i s t i c abel ian subgroup o f M i s

c e n t r a l i n G. I n conclusion we have proved tha t :

= G/Z(G) and l e t M i G such t h a t M/Z(G) = M.

G

( 2 ) It‘ - $1 !nlRc - I ) + Rc c y1 , = I N ( and c = ( P n Z ( G ) ( . From (2), i t fo l l ows t h a t t - l j c since n>tc and

form [x ,g ] . Moreover, i f T 9 6 # T then T T g n T T 6 = T and hence, the R-7 commutators

(I) ( I M I , I T I ) = I ,

(11) [ M , T l = M,

(111)

By a r e s u l t of Thompson [ L l ] , I I I , Satz 13.6, we then have t h a t M i s a nonabelian

T cen t ra l i zes every c h a r a c t e r i s t i c abe l i an subgroup o f M.

S-Purtitioris of Groiqs arid Steirier Systems 79

special p-group. Moreover, since Z(M) is a characteristic abelian subgroup of M,

we have that Z(M) = Z(G). Let lZ(G)I = t - 1 = p , h21, and let l M l phtn. Since a Frobenius complement of G/Z(G) has order . t=ph+I , it follows that p +llp"-l. From

ph+ 1 1 p n - 2 h - I . Let b E P such that bhSn< Ib+ lih. By iterating the above procedure, it

is easy to prove that b must be even and P ' - ~ ~ - I = O . This completes the proof.

A Lingenberg S-partition of type I satisfying conditions ( 1 ) and (2) of Proposi-

tion 3.3 and its associated A-structure will be called 4peciu.e. An example is given below.

EXAMPLE VI. Ue assume the reader is familiar with [14],V,§32. Let ~1 be the projec-

tive plane over GF(qz), q=p , and let p be a hermitian polarity of TI. It is well

known that the absolute points and non-absolute lines of p make a Stciner system

u with parameters b=q+l and v = q 3 + 1 , which is usually called the d u b b i c d unitul.

Moreover the group P(U) consisting o f the projectivities of TI leaving U invariant

is isomorphic to PGU(3,qZ). According to Bose [5],§6, for each absolute line p of [I , we may define a parallelism among the lines of U as follows: a class of paral-

lel lines consists of a non-absolute line fi through p(p ) and the non-absolute

lines through p ( h ) . Note that p ( p ) E h implies p ( 8 ) 6 p. Therefore, the group T ( h )

consisting o f all ( p ( h ) ,&)-homologies lying in P(U) preserves the parallelism just

defined in U , because it fixes the line p. The group T(a) fixes each line i? paral-

lel to h and acts regularly on the points o f L lying in U because T(h) has order

9 t J . Moreover, there exists a unique Sylow p-subgroup M of P(U) which fixes p ( p ) and so p itself and acts transitively on the Q' non-absolute lines through p ( p ) . It follows that U - {PI is a quasi-translation A-structure with respect to the pencil 0 of non-absolute lines through p(p ) and to the family 8 = { T ( h ) : h E O}. It

is easily seen that:

- T(A) acts semiregularly on o - [A} ; - IZ(M)I= q and Z ( M ) consists of all (p(p),p)-elations lying in P(U) and therefore

it fixes every line of 0;

h

h n-h- h h n-2h- I . so this, we have that ph+ I I pn+ph=ph (pn-h+ 1 ) and hence ph+ 1 I p P =P (P

h

- G = < T ( h ) : = M X T ( b ) , b E @ ;

- I M I = q 3 and M/Z(M) scts regularly on 0.

From this, it follows that Z ( M ) = Z(G) and G/Z(G) is a Frobenius group. Therefore,

(G,T(h),T(h)) determines a special Lingenberg S-partition.

We now consider case (a) of Theorem 3.2. Assume,

141 Z 0 u L i ~ g e ~ b e h g S - p ~ ~ h L i . t i o ~ 0 6 G 0 6 .type I ~ i R h hebpecR 20 .the bubghoup T

w i t h IT I t 3 and (44) G = CT, whefie C 0 bo.&ubi?e and C Q G.

L e t 0 be the pencil of lines of [G,S,T]. By Propositions 2.2 and 2.4, we have that

( / T I , J Z ( c ) ) ) = 1 and hence Z(c) < ?. Let c/Z(E) be a minimal normal subgroup of G/Z(G) contained in e/Z(c). Since e $ Z ( E ) and E/Z(c) acts transitively on 0, then,

80 M . Biliotti

i f 4 = NG(T)C 0 and

since k ( E ) i s elementary abel ian f o r ? i s solvable, i t fo l l ows t h a t LR i s elemen-

t a r y abel ian. I t fo l lows now t h a t acts r e g u l a r l y on R. Moreover, ?R = 7 leaves

R i n v a r i a n t and acts semiregularly on s1 - [ k } . From t h i s , we i n f e r t h a t ?ITn i s a

Frobenius group. Now se t F = <Tx : x c L > , 5, = SnF , No = NG(T)nF, I S o \ = h a ,

IN,( = n o , ( T I = t and I F ( = 6 . Since C i s a Lingenberg 5 - p a r t i t i o n o f G o f type I,

the fo l l ow ing r e l a t i o n holds:

( 3 )

From this, i t fo l l ows t h a t n o / b o t t-1. On t h e other hand we have t h a t [N,:S,] S

i s the o r b i t o f h under r, we have t h a t Is11 > 1 . Moreover,

( , t ~ o - bo)(d/na - I ) t n o 2 6 .

S [N:S] = . t - 1 . Therefore, n o / b o

i t i s no t d i f f i c u l t t o see t h a t

then C I = { T f : x t F1} u {NFl(T1)

respect t o the subgroup TI. But

b p e c i d l ingenbekg S-patL iL iun.

= t-f and ( 3 ) holds as an equa l i t y . Using t h i s ,

if we set R = GFs;, s l = SJR, T ~ = IRA, F,= F/R,

i s a Lingenberg S - p a r t i t i o n o f F 1 o f type I wi th -R -R-R F I = I- T and hence, by P ropos i t i on 3.3, X I i s a

From a geometrical p o i n t o f view, the previous r e s u l t can be expressed as fo l lows.

PROPOSITION 3.4. L e t C be. a Lingenbehg S-pa/ttLtitian a6 G 0 6 type I w c t h k e ~ p e c t t o

-the hubghaup -i w i t h T 2 3. Adbume G = CT, whem C i h a hoLwabLe n u m d hubghoup

ud G. Then t h e A-bRhuctwle [G,S,T] cuntaim a bubbpace wh ich i~ a b p e c i d A-btkuc-

t u k e .

Pkoo6. L e t [FI,S1,T1] be the spec ia l A-structure r e l a t e d t o the spec ia l S, -par t i -

t i o n & o f Fi described above. I f SIX i s a p o i n t o f [F,,Sl,T1] and x = Kx, l e t q be

the map from the set o f po in ts o f [FL,SI,T1] i n t o the se t o f p o i n t s o f [G,S,T]

defined as fo l l ows -

n : SIX -t sx . I t i s s t ra igh t fo rward t o show t h a t 11 i s w e l l def ined and g ives an embedding o f

[FI,Sl,TIl i n t o [G,S,Tl.

4 . LINGENBERG 5-PARTITIONS OF DOUBLY TRANSITIVE PERMUTATION GROUPS

Assume ( G , I , T ) determines a Lingenberg S -pa r t i t i on . Since i n Examples I - V vie have

tha t :

( a ) t h r gnoup

then the n a t u r a l question a r i ses whether i t i s poss ib le t o c l a s s i f y a l l the t I i p l e s

(G,S,T) which determine Lingenberg S -pa r t i t i ons s a t i s f y i n g cond i t i on (a ) . I n the

fo l lowi i ig , we s n a l l prove t h a t a r a t h e r s a t i s f a c t o r y answer t o t h i s question may

be g iven provided t h a t the c l a s s i f i c a t i o n o f doubly t r a n s i t i v e permutation groups

i s assumed. As i t i s w e l l knowri, such a c l a s s i f i c a t i o n fo l l ows f r o m t h a t o f F i n i t e

simple groups.

THEOREM 4.1. Annwrie (G,S,l) deXehmAneb a Lingenbehg S-pamLCi.on C b a t ~ A 6 y ~ n g cun-

&Lion ( u l . 16 C 0 ad t y p e I then une 06 t h e ,joXCawing h d h :

c i c i b Z-tcanoiLivek?y un t h e pen& 06 f i n e d 0 06 .the A-~ihuc. tuce

[ G , S , T l .

( I ] G

( 2 1 G 2 Sz(q), Q =

( 3 1 G 2 PSU(3,qz), q = 2 ' , h t 2 ; T = Z(P) wiRh PE S2(G), S = P A CI, wh&he I C I I =

(4) G

18 C 0 0 6 t y p e I1 lhen PSL(n,q) S G/Z(G) S PTL(n,q) iyhehe n23.

Pk006. Assume (G,S,T) determines a LinyenbErg 5 - p a r t i t i o n o f type I. By Propos i t i on

2.7, t l ie group G

(h) f o r each h e @ , the s t a b i l i z e r o f 5 i n 6' conta ins a normal subgroup which ac ts

SL(Z,q), q = p c l , ph22; T = S = P l ~ i h i h PES (C); P

ntl; T = Z(P), S = P wi2h P6S2(G) ; 2n+ I

= ( q + l ) / d , d=(3,y+ll;

R ( q ) , q = 3 2 n C 7 , n 2 i ; T = Z(P), S = P w a h PES3(G).

-,@ - * G/Z(E) s a t i s f i e s the fo l l ow ing condi t ion:

semiregular ly on o - {t}. From the c l a s s i f i c a t i o n theorem o f f i n i t e doubly t r a n s i t i v e permutation groups, we

h a ? t h a t the so c a l l e d "Hering conjecture" (see [ 4 ] ) i s a c t u a l l y a theorem (see

[19],p.302) asse r t i ng t h a t i f c0 i s 2 - t r a n s i t i v e on 0 and s a t i s f i e s (h) then one

o f the fo l l ow ing holds:

(j) c0 conta ins a regu la r normal subgroup,

(jj) Eo -1 PSL(Z,(i), q L 4 , Sz(q), PSU(3,q"), q > 2 , or R ( y ) , 4 '3 , aiid Go ac ts on 0 i n

i t s usual doubly t r a n s i t i v e representat ion.

We s h a l l i n v e s t i g a t e these cases separately.

C~c.14 ( j ) . Le t E0 be the regu la r rlorrnal subgroup o f cO. We have t h a t Go =

hence Go = "GZ(e ) i s a Frobenius group. If ( T I = 2 then EB = = SL<2,2). I f ( T I 1 3

then by Propos i t i on 3.3, we must have I N I = q Z m , where q i s a prime power and m 2 1

tdureoveL, ( T I = q + ~ . Since Go i s 2 - t r a n s i t i v e on B, i t i o l l o w s t h a t q 2 m - l = q + l .

This imp l i es q = 2 , m = l and i t i s very easy t o show t h a t G = SL(2,3).

Cane ( j j ) . Note t h a t i f L 5 G and LZ(G) = G then L = G s ince ( \ T l , l Z ( G ) l ) = I and

G i s generated by T and i t s conjugates. Therefore, G i s a c e n t r a l i r r e d u c i b l e ex-

tens ion o f $). Since, i n the case under considerat ion, G i s a simple group ( see

[11],[12]) then there e x i s t s a unique representat ion group H o f ? and G = H/Z,

for some subgroup Z, of Z(H). Moreover, Z(G) = Z(H)/Z,. I f M(E') denotes the Schur

m u l t i p l i e r o f EB then IZ(H)I = IM(CO)I ( s e e [21],Ch.2,59). Since, by P ropos i t i on

2.2, we have t h a t (Z(G)I I I T 1 - I , i t fo l l ows t h a t IZ(G)I I ( l M ( ~ O ) ~ , ~ T l - J ) . Now

assume :

G"' = PSL(p,(i), q=p , h 2 2 . I f p i s odd then we cannot have G

t h i s case every normal subgroup o f NG(P), where P e S (C), conta ins P and hence re-

l a t i o n (3 ) o f P ropos i t i on 2.2 cannot be s a t i s f i e d fo r \ P I = q and ING(P)I = % q [ q - I I .

Therefore, when q:J,9 by [11],V,25.7, we must have G = H = SL(2,q) and the t n e s i s

fo l l ows from Lemma 3.1. If 9 - 4 then T has even order and, by Theorem 3.2, G =

= SL(2 ,4 ) . Le t q - 9 . Since NG(T)/Z(G) is a Frobenius group o f order 9.4 with Fro- benius ke rne l o f order 9 , i t fo l l ows t h a t I T 1 I 9 . On the other hand, I T 1 I I01-J

where 101= I0 and hence IT I = 9 . By [8], Table 1, we have t h a t M(Co) = Z6. There-

fore, /Z(G)I = 2 and G - SL(2,9).

and

-,o

h PSL(2,q) s h c e i n

P

82 M . Riliotti

Le t Go -- Sz(g). I n t h i s case T has even order and (2) f o l l ows from Theorem 3.2.

Le t to = PSU(3,q2), q > 2 . By [ 9 ] , Theorem 2, we have e i t h e r G 2 PSU(3,q') o r G 2

= SU(3,q'). The case G

proof o f Theorem 3.2. Assume G 1̂ PSU(3,qZ), y = p , p an odd prime. By Lemma 3.1, we

have t h a t T = Z(P) with P t S (G) and S = P X C1, where C I i s c y c l i c of order g+J/d, P

d=(3,q+J). Moreover C , < C, where C i s c y c l i c o f order q2-J/d and C < NG(P). L e t

C2 be the subgroup o f C of order q - J . As i t i s w e l l known, the re e x i s t s a subgroup

M o f G such t h a t MnNG(P) = TC2 and SL(2,q) = M = <T,TX> for a s u i t a b l e conjugate

subgroup Tx o f T, where TX may be chosen i n such a way t h a t Cz < NM(Tx). L e t b =

= .ti, where I f - t e T and i is t he unique i n v o l u t i o n i n C2. As was shown i n Example I,

(M,TX,TX) determines a Lingenberg S - p a r t i t i o n o f type I. Since NM(T)nNM(TX) = C 2 ,

we have t h a t b { NW(TX). Therefore, there e x i s t s Ty < M with T#TY#Tx such t h a t

b E TXTY and hence T x T y n S # < l > since b E S. Note t h a t i E C i f o r q odd. I t fo l l ows

t h a t (G,S,T) does no t determine a Lingenberg S -pa r t i t i on . The thes i s can now be achieved by using Lemma 3.1.

Le t Go lows from Lemma 3.1.

Now assume (G,S,T) determines a Lingenberg S - p a r t i t i o n o f t y p e I I . I f NG(T)x i s a

l i n e o f 0 then the s t a b i l i z e r NE(T"X)' o f t h i s l i n e i n Eo conta ins the normal sub-

SU(3,q2), 3 1 q + l , may be excluded by arguing as i n the h

R(g) , q>3. By [l], Theorem 1, we have t h a t M(?) = clz and ( 4 ) again f o l -

- - group Tx' s a t i s f y i n g the cond i t i on

(1) yX0f3?@ = <l> for each ?@ such t h a t T"yo#T"xxo. Indeed, assume (1 ) does not hold. Then TXZ(G)i7TYZ(G) > Z(G) f o r some Ty with TX#

f T y . I t fo l lows t h a t TXTYnZ(G)#<l> since TXnTY = <l> by Lemma 2.1. So the re ex-

s i s t s z E Z(G) , ~ $ 1 , such t h a t z c T T Y X E STY'-'. Since z E NG(T) and STY'-' n NG(T)=

= S then z E 5, bu t t h i s i s a c o n t r a d i c t i o n because Sn Z(G) = <1>. Moreover, by

P ropos i t i on 2.2, we have:

(2) 3' does no t ac t semiregularly on Q - {NG(T)x].

By a w e l l known r e s u l t o f O'Nan [15] , Theorem A, condi t ions (1) and ( 2 ) imply

PSL(n,q) 5

As a f i n a l remark, we note t h a t t he c l a s s i f i c a t i o n theorem o f doubly t r a n s i t i v e

permutation groups i s requ i red only when (G,S,T) determines a Lingenberg S - p a r t i t i o n

o f type I and T has odd order.

- 1

6 PrL(n,q) with n23. So the t h e s i s fo l lows from Propos i t i on 2.4.

REFERENCES.

[l] Alper in , J.L. and Gorenstein, D., The m u l t i p l i c a t o r s o f c e r t a i n simple groups,

[ 2 ] Andre, J., Uber Para l l e l s t ruk tu ren , T e i l I : Gundbegriffe, Math. Z. 76 (1961), Proc. Am. Math. SOC. 17 (1966), 515-519.

85-102.

S-Partitions of Groups and Steirier Systems 83

[ 31 Andr6, J., Uber Para l l e l s t ruk tu ren , T e i l I1 : Translat ionsst rukturen, Math. Z.

[41 Aschbacher, M., F-sets and permutation groups, J. Algebra 30 (1974), 400-416. [5] Bose, R.C. , On the a p p l i c a t i o n o f f i n i t e p r o j e c t i v e geometry f o r d e r i v i n g a

c e r t a i n se r ies o f balanced Kirkman arrangements, in:The Golden Jub. Comm., Calcut ta Math. SOC. (1958-591, 341-354.

[6] B r e n t i , F., S u l l e S - p a r t i z i o n i d i Sylow i n alcune c l a s s i d i gruppi f i n i t i , Boll. Un. Mat. It. (6) 3-8 (1984), 665-685.

[ 71 Burkhardt, R., Uber d i e Zerlegungszahlen der un i ta ren Gruppen PSU(3,2 J. Algebra 61 (19791, 548-581.

[ 8 ] Griess, R.L.,Jr., Schur M u l t i p l i e r s o f the known f i n i t e simple groups, B u l l . Am. Math. Soc. 78 (19721, 68-71.

[9] Griess, R.L.,Jr., Schur M u l t i p l i e r s o f f i n i t e simple groups o f L i e type, Trans. Am. Math. SOC. 183 (1973), 355-421.

[ 101 Hering, C . , On subgroups w i t h t r i v i a l normalizer i n t e r s e c t i o n , J. Algebra 20

[ll] Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin-Heidelberg-New York, 1979).

[ 121 Huppert, B. and Blackburn, N., F i n i t e Groups I11 (Springer-Verlag, B e r l i n - -Heidelberg-New York, 1982).

[ 131 Lingenberg, R. , Uber Gruppen p r o j e c t i v e r Kol l ineat ionen, whelche e ine perspec- t i v e D u a l i t a t i n v a r i a n t lassen, Arch. Math. 13 (1962), 385-400.

[ 141 Luneburg , H., T rans la t i on Planes (Springer-Verlag, Berlin-Heidelberg-New York, 1980).

[ 151 O'Nan, M.E., Normal s t r u c t u r e o f the one-point s t a b i l i z e r o f a doubly-tran- s i t i v e permutation group. I, Trans. Am. Math. SOC. 214 (19751, 1-42.

[16] Rosati , L.A., S u l l e S - p a r t i z i o n i n e i gruppi non a b e l i a n i d 'o rd ine pq, Rend. Sem. Mat. Univ. Padova 38 (19671, 108-117.

[17] S c a r s e l l i , A., S u l l e S - p a r t i z i o n i r e g o l a r i d i un gruppo f i n i t o , A t t i ACC. Naz. L ince i , Rend C1. Sc i . F i s . Mat. Nat. ( 8 ; 62 (1977), 300-304.

[18] Schulz, R.H., Zur Geometrie der PSU(3,q ) , i n : Bei t rage zur Geometr. Algebra, Proc. Symp. Duisburg, 1976 (Birkhauser, Basel, 1977), 293-298.

[19] Shult, E.E., Permutation groups with few f i x e d po in ts , i n : Geometry - von Staudt 's P o i n t o f View, Proc. NATO Adv. Study I n s t . Bad Windsheim, 1980 ( 0 . Reidel P.C., Oordrecht, 19811, 275-311.

[20] Suzuki, M., On a c lass o f doubly t r a n s i t i v e groups, Ann. Math. 75 (19621,

[ 211 Suzuki, M., Group Theory I (Springer-Verlag, Berlin-Heidelberg-New York, 1982) [22] T i t s , J., Les groupes simples de Suzuki e t de Ree, in: Sem. Bourbaki, 13e an-

1231 Ward, H.N., On Reels se r ies o f simple groups, Trans. Am. Math. SOC. 121

76 (1961), 155-163.

2 f 1 ,

(19721% 622-629.

104-145.

nCe, 210 (1960/61), 1-18. ~-

(19661, 62-89. 1241 ZaDOa. G.. S u a l i sDazi aenera l i quasi d i t ras laz ione , Le Matematiche (Cata- i ,

n i a ) i9 (i9647, 127-143: 1251 Zappa, G., S u l k S - p a r t i z i o n i d i un gruppo f i n i t o , Ann. Mat. Pura Appl. (4) . .

74 (19661, 1-14. [26] Zappa, G., P a r t i z i o n i genera l izzate ne i gruppi, i n : C o l l . In t . Teorie Comb.

1973 (Acc. Naz. L ince i , Roma 1976), 433-437.

COLLINEATION GROUPS STRONGLY IRREDUCIBLE ON AN OVAL

Mauro Biliotti Gabor Korchmaros

Dipartimento di Matematica Universitl degli Studi di Lecce

via Arnesano, 73100-LECCE via N. Sauro, 85, 85100-POTENZA

Istituto di Matematica Universitl degli Studi della Basilicata

Italy Italy

In recent years, Hering has written several papers concerning the composition se-

ries of collineation groups of a finite projective plane. Prominent in his studies

is the noIion of o&zongcy u u z e d u u b l e cu.Umea.t*on p o u p on a pnoaectcve p4me. one

which does not leave invariant any point, line,triangle or proper subplane. There

is a well developed theory of strongly irreducible collineation groups containing

perspectivities, which has significant applications (see [4],[5],[15]). However,

it should be noticed that only isolated results are known for such groups in the

general case.

It should be interesting to investigate also "local" versions of the concept of

irreducibility. In this connection, here we consider a finite projective plane TI

of even order with a collineation group r and a r-invariant oval n such that r does not leave invariant any point, chord or suboval of n. Here a suboval o f n is

a subset of points of R which is an oval in a proper subplane of n. We say that r

is 4.t/Lony4y uzneduc~64e on f i e o v a l n. Clearly r is not strongly irreducible on TI

since it fixes the knot K of 0 .

Our main result states that if r has even order then r contains some involutorial

perspectivities, i.e. elations. The subgroup < A > generated by all involutorial e-

lations is essentially determined. If r has a fixed line then < A > is the semidi-

rect product of O(<A>) with a subgroup of order two generated by an elation. If r has no fixed line then r acts as a "bewegend group" [6] on the dual affine plane

of ~i with respect to the line at infinity K. From Hering's result [6] on bewegend

groups containing involutorial elations, it then follows that < A > is isomorphic to

one of the simple groups: SL(Z,q), Sz(q), PSU(3,q2), where q is a power of 2 and

4'4 *

Clearly any collineation group of TI mapping n onto itself and acting transitively

on its points is strongly irreducible on n. As we shall prove in Section 5, such a

86 M . Biliotti and G. Korclirnaros

group cannot i n v o l v e PSU(3,q‘). Hence, t h e o n l y non-solvable c o l l i n e a t i o n groups o f

H a c t i n g t r a n s i t i v e l y on R are t h e groups 1 f o r which e i t h e r SL(2,q)c Z P r L ( 2 , q )

o r Sz(q)c z cAut S z ( q ) . The groups a re always 2 - t r a n s i t i v e on R . Furthermore, i n

t h e former case, IT i s a desarguesian p lane o f o rde r q and i s a con ic . For the

l a t t e r , we may on ly a s s e r t t h a t , a t t he present s t a t e o f our knowledge, t h i s s i -

t u a t i o n occurs i n t h e dua l Luneburg p lane of o rde r qz (see ~ l l ~ , [ 1 3 ] , [ 1 4 ] ) .

2. NOTATION AND PRELIMINARY RESULTS

F a i r l y s tandard n o t a t i o n i s used. A c e r t a i n f a m i l i a r i t y w i t h f i n i t e p r o j e c t i v e

planes as w e l l as w i t h f i n i t e groups i s assumed. For t h e necessary background t h e

reader i s r e f e r r e d t o [ 2 ] , [ 9 ] .

Throughout t h i s paper, 11 denotes a p r o j e c t i v e p lane o f even orde r n c o n t a i n i n g an

ova l R . Here an ova l i s de f i ned as a s e t o f n+ l p o i n t s no t h r e e o f which a r e c o l -

l i n e a r .

The f o l l o w i n g elementary r e s u l t s a r e used i n the p roo fs .

Through each p o i n t o f R t h e r e e x i s t s e x a c t l y one tangent o f a. The tangents a r e

concur ren t ; t h e i r common p o i n t K i s c a l l e d t h e kno t o f 0. Each l i n e th rough K is a

tangent o f n. A l i n e o f IT i s an e x t e r n a l l i n e o r a secant l i n e o f R accord ing

t o whether Ir flnl=O o r 2. There a r e e x a c t l y n (n-1) /2 e x t e r n a l l i n e s and n(n+1) /2

secants o f n i n H . A chord o f R i s t h e p a i r o f p o i n t s which n has i n common w i t h a

secant.

L e t G be a c o l l i n e a t i o n group o f TI mapping 62 on to i t s e l f . Then G f i x e s K. I f G has

no f i x e d p o i n t on R t hen i t has no f u r t h e r f i x e d p o i n t i n n . The on ly element o f G

w i t h a t l e a s t Jn+2 f i x e d p o i n t s on B i s t h e i d e n t i t y c o l l i n e a t i o n o f H . The r e s t r i -

t i o n map o f G on R i s a f a i t h f u l rep resen ta t i on .

Any n o n - t r i v i a l e l a t i o n o f G i s i n v o l u t o r i a l . I t s cen te r does n o t belong t o R U ( K 1 .

Two d i s t i n c t e l a t i o n s o f G do n o t have t h e same cen te r . The a x i s o f any e l a t i o n i s

a tangent o f 8. Any i n v o l u t o r i a l c o l l i n e a t i o n o f II i s e i t h e r an e l a t i o n o r a Baer-

i n v o l u t i o n . The s e t o f a l l f i x e d p o i n t s and l i n e s o f a B a e r - i n v o l u t i o n f i s a sub-

p lane o f o rder hi, c a l l e d t h e Baer-subplane o f f .

BAER INVOLUTIONS MAPPING R ONTO ITSELF

Collineation Groups 87

F denote t h e Buen-4ubpLune u{ f. Thm

?noo[. Since n i s even, n has an odd number o f p o i n t s . So, f has some f i x e d p o i n t

on 0. Given any f i x e d p o i n t P on R , t h e s e t o f f i x e d p o i n t s o f f on Q c o n s i s t s o f P

and a l l p o i n t s Q f o r which t h e l i n e PQ be longs t o F. Other than t h e tangent o f R

a t P, t h e r e a re e x a c t l y J E l i n e s th rough P be long ing t o F. There fore , I R I=,'ii+l.

Th is proves ( 1 ) .

L e t R be any p o i n t o f R- R

Moreover, f does n o t f i x R and so r i s n o t a tangent o f R . L e t ( R , S } = r n a . Then a l -

so SEn - R

1.Q - R I=n-Jii, we o b t a i n i n t h i s way each e x t e r n a l l i n e o f nF i n t h e subplane. T h i s

proves ( 2 ) .

F

There i s a unique l i n e r th rough R be long ing t o F. F '

There fore , r i s an e x t e r n a l l i n e o f RF i n t h e subplane F. Since F '

F

$.zoo{. By way o f c o n t r a d i c t i o n , assume F=G. Choose a l i n e r be long ing t o F which i s

an e x t e r n a l l i n e o f R By ( 2 ) o f Prop. 1, I r n n l = 2 . L e t P E r n n . Since f ( r ) = g ( r )

and f(P)+P, g(P)+P, i t f o l l o w s t h a t f (P)=g(P) . Hence fg (P)=P wi th P B F. T h i s i m -

p l i e s t h a t f g i s t h e i d e n t i t y c o l l i n e a t i o n o f TI which is a c o n t r a d i c t i o n .

F '

?mw,L L e t F ( resp . GI be t h e Baer-subplane o f f ( resp . 9). Since fg=g f , then f l e a -

ves G i n v a r i a n t . L e t f ' denote the i n v o l u t o r i a l c o l l i n e a t i o n induced by f on G. S i -

m i l a r l y , l e t g ' denote t h e i n v o l u t o r i a l c o l l i n e a t i o n induced by g on F. Accord ing

t o [ 2 ] 4.1.11, we have e i t h e r

I c l f ' wid g ' u.te both Buen-uzvo4uicon4 d F n G c4 u 4 u b p l m e OL onden 4fi uz

both 4itbplunea F und G , ua

88 M. Biliotti and G. Korchmaros

We prove that the former possibility cannot occur. Suppose that H = F n G is a subpla-

ne of order ' / A . So 0 =H n 0 is an oval of H by (1) of Prop. 1 , Choose a line t of

H such that It n It n n 1=2. Similarly. It n a 1=2. This yields I t nlilt n Q I+lt n n I+lt n n I=

H I=O. Applying ( 2 ) of Prop.1 to G, f' and nG, we can infer that

H

G F F G H =4. A contradiction, since n is an oval.

So we may assume that (ii) holds. In this case nF n aG=tr n a ] . The lines through C

which are secants of either n or belong to F fl G. Since R is an oval, such li-

nes are pairwise distinct. Thus F G

Suppose there is a point P E Q - (n fl r) fixed by fg. Then P#nF U QG. Set Q=f(P)=

g(P). Again, Q # OF U n The line t joining P and Q meets

ints. In particular, tfr. Both f and g leave t invariant. Thus, t belongs to H.

By ( 4 1 , It n ( n U a ) I = 2 . It follows that t has four common points with a . Since n

is an oval, this is impossible.

Therefore, we have that f g has a unique fixed point on n. By ( 1 ) of Prof. 1 , this

implies that fg is an elation.

R - ( a U R 1 in two po- G' F G

F G

P m m F . Let g ' denote the involutorial collineation induced by g on G . By way of

contradiction, assume that g ' is either an elation or the identity. Choose an exter-

nal line r of n in the subplane G such that r is fixed by g'. Applying ( 2 ) of Prop.

1 to G , g2 and nG, it follows that r meets n - n

ves r invariant, then g2 fixes P and Q. On the other hand g 2 fixes G pointwise. Sin-

ce P , Q $ G , it follows that g* is the identity collineation, contrary to our assump-

tions.

G in two points P and 8. As g lea- G


Paou,f. By Prop. 3, S c o n t a i n s a unique i n v o l u t i o n . Then by [ 9 ] , 112. Satz 8 . 2 , S i s

e i t h e r a c y c l i c o r a genera l i zed qua te rn ion group. We s h a l l prove t h a t t h e l a t t e r

p o s s i b i l i t y cannot occur .

Denote by F t h e Baer-subplane o f t h e unique i n v o l u t i o n f o f S . Assume t h a t S is a

genera l i zed qua te rn ion group. Then t h e c o l l i n e a t i o n group 5 induced by S on F admi ts

an elementary Abe l i an subgroup 7 o f o rde r 4 . By Prop. 4 , each o f t h e t h r e e i n v o l u -

t i o n s i n f i s a B a e r - i n v o l u t i o n i n F. By Prop. 2 , t h e i r subplanes a r e p a i r w i s e d i -

s t i n c t . Bu t such a s i t u a t i o n is excluded by a p p l y i n g Prop. 3 t o F , n and any two

i n v o l u t i o n s o f 7. F i n a l l y , t h e statement concern ing t h e o rde r o f S f o l l o w s f rom [ 2 ]

4. '1.10.

F

Piroof. L e t P be t h e s e t o f f i x e d p o i n t s o f Y on a . As r l eaves i n v a r i a n t no p o i n t

o r chord of a we have t h a t e i t h e r ~ = 0 o r 1 ~ 1 2 3 . I f 1~123, Ly f i x e s a quadrangle

s ince the kno t K o f 0 is a l s o f i x e d by Y . Thus, t h e f i x e d elements o f Y i n n fo rm a

subplane TI' and p = n 11 ill i s a suboval o f 0. Since Y is a normal subgroup o f r , then

r l eaves P i n v a r i a n t . As r i s s t r o n g l y i r r e d u c i b l e on n, t h i s i s imposs ib le . Thus P

i s empty. As Y is an elementary Abe l ian p-group, t h i s i m p l i e s t h a t p d i v i d e s 10.1.

Hence p In+ l .

Now we s h a l l p rove t h a t r f i x e s e x a c t l y one l i n e i n t h e s e t E o f a l l e x t e r n a l l i n e s

of R . Since IE i=n(n-1) /2 and ( n + l , n ( n - l ) / Z ) = I , then Y f i x e s a t l e a s t one l i n e o f E .

The common p o i n t o f any two l i n e s o f E is d i s t i n c t f rom t h e kno t K o f 0. As P is

empty, i t f o l l o w s t h a t Y cannot have f u r t h e r f i x e d l i n e s i n E.

L e t r be t h e unique f i x e d l i n e o f Y i n E. As 'Y is a normal subgroup o f r , then r f i -

xes r . But then, by ( 2 ) o f Prop. 1 , r has no Baer i n v o l u t i o n .

Since a f i n i t e group w i t h c y c l i c Sylow 2-subgroups i s s o l v a b l e (see [ 9 ] , I V . Satz

2 . 8 ) , then P r o p o s i t i o n s 5 and 6 y i e l d t h e f o l l o w i n g r e s u l t :

90 M . Biliotti and G. Korchmaros

4. COLLINEATION GROUPS STRONGLY IRREDUCIBLE ON AN OVAL

P ~ c J u ~ . We d i L t i n g u i s h two cases accord ing t o whether r f i x e s e x a c t l y one l i n e o r i t

has no f i x e d l i n e .

Assume r f i x e s a l i n e r o f n. C l e a r l y , r i s an e x t e r n a l l i n e o f a . By ( 2 ) o f Prop.

1, r con ta ins no Baer - i nvo lu t i on . Hence, any i n v o l u t i o n of r i s an e l a t i o n whose

center belongs t o r . But then two d i s t i n c t i n v o l u t i o n s o f r cannot commute s i n c e i t

i s e a s i l y seen t h a t t h e i r cen te rs as w e l l as t h e i r axes must be d i s t i n c t . So any two

d i s t i n c t i n v o l u t i o n s i n r generate a d i h e d r a l group w i t h c y c l i c stem o f odd order .

By [ 3 ] , C o r o l l a r y 3, i t f o l l o w s t h a t < A > i s t h e semid i rec t p roduc t o f O(<A>) by a

group o f o rder two generated by an i n v o l u t o r i a l e l a t i o n .

Assume t h a t r has no f i x e d l i n e . By Theorem A , A i s non-empty. So we can app ly He-

r i n g ' s main theorem on bewegend groups [6]. As t h e kno t K of n cannot be t h e cen te r

o f any e l a t i o n i n r , our s i t u a t i o n corresponds, up the d u a l i t y , w i t h t h a t conside-

red i n Theorem 1 o f [ 6 ] . l t remains t o exclude the p o s s i b i l i t y t h a t <A>=SU(3,qz),

where q i s a power o f 2 and q24. I n such a s i t u a t i o n , Z(<A>) has order 3 and f i x e s

t h e a x i s o f each e l a t i o n i n A . Thus, Z(<A>) has some f i x e d p o i n t s on P. L e t p be t h e

s e t o f a l l f i x e d p o i n t s o f Z(<A>) on R . r l eaves p i n v a r i a n t . As r leaves i n v a r i -

an t no p o i n t o r chord o f n then 1~123. But as we have shown i n the proo f o f Prop, 6

1 ~ 1 ~ 3 i m p l i e s t h a t P is a suboval o f P . Since r i s s t r o n g l y i r r e d u c i b l e on R , t h i s

i s imposs ib le .

A s i m i l a r argument shows t h a t t h e

r 5 Aut <A>.

c e n t r a l i z e r o f < A > i n r i s t r i v i a l . There fore ,

In the following, we shall be concerned with some geometrical properties of the set

D o f all points which are centers of involutorial elations of A. Also the set D U S U

U I K 1 will be considered. Here S denotes the subset of n consisting of those points

which are fixed by some involutorial elation

o f Prop. 6 and Theorem 6 , the following statement holds:

of A . As we have seen in the proofs

Now we shall prove

B 4 4 . In our situation, < A > has exactly one class of involutions. So all involu-

tions in < A > are elations.

Given any point P t S , let a denote an involutorial elation of A fixing P. Let z(Z)

be the center of the unique Sylow 2-subgroup L of < A > containing u. The involutions

o f < A > commuting with o are exactly those belonging to Z ( L ) . Each o f them fixes P

and has axis PK. Conversely, any two involutions of < A > with the same fixed point P

on S commute because they have the same axis PK. Thus, < A > acts on S as the corres-

ponding simple group acts on the set of its Sylow 2-subgroups. This completes our

proof.

Assume <n>~SL(2,q), q=2' and q24. By a result of Hering [7], Theoren 2.8.c, D U S U

U i K 1 is a desarguesian subplane n ' of order q o f n and S is a suboval of n. Since

r is strongly irreducible on n, then we must have S=n . Hence, n'=n . Moreover, n

is a conic. Therefore, we have

Assume either c a > - S z ( q ) , q=Za q>4, or <A>zPSU(3,q2), q=2' q24. The present state o f

92 M. Biliotti and G. Korchmuros

our knowledge does n o t a l l o w us t o determine t h e u n d e r l y i n g p lane TI. For a geomet-

r i c a l

o f D as w e l l as t h a t o f D U S U I K } . Here, a s e t U o f p o i n t s o f a p r o j e c t i v e p lane

has c l a s s [XI , ..., x ] when I r I l U l belongs t o the i n t e g e r s e t I x l , ..., x 1 f o r any

l i n e r i n the plane. Resu l t s about s e t s w i t h p resc r ibed c l a s s a re g i ven i n [ E l ,

approach t o t h i s e s s e n t i a l ques t ion , i t may be o f i n t e r e s t t o know t h e c l a s s

K

~ 1 7 1 , [IE

P / L U O [ . L e t u 1 and a 2 be any two d i s t i n c t i n v o l u t o r i a l e l a t i o n s be long ing t o A. We

s h a l l denote t h e i r cen te rs by R . ( i = 1 , 2 ) and the l i n e through them by r. We want t o

determine 1 r n D I . We a l ready remarked t h a t each i n v o l u t i o n o f < A > i s an e l a t i o n . Moreover, as i t was

shown i n the p roo f o f Theorem B, < A > does n o t leave r i n v a r i a n t . Hence D g r n D.

Assume f i r s t a1a2=a2a,. An argument s i m i l a r t o t h a t used i n t h e p roo f o f Prop. 8

shows t h a t r i s a tangent o f 0 such t h a t rfl R E S , and r f l D c o n s i s t s o f t h e q-1 cen-

t e r s o f t h e i n v o l u t i o n s i n Z ( E ) , where Z i s t he Sylow 2-siibgroup o f < A > c o n t a i n i n g

011a2.

Assume now ala2fa2a1. Then r i s n o t a tangent o f R. By [14 ] , Lemma 5.1, r i s

the unique f i x e d l i n e o f u 1 a 2 which does n o t pass th rough K. L e t A be any d i h e d r a l

subgroup o f < A > which con ta ins u 1 a 2 . Then A a l s o l eaves r i n v a r i a n t and f i x e s K .

Thus, t h e cen te rs o f i n v o l u t i o n s o f A be long t o r , a l so . If A i s such t h a t no < A , C J >

w i t h a E A , u B A , l eaves r i n v a r i a n t then r n 0 c o n s i s t s o f t h e cen te rs o f t h e i nvo -

l u t i o n s i n A and I r n D l = l A l / Z . We p o i n t ou t t h a t t h i s s i t u a t i o n occurs when A i s a

maximal subgroup o f <A>. We s h a l l prove t h a t such a d i h e d r a l group A e x i s t s i n bo th

o f cases under cons ide ra t i on .

Assume <A>zsZ(q), q=Za and q>4. Then < A > admits e x a c t l y t h r e e con iugate c lasses o f

d i h e d r a l subgroups which a r e n o t p r o p e r l y conta ined i n any o t h e r d i h e d r a l subgroup.

They have o rde rs 2(q-1) o r 2 ( q V G + l ) . Those o f o rde r 2(q-1) a r e maximal subgroups

o f < A > . Moreover, each o f those o f o rder 2 ( q + / q + l ) i s con ta ined i n e x a c t l y one ma-

1

x ima l subgroup o f o rde r 4 ( q + J q + l ) , which has c y c l i c Sylow 2-subgroups (see [161,

Theorem 9 ) . Thus, e i t h e r I r ; lD I=q - l o r l r n D l = q + d z + l .

Assume <A>=PSU(3,q2), q=2a and 924. Then f o r any two d i s t i n c t Sylow 2-subgroups E l ,

6, we have <Z(6 , ) ,Z (zz)>=SL(2 ,q) (see [I.?], Satz 4 . 3 . v i i ) . There fore , each d i h e d r a l

subgroup o f <A> i s con ta ined i n some subgroups o f < A > i somorph ic t o SL(2,q). By [ 9 1

11.8), t h e r e a r e e x a c t l y two con iugated c lasses o f d i h e d r a l subgroups i n SL(2.q)

which a r e n o t p r o p e r l y conta ined i n any o t h e r d i h e d r a l subgroup o f SL(2,q). These

groups have o r d e r s 2(q-1) o r 2 ( q + l ) acco rd ing t o whether I r n S 1 = 2 o r 0. I n PSU(3,q')

t h e subgroups isomorph ic t o SL(2,q) a r e con iugate . Thus, t h e above a s s e r t i o n h o l d

f o r PSU(3,q2), also. From these f a c t s we can i n f e r t h a t < a l , a 2 > i s con ta ined i n a

d i h e d r a l subgroup A o f o rde r 2 (q- I ) o r 2 (q+ l ) acco rd ing t o whether l r n S I = 2 o r 0.

Assume q=8. L e t / r n S I = 2 . L e t us cons ider t h e subgroup Y o f < A > which leaves r n S

i n v a r i a n t . By [ 9 ] , 11.8, I i s t h e d i r e c t p roduc t o f A with a c y c l i c group o f o rde r

3. There fore t h e i n v o l u t i o n s o f Y a r e e x a c t l y those o f A . We may assume t h a t r and

S a re d i s j o i n t . L e t M=SL(2,8) be t h e subgroup o f <A> c o n t a i n i n g A , Denote by 0 t h e

subgroup o f o rde r 3 o f < A > which c e n t r a l i z e s M. I f 2 i s t h e subgroup o f o rde r 9 o f

A then 0 x 5 i s a Sylow 3-subgroup o f < A > . Since t h e c e n t r a l i z e r o f 3 i s con ta ined

i n O X M then 0x5 i s t h e un ique Sylow 3-subgroup o f < A > which c o n t a i n s f. L e t R de-

no te t h e s e t o f a l l i n v o l u t i o n s i n A whose cen te rs l i e i n r. For any P , , P ~ E R w i t h

p , 6 p 2 , we have t h a t cp,,p,>is conta ined i n a d i h e d r a l subgroup o f o rde r 2.3'. Thus

R i s a f u l l c l a s s o f con juga te i n v o l u t i o n s i n < A > and Glauberman's theorem (see [ 9 ]

Cor. 3) may be a p p l i e d . I t f o l l o w s t h a t 1<R,1=2-3 and I<RR>/=3 . Moreover, O x A has

o rde r 2-33 and c o n t a i n s e x a c t l y 9 i n v o l u t i o n s , which a r e e x a c t l y those o f A . Since

4 i s con ta ined i n <RR>, we have e i t h e r <RR>=E o r <RR>=@xf. Thus, < R k A o r <R>=@xA

ho lds . But the l a t t e r p o s s i b i l i t y c.annot ,occur because a l l i n v o l u t i o n s o f O x A a re

i n A .

Assume qf8. L e t < a 1 , a 2 > be any d i h e d r a l subgroup o f <A> o f o rde r 2(q-1) o r 2 ( q + l ) .

I n o rde r t o p rove t h a t each i n v o l u t i o n O E A , with c e n t e r on r, belongs t o A , we

s h a l l show t h a t i f we deny t h i s then t h e r e e x i s t two commuting i n v o l u t i o n s i n <A>

bo th l e a v i n g r i n v a r i a n t , which is a c o n t r a d i c t i o n . I n f a c t , such e l a t i o n s must

have d i s t i n c t cen te rs , s i n c e they map i? on to i t s e l f . There fore , t h e y must have t h e

same a x i s t. But then, they bo th cannot leave r i n v a r i a n t s i n c e t, be ing a tangen t

o f a , i s d i s t i n c t f rom r.

L e t I:, and z denote t h e Sylow 2-subgroups c o n t a i n i n g a , and 0 , r e s p e c t i v e l y . I f I:,=

B B

M. Biliotti and G. Korchmaros 94

=Z , t h e r e

Z ( Z ) > . I f

(2,q) (see

s n o t h i n g t o prove. Otherwise, assume f i r s t t h a t < Z ( Z , ) , Z ( Z , ) > = < Z ( Z , ) ,

u g h then <aIA>=SL(2,q), as A i s a maximal subgroup o f < z ( Z , ) , z ( Z ) > = s L

[9], 11.8.27). Hence, < o , A > con ta ins two commuting i n v o l u t i o n s . Assume

now <Z(z,),Z(z,)>#<Z(e,),Z(z)>. Then N<*>(Z,) con ta ins two d i s t i n c t c y c l i c subgroups

0 and o 2 o f o rder ( q + l ) / d wi th d=(3,q+l) which c e n t r a l i z e < Z ( Z , ) , Z ( Z ) > and < Z ( Z , ) ,

Z ( z 2 ) > , r e s p e c t i v e l y (see [ 1 2 ] , Satz 4 . 3 . v i i ) . These c y c l i c subgroups bo th l eave r

i n v a r i a n t , s ince t h e cen te rs o f a , 0 , and a, l i e on r . We prove t h a t admits

two commuting i n v o l u t i o n s by showing t h a t I<O,o ,>n Z(Z,)1>2. By [121, Satz 4 .3 .v -

V ~ ~ , @ ~ < Y = Z , O

over, 0 and 0, a r e Froben ius complements o f Y . Since 1Z11=q2, IBI=l@,l=iq+l l /d, qf8,

i t i s n o t d i f f i c u l t t o show t h a t <e,o,>=!. I f <EI),@~>=Z,O w i t h z,<z,then /I,, f l z ( E , ) l

=2 , where ~ 2 1 s ince each element o f z,-Z(Z,) has order 4 and i t s square l i e s i n

Z ( Z , ) . Since the q-1 i n v o l u t i o n s i n Z ( Z , ) a re con jugated under N<A,(Z,) then each

o f them i s t h e square o f e x a c t l y q (q+ l ) elements of Z , - Z ( Z , ) . But , C, con ta ins 2''

( q2 -1 ) elements o f C,-Z(C,). Hence l E , f l Z ( Z , ) 1 > 2 .

From t h a t we have seen so f a r , i t f o l l o w s t h a t I r n D l = q - l o r I r n D l = q + l .

Now cons ider D U S U I K I . A l i n e r through t h e kno t K con ta ins q-1 p o i n t s o f D and

one p o i n t o f 5. Hence, Ir f l ( 0 U S U I K 1 ) I=q+l . Assume r fl S=(A,B), wi th A&. Since<A>

i s a a c t s on S i n i t s usua l doubly t r a n s i t i v e rep resen ta t i on , we have t h a t < A >

d i h e d r a l group w i t h c y c l i c stem o f o rder q-1 o r ( q * - l ) / d accord ing t o whether <A>?

Sz(q) o r PSU(3,q2) (see [ 1 6 ] and [ 1 2 ] , Satz 4 . 3 . v i ) . I n t h e former case, I r n D I = q - I

con ta ins e x a c t l y q - I i n v o l u - I n the l a t t e r , we have aga in Ir n DI=q-I s ince <A>

t i o n s , namely those l y i n g i n i t s d i h e d r a l subgroup

Conversely, i f I r f l O l = q - l then r i s f i x e d by a d i h e d r a l group H o f o rder 2(q-1)

which, o f course,does n o t f i x any o t h e r l i n e . But , H in te rchanges two p o i n t s o f S

and hence i t f i x e s t h e l i n e through them. I t f o l l o w s t h a t such a l i n e must be r and

so l r f l S l = 2 . T h i s completes the p roo f o f P r o p o s i t i o n 10.

and k , o / Z ( z , ) i s a Frobenius group w i t h Froben ius k e r n e l z , . More-

Y

IA,Bl

iA,B) o f o rde r 2(q-1).

5 . COLLINEATION GROUPS OF EVEN ORDER WHlCH ARE TRANSITIVE ON AN OVAL

ColIimeation Groups 95

P/iou[. Clearly r is strongly irreducib acts transitively on R , every point o f

e on 0. So we can apply Theorem B. Since r R is fixed by an involution of A. But then,

by Gleason’s lemma (see [2], 4.3.15). < A > also acts transitively on il. So, with

the notation o f Section 4, we have S=R.

If r leaves a line r invariant then every point o f r is the center of an involu-

tion o f Aand Propositions 6 and 7 yield (i).

If r does not leave any line invariant then it turns out that either (ii) or (iii)

holds. In fact, actually r cannot involve PSU(3,q’). To see this, assume, by way

of contradiction, that <A>zPsu(3,q2) holds. Since S=R, we have n=q3 with q=2a, a22.

With the notation of Section 4, let P be any point P o f DUSUIK). By Prop. 10, for

any line r through P, Irn ( D U S UtK):))_2 implies Irn [ D u s U{Kl)I=q+l. Since I D U

U S U{K}I=(q +l)(q+l). it follows that no line in the plane meets D U S U t K l in a

unique point. Hence, D U S U i K 1 is actually of class [U,q+l], i.e. it is a maximal

((q’+l)q+l,q+l)-arc. By [ 8 ] 12 .2 .1 , this implies (q+1)l(q3+l)q+l, a contradiction.

Notice that the possibility <A>+‘SU(~,C,~) can also be excluded by applying [I].

Research partially supported by G.N.S.A.G.A o f C.N.R. and by M.P.I.

REFERENCES

(1; Biliotti, M. and Korchmaros, G . , On the action of PSU(3.q’) on an affine plane

o f order q, Archiv Math. 44 (1985) 379-384.

[ Z ] Dembowski, P., Finite Geometries (Springer Verlag, Berlin-Heidelberg-New York,

1968) .

96 M. Biliotti and G. Korchmaros

[ 3 ] Glauberman, G . , Cen t ra l elements i n co re - f ree groups, J . Algebra 4 (1966) 403-

420.

[ 4 ] Her ing , C . , On t h e s t r u c t u r e o f f i n i t e c o l l i n e a t i o n groups o f p r o j e c t i v e p l a - nes, Abh. Math. Sem. Hamburg 49 (1979) 155-182.

[ 5 ] Her ing , C . , F i n i t e c o l l i n e a t i o n groups o f p r o j e c t i v e p lanes c o n t a i n i n g n o n t r i -

v i a l p e r s p e c t i v i t i e s , i n : F i n i t e Groups, Santa Cruz Conf. 1979, Proc. Symp. Pu-

r e Math. 37 (1980) 473-477.

[ 6 ] Her ing C., On Beweg l i chke i t i n a f f i n e planes, i n : F i n i t e geometr ies, Proc. Conf.

Hon. T.G. Ostrom, Wash. S ta t . Univ. 1981, Lec t . Notes Pure Appl . Math. 82 (1983)

197-209.

[ 7 ] Her ing , C . , On p r o j e c t i v e planes o f type V I , i n : Co l l oq . i n t . Teor ie comb., Ro-

ma 1973, A t t i d e i Convegni L i n c e i 17 Tomo I1 (1976) 29-53.

[ 8 ] H i r s c h f e l d , J.W.P., P r o j e c t i v e geometr ies over f i n i t e f i e l d s (Clarendon Press,

Oxford, 1979).

[ 9 J Huppert, B . , End l i che Gruppen 1 (Spr inger Ver lag , Ber l in-Heidelberg-New York,

1967).

[ l o ] Huppert , B. and Blackburn, N., F i n i t e Groups 111 (Spr inge r Ver lag, Be r l i n -He i -

delberg-New York, 1982).

[ l l ] Kantor , W.M., Symplec t ic groups, symmetric des igns and l i n e ova ls , J . Algebra

33 (1975) 43-58. f 2 f

[ 1 2 ] Klemm, M., Charak te r i s ie rung de r Gruppen PSL(2,p ) and PSU(3,p ) durch i h r e

C h a r a c t e r t a f e l , J . Algebra 24 (1973) 127-153. 2 r

[ 1 3 ] Korchmaros, G . , Le o v a l i d i l i n e a d e l p iano d i Luneburg d ' o r d i n e 2 che pos-

sono v e n i r mutate i n se da un gruppo d i c o l l i n e a z i o n i i somor fo a1 gruppo sem-

p l i c e S Z ( ~ ~ ) , A t t i Accad. Naz. L i n c e i , Memorie, C 1 . Sc i . F i s . Mat. Nat. , ( 8 )

15 (1979) 295-315.

[ 1 4 ] Luneburg, H., T r a n s l a t i o n planes (Spr inger Ver lag , Ber l in-Heidelberg-New York,

1980).

1151 S t r o t h , G . , On Cheval ley-groups a c t i n g on a p r o j e c t i v e planes, J . Algebra 77

( 1 982) 360-381 . [161 Suzuki, M., On a c l a s s of doubly t r a n s i t i v e groups, Ann. Math. 75 (1962) 105-

145.

[ 1 7 ] T a l l i n i , G., Problemi e r i s u l t a t i s u l l e geometr ie d i Ga lo i s , Relazione n.30

1s t . Matem. Univ. Napo l i (1973).


[I81 Tallini-Scafati, M . , Sui (k,n)-archi di un piano grafico finito, con partico-

lare riguardo a quelli a due caratteri, Atti dell’Accad. Naz. Lincei, Rendi-

conti, C1. Fis. Mat. Nat. (8) 40 (1960) 812-818 and 1020-1025.

ori SETS OF PLUCKER CLASS TWO IIJ P G ( ~ , Q )

-2)

P a o l a R i o n d i and r J i c o l a Me lone

D i p a r t i m e n t o d i M a t e m a t i c a e A p p l i c a z i o n i "R. Cacc i o p p o l i"

U n i v e r s i t d d i N a p o l i ITALY

I n t h i s p a p e r t h e c o n c e p t o f t h e P l u c k e r c l a s s o f a k - s e t i n PC(n,q) i s i n t r o d u c e d and c e r t a i n k - s e t s o f P l u c k e r c l a s s two in PG(3,q! a r e c h a r a c t e r i z e d .

IPJTt?ODlJCTIOfJ

The c o n b i n a t o r i a l c h a r a c t e r i z a t i o n o f g e o m e t r i c o b j e c t s embedded i n P G ( n , q ) , t h e n - d i m e n s i o n a l p r o j e c t i v e space o v e r t h e G a l o i s f i e l c l G F ( q ) , i s one o f t h e most i n t e r e s t i n g p r o b l e m s in c o m b i n a t o r i a l geomet r i es .The t h e o r y o f k - s e t s , i . e . t h e i n v e s t i g a t i o n o f s u b s e t s o f s i r e k i n PG(n,q) w i t h r e s p e c t t o t h e i r p o s s i b l e i n t e r s e c t i o n s w i t h a l l subspaces o f a g i v e n d i m e n s i o n ( s e e f o r i n s t a n c e

[?Oil t u r n s o u t t o be q u i t e a p o w e r f u l and u s e f u l t o o l i n such c h a r a c t e r i z a t i o n s .

A k - s e t I< i n P C ( n , q ) i s s a i d t o be o f c l a s s [ml,m 2,...,ns 1, i f f o r any

d-suhspnce s

non-nega t i ve i n t e q e r , % = ( q d " - l ) l ( q - 1 )

n . - s e c a n t d - s p a c e . I f f o r a l l j = l , Z , ..., s some m. -secan t d-space e x i s t s , t h e n -J J K i s o f type .The k - s e t t h e o r y c l a s s i f i e s a l l s e t s i n a g i v e n c l a s s .

The f i r s t r e s u l t s f r o m t h i s p o i n t of v i e w were o b t a i n e d by 9.Segre LO], El], k21, whose c h a r a c t e r i z a t i o n o f c o n i c s i n PC(2 ,q ) ((1 o d d ) , a s k - s e t s o f c l a s s [0,1,2], i s w e l l known.

T h i s r e s u l t was g e n e r a l i z e d t o q u a d r i c s by G . T a l l I . n i F.Buekenhout I?] ,as a r e s u l t o f t h e i n v e s t i g a t i o n o f s p e c i a l s e t s o f c l a s s b , 1 , 2 ,

q + l I 1 s e t s was c a r r i e d o u t by

b5], DG],

we tiave l K n s 1 E{m,,m2 ,..., ms] , (0<ml<m2< ... m s < % , m , a rl J

.If IK n S d l = m . , :;d i s c a l l e d an J

(m,,m2, ..., m s d

E3], b4] and l a t e r on by

E , n , q + d 1 J .W.P .H i r sch fe ld & J.A.Thas

i n P G ( n , q ) . F u r t h e r m o r e , a t h o r o u g h i n v e s t i g a t i o n o f t h e c l a s s M . T a l l i n i S c a f a t i E7] and

Whereas t h e l i t e r a t u r e i s f a i r l y abundan t f o r d = l , t h e r e a r e n o t so many r e s u l t s when d:l .Fo r i n s t a n c e , s e t s o f t y p e (rn,n) were i n v e s t i g a t e d by A . D i c h a r a [l] , '.4.J.de Resmin i arld hl.de F i n i s [4 ] ,M. feFin is k ] , G . T a l l i n i h 4 . T a l l i n i S c a f a t i [lU], Lg], J.A.Thas 1211. F i n a l l y , r e s u l t s on more t h a n t w o c h a r a c t e r s e t s , a q a i n when and r.l.hlelone [ g ] :

A c o m p l e t e c h a r a c t e r i z a t i o n o f k - s e t s w i t h more t h a n t w o c h a r a c t e r s w i t h r e s p e c t t o d-spaces, d > l , seems t o be e x t r e m e l y d i f f i c u 1 t . t h e r e f o r e we s h a l l add some

n a t u r a l a s s u m p t i o n s on t h e s e t s i n o r d e r t o c l a s s i f y them.Such a s s u m p t i o n s may be

o f e i t h e r a r i t h m e t i c o r s t r u c t u r a l n a t u r e .

d > l ,were o b t a i n e d e.g. by A . B i c h a r a [ 2 ] , 0 . F e r r i 161

___------------ ( " ) kiork s u p p o r t e d by I t a l i a n M.P.I.

100 P. Riondi and N. Melone

From t h i s p o i n t o f v iew we d e f i n e t h e P l u c k e r c l a s s o f a s e t i n PG(n,q) a s f o l l o w s . D e n o t e by K a k - s e t i n PG(n,q).A h y p e r p l a n e H i s s a i d t o be t a n g e n t t o K a t i t s p o i n t p i f any l i n e i n H t h r o u g h p i s e i t h e r a 1 -secan t o r ( q + I ) - s e c a n t o f K .For any (n-21-space S i n PG(n,ql we d e f i n e m(K,S ) a s t h e number o f t a n g e n t h y p e r p l a n e s t o K ":$rough S . I f , f o r any i n PG(n,q) , m(K,Sn-2)E [ O , l , ..., m,q+l} w i t h m<q+ln;?hen we say t h a t P l u c k e r c l a s s m .

n-2

K has 'n-2

I n t h i s paper we d e a l w i t h k - s e t s i n PG(3,q) o f and P l u c k e r c l a s s m=2 .Tak ing i n t o a c c o u n t t h e r e s u l t s i n t y p e s f o r such a s e t a r e (1 ,q+1,2q+1)2 , ( l , q + l ) and ( +1,2q+1I2 .S ince t h e s e t s o f t y p e ( l , q + l ) c o n s i d e r t h e s e t s o f Zypes two.The n e x t s t a t e m e n t sums up t h e o b t a i n e d r e s u l t s .

Theorem.Let K be a k - s e t i n PG(3,q) two c o n t a i n i n g p 1 ines.Denote by t t h e number o f t a n g e n t p l a n e s t o K.Then one o f t h e f o l l o w i n g o c c u r s . ( i ) I f K i s o f t y p e ( l ,q+1,2q+1)2 w i t h P 2 2 , t h e n p 2 3 and K i s a cone p r o j e c t i n g a ( q + l ) - a r c o f a p l a n e f r o m a p o i n t . F u r t h e r m o r e P = 3 i f f (ii) If K i s o f t y p e ( 1 , q + 1 I 2 , t h e n K i s e i t h e r a l i n e o r a (q2+1) -cap ( i . e . an o v o i d ) . (iii) I f K i s o f t y p e (q+1 ,2q+1 I2 w i t h t > O and pa5 , t h e n K c o n s i s t s o f t h e p o i n t s on q + l p a i r w i s e skew 1 ines .wh ich e i t h e r have one o r two t r a n s v e r s a l s o r f o r m a h y p e r b o l i c q u a d r i c .

a r e t h e l i n e s and t h e (q2?1)-caps hl1,it rema ins t o ( l , q+1 ,2q+1)2 and (q+1,2q+1)2 and o f P l u c k e r c l a s s

o f c l a s s E,q+l,Zq+1J2 and P l u c k e r c l a s s

q=2.

1. k-SETS OF TYPE ( l , q + 1 , 2 q + 1 ) 2

I n t h i s s e c t i o n K i s assumed t o be a k - s e t i n PG(3,q) o f P l u c k e r c l a s s t w o and t y p e w i t h P 2 2 , i . e . i n wh ich t h e r e a r e a t l e a s t two l i n e s .

Cons ide r a 1-secant p l a n e H and l e t p = K n H .S ince any l i n e i n K i n t e r s e c t s H i n a p o i n t , i t must pass ehrough p . f i o reovg r , s ince any p l a n e has a t most 2:+1 p o i n t s i n common w i t h K,no t h r e g l i n e s c o n t a i n e d i n K can be c o p l a n a r . C o n s i d e r f i r s t t h e case P 2 4 .Take f o u r l i n e s 11,12,13,14 l y i n g on K .Then t h e t h r e e d i s t i n c t p l a n e s < 11,1 > , <1,,13> , (1 ,1 7 a r e t a n g e n t t o K.Since K i s o f P l u c k e r c l a s s two, any psane t h r o u g h 1 !s f a n g e n t t o K.Hence, for any p o i n t x on K \ 1 , t h e p l a n e j o i n i n g l1 and x1 i s t a n g e n t t o K and meets K i n t w o lines,name!y l1 and x ,The re fo re , K i s t h e u n i o n o f P l i n e s t h r o u g h p . L e t t i be a p l a n e :ot t h r o u g h p .Then K n H c o n s i s t s o f

t h e p o i n t s i n which'the P l i n e s on t h e s e l i n e s a r e c o p l a n a r , K n H i s a p-arc .0n t h e o t h e r hand, H i s e i t h e r a ( q + l ) - s e c a n t o r a ( 2 q + l ) - s e c a n t p l a n e o f K ; t h u s , p = q + l and K i s t h e cone p r o j e c t i n g f r o m po t h e ( q + l ) - a r c K n H . Next,we c o n s i d e r t h e case p=2 . L e t l,m be t h e two l i n e s c o n t a i n e d i n K,and deno te by H1 t h e p l a n e t h r o u g h 1 and m .S ince P=2 , no ( 2 q + l ) - s e c a n t p l a n e d i f f e r e n t f r o m H i s a t a n g e n t p lane.Thus,any o t h e r p o s s i b l e t a n g e n t p l a n e t h r o u g h 1 o r m is a ( q + l ) - s e c a n t plane.\rJe c l a i m tha t n e i t h e r 1 n o r m i s on t h r e e t a n g e n t p l a n e s ; o t h e r w i s e , a n y p l a n e t h r o u g h 1 ( o r m , r e s p e c t i v e l y ) d i s t i n c t f r o m t i i s a ( q + l ) - s e c a n t p l a n e wh ich i s t a n g e n t t o K .Thus, K c o n s i s t s o n l y o f t h e p o l n t s on 1 and m , a c o n t r a d i c t i o n s i n c e K i s o f t y p e ( l , q + 1 , 2 q + l ) . Assume t h a t t h e r e i s a un ique t a n g e n t p l a n e L f H1 t h r o u g h 1 .Then a l l o t g e r p l a n e s t h r o u g h 1 a r e ( 2 q + l ) - s e c a n t planes.Hence,

( 1 . 1 ) k = q +q+ l .

( l , q + l , 2 q + 1 ) 2

and t h e l i n e t h r o u g h p

K t h r o u g h po mee? H .S ince no t h r e e o f

1 .

1

2

o f ( 2 q + l ) - s e c a n t p l a n e s o f K t2q+1

Consequen t l y ,by b5] eq. ( 1 8 ) , t h e number e q u a l s

Sets of Plucker Class Two in PG(3,q) 101

= q ( q + 1 ) / 2 . ( 1 . 2 ) t z q + 1

N e x t , t a k e a 1 - s e c a n t l i n e r o f K a n d d e n o t e b y u , s , b t h e n u m b e r s o f l-secant,(q+l)-secant,(2q+l)-secant p l a n e s o f K t h r o u g h r , r e s p e c t i v e l y . C o u n t i n g t h e p o i n t s o f K o n t h e p l a n e s t h r o u g h r , we g e t i n v i e w o f ( 1 . 1 )

( 1 . 3 ) b = u .

A n a l o g o u s l y , i f r ' i s a n n - s e c a n t o f K , n 2 , a n d s ' a n d b ' d e n o t e t h e n u m b e r s o f ( q + l ) - s e c a n t a n d ( Z q + l ) - s e c a n t p l a n e s o f K t h r o u g h r ' , r e s p e c t i v e l y , t h e n

( 1 . 4 ) b ' = n - 1 . E a c h f Z q + l ) - s e c a n t p l a n e o t h e r t h a n H, m e e t s H1 i n a l i n e t h a t , b y ( 1 . 4 ) , p a s s e s t h r o u g h t h e P o i n t P o = l m . c o n s i d e r now a 1 - s e c a n t l i n e r " t h r o u g h p o i n , T h e n , b y ( 1 . 3 1 , t h e r e i s a t l e a s t o n e 1 - s e c a n t p l a n e t h r o u g h r " !'On t h e o t h e r h a n d , i f t h e r e w e r e t w o 1 - s e c a n t p l a n e s t h r o u g h r " , t h e n a n y p l a n e t h r o u g h r " w e r e a t a n g e n t p l a n e a n d we w o u l d g e t a c o n t r a d i c t i o n as a b o v e . C o n s e q u e n t l y , t h r o u g h a n y l i n e t h r o u g h p i n H o t h e r t h a n 1 a n d m t h e r e is a u n i q u e 1 - s e c a n t p l a n e . t i g n c e , b y 7 1 . 3 ) , t h i s e q u a l i t y w i t h ( 1 . 2 ) , w e h a v e q=2 ; m o r e a v e r , K c o n s i s t s o f s e v e n p o i n t s i n P G ( 3 , 2 ) . F i v e o n e o f t h e s e p o i n t s a r e o n t h e l i n e s 1 a n d i n , t h e r e m a i n i n g t w o a r e o f f t h e p l a n e H1 a n d n o n - c o l l i n e a r w i t h p . S i n c e t h i s c o n f i g u r a t i o n i s n o t t y p e ( l , q + 1 . 2 q + 1 ) 2 ,we h a v e a c o n t r z d i c t i o n .

Idow we a s s u m e t h a t H i s t h e o n l y t a n g r n t p l a n e t h r o u g h 1 . T h u s , e a c h p l a n e t h r o u g h l1 i s a ( 2 q + l ) - s e c a n t p l a n e a n d t h e n

( 1 . 5 ) k = ( q + l ) . N o w , e q u a t i o n ( 1 8 ) i n 1 5 i m p l i e s t h a t t h e r e i s n o 1 - s e c a n t p l a n e , a c o n t r a d i c t i o n .

F i n a l l y , w e d e a l w i t h t h e c a s e = 3 . L e t l , m a n d r b e t h e t h r e e l i n e s c o n t a i n e d i n K ; d e n o t e b y H1 , H 2 , H3 t h e p l a n e s j o i n i n g 1 a n d m , 1 a n d r a n d m a n d r , r e s p e c t i v e l y . A s s u r n e t h a t t h r o u g h o n e o f t h e s e l i n e s , s a y 1 , t h e r e i s a ( q + l ) - s e c a n t p l a n e . S i n c e s u c h a p l a n e i s t a n g e n t t o K , a n y p l a n e t h r o u g h 1 d i s t i n c t f r o m H a n d 1i2 i s a t a n g e n t ( q + l ) - s e c a n t p l a n e . T h u s , K i s t h e u n i o n 07. t h r e e c o n c u r r e n t n o n - c o p l a n a r l i n e s , if a n y $ l a n e t h r o u g h 1, m a n d r i s a ( 2 q + l ) - s e c a n t p l a n e , t h e n k = ( q + l ) a n d a c o u n t i n g a r g u m e l t s h o w s t h a t t h e r e a r e n o 1 - s e c a n t p l a n e s , a c o n t r a d i c t i o n ( c o m p a r e 1 5 e q . s ( 1 8 ) 1 .

T h u s we h a v e p r o v e d t h e f o l l o w i n g s t a t e m e n t

T h e o r e m I. L e t K b e a k - s e t o f c l a s s ( l , q + 1 , 2 q + l ) a n d P l u c k e r c l a s s t w o i n P G ( 3 , q ) s u c h t h a t 2 . T h e n 3 a n d 'K i s t h e c o n e p r o j e c t i n g f r o m a p o i n t a ( q + l ) - a r c . F u r t h e r m o r e , = 3 i f f q = 2 .

t 2 +, = 2 q + l , C o m p a r i n g

2

a n d s o q = 2 . On t h e o t h e r h a n d

2 . SETS OF TYPE ( q + 1 , 2 q + 1 I 2 . I n t h i s s e c t i o n , K d e n o t e s a k - s e t o f t y p e ( q t 1 , 2 q + 1 ) 2 a n d P l u c k e r c l a s s t w o i n P G ( 3 , q ) s a t i s f y i n g 0 . P r o p o s i t i o n 2 . 1 .We h a v e k = ( q + 1 ) 2 . F u r t h e r m o r e , e a c h t a n g e n t p l a n e is a ( 2 q + l ) - s e c a n t p l a n e .

102 P. Biondi and N . Melone

Proo f .S ince t h e t a n g e n t p l a n e s o f K meet K i n e i t h e r one o r two l i n e s , i t is s u f f i c i e n t t o p r o v e t h a t t h e p l a n e s t h r o u g h a l i n e o f K a r e ( 2 q + l ) - s e c a n t p l a n e s . I n o u r s i t u a t i o n , t h e e q u a t i o n ( 2 2 ) o f [15] r e a d s

= 0 .

-

2 2 q k - ( ( m + n ) 8 - q )k tmn

2 2

F o r rn=qt l , n=2q+l , t h i s e q u a t i o n has t h e u n i q u e i n t e g r a l s o l u t i o n k = ( q + l ) . C o u n t i n g t h e p o i n t s o f K on t h e p l a n e s t h r o u g h a l i n e i n K we g e t t h a t a l l t h e s e p l a n e s a r e ( 2 q + l ) - s e c a n t ,

P r o p o s i t i o n 2.2 .Through any n-secant l i n e 1 t h e r e a r e p r e c i s e l y n ( 2 q + l ) - s e c a n t p l a n e s ( o l n l q + l ) .

P r o o f . L e t b be t h e number o f ( 2 q + l ) - s e c a n t p l a n e s t h r o u g h 1 .Coun t ing t h e p o i n t s o f K on t h e p l a n e s t h r o u g h 1 ,we g e t k=q2+(b+2-n)q+l .It f o l l o w s b=n, i n v iew o f p r o p o s i t i o n 2.1 . P r o p o s i t i o n 2.3 .Through any p o i n t on K t h e r e a r e a t most two l i n e s c o n t a i n e d i n K.

Proof.Assume t h a t t h e r e a r e t h r e e c o n c u r r e n t l i n e s l , m , r c o n t a i n e d i n K . L e t p be t h e i r common p o i n t and deno te b y H t h e p l a n e t h r o u g h 1 and m .Any l i n e r ' f 1 , m t h r o u g h p i n H i s a 1 -secan t .Hence,by p r o p o s i t i o n 2.2 , H i s t h e u n i q u e ( Z q + l ) - s e c a n t p l a n e t h r o u g h i t .Thus,the p l a n e H' t h r o u g h r and r ' i s a ( q + l ) - s e c a n t p l a n e . C o n s i d e r i n g t h e p l a n e s t h r o u g h r ,we g e t t h e r e f o r e k = 3 q + l , wh ich c o n t r a d i c t s p r o p o s i t i o n 2.1 .

-

Plow we a r e ready t o p r o v e t h e f o l l o w i . n g

Theorem I1 . L e t K be a k - s e t i n PG(3,q) of t y p e ( q + 1 , 2 q + l ) and P l u c k e r c l a s s two w i t h t >O.Then k = ( q t l ) 2 and a l l t h e t a n g e n t p l a n e s a r e 2 ( 2 q t l ) - s e c a n t p l a n e s . F u r t h e r m o r e , i f P 2 5 t h e n e i t h e r K c o n s i s t s o f q + l m u t u a l l y skew l i n e s w i t h e i t h e r one o r t w o t r a n s v e r s a l s , o r K i s a h y p e r b o l i c q u a d r i c .

Proof .Denote by D t h e s e t o f p o i n t s on K w h i c h a r e on two l i n e s c o n t a i n e d i n K. F i x a t a n g e n t p l a n e Ho wh ich meets K a t t h e l i n e s 1, and mo and l e t po = lonmo .Set Do = D n H o and do = IDo I .S ince p0€ Do , d o > l . S i n c e p > 5 , t h e r e a r e a t l e a s t t h r e e l i n e s 11,12,13 d i f f e r e n t f r o m lo and mo c o n t a i n e d i n K. These l i n e s meet 1, o r mo ;moreover ,by p r o p o s i t i o n 2.3 , t h e p o i n t s o f i n t e r s e c t i o n a r e d i s t i n c t and d i s t i n c t f r o m po .Consequen t l y , d o 2 4 . S o , t h e r e a r e a t l e a s t two p o i n t s a,b i n Do ,wh ich a r e c o l l i n e a r w i t h po ,say a , b € l o . S i n c e K is o f P l i j c k e r c l a s s two , each p l a n e t h r o u g h 1, i s tangent .Hence,by p r o p o s i t i o n 2.3 , loC Do w h i c h i m p l i e s d o 2 q + l .If d =q+ l , t hen ,by p r o p o s i t i o n 2.1 , K c o n s i s t s o f q + l m u t u a l l y skew l i n e s w i t h a u n i q u e t ransve rsa1 ,name ly lo. If d o = q+2 , t h e n Do c o n s i s t s o f t h e p o i n t s on 1, and a u n i q u e p o i n t p1 o t h e r t h a n po on (11 .Denote by m the l i n e t h r o u g h p1 l y i n g on K and o t h e r t h a n m0 ; 1, and mo a r e skew and t h e q + l l i n e s of K on t h e p l a n e s t h r o u g h 1, meet b o t h lo and m .Consequent ly , K c o n s i s t s o f q + l p a i r w i s e skew l i n e s h a v i n g j u s t two t r a n s v e r s a l s , n a m e l y 1, and m . F i n a l l y , s u p p o s e d 2 q + 3 .Then ,a l so moC_Do lMoreover,any l i n e 1 d 1, i n K t h r o u g h a p o i n t o f lo mgets any l i n e mfmo i n K t h r o u g h a p o i n t o f mo .Hence, K i s a h y p e r b o l i c q u a d r i c .

-

REFERENCES

[l ] Bichara ,A . , S u i k - i n s i e m i d i S d i t i p 0 ( ( n - l ) q + l , n q + l I2,Rend.Acc.Naz. L i n c e i , ( 8 ) , 6 2 ( 1 9 7 7 ) 480-485 .3 'q

[2] B i c h a r a , A . , S u i k - i n s i e m i d i PG( r , q d i c l a s s e k, 1,2, n J *,Rend.di Mat. ,Roma, 1 3 ( 1 9 8 0 ) .

[3] Quekenhout,F. ,Ensembles q u a d r a t i q u e s des espaces p r o j e c t i f s , M a t h . Z . ,110 (1969)306-318.

Sets of Plucker Class Two in PG(3.y) 103

[4] de Finis,M. and de Resmini,M.J. ,On a characterization o f subgeometries PG(r, q) i n PG(r,q),q a square lEurop.Jrn l .Cornb. ,3(1982) 319-328.

de Finis,M.,On k-sets in PG(3,q) of type ( m , n ) with respect to planes,to appear.

Ferri.0. ,Su di una caratterizzazione grafica della superficie di Veronese di un S5,q ,Rend.Acc.l\laz.Lincei,VIII,vol.LXI,6~1976~603-610.

Hirschfeld,J.W.P. and Thas,J.A. ,The characterization of projection of quadrics over finite fields of even order,Jrnl.London Math.Soc.,22(1980) 226-238.

Hirschfeld,J.W.P. and Thas,J.A.,Sets of type (l,n,q+l) in PG(d,q),Proc.London Ilath.Soc.,41(1980)254-278.

Melone,N.,The linear line geometry in PG(3,q) from a synthetic point of view, Pubbl . 1st .Mat. "R. Caccioppol i" , Un iv . Nap01 i ,30 ( 1983 ) 7-1 5. Se g re, '2. , 0 Ja 1 s in a finite pro j ec t i v e plane , Canad. J r n 1. Math . ,7 ( 1 955 ) 4 14-4 1 6. Segre,B. ,Curve razionali normali e k-archi negli spazi finiti,Ann.Mat.Pura e A p p l . , ( 4 ) , 3 9 ( 1 9 5 5 ) 3 5 7 - 3 7 9 .

Segre,R. ,Le geometrie di Galois,Ann.Mat., (4),48(1959)1-97.

Tallini,G. ,Sulle k-calotte di uno spazio lineare finito,Ann.Mat. (41,42( 1956) 119-1 64.

Tallini,G. ,Caratterizzazione grafica delle quadriche ellittiche negli spazl finiti,Rend.Mat.,Rorna,l6(1957)328-351.

Tallini,G.,Problemi e risultati sulle geometrie di Galois,Pubbl.Ist.Mat. "R. Caccioppol i" , Un iv . PJapoli ,30( 1973 ) .

[ 5 ]

[6]

[7]

[ 8 ]

[!I]

rl 01 Ed

114 L14 -

[14]

[15]

bd Tallini,G.,k-insierni e blocking sets in PG(r,q) e in AG(r,q),Sem.Geom.Cornb. Ist.Mat.Appl.Univ.Aquila,l(l9&?2).

[17] Tallini Scafati , P l . ,Caratterizzazione grafica delle forme hermitiane di un S ,Rend.Mat.,Roma,26(1976)273-303. r,q

k q Tallini Scafati,M.,Calotte d i tipo ( m , n ) i n uno spazio d i Galois S r,q ' r?end.Acc.Naz.Lincei,(8),53(1973)71-81.

b9] Tallini Scafati,M.,Sui k-insierni di uno spazio di Galois S a due carat- ter i nel la dirnens ione

Tallini Scafati,fA. ,La teoria dei k-insiemi negli spazi di Galois,Sem.Geom. Cornb.Univ.Rorna,40(1982).

Thas,J.A. ,A combinatorial problern,Georn.Ded. ,1(1973)236-240.

d , Rend. Acc . Naz . Lincei , ( 8 1 ,40( 1976 ) 7fi2q-788.

[20]

bl]


A FREE EXTENSION PROCESS YIELDING A PROJECTIVE GEOMETRY

Flavio Bonetti Dipartimento di Matematica - Via Machiavelli, 35 - 44100 Ferrara

Nino Civolani FacoltA di Scienze - UniversitA della Basilicata - 85100 Potenza

Summary. Presented is a free extension process, mainly based on the configuration of Veblen and yielding a projective geometry.

The basic definitions can be found e.g. in the following sources: partial plane, Desargues' condition (resp. configuration), projective plane, in [ I ] ; (reducible) projective geometry, dimension, in [ 21; free extension (resp. completion) process, free projective plane, in [3] .

Let B = ( P , f , I) be a partial plane. The free projective plane T( B ) can be associated to B by the well-known free completion process 1 . We will describe another free extension process, 3 " , yielding a projective geometry T pv(iJ), possibly reducible with dimension 2 3, hence mostly)) different from the free projective plane1 (C).

To this end we utilize the notion of Veblen configuration { c, a, a', b, b', A, B, C, C'}(see Fig. l ) , i.e. five distinct points c, a, a', b, b' E P and four distinct lines A, B, C, C' E € such that: c, a, a' I A; c, b, b' I B; a, b, I C; a', b' I C'. The two lines C, C' are the enrering ones of the Veblen configuration, which is closed if they meet in a point.

Fig. 1

Now we define the free extension process 1'' = (C:'),, o :

B y = B

ic:y I = ( P : y l , € I:yI), where p : y l = 0,'" u { (C, C' 1 c, C' f f ;v

f : y 1 = € :v u {(P, P'}I P, P' E 0 y are the two entering lines of a nonclosed Veblen confeuration of a!"};

are distinct points not joined by any line of L ;

106 F. Bonetti and N . Civolani

1;: consists of the pairs of 1:" and also of those originated by the new elements

i: Xy 1, namely ((C, C'}, C), ({C, C'\, C'), (p, { P. P'}), (P',{P,P'}~ of P t;y

Clearly 3 "(6) is a projective geometry, and 3 '"( 6) = 6 if and only if B is a projective geometry.

It is also easily seen that, if D is a non-closed Veblen configuration, then at each stage D Z n - l (n > 1) there appears at least one new point.

The next results about this extension process are summarized by the following

THEOREM. Let D beapartialplane, with Fpv(D) # & . Then:

i) 3 ' ii) dim T p v ( 3 ) 2 3 - TPV(D) is reducible;

iii) dim 3 ' " ( D ) = 2 = + T P V ( D ) =F(&).

Proofs are straightforward.

has infinitely many distinct stages 5 x" ;

REFERENCES

[ 11 Dembowski,P., Finite Geometries (Springer, Berlin-Heidelber-New York; 1968).

[2] DubreilJacotin,M.L., Lesieur, L. and Croisot.R., Lqons sur la Thborie des Treillis, des Structures algbbriques or-

[3] Siebenmann, L.C., A Characterization of free projective Planes, Pacific J . Math. 15 (1965) 293 - 298.

donndes et des Treillis g6omCtriques (Gauthier-Viars, Paris, 1953).

Annals of Discrete Mathematics 30 (1986) 107-1 14 0 Elsevier Science Publishers B.V. (North-Holland) 107

SYMMETRIC FUNCTIONS AND SYMMETRIC SPECIES

FLAVIO BONETTI Universitrl di Ferrara GIAN-CARLQ ROTA (*) M.I.T. Boston DOMENICO SENATO Universitrl d i Napoli ANTONIETTA M. VENEZIA Univenitrl di Roma 1

INTRODUCTION

The idea of proving identities for symmetric functions by bijective arguments is quite old; it goes back to Lucas (Theorie des nombres, 1891) and probably earlier. To the best of our knowledge, the first glimmerings of a systematization of such bijective arguments goes back to one of the present author (cf. 191); the idea was further developed by R.P. Stanley, wo gave a bijective proof of Waring’s formula by Mobius inversion on the lattice of partition of a set, and later by Doubilet, who gaves bijective proofs of several identities in the theory of symmetric functions.

Joyal’s theory of species led us t o develop a systematic setting for such bijective proofs. We introduce here the notion of synirnetric species, which can be viewed as a set-theoretic (a category-theoretic) counterpart of the notion of a symmetric function. To each of the classical classes of symmetric functions we associate a symmetric species. Operations on species, as introduced by Joyal, are generalized to symmetric species, and simple categorical operations yielded bijective proofs of all identities among elementary symmetric functions. By way of example, we give a bijective proof of Waring’s forniula, which we believe to be new, and dispenses altogether with Mobius inversion, as well as bijective proofs of several related identities.

This note is part of a communication presented in Bari at ctCombinatorics 84)).

I . DEFINITIONS AND PRELIMINARIES

We denote by 9 the category of finite sets and bijections, and we denote by: I* :9 -+a

the contruiwiunt identify ficnctor. mapping every finite set E to itself, and such that, i f

u : E + F

is a bijection, then I*[u] = u-’

Recall that a species (Joyal) is a functor

M : @ + @ .

We shall follow Joyal’s terminology for species.

Let X be an infinite set, which will remain fixed throughout, whose elements will be called vuriubles. To agree with current usage, we may occasionaly list the variables in linear order xl, x 2 , . . . through this listing is strictly speaking irrelevant.

Recall, that, if N* is a contravariant functor of @ to 9, the functor N from 9 to the category of sets whose objects are

-

(*) Research supported by N S F contract n. MCS 8104855.

108 F. Bonetti et al.

N[E] = Hom (N*[E], X)

is covariant.

Indeed, if u E Horn (E, F) and if N*[u] : N*[F] +. N*[E] then N[u] : NLE] +. N[F] is defined by N[u](f) = = f o N*[u].

Let M be a species, and let N* be a contravariant functor of CP to a . We denote by Pol (M, N*) the functor defined as:

Pol (M, N*) [El = M[E] x Horn (N*[E], X)

and whose morphisms are, for u : E +. F, (M[u], N[u]) : M[E] x Horn (N*[E], X) +. M[F] x Horn (N*[F], X).

A polynomial spades P is a subfunctor of Pol (M, N*), that is, for every object in Pol (M, N*) it is a subset

P[E] C M[E] x Horn (N*[E], X)

such that if u E Hom (E, F) and if (s, f) E P[E] then

M u 1 (9, f 0 N*Lul) E P [Fl. In other words, the subset PIE] is functorially assigned.

When N* = I*, we say that the polynomial species is ordinary.

Let 9 : X -+ X be an isomorphism and let P be a polynomial species. We denote by Pv the polynomial species defined by:

( s , f )EPv[E]C M[E]xHom(N*[E],q(X))-(s , cp-’ of )EP[E] . Clearly, P +. Pv is a natural trasformation of functors.

A polynomial species is a symmetric species when P = Pq for every isomorphism 9 : X +. X.

EXAMPLE 1 . The elementary symmetric species E.

E[E] = the set of all monomorphisms from E to X C Pol (I, I*)

where I is the identity functor.

EXAMPLE 2 . The powermm species s

S[E] = the set of all functions from E to X of constant value C Pol (I, I*).

EXAMPLE 3 . The disposition species H.

Let S[E] be the set of all permutations on E. Take M = Exp (S) and let r E Exp ( S ) [El (i.e. r consists of a partition II and a permutation on each block of n).

We let H[E] be the subset of Exp ( S ) [El x Horn (E, X)

of all pairs (r, f) such that 1~ is the kernel of the function f : E +. X, that is, such that the blocks B E n are the sets f’(x)whenever f ’ (x) is non-empty, as x ranges on X.

This defines the species of dispositions.

EXAMPLE 4 . K, = the monomial elementary species of class A.

Symmetric Functions and Symmetric Species 109

Let IT = { B, , . . . , Bk} be a partition of E, I E 1 = n, and let ri (1 Q i Q n j be the number of blocks of n with i elements. The class of IT is the partition of the integer n defined by:

c l ( n ) = ( l r l , 2 2 , . . . , n r n j = (h,,X, , . , , h k j

where Xi = I Bi 1.

Let h be a class (i.e. a partition of n), and let M,[E] be the set of all partitions of class X on set E.

We define K,[E] to be the symmetric species of all pairs (n, f) where n is the kernel of the function f. We call this the monomial elementary species of class A.

EXAMPLE 5 . The cyclic species C.

Let C[E] be the set of all cyclic permutations on E. We set C[E] to be the cyclic polynomial species on Pol (C, I*) of all pairs ( p , f) where p E C[E] and f is constant.

EXAMPLE 6. The species Hx

Hx[El ={(o, f) I u E S[E] and f(e) = x for ail e E E}C S[E] x Hom(F[E], X).

2. THE POLYNOMIAL OF POLYNOMIAL SPECIES

Let Z[X] be the ring of all polynomials in the variables x E X.

We again denote by x the canonical image of x in the ring Z[X].

Let C be a cofinite subset of X. We write, if p E Z[X], p/,-, p by setting to zero all x E C. If p E Z[X; and C is a cofinite subaet of X, we let

to denote the polynomial obtained from

A(P; C)

be the set of all q E Z[X] such that

P/c=o = q/c=o.

This defines a topology on the ring Z[X]. The completion of Z[X] in this topology is denoted by Z[[X]]. Thus, an element r E Z[[X]] is an infinite sum of polynomials such that for every cofinite set C, dC= is an ordinary polynomial in Z[X].

Let { pc be a set of polynomials in Z[X]. We write

lim pc = r C

when the set { pc } converges to r in Z[[X]] along the filter of all cofinite sets.

A sufficient condition that ensures that ( p c ) converge is that, for C ’ C C

PC’/C=O = Pc.

The element of Z[[ X]] defined as

gen (0 = 11 ke) = n xlf”(x1 e € E X€X

is called the generafing polynomial of the function f (cf. [9]).

If P is a polynomial species on Pol (M, N*) we write:

110 F. Bonetti et al.

gen(P[EI) = sen(f). (s,ne PI E 1

Noting that gen (PLE]) depends only on the cardinality n of E, we write

gen (P, n) = gen (PIEI) for any set E of cardinality n. We call this the n-th polynomial coefficient of the polynomial species P.

Thegeneratingfiction of a polynomial species P is the element

of the algebra of formal power series in the variable z over Z[[X]].

PROPOSITION 1 . Let P be a symmetric species. Then the polynomial gen (P, n) is Symmetric, in other words gen (P, n) is invariant for any bijection of X.

EXAMPLE 1 . (cont. d.), The polynomial of the elementary symmetric species E is

gen(E,n) = n! y7 xi xi . . . xi, = n! e n ( x l , . , ,) L L ’ i , < ... < i n

that is, except for the factor n!, it is the n-th elementary symmetric function. rn

EXAMPLE 2 . (cont. d.). The polynomial of the power sum species S is:

gen (s, n) = x x: i

that is the powersum symmetric function s,

EXAMPLE 3 . (cont. d,). The polynomial of the disposition species H is:

gen (H, n) = n! h,(xl , . . J

with

h,(xl , . . .) = x xi, . . . xi, i, G ... G i n

.

that is h,, is the elementary homogeneous function of degree n

EXAMPLE 4 . (cont. d.). The polynomial of the monomial elementary species Kh of class X is:

where p(n, A) is the number of the partitions of class A on E, k, is equal to Zn:I . . . q?, and the sum ranges over all distinct monomials.

Except for a coefficient, gen ( K h , n) is a monomial elementary symmetric function w

EXAMPLE 5 . (cont. d.). The polynomial of the cyclic species Cis

gen(K, , n ) = p(n, A) r l ! r2! . . . k,

and the generating function of the polynomial species C is:

Symmetric Functions and Symmetric Species 1 1 1

EXAMPLE 6. (cont. d.) The polynomial of the species Hx is:

gen(H, ,n)=n!x" Thus

1 Gen(Hx z) = -

1 - xz

Sum, product and exponential of polynomial species are defined as in Joyal.

We recall the definition of product and exponential for ordinary species.

Let Pi be a species on Pol (Mi, I*) (i = 1 , 2 ) . We let the product P, x P, be the polynomial species on Pol (M, x M, , I*) such that:

I I s = (s l , s2) and f l E l = f , , f I E 2 = f,

where E, + E2 = E ranges the set of all 2-scomposition of E.

PROPOSITION 2.

Gen (P, x P2,z) = Gen (P, , z) . Gen (P2. 2).

Let P be a polynomial species on Pol (M, I*) without constant term (i.e. P[@] = 4) and let i~ = ( B , , . . . , B, } be a partition of E in k blocks. An assembly on E of order k of species P is defined as the set of all pairs (s, f) such that, if si represents the structure induced by s on Bi, we have

(si, f/Bi) E P [Bi], for i = 1, . . . , k.

The species of assemblies of species Pof order k is the polynomial species Expk(P) on Pol (Exp,(M), I*) defined as follows:

Expk(P) [El = the set of all polynomial assemblies of species P on E of order k,where the partition ranges

over the set of all partitions of E into k blocks.

The species of fhe assemblies of species P is the polynomial species on POI (Exp(M), I*) defmed by:

PROPOSITION 3. (Theorem of the assemblies)

Gen (Exp (P), z) =

For present purposes, a family { Px ! x E

conditions are satisfied:

(i) Px C Pol (M, I*);

(ii) P,[@J] ={ ( s , f :@-X)J,sEM[$];

(iii)

of polynomial species will be called multipliable if the following

(s, f) E Px[E] iff f(E) = { x}.

1 1 2 F. Bonetti et al.

If Comp (C) denotes the complement of the cofinite set C and Pc = r i p W

o f t h e family { P x } x e x as the lim Pc.

we define the product xEComp(C1

C

PROPOSlTlON 4.

Gen (lim P,, z) = lim (Gen (P, 2)) C C

EXAMPLE. There exists a natural transformation from the species H of the disposition to the product species of the family { H, I,, x .

As a consequence we obtain the classical identity:

1 h,z". 11 -=

W€X 1 - - z n,O

3. A PROOF OF WARING'S FORMULA

PROPOSlTION 5 . There is a natural transformation between the species H of dispositions and the species Exp (C) of assemblies of cyclic species, i.e.

H = Exp (C)

Proof.

The following algorithm gives a canonical bijection between dispositions over a finite set E and the cyclic species assemblies on the same finite set.

STEP 1 .

Let ( ( u i } , f) E H[E] be a disposition on E. We shall write pi for to denote the cycles of ui and c(( the supports of these cycles. The assembly of associated. cyclic species is obtained in the following way: the partition of E is that in which the blocks are the Fj, on each block the cyclic permutation p/ and the function f/-. with constant value xi are defined.

STEP 2.

Given an assembly of cyclic species (s, f) relative to a partition II = { T I , . . , , F,,,. More explicitly, on each block i7. a cyclic permutation vj and a constant function are defined. The disposition associated with (s, n i i thc fair ({(I%}, f) with ux a permutation whose cycles are those v j on the supported of which the fiiriction f assumes the constant value.

4

As a special case of propositions 3 and 5 we have the following:

THEOREM (Waring's formula)

REFERENCES

[ 1 ] Aigner. M.. Coinbinatorial theory. Springer-Verlag. New York (1979). [ 2 ] B0urbaki.N.. Element de Mathimatique: A l p d m Commutative, Hermann, Paris (1965). 1-71 Coiiitrt,L., Advaiccd Combinatorics. Reidel. Dordrecht.Holland, Boston (1974).

Symmetric Functions and Symmetric Species I I3

[4] Doubilet, P., On the foundations of Combinatorid theory. VII: Symmetric functions through the theory of Di-

[5] Cratzer,G., Universal Algebra,D. Van Nostrand,Princeton,NJ. (1968). [6] Joyal,A., Une th6orie Combinatoire des &ries formelles, Adv. in Math. v. 42 (1981). [7] MacDonaldJG., Symmetric functions and Hall polynomials, Clarendon Press, Oxford 1979. [8 ] Metropolis,N., Rota,G.C., Witt vectors and the algebra of necklace, Adv. in Math. vol. 50 (1983). [9] Rota,G.C., Baxter algebras and combinatorial identities I e 11. Bull. Amer. Math. Soc. (1969).

stribution and Occupancy, Studies in App. Math. Vol. 51 (1972).

[ lo] Rota,G.C., Finite operator Calculus, Academic Press, Inc. (1975).

Annals of Discrete Mathematics 30 (1986) 115-124 @ EIsevier Science Publishers B.V. (North-Holland)

ON 'THICK (Q+2)-SETS

Rita Capodaglio Di Cocco

Universita' di Bologna

Summary: Un k-insieme K di un piano proiettivo finito viene detto denso se da ogni punto del piano esce almeno una s-secante di K, con s>l. Qui si studiano alcune proprieta' dei (q+2)-insiemi densi di un piano proiettivo d'ordine dispari.

INTRODUCTION

About forty years ago, Bose [ 5 \ and Qvist 1201 introduced some subsets of a finite projective plane, called " non collinear systems of points". B. Segre and his school studied the same subsets, renamed k-arcs, and found very important results about them. In particular, if C is a k-arc of the plane PG(2,q) with q odd, we recall (see [28] p. 270-298) :

2) if k=q+l, then C is a conic 3 ) a q-arc cannot be complete (this proposition is still true if q is even [ 3 3 ] , while it is wrong [21 in a non-desarguesian plane). 4) if k=q-c, where c>O, and is "large enough relative to c", then every k-arc i s contained in a conic, 5) if C is complete, then its secants fill up the plane, i.e. each point of the plane is contained in at least one secant of C.

1) k<q+2

There arises the problem: for which values of k and q do complete k-arcs in PG(2,q) exist? As an answer, L. Lombard0 Radice constructed complete (q+5)/2 -arcs,

and Pellegrino 1191 showed that if q constructed complete [ k-arcs] with then complete k-arcs with k <(q+1)/2 exist. Among the k-arcs which are not necessarly complete but are not contained in a conic, we recall the k-arcs contained in a cubic (see Di Comite [ 1 3 ] , 1141, 1151 and Zirilli [37] ) .

It easy to see that the order of the known complete k-arcs is very "large" with regard to the theoretical valuation k > ( 3 + 1 / 1 t U q ) / 2 (see 1281) and we are still a good way from finding a solution of the following problem: I f q is fixed, what is the smallest number k, such that a complete k-arc of PG(2,q) exists?

-

116 R. Capodaglio di Cocco

Recently, in order to obtain new resul.ts, the definition of k-arc has been generalized as follows (see [3]).

DEFINITION 1: A k-set K of, a finite projective plane 9 (not necessary desarguesian) is called thick if V P E 9 there exists a s-secant of K passing through P with s>l.

DEFINITION 2: A thick k-set K is called minimal if every proper subset of K is not thick,

Obviously every complete k-arc is a minimal thick k-set, but the next examples show that the converse is wrong: 1) Let r and s be distinct lines of a projective plane of order q > 5 . If P=rns, let AlrA2,...,Ac., be points of r and F , ,B2 be points of s , with Ai:bPkBi (i=1,~,...,q-1;j=1,2~. Then the set K = {A11A21 . . . IAq- l ,B , ,B2} is a minimal thick (q+l)-set. 2) Let r,s,t be three non-concurrent lines of Pc(2,q). If we embed PG(2,q) in PG(2,q2), the set of the points of r,s,t is a minimal 3q-set of PG(2,q2). (E. Ughi's example).

In the sequel, we will be interested in the minimal thick k-sets for which k takes the maximal value, i.e. qt2 (see [3] ) .

We point out that minimal thick (qt2)-sets exist: a (qt2,qtl)-arc is a trivial example of such sets. Moreover we will show that a minimal thick (q+2)-set can be represented by a permutation polynomial, and so the study of these sets is connected to a subject to which many important papers have been devoted (see [ 6 1 r [8I [91 I101 i r [12I).

DEFINITION 3: A point N of a k-set K is called a nucleus of K if every line through N is a s-secant of K with s<3.

A (qt2)-set with a nucleus is obviously thick, but not necessarly minimal: for example let q=ph, with p*2,3 odd and qrl (mod. 3). In PG(2,q) assume r is an irreducible cubic with an isolated double point N. It is easy to see that r is a non-minimal (qt2)-set with nucleus N. Moreover if q=5, and F is a point of inflection of r ,then r- {F} is a minimal thick 6-set. Remark: Irreducible cubics with an isolated double point are used in [IS] to construct (qt9)/2-arcs. So it seems that the following problems are the most important in the theory of minimal thick (qt21-sets: I: Has every minimal thick (q+2)-set one and only one nucleus? 11: For which number n is a minimal thick (qt21-set K a (qt2,n)-arc? 111: When is a (qt2)-set K with a nucleus minimal? In the following we suppose that the order of the plane is odd and we give partial answers to these problems. In particular in: problem I: we show that a minimal thick (qt2)-set has at most one nucleus, Moreover, if the plane is PG(2,q), we find conditions for the nucleus to exist.

On Thick (Q+2)-Sets I17

problem 11: we show that if K has a nucleus, then either n=q+l or n<q-1. Moreover if the plane is PG(Z,q), then 4535qq/3. problem 111: we find necessary conditions for K to be a minimal (q+2)-set of PG(2,q).

1

Let 9 be any finite projective plane of odd order q.

DEFINITION 4 : A point A of a thick k-set K is called essential if the set K - { A ] is not thick.

Obviously, a thick k-set K is minimal if and only if every point o f K is essential. Suppose now that K is a minimal thick (q+2)-set, then for each X E K there esists at least a point Px such that every line through Px is an 1-secant for K - { X } . If X is not a nucleus o f K , then P,FX, whereas if K has a nucleus N and X#N, then any point Px is on the line NX. In this section we are interested in problem I, first we shall show that a minimal thick (q+2)-set has at most one nucleus.

THEOREM 1: If a (q+Z)-set K has two distinct nuclei, then it is not minimal. Proof: The plane Y contains no (q++arc, s o K has at least a s-secant r with s > 2 . Then every point A of K n r is not essential.

Corollary; If the minimalqhick (q+2)-set K has a nucleus N and X E K , X Z N , then there exists at least one s-secant r3X of K with s > 2 .

REMARK: We point out that some (q+2)-sets with two nuclei exist. In fact, if N1 and N are any two distinct points and r is the line N N , let ,ti differen$ from k, moreover let is easy to prove that K= { snf(s); sEgl}u { N1, N2 ] is a thick (q+Z)-set and Nl, N2 are nuclei of K . So the number of the (q+2)-sets with nuclei N1,N2 is equal to the number of the bijections from El to , i.e. q!. On the other hand , if .iP = =PG(Z,q) , in[4] it is siown that a (q+2)-set can have more than two nuclei only if q is even. Now we find conditions for a minimal thick (q+2)-set to have a nucleus.

2 (i=1,2) be the set o f the lines through N. and 1 1 f: E l + E 2 be any bijection. It

THEOREM 2 : Let K be any minimal thick (q+2)-set o f PG(Z,q), with q odd; then K contains two points C and D, such that, if the frame is conveniently chosen, it is possible to represent the set W = K - { C , D )

118 R. Capodaglio di COCCO

by an equation y=f(x), where f(x) is a permutation polynomial with 1) the polynomial f(x)-x has no root in G F ( q ) . Moreover if K has a nucleus, we have 2)vm€GF(q), m-1-1, the polynomial f(x)-mx has only one root in GF(q). Proof : In the first place we suppose that K has no nucleus and we define an application r:K--->PG(2,q) in the following way: we choose a point X l € K and we pose r ( X 1 ) = P where Pq is one of the above-stated points. The line X , C ( X l f l 'intersects K at XI and at an other point, say X 2 . We pose z ( X 2 ) = Z ( X l ) . Then we choose a point X 3 f X 1 , X 2 and. we call z ( X 3 ) one of the points P The line X 3 r ( X 3 ) intersects K at X 3 and at an Other point, say X 4 . If either X 4 = X 1 or X 4 = X 2 , we have nothing to define; if X 4 k X 1 1 X 2 we pose r i X 4 ) = r ( X 3 ) and so on. Since qt2 is odd, there must exist at least two points A and B of K such that the distinct lines A r ( A ) and B r ; B ) intersect K at the same point C, because otherwise the set K would have a partition in disjoint pairs. Obviously z ( A ) , r ( B ) and C are not collinear, so we can choose r ( A ) as the improper point of the axis x, r ( B ) as the improper point of the axis y and C as the point ( 0 , O ) . Let D be the only point of K on the improper line, then, using the terminology of 1311, the set W=K-(C,D} is a diagram relative both to z ( A ) and to z(B) and so it can be regresented by an equation y=f(x), where f(x) is a permutation polynomial. If we choose the point (1,l) on the line CD, we obtain the cond. 1). Now let K have a nucleus N and A,B be distinct points of K, with AcN-fB. We choose a point PA (resp. a point PB) as the improper point of the axis x (resp. of the axis y) and we call D the only improper point of K. If we pose C=N, w e can repeat the above proof. The cond, 2 ) is satisfied, because C is the nucleus of K.

x3'

THEOREM 3 : In P G ( Z , q ) , with q odd,let W be the set represented by the equation y=f(x), where f(x) is a permutation polynomial with 1) the polynomial f(x)-x has no root in GF(q) 2 ) vmEGF(q), mtl, the polynomial f(x)-mx has only one root in GF(q). If C is the point ( 0 , O ) and D is the improper point of the line y=x, then the set K=Wu{C,D} is a (qt2)-set with nucleus C. Proof: Self evident.

I1

Now we shall deal with problem 11. In the plane 9 let K be a (qt2)-set with a nucleus N. In conformity with the terminology of ( 3 6 1 , K is a (q+2,n)-arc for a convenient number n, and it has at least three characters, because i t has s-secants for s=1,2,n. Let t be a line which intersects K exactly in the n points B1,

B 2 t . . . 10, .

On Thick lQ+2)-Sets 119

THEOREM 4: Suppose the minimal (qt2)-set K has a nucleus N, then either n=q+l or n<(q-l) and so q > 4 . Proof: If n=qtl, we have nothing to prove. Suppose n=q; if C ~t is

point different from Bl,B2,. . . ,Bnr we have line

but this means that A is another nucleus for K, in contrast with th. 1. Suppose n=q-1; if C1 and C2 a r the points of t different from B l r B 2 , . . . ,Bn, we have K= N,B1 ,B?, . . . ,Bn,A1 , A # where A , is a convenient point of NC, (i=l,2), Since A1 cannot be a nucleus, the line A,A2 must pass through one of the points B ~ : but this is impossible because this point would be not essential. So we have n<q-1.

Ifp=PG(2,q), the above result is improved by the next theorem and its corollary.

N,B1 ,B2,...,BnrA} , where A is a convenient point of the

THEOREM 5 : Suppose K is a minimal thick set of PG(2,q) with a nucleus N . Then any line A1 A2, Ai E K, A, 3:N , A1,/zA2, is at least a s-secant df K with s>3. Proof: Let A1kN:bA2 be two disinct points of K. Let PA, be the point of the line NA, definied in the section I (i=l,2). For the sake of brevity we write P, instead of P ~ ~ . A f t e r a B. Segre's scheme of proof (see 1281 ) , we chose P2 ,PI ,N as the fundamental triangle of a homogeneus coordinate system in PG(2,q). The line PIP2 contains only one point of K, say A3. Let U=P2A1 n N A 3 ; we choose U as (l,l,l), so we have A l = ~ O , l , l ~ , A3=(l,1,0) and A2=(l,Ora) with a.kO. For each point CEK, C#.N,AlrA2,A3, the lines NC, PIC, Pf are represented respectively by the equations x1 =m2x0, x ~ = ~ ~ x ~ , x2=mOx1 with m,-=mo m2 If we consider all the points C of K, C=FN,Al,A2,A3, we obtain that m 2 and mo take all the values of GF(q) different from 0 and 1, while m 1 takes all the values of GF(q) different from 0 and a. Since the product of all non-zero elements of GF(q) is equal to -1, we have a=-1. This means that the points A1,A2,A3 are collinear . Starting from the points A1 and A3 and putting

the only point of K ne A1A3 . If this intersects K in at

intersection of K A 1 and lies on the

P=P by the above arguments vhich is on the line PI P3 is a point is distinct from A2, then least four points; otherwise the with the line P2P3, is certainly line AIA2.

3 $3 ' , we have that so on the 1 the line A1A2 point A4, the distinct from

Corollary: Suppose the minimal thick set K of PG(2,q) has a nucleus N, then K is a (qt2,n)-arc with either n=qtl or 4<n<q/3. Proof: If n<qtl, let the line t intersect K exactly in the n points B1,Bpr . . . , Bn, and let A E K , APN, be not on t. By th. 5, each line ABj (j=l,Z ,...,n) contains at least three points of K different from A and N . So 3 n Q .

I11

We need a new definition. Let q be an odd prime power. It is known [12] that a permutation polynomial for GF(q) is in reduced form if its degree is <q, otherwise it is called "crude". Let @ be the se't of all permutation polynomials for GF(q) in reduced form and let be the set of all permutations of GF(q). Then there exists a well-known bijection a : @ --->q , For the sake of brevity we put U(f(x))=vf.

DEFINITION 5: A permutation polynomial is called pseudolinear if there exists a group d , subgroup of AGL(l,q), transitive on the elements of GF(q), except for at the most one element, and such

1 that V d e A the permutation polynomial a-'( vf 6 v ; ) is of the first degree.

All the permutation polynomials xk , with GCD(k,q-l)=l, are pseudolinear because for them we can choose A = { x ---> ax; a E G F ( q ) , a-kO}. On the contrary,the polynomial corresponding to the permutation x--->x if'x-l.0~1; 0--->1:1--->0 is not pseudolinear.

THEOREM 6: In PG(2,q) let H be the set represented by the equation y=f(x), where f(x) is a pseudolinear permutation polynomial and assume n is the group of the affinities which map H onto itself. Then 17 is transitive on H, except for at the most one point. Proof : It is sufficient to point out that I1 cointains all the affinities x= a - $ 6 ) (xl ) ly= a- ( vf 6 v;' )(y') where ~ E A .

Now we treat the problem 111. Suppose the (q+2)-set K of PG(2,q) has a nucleus N. We choose a coordinate system different from the one we used in the proof of th. 2. Let A k N be a point of K, then we choose A N as the improper line and N as the improper point of the axis y . Since the set H=X-{N,A} is a diagram relative to N, so it can be represented by an equation ( 1 ) y=f(x) where f(x) is a polynomial.

THEOREM 7 : Suppose K is minimal, then it is possible to choose the improper point of the axis x such that f(x) is a permutation polynomial, Proof: self-evident.

On Thick (Q+2)-Sets 121

If in (1) f(x) is a pseudolinear permutation polynomial, then, by th. 6 , the group n of the affinities that map H onto itself is not trivial. Now we suppose K is a (q+2,n)-arc with ncq+l and we prove that if some conditions are satisfied, then K is not minimal.

THEOREM 8: Let f(x) be a pseudolinear permutation polynomial. Assume 17 contains a subgroup 17, of the group of the translations and nq is transitive on H, then A and N are the only essential points of K . Proof: Let B be any point of H and t E 0 be a s-secant of K with s>l; assume T is the improper point of t. Since n<q+l, then there exists Q such that o(t)=t'+t. Obviously t' passes through T and it is likewise a s-secant of K . On the other hand, since any affine line through T is either external or s-secant of K , then no 1-secant of K passes through T. Let now Q be any point of the line BN: since the line QT is not a 1-secant of K - { B ) - , then the point B is not essential.

THEOREM 9 : Let f(x) be a pseudolinear permutation polynomial. Assume any element of n fixes the improper points of the axes and a proper point 0 (obviously 0 is on H ) ; moreover assume 17 is transitive on the points of H different from 0, then 0 is not essential. Proof: Let X be the improper point of the axis x; since the line XO is an 1-secant of K, then there exists at least a line 230 that is a s-secant of K with s>2. The line r is a m-secant of H, where m=s-1 if A is on r, m=s otherwise. Since is transitive on H - { O } , then the q points of H are shared among d lines which pass through 0 and are m-secant H , with obviously q-l=d(m-l). It is easy to see that is transitive on the lines passing through X and different from XN and XO, and therefore also on the points of the line ON different from 0 and N. Suppose the line ON contains a point Q:bO,N such that any line u3Q is t-secant H with t<2, then the same is true for every point R of ON (RI0,N). But evidently this is absurd, So the point 0 is not essential.

EXAMPLES :In PG(Z,q), with q=phodd, let N and A be respectively the improper point of the axis y and the improper point of the line y=x. Moreover let he the set represented by the equation

where c i EGF(P) and GCD(cotclx +. . .+ck1xh-l,1-xh )=l. y=f (x )'COX

It is known [12] that f(x) pseudolinear because we can K=Hu{A,N}. Since the

xpt. . . tch- xp %-I

satisfied, no point of H is essential for K . 2) Let A,N,K have the same meaning d s in l), while H is the set represented by the equation

where GCD(k,q-l)=l. As we have already proved, x k is a pseudolinear permutation polynomial. In this case the conditions

y = x k


required by th. 9 are satisfied, so the point ( 0 , O ) of H is not essential for K.

LITERATURE

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161 A.Bruen: Permutation functions on a finite field. Canad. Math.Bull.15 (1972)

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[8] L.Carlitz:Permutations in a finite field. Proc.Amer.Math. Soc.4 (1953) 538.

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[Ill L.Carlitz: Some theorems on permutation polynomials. Bull, Amer.Math. 68 (1962) 120-122.

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Math-Sze-

ane. Rend.

On Thick (Q+Zl-Sets 123

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[27] B.Segre: Le geometrie di Galois. Ann.Mat.Pura Appl. 48 (1959) 1-96

[ L 8 ] B.Segre: Lectures on modern Geometry. Cremonese,Roma 1961

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[33] G.Tallin teristica p=2

Galois caratterizzante quelli formati dalle singole ad una conica,Rend.Acc.Naz.Lincei 62 (1977) 613-618.

: Sui q-archi di un piano lineare finito di carat- Rend.Acc.Naz.Lince1 23 (1957) 242-245.

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Annals of Discrete Mathematics 30(1986) 125-136 0 Elsevier Science Publishers B.V. (NorthHolland) 125

ON A GENERALIZATION OF INJECTION GEOMETRIES

P i e r V i t t o r i o Ceccher in i and N a t a l i n a Venanzangeli D ipar t imento d i Matematica "G. Castelnuovo"

U n i v e r s i t a d i Roma "La Sapienza" C i t t a U n i v e r s i t a r i a , 00100 Roma, I t a l y

I n j e c t i o n geometr ies have been in t roduced i n [ 3 ] as a gen- e r a l i z a t i o n o f permuta t ion geometr ies s t u d i e d in [ 1 1 . We presen t a g e n e r a l i z a t i o n o f i n j e c t i o n geometries, namely 7-geometr ies, improv ing i n some cases p r o p e r t i e s o f i n j e c - t i o n geometr ies s t a t e d i n [ 3 ] . 7-geometr ies have been in t roduced i n [ 9 1, and a l s o prean- nounced i n [ 3 1 , i n a "Concluding remark" unknown t o t h e au thors , F g e o m e t r i e s have been r e c e n t l y cons idered a l s o i n [ 4 1, under t h e name o f "squashed geometr ies" and i n [ 6 1 , [ 8 ] .

1. INTRODUCTION

I n what 2

N = N x N .

= I ( x , y ) EN 2

G 1 = I g l (a )

f o l l o w s a l l s e t s w i l l be f i n i t e . L e t N be a non empty s e t and

f aEN, we ge t t h e genera tors g ( a ) = { (x ,Y)E N2 : x = a) and g ( a ) =

: y = a]; t h e f i r s t ( resp. second) system o f genera tors i s 1 2

aEN1 ( resp . G = g2(a) : a € N 1. A ( p a r t i a l ) correspondence o f N 2

z i s any subset o f N ; a p a r t i a l a p p l i c a t i o n ( resp . c o a p p l i c a t i o n ) o f N i s any sub-

s e t o f N which i s 0- o r 1-secant each g E G ( resp . g E G ) ; a subpermutat ion F

o f N i s any subset o f N2 which i s 0- o r 1-secant each genera tor ; i f dom F = N,

then F i s a permuta t ion o f N. I n [ 7 1 s e t s and groups o f permuta t ions o f N a re

s t u d i e d ( w i t h s p e c i a l a t t e n t i o n t o t r a n s i t i v i t y p r o p e r t i e s ) f rom t h i s geomet r ica l

p o i n t o f wiew. I n [ 2 ] c e r t a i n semigroups o f subpermutat ions a re cha rac te r i zed i n

t h e c l a s s o f a l l semigroups. In [ 1 1 spec ia l se ts o f subpermutat ions o f N (namely

permuta t ion geometr ies, c f . no. 2) a r e i n t roduced by means o f axioms s i m i l a r t o

ma t ro id axioms; more genera l r e s u l t s a r e ob ta ined i n a s i m i l a r way i n ( 5 1 , where

se ts o f p a r t i a l a p p l i c a t i o n s ( o r c o a p p l i c a t i o n s ) i n s t e a d o f subpermutat ions a r e

considered. Other g e n e r a l i z a t i o n s o f permuta t ion geometr ies a re i n j e c t i o n geome-

t r i e s , c f . [31. I f d a l i s an i n t e g e r , i n t h e s e t N we have d systems o f genera-

tors :

2 1 1 2 2

d -

126 P. V . Ceccherini and N. Venanzangeli

d G . = t g . ( a ) : a e N ) , where g . ( a ) = I ( x l ,..., x d ) e N

a subset o f N i s c a l l e d i n j e c t i v e i f i t i s 0- o r 1-secant each genera tor ; an

i n j e c t i o n geometry i s a s e t o f i n j e c t i v e se ts o f N s a t i s f y i n g s u i t a b l e axioms,

which a r e s i m i l a r t o ma t ro id axioms and which reduce t o those f o r d = 1 and t o

permuta t ion geometry axioms f o r d = 2.

: x . = a1 ( i = l y . . . y d ) ; 1 1 1 1

d .

d

Th is t a l k concerns a g e n e r a l i z a t i o n o f i n j e c t i o n geometries (namely 9-geo-

me t r i es ) g iven i n [ 9 ] i n a very a b s t r a c t way which i nc ludes a l s o p a r t i a l app l i ca -

t i o n ( o r c o a p p l i c a t i o n ) geometries ( c f . no. 2 ) .

P r o f . M. Deza in fo rmed us d u r i n g t h i s conference t h a t t h e concept o f 9-geo-

me t r i es i s c la imed i n a conc lud ing remark (added i n p r o o f s ) i n [31 and t h a t i t

i s going t o be developed i n [ 4 1 , where our 3-geometr ies a re c a l l e d squashed geo-

me t r i es . A d i f f e r e n t and ve ry e legant approach t o squashed geometr ies as "bou-

quets o f mat ro ids" i s g i ven i n [81, where i n s t e a d o f our s e t 7 an a n t i c h a i n C i s

cons idered s a t i s f y i n g s u i t a b l e axioms ( i n our language C i s t h e s e t o f maximal

elements o f A ) ; t h a t approach seems t o be very e f f i c i e n t .

I n what f o l l o w s we ske tch a theo ry o f 9-geometr ies, improv ing i n some cases

p r o p e r t i e s o f i n j e c t i o n geometries s t a t e d i n [31.

2. DEFINITION OF 9-GEOMETRIES

i o f

such

A ' 3 -

Let us s t a r t w i t h t h e f a m i l i a r concept o f mat ro id .

DEFINITION 2.1. A ma t ro id Mr(X), o r rank r on se t X, i s a p a i r Mr(X) = (X,A)

where A i s a se t o f subsets o f X, p a r t i t i o n e d i n t o A = A,,u... UAr w i t h Ar # 0,

s a t i s f y i n g t h e f o l l o w i n g axioms ( t h e elements A .EA. a re c a l l e d t h e f l a t s o f rank 1 1

M ( X ) , O _ A iu Ib1 ; moreover A

AU [ b l .

We r e c a l l now t h e d e f i n i t i o n s o f t h e s t r u c t u r e s mentioned i n $ 1 .

DEFINITION 2.2 ( [ 1 1 ) . L e t N be a non empty s e t and X - c N x N. A permuta t ion

r

then t h e r e e x i s t s an un ique AitlE Aitl i'

i t1 i s i n c l u d e d i n each A ' E A such t h a t

On a Generalization offnjection Geometries 127

geometry P (X) o f rank r on X, i s a p a i r P ( X ) = (X ,A) where A i s a s e t o f subper-

w i t h A # 0, s a t i s f y i n g t h e muta t ions o f N, p a r t i t i o n e d i n t o A = A o u ... axioms ( 1 ) - ( 3 ) w i t h the r e s t r i c t i o n t h a t axiom ( 3 ) ho lds f o r those b E X \ A . such

t h a t AiU t b l i s a subpermutat ion of N. (Note t h a t t h e o r i g i n a l d e f i n i t i o n g i ven

i n [ 1 1 concerns t h e p a r t i c u l a r case X = N ; 1.e. o u r d e f i n i t i o n i s s l i g h t l y more

general 1.

r r

UAr r

1

2 .

DEFINITION 2.3 ([5]). L e t N be a non empty s e t and X - C N x N. A p a r t i a l ap-

p l i c a t i o n ( resp . c o a p p l i c a t i o n ) geometry o f rank r on X i s a p a i r ( X , A ) , where

A i s a s e t o f p a r t i a l a p p l i c a t i o n s ( resp . c o a p p l i c a t i o n s ) o f N, p a r t i t i o n e d i n t o

A = A, U... U A wi th A # 0, s a t i s f y i n g t h e axioms ( 1 ) - ( 3 ) w i t h t h e r e s t r i c -

t i o n t h a t axiom ( 3 ) ho lds f o r those b E X \ A . such t h a t A . u i b 1 i s a p a r t i a l a p p l i -

c a t i o n ( resp . c o a p p l i c a t i o n ) o f N.

r r

1 1

d DEFINITION 2.4 “ 3 1 ) . L e t d > 1 be an i n t e g e r , N a non empty s e t and Xc - N .

An i n j e c t i o n geometry o f rank r on X i s a p a i r I ( X I = (X,A) where A i s a s e t o f d

i n j e c t i v e subsets o f N , p a r t i t i o n e d i n t o A A , U ... U A w i t h Ar # 0, s a t i s f y - r

r the r e s t r i c t i o n t h a t axiom ( 3 ) ho lds f o r those b E X \ Ai

i n j e c t i v e subset o f N . The number d w i l l be c a l l e d t h e d i n g axioms ( 1 ) - ( 3 ) w i t h

such t h a t A . u t b l i s an

dimension o f I r ( X ) . 1

We g i v e now t h e de i n i t i o n o f 3 -geomet r ies .

DEFINITION 2.5 ( c f . [9], and a l s o [ 3 ] , [ 4 ] , [8] where t h e d e f i n i t i o n i s g i ven

i n a s l i g h t l y d i f f e r e n t fo rm) . An 9-geometry o f rank r on a s e t X, i s a quadruple

G ( X ) = (S,3,X,A) where S i s a non empty set , I i s a s i m p l i c i a 1 complex o f d i s t i n -

guished subsets o f S ( i . e . Z C Z ’ E ~ i m p l i e s Z E I ) , A i s a subset o f 9 p a r t i t i o n e d

i n t o A = A,u ... u A w i t h A # 0 and X = u A, s a t i s f y i n g t h e axioms ( 1 ) - ( 3 )

w i t h t h e r e s t r i c t i o n t h a t axiom ( 3 ) ho lds f o r those b € X \ Ai such t h a t

A . U t b l E 3.

r

r r AEA

1

The elements AiE Ai a r e c a l l e d t h e f l a t s o f rank i o f t h e geometry G (X I .

We s h a l l w r i t e A

A ma t ro id M (X) i s a geometry Gr(X) = (S,Y,X,A) w i t h 9 = 2 and X = S, and

r = A. v i b l f o r t h e s e t Ai+l mentioned i n axiom ( 3 ) .

S i + l 1

2 r

converse ly . A permuta t ion geometry Pr(X) = (X,A), w i t h XcN , i s a geometry Gr(X) =

= (S,3,X,A) w i t h S = N and 3 = IF - c N2 : F i s a subpermutat ion o f N l , and conver-

se l y . P a r t i a l a p p l i c a t i o n ( resp . c o a p p l i c a t i o n ) geometr ies can be e a s i l y charac te-

2

128 P. V, Ceccherini and N. Venanzangeli

r i z e d i n a s i m i l a r way between %geometries. An i n j e c t i o n geometry I r (X ) = (X,d)

w i t h XsNd i s a geometry Gr(X) = (S,Y,X,A) w i t h S = N and 9 = I F 2 Nd : F i s

i n j e c t i v e ) , and conversely. Several examples o f geometries G ( X I can then be dedx

ced f rom [ l ] , [5 ] , [ 3 ] where examples o f permutat ion geometries and o f i n j e c t i o n

geometries are given. We now g i v e some o the r examples.

EXAMPLE 2.6. Free 9-geometries. The f r e e geometry Gr(X) = (S,9,X,A)

d

r

i s

= t A E 9 : ( A ( = il, def ined by assuming X = S, 9 a s i m p l i c i a 1 complex o f S, A

O < i < r , r a l . i

EXAMPLE 2.7. S ta r 9-geometries. A s t a r geometry G (X) = (S,9,X,A) w i t h 1

center C i s def ined by assuming CCS, A, = t C l , A , = t D l,...,Dt where C C D . c S

and D.n 0 . = C ( t a 2 , i,j=l ,..., t, i# j ) , X = Di, 5’= u 2Di. 1 J i =1

J t t

REMARK 2.8. S ta r %geometries are 9-geometr ies o f rank 1 w i t h IA I > 1 and 1

conversely ( c f . Theorem 5 . 3 ) .

EXAMPLE 2.9. Truncat ions o f an 9-geometry. I f Gr(X) = (S,9,X,A) i s an 9-geo -

metry w i t h A = A , U ... u A and i f 1 6 k < r , we can consider A ( k ) = A , u . . . UAk. r

) i s an 9-geometry o f rank k, c a l l e d t h e k - t r u n c a t i o n o f ( k ) . Then Gr l k ) = (S,9,X,A

Gr’

EXAMPLE 2.10. Cotruncat ion o f an 9-geometry. I f Gr(X) = (S,S,X,A) i s an

9-geometry and i f Ah€ Ah (06 h c r ) i s i nc luded i n some Are Ar, then we can

consider 2 = g, (A,,) = tAhti€Ah+i : Ah c_ Ahti, i = O , l , . . . , r -h ] . Then G ( A 1 =

= (S,g,X,A) w i t h

G (A,) = Gr, c f . 5.1 ( a ) .

i i r h = i, U... ~ f i ~ - ~ , i s an 9-geometry o f rank r -h , Note t h a t

r

EXAMPLE 2.11. B i t runca t ions o f an 9-geometry. I f Gr = (S,Y,X,A) i s an 9-geo -

metry and i f A h € Ah, w i t h O < h c k c r and A h c A f o r some Ak€Ak then we can con- k s i d e r Ji = i . ( A ) = {Ahti E Ahti : AhCAh+i, i=O, ..., k-h}. Then G = (S,Y,X,#) w i t h

A = A, u.. . uA;(-~ i s an 9-geometry o f rank k-h. i h - -

EXAMPLE 2.12. I n t e r v a l s o f an 9-geometry. I f Gr = (S,g,X,A) i s an %geometry

and i f Ah€ Ah, Ak€ Ak w i t h Ahc Ak, O d h < k < r, then we can consider Ai(A A ) =

= {Ahti E Ahti : AhL AhtiCAk, i = O ,..., k - h l . Then i i , u . . . u f i

o f rank k-h on Ah.

h’ k g ives a ma t ro id

k -h

On a Generalization of Injection Geometries 129

EXAMPLE 2.13. R e s t r i c t i o n s o f J-geometries. l e t Gr(X) = (S,9,X,A) be an 9-

Z A ' , r ' = max t i : A i E A ' l . Then

geometry and l e t be S ' such t h a t A € S ' C X f o r some A E d Def ine A' = 1 1 1'

= tAEA: AcSIl, X I = u ,A, 9' =

Gr , (X ' ) = (S'J',X',A') i s an 3-geometry o f rank r ' on X ' .

B - c A such t h a t D n A l # 0; i n t h i s case r ' = max l i :d in B f 01,

A' = t A E A : ACE f o r some B C B } and 9' =

LJ A€ A A ' €4 '

A spec ia l i n t e r e s t i n g case i s obta ined when S 1 = X ' = U A f o r some f i x e d AE B

2A. A € D

3. D I R E C T SUMS OF 9-GEOMETRIES

By s t a r t i n g f rom two g i ven 9-geometr ies, i t i s p o s s i b l e t o c o n s t r u c t a new

one (namely t h e i r d i r e c t sum), i n t h e standard way descr ibed below.

DEFINITION 3.1. D i r e c t sum Q of 9-geometries. L e t G I r , = (S ' ,9 ' ,X1 ,d ' ) and

G",,, = ( S " , ~ " , X " , ~ " ) be two 9-geometr ies; we can suppose w i thou t l o s s o f genera-

l i t y t h a t S'n S" = 0. Assume:

A. = [ A ! , uA" : A ! , € A ! , , A!,,€ A'.',,, i ' t i " = i } ,

A = A, u.. . u A

1 1 i " 1 1 1 1

r = r ' t r". r ' Then (S,9,X,A) w i l l be c a l l e d t h e d i r e c t sum G ' 8 GI, , o f G;, and G I , , . It i s

r ' easy t o prove t h e f o l l o w i n g :

THEOREM 3.2. The d i r e c t sum G;, b G'',, o f two 9-geometries

o f rank r = r ' t r".

DEFINITION 3.3. F u l l d i r e c t sum 6 o f i n j e c t i o n geometries.

i s an 9-geometry

Le t G;,

= (S ' , 9 ' ,X ' ,A ' ) and G'',, = (S",?',X",A") be two i n j e c t i o n geometries o f t h e ~ a m e dimension d. A c t u a l l y X ' c - S ' = N o d and X " c - S" = NtId. We can suppose w i thou t l o s s

o f g e n e r a l i t y t h a t N'n N" = 0, so t h a t S'n S" = 0 . Assume: -

- d N = N ' U N " , s = N , x = x ' u x " , T = r 1 E 2 : I i s i n j e c t i v e ) ,

A. = ( A ' u A;,, : A i l € A ; , , A;,, E A;,,, i' t i" = il,

A = A, u. . .uAr ,

Then (S,?,X,A) w i l l be c a l l e d t h e f u l l d i r e c t sum cr = G;, s G I , , o f G;, and GF,,.

1 i'

r = r ' t r " .

130 P. V, Ceccherini and N. Venanrangeli

THEOREM 3.4. The f u l l d i r e c t sum GAI & GFII o f two i n j e c t i o n geometries o f d i -

mension d i s an i n j e c t i o n geometry o f rank r = r ' t r " and dimension d.

Proo f . L e t Gr = (S,5',XP)=G', B G" be t h e d i r e c t sum o f G;, and G;,, cons id -

ered as 9-geometr ies ( c f . Theorem 3.2). A c t u a l l y S = S ' U S " = N ' u N1Idc Nd = 3 and 9 C T . I t f o l l o w s immediately t h a t Er i s an i n j e c t i o n geometry o f rank

r = r ' t r " and dimension d.

d r r "

For d = 2 we have

COROLLARY 3.5. The f u l l d i r e c t sum G L I 5 G;,, o f two permuta t ion geometr ies

(cons idered as i n j e c t i o n geometr ies o f d imension d.2) i s an i n j e c t i o n geometry

o f dimension two, i . e . i t i s a permuta t ion geometry. 0

REMARK 3.6. Le t GAI and G",, be two "permuta t ion geometr ies" i n t h e r e s t r i c -

G;,I i s a permuta t ion geometry -- t e d meaning o f ( 1 1 . Then t h e f u l l d i r e c t sum GAI

( i n our meaning), b u t i t i s n o t a "permuta t ion geometry" i n t h e meaning o f [l],

because ( w i t h t h e n o t a t i o n o f D e f i n i t i o n 2.2) XcN , 2

4. REGULAR 9-GEOMETRIES

A geometry G (X) = (W,9,XyA) i s c a l l e d r e g u l a r i f each A Ed i s i nc luded i n

some A E A . Every s t a r 9-geametry i s regu la r . The f o l l o w i n g G2(X) = (S,S,X,A)

i s no t r e g u l a r :

r

r r

X = S = ta,b,c,d), A, = t A , : [ a l l , A = [ A = I a ,b l , A ' = I a , c l , A" = ta,dl.}, 1 1 1 1

A = [ A = Ia,b,c}}, A = A, u A u A 2 2 1 2'

1 A1, A;, A" , {b,c l , A21. We no te t h a t A;' i s no t i nc luded 9 = (A,, t b ) , {c}, t d l ,

i n A 2'

n o t n e c e s s a r i l y t h e f a m i l y o f t h e independent s e t s o f a m a t r o i d on S . It i s a l s o

easy t o g i v e examples o f r e g u l a r Gr(X) w i t h t h e same p roper t y (191) .

r each A. E A , , w i t h i < r , t h e r e e x i s t s b e X \ A. such t h a t A. u [ ~ } E 1.

The same example shows t h a t i f G (X) = (S ,S,X,A) i s a geometry, then 9 i s r

PROPOSITION 4.1. A geometry G (X) = (W,Y,X,A) i s r e g u l a r i f and o n l y i f f o r

1 1 1 1

Proof . Suppose G ( X I r egu la r , A. E Ai, i < r . L e t A . c A E dr. We have r 1 1 - r

A.cA s ince i < r ( c f . Prop. 5.1. ( e ) ) . I f b E A r \ A

A . W b I E 9 , s ince A.u:bl C_ A r E 9 and 7 i s a s i m p l i c i a 1 complex.

then b E X \ A i and i r i'

1 1

Conversely, i f f o r each A.E Ai, w i t h i < r. t h e r e e x i s t s b € X \ A. such t h a t 1 1

A.U I b I E 9, then A . i s i nc luded i n some ArE A : t h i s f o l l o w s by app ly ing ( 3 ) r - i

t imes. 0 1 1 r

P R O P O S I T I O N 4.2. A geometry Gr(X) i s r e g u l a r w i t h lAr l = 1 i f f Ar = [ X I .

P roo f . I f A = ( X I , then each A. E A. w i t h i < r i s i n c l u d e d i n A = X = u A

( I n t h i s case G ( X I i s e x a c t l y a m a t r o i d ) . r

r 1 1 r AEA so t h a t G

A = ( A I . I f X E X = u A, t hen XE A . f o r some A. E A , , so t h a t X E A . c A

by r e g u l a r i t y . There fore X c Ar, i .e. X = A .

i s r e g u l a r w i t h IA 1 = 1. Conversely l e t G (X) be r e g u l a r w i t h r r r

r r A E A 1 1 1 1 - r

r -

5. FIRST PROPERTIES OF .?'-GEOMETRIES

PROPOSIT ION 5.1. L e t Gr = (S,S,X,A) be an 3-geometry.

( a ) IA,1 = 1 and t h e element A,EA, i s t h e minimum o f A .

( b ) I f Ah E Ah, A E A w i t h A c A and 0 s h < k-< r. then t h e r e e x i s t s a

( c ) Each A. E A . ( O < i s r ) i s i nc luded i n a cha in A,C A C . . C A. w i t h

k k h k cha in AhC Ah+l c...c A w i t h A E A ( s = h , h t l , ..., k ) .

k s s

A E AS (s=O,l, ..., i ) .

A, c...c Ai C...C A w i t h A E A r s s

where Ai = A . i f and o n l y i f i=j.

A E A 1

then A,, C - A:. Then A, = A;; o therw ise i f bEA:\ A,,, then f rom ( 3 ) i t f o l l o w s t h a t

A, v { b } = A C A; w i t h AIEA1, c o n t r a d i c t i n g ( 2 ) .

C A w i t h Ah+l~Ahtl. I f h t l = k ,

then Ahtl = Ak by (3). I f h t l < k , then Ahtl C A and i t e r a t i o n o f t h e same argu-

ment leads t o a cha in A h C ... C Ak-l C A," A,, w i t h A S e A ( h < s < k ) and A; (€Ak.

A c t u a l l y k -1 <r , so t h a t ( 3 ) i m p l i e s A;( = Ak, and Ahc ... C A ~ - ~ C A k i s a cha in as

requ i red .

1 1 1 1

S

( d ) I f Gr i s regu la r , each A.E A . ( O < i < r ) i s i nc luded i n a cha in 1 1

( s = O ,..., r ) .

( e ) I f Aie A i , A . € A . ( w i t h i , j = 0,1, ..., r ) , and i f A . L A . , then i C j , J J 1 J

J Proof . ( a ) L e t 1.) A = A . E A . . From ( 2 ) i t f o l l o w s t h a t i = 0. I f AAEA,,

1 - ( b ) L e t b E A k \ A From (31, Ah v I b } = A h ' h t l - k

k

S

( c ) Take h = 0 and k = i i n (b).

( d ) If i = r, ( d ) f o l l o w s f rom ( c ) . I f i c r , t hen A.C A f o r some A E A by

r e g u l a r i t y . By ( c ) we have a c h a i n A, C...C A. and by (b ) we have a cha in

S ' Ai C...C Ar-lC Ar, so t h a t we ge t a cha in A,C ... C A . C. . .C Ar (AS€ A

s=O, ..., r ) .

i r r r

1

1

132 P. V. Ceccherini and N . Venanzangeli

A. = 1

then

A . ( j J

( e ) We have i c j by ( 2 ) . From Ai = A . i t f o l l o w s obv ious l y t h a t i = j. Sup- J

pose now A. C A ! w i t h Ai, A;EAi. I f i = O , then A, = A: by ( a ) . I f i f O , then by ( c )

we ge t a cha in A,C . . .CAi- lCAi C_ A;, so t h a t c o n d i t i o n ( 3 ) i m p l i e s 1 - 1

v {b ] = A ! where b EA. \ A . So A . C A! i m p l i e s A . = A ! . 0 Ai-l 1 1 1 - 1 - 1 - 1 1 1

From 5.1 ( e l f o l l o w s

COROLLARY 5.2. L e t Gr be an 9-geometry. I f Ai, A; €Ai w i t h Ai # A;(O c i s r ) ,

A i n A ; E A . w i t h O s j c i - 1 .

THEOREM 5.3. Le t Gr = (S,3',X,A) be an %geometry.

( a ) Each X E X \ A,is i nc luded i n a unique A E A

> O ) , then x i s i nc luded i n some Ai E Ai f o r every O < i c j . I n p a r t i c u l a r i f

J

I f x i s i nc luded i n some 1 1 '

G i s regu la r , then each x i s i nc luded i n some A. E A. f o r every 0 < i c r . r 1 1

( b ) I f 0 < i < j < r , each A . E A . i s t h e un ion o f t h e elements o f A . which J J 1

j ' are i nc luded i n A

( c ) I f Ai E A i ( O 6 i < r ) i s i nc luded i n two d i s t i n c t element A ! A'.' E Aitl, l t l ' l t l

then A . = A ' n A:' = n A. 1 i t 1 i t 1 Ai 5 A €Ait l

( d ) The f o l l o w i n g cond i t i ons a re equ iva len t :

(d l ) f o r a l l 0s i < r, each A . E Ai i s i nc luded

( d ) f o r a l l 0 c i < r , each A . E A . i s i nc luded

1

Aitl;

2 1 1

i n two d i s t i n c t elements o f

i n two d i s t i n c t elements o f

( d 1 f o r a l l 0 s i < r , each A. E A. i s t h e i n t e r s e c t i o n o f elements o f A * 3 1 1 r '

( d f o r a l l O c i < j 6 r , each A. E A . i s t h e i n t e r s e c t i o n o f t h e elements 4 1 1

J Proof . ( a ) Suppose x E X \ A , . Since X = U A, we have X E A . f o r some A.EA

i' o f A. c o n t a i n i n g A

AE A J J j ' Then xEA, v {XI = A C A , and A i s t h e unique element o f A i n c l u d i n g x. I f

x E A . and O < i < j , t h e r e i s a cha in A,CA C.. .C Aic ... c A . ( A S € A s , O t s < j ) ,

w i t h x E A1 = A, v {XI, so t h a t x E A cA

( b ) It i s enough t o prove t h a t i f x E A . \ A,, t h e r e e x i s t s some A. E A . such J 1 1

t h a t xEAi C A

1 - j 1 1

J 1 J

1 i'

This f o l l o w s from ( a ) . j '

( c ) Ai 5 A i t l n = A . E A ., w i t h i s j s ( i t l l - 1 by C o r o l l a r y 5.2, so t h a t J J j = i and A = A ! n Uyt1 by P ropos i t i on 5.1 ( e ) . i i t 1

( d ) ( d l ) * ( d 2 ) . From ( d i t fo l l ows t h a t each A.E A . ( O < i < r ) i s i nc luded 1 1 1

i n a cha in A . C

(d2 ) * ( d

ments o f Ar. I f

1 and Ar-l i s i nc luded i n two d i s t i n c t elements o f A

. e C Ar-1 r ' 1. From ( d

x ' EA; \ A;: and x" EA; \ A;, then Ai 2 A;+1 n AYtl w i t h A!

we have t h a t A . C A ' n A " w i t h A;, A" d i s t i n c t e l e - 2 1 - r r r =

1 + I = A. v t x ' l and A! = A. v t x " 1 , A ' # AYtl. 1 l + l 1 i t 1

L 1 2 1 2

(dl) =, ( d 3 ) . From ( d 1 we have A. c A. "Aitl w i t h Ai+l # Ai+l and f rom ( a ) - 1 1 - 1 + 1 L

we get Ai = A : nAit,. I f i t 1 = r then ( d ) i s proved. I f i t 1 < r then f rom ( d ) 1 +1 3 1

11 12 2 21 and f rom ( a ) we get A1 = Ait2 nAi+2 and Ai+2 = A. nA:f2 so t h a t i t 1 1 t 2

hk A. = /--\ Ait2, and so on. i l ~ h , k c 2

(d4) * ( d 1. Obviously.

(d l ) * (d4 ) . I n d u c t i o n on j-i. If j-i = 1, then, by ( a ) , (d 1 i m p l i e s ( d ) .

i n c l u d i n g

3

1 4 Suppose t h a t each Ai E Ai i s t h e i n t e r s e c t i o n o f t h e elements o f A

A.. By t h e p rev ious argument each such A . i s t h e i n t e r s e c t i o n o f t h e elements 1 J -1

o f A. i n c l u d i n g A * a l l such A . a re p r e c i s e l y t h e elements o f A. i n c l u d i n g A J - j - 1 ' J J i

(because i f AiC A. f o r some A . E A . , t hen t h e r e e x i s t s Aj-, E Aj-l w i t h

A.C A .

j - 1

J J J C J \ . , by P r o p o s i t i o n 5.1 ( b ) ) .

1 J - 1 J

I f A E A h and A E A we say t h a t A and A a re j o i n a b l e i f they a re i n - h k k ' h k

c luded i n some A E A . I f A and A a re j o inab le , we d e f i n e Ah v A = n A; h k k A E A

k A 2 A h U A

= Ac E Ac and P R O P O S I T I O N 5.4. I f A,,€ Ah, Ak E Ak are j o i n a b l e w i t h Ah v A

Proof. I n d u c t i o n on k - i 2 0 . I f k = i , then A = A . C_ A and A = A v A =

= A v Ai = Ah; i t f o l l o w s h t k = c t i .

if k & i + l , l e t be A E A

then Ah v A = A E A f o r some c >, max [h , k l . k c c

k A nA = A.EA. then h+k>I i+c.

h k 1 1

k i h c h k

h w i t h Ai 5 Ak-lCAk. Then

v {XI) = (Ah v A

k-1 k -1 A c , = Ah v Ak-l c_ Ah v Ak = Ac . Then c ' t c . I f c ' c c t h e n t d A c , ; i f x E A k \ A c , , then

x ~ A ~ - ~ so t h a t A = A v Ak = Ah v (A

I n conc lus ion c ' t l > c . By t h e i n d u c t i o n hypothes is h t k - 1 2 i + c ' , so

) v Ix) = A v I x ) = c h k- 1 k- 1 C '

A c l + l '

0 t h a t h t k %i t c ' t 1 > i t c .

THEOREM 5.5. L e t Gr(X) = (S,S,X,A) be a geometry o f rank r 6 3 . Then Gr(X)

s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n :

( * ) i f Ah€ $, AkC Aky A h n A k = A . w i t h h t k s i t r and i f AhU A E 7, then Ah

and A a re j o i n a b l e , 1 k

k

134 P. V. Ceccherini and N . Venanzangeli

Proof. We can suppose w i thou t l o s s o f g e n e r a l i t y t h a t h s k . The theorem i s

t r u e f o r r = 1.

k If r = 2, then a c t u a l l y h t k - i s 2. We can ObViOUSlY assume t h a t Ah and A

a re no t i n i n c l u s i o n . I n t h i s case, t h e poss ib le values o f h, k and i are : i = O ,

h=k= l . L e t A1, A ' € A be such t h a t AIUA' E 7 w i t h A nA' = A,€ A,. Then A1 # A'

and i f x E A ' \ A i t i s easy t o prove t h a t A v (X I i s an element o f A i n c l u d i n g 1 1 1 2

A1 and A;.

a re no t i n i n c l u s i o n . I n t h i s case, t h e poss ib le values o f h, k and i, which are

d i s t i n c t f rom t h e cho ice i = O , h=k= l a l ready considered, a r e I i = O , h= l , k=21 and

li =1, h=k=2) ,

1 1 1 1 1 1'

k I f r = 3 then a c t u a l l y h t k - i -<3. kk can obviously assume t h a t Ah and A

When i = O , h = l and k.2, we have A E A1, A2€A2, A1 n A2 = A,, A U A € 9 . It 1 1 2

2 1 i s easy t o check t h a t t h e element A3€A3 de f ined by A v {XI where x E A \ A o ,

con ta ins A and A 2 '

When i = l , h=k=2, we have A

i s easy t o prove t h a t t h e element A E A

1 A;EA2, A nA' = A € A and A2UA' € 9 , Then i t

2' 2 2 1 1 2 de f i ned by A3 = A v {XI where

3 3' 2 xEA ' \ A = A ' \ A con ta ins A and A '

2 2 2 1 ' 2 2'

REMARK 5.6. When r ,4, c o n d i t i o n (*) i s no t n e c e s s a r i l y s a t i s f i e d ( c f . Theo-

rem 5.8). I t can be proved t h a t t h e f i r s t values o f i, h and k , f o r which the

prev ious argument f a i l s , a r e I i = O , h=k=2).

LEMMA 5.7. There e x i s t s an 9-geometry G ( X ) = ( S , 9 , X , A ) , namely a permuta t ion 4

geometry o f rank 4, f o r which c o n d i t i o n ( * ) i s n o t s a t i s f i e d . 2 Proof . L e t be N = {1,2, ..., 61. An element (h ,k )EN w i l l be w r i t t e n hk. L e t

X = U l , 22, 33, 44, 45, 35, 26, 15), A,, = 0, A, = IAJ ,

A. = I t h e se t o f s ing le tons o f X I = tIxy1 : xyEX1, I

10 A = I t he se t of a l l t h e i n j e c t i v e p a i r s o f X I , A = U A(i) w i t h : 2 3 i = l 3

A:')= t l l , 22, 33, 451, A:') = t l l , 22, 44, 351,

A:3)= [33, 44, 11, 261, A:4) = I33, 44, 22, 151,

A:51= Ill, 45, 261, A i6 ) = (11, 35, 26), A:7) = I33, 45, 261,

A!) = (33, 26, 151, A:') = (44, 35, 261, A:") = t44, 26, 151,

3

A = ." A(i) w i t h : A:') = (11, 33, 45, 261, 4 1 = 1 4

A:')= ill, 44, 35, 26), A t 3 ) = f33, 44, 26, 15). L e t be

9 = I1 c N2 : I i s i n j e c t i v e 1 and A = 4 u A - i = o i'

It i s easy t o check t h a t (S,y,X,A) i s a permuta t ion geometry G ( X ) o f rank 4 4

on x. L e t us cons ider t h e elements o f A . A = ill, 221 and A; = I33, 441. Actual 2 ' 2 -

l y A 2 n A ; = 0 = A,, 2 + 2 - 0 6 4 , A p U A ; € 7 , b u t A and A ' a re no t j o i n a b l e , i . e .

G ( X I does n o t s a t i s f y c o n d i t i o n [ * I . 2 2

4

THEOREM 5.8. For each r >, 4 t h e r e e x i s t s an 9-geometry (resp. a permuta t ion

geometry) Gr(X) o f rank r, such t h a t c o n d i t i o n (*) i s no t s a t i s f i e d .

Proo f . I n d u c t i o n on r 3 4 . For r = 4, t h e theorem reduces t o Lemma 5.7.

Suppose t h a t t h e r e e x i s t s an 9-geometry ( resp . a permuta t ion geometry) G;-l =

= (S ' ,S ' ,X ' ,A ' ) o f rank r - 1 2 4, f o r which c o n d i t i o n ( * ) i s no t s a t i s f i e d ; i n o t h e r

words, t h e r e e x i s t A ' E A ' such t h a t : A ' n A ' = A!, h t k - i s r - 1,

A ' UA' E 7' and A' and A ' a r e n o t j o i n a b l e . k k h k i

h k h k L e t GI' = (S",7",X",d") be an F g e o m e t r y ( resp . a permuta t ion geometry) of

1 rank 1 w i t h A: = (A:). Then we c l a i m t h a t Gr = G A q 1 b G; = (A,9,X,A) ( resp .

Gr = G '

t h a t c o n d i t i o n ( * ) i s no t s a t i s f i e d . Indeed

- b G;) i s an 9-geometry ( resp . a permuta t ion geometry) o f rank r such

r - 1

Ah = A ' U A t E A

A h n A k = (A; n A ; ( ) u A : = A;UA: €A i , h t k - i s r - I < r ,

A h U A k = ( A ; u A t ) u A : € 9 ( r e s p . E 3 ) , bu t A and A a re no t j o i n a b l e , because i f k h k

A h U A k 5 ( A ' U A " ) E A then A,',uA' C A ' E A ' , which i s imposs ib le .

A = A' U A: E A a r e such t h a t : h h ' k k k

0 k -

REMARK 5.9. From Theorem 5.8. we g e t a counterexample f o r P r o p o s i t i o n 2.3.

o f [ 31 f o r a l l r 5 4. The p r o o f o f P r o p o s i t i o n 2.3. i s i n c o r r e c t because t h e f l a t s

" A " and " B ' " a r e no t n e c e s s a r i l y such t h a t A u B ' i s an i n j e c t i v e se t .

We no te t h a t P r o p o s i t i o n 2.3. o f [ 3 ] i s t r u e f o r r 4 3 , by Theorem 5.6.

136 P. V. Ceccherini and N. Venanzangeli

Other p r o p e r t i e s o f 3'-geometries, s t a t e d i n [ 9 ] , w i l l be developed i n ano-

t h e r paper.

ACKNOWLEDGEMENT. This research was p a r t i a l l y supported by GNSAGA o f CNR and

by M P I .

BIBLIOGRAPHY

[l] P.J. Cameron and M. Deza, On permuta t ion geometries, J. London Math. SOC. ( 2 ) 20 (1979) 373-386.

[ 2 ] P.V. Ceccher in i and G. Ghera, A Vagner-Preston t ype theorem f o r semigroups w i t h r i g h t i d e n t i t i e s , Quad. Sem. Geom. Comb. 1s t . Mat. Appl. Univ. L ' A q u i l a 4, (1984).

[ 3 ] M. Deza and P. F rank l , I n j e c t i o n geometr ies, J . Comb. Theory (B) 37 (1984) 31-40.

[ 4 1 M. Deza and P. F rank l , On squashed designs, ( t o appear) .

[51 G. Ghera, Algebra e geometr ia d e l l e corr ispondenze p a r z i a l i d i un insierne i n - se, Tesi, Roma, 1s t . Mat. " G . Castelnuovo", l u g l i o 1983.

[61 M. Laurent, Geometries laminges: aspects a lgebr iques e t a lgor i thmiques , These Univ. P a r i s V I I ( t o appear).

I 7 1 6. Segre, I s t i t u z i o n i d i geometr ia super io re (a.a. 1963-641, Appunti d i P . V . Ceccher in i Vol . 111: Complessi, r e t i , d i segn i , Roma, 1 s t . Mat. "G. Castelnuo- vo", 1965.

[81 M.C. S c h i l l i n g , GBometries laminees e t bouquets de mat ro ids , These, Univ. P a r i s V I ( t o appear) .

[ 9 ] N. Venanzangeli, Geometrie d i permutazioni , geometr ie i n i e t t i v e e l o r o gene- r a l i z z a z i o n e , Tesi, Roma, 1s t . Mat. "G. Castelnuovo", l u g l i o 1984.


A NEW CHARACTERIZATION OF HYPERCUBES

P i e r V i t t o r i o Ceccher in i and Anna Sappa Oipar t imento d i Matematica "G. Castelnuovo"

U n i v e r s i t i d i Roma "La Sapienza" C i t t i U n i v e r s i t a r i a , 00100 Roma, I t a l y

By u s i n g a theorem o f S. Foldes [ Z ] , we prove t h a t a f i n i t e graph G i s a hypercube i f f i t i s connected, b i p a r t i t e , and t h e number o f geodesics between any two v e r t i c e s o f t h e graph GxK2 depends o n l y on t h e i r d i s tance . Graphs o f t h e t y p e GxK, a re a l s o cons idered.

1. INTRODUCTION

I n what f o l l o w s , a l l graphs w i l l be f i n i t e w i t h o u t loops o r m u l t i p l e edges.

I f G = ( V , E ) i s a graph and i f two v e r t i c e s x , y ~ V a r e j o i n e d by a path, t h e - d i s -

tance d (x , y ) i s d e f i n e d as t h e number o f edges i n a geodes ic ( s h o r t e s t p a t h ) be-

tween x and y . We denote by u ( x , y ) = vG(x ,y ) t h e number o f d i s t i n c t geodesics o f

G between x and y . I f x=y, we pu t d(x,y)=O and y ( x , y ) = l .

We say t h a t a connected graph G i s a graph w i t h a geodet ic f u n c t i o n i f t h e r e

e x i s t s a map F:[O,l,.., diam G I - N such t h a t u (x ,y ) = F (d (x , y ) ) ; i n t h i s case we

s h a l l say a l s o t h a t G i s F-geodet ic. A s tudy o f F-geodet ic graphs i s developed i n

[8] i n a more general con tex t .

-

For each p o s i t i v e i n t e g e r n, t h e n-cube Q i s de f i ned (un ique ly up t o i s o - n

morphism) as t h e graph whose v e r t i c e s a re t h e subsets o f a s e t S w i t h n elements

and two v e r t i c e s a r e j o i n e d by an edge i f and o n l y i f t h e y d i f f e r f o r e x a c t l y one

element. I n o t h e r words t h e v e r t e x s e t o f Qn i s t h e s e t o f o rdered n -p les o f 0 and

1; two v e r t i c e s a r e j o i n e d by an edge i f and o n l y i f they d i f f e r f o r e x a c t l y one

d i g i t . A hypercube i s a graph isomorphic t o some 9,.

Several c h a r a c t e r i z a t i o n s o f hypercubes were g i ven i n [11 - [ 5 1 ; i n [ 5 1 t h e

prev ious c h a r a c t e r i z a t i o n s a re a l s o summarized.

We ment ion here t h e f o l l o w i n g c h a r a c t e r i z a t i o n g i ven by S. Foldes.

THEOREM 1.1 (S. Foldes [ 2 1 ) . A graph G i s a hypercube i f and o n l y i f

138 P. V. Ceccherini and A . Sappa

( 1 ) G i s connected and b i p a r t i t e ,

( 2 ) G i s F-geodet ic w i t h F ( k ) = k ! .

Theorem 1.1 has been extended i n [ 6 ] t o a q-analogous r e s u l t . A q-hypercube

Qq i s de f i ned as t h e graph whose v e r t i c e s are t h e subspaces o f a g raph ic space S

o f o rde r q and dimension n-1, and two v e r t i c e s a r e j o i n e d by an edge i f f one o f

them i s covered by t h e o the r i n t h e l a t t i c e o f a l l subspaces. For q= l , we have

n

1 Qn 9,.

I n t h i s paper we g i v e a c h a r a c t e r i z a t i o n o f hypercubes which i s based on

theorem 1.1 and on t h e concept o f t h e t r a n s l a t i o n graph TG' o f a graph G ' =

(V ' ,E ' ) . The graph TG' i s t h e permuta t ion graph ( G ' , i d V , ) , i . e . TG ' = G'xK

where K i s t h e complete graph on two v e r t i c e s ( 1 and 21, ( c f . a l s o [ 7 1 ) . 2'

2 The d e s c r i p t i o n o f TG' = (V,E) as t h e permuta t ion graph ( G ' , i d V , 1 i s t h e

f o l l o w i n g . L e t G" = (V",E") be a copy o f G ' = (V ' ,E ' ) and l e t x " E V " denote t h e

copy o f X ' E V ' ; more genera l l y , i f A 'CV ' , - then A"cV" - w i l l denote t h e copy o f A ' ,

i . e . A" = { ~ " E V " : x ' E A ' I . Then T G ' = (V,E) where V = V'uV" and

E = E ' uE"L I { (x ' ,X I ' )€ V ' X V " : X ' E V ' l . The d e s c r i p t i o n o f TG' = (V,E) as t h e graph G'xK i s t h e f o l l o w i n g :

V = V 'x t1 ,Z I = t ( x ' , h ) : x ' E V ' , h=1,2},

E = t ( ( x ' , h ) , ( y ' , k ) ) E V x V : [ ( x ' = y ' and ( h , k ) E E ( K 2 ) ) o r ( ( x ' , y ' ) E E '

and h=k) 11.

2

We s h a l l prove t h e f o l l o w i n g

THEOREM 1 . 2 A graph G ' i s a hypercube i f and o n l y i f

( 1 ) G ' i s connected and b i p a r t i t e ,

( 2 ) G ' x K i s F-geodet ic, for some F. 2

2. PROOF OF THEOREM 1.2.

We s h a l l s t a r t w i t h t h e f o l l o w i n g lemma.

LEMMA 2.1. L e t G ' be a connected graph and l e t G = G ' x K 2 = TG' be t h e t r a n s -

( a ) G ' i s F ' -geode t i c w i t h F ' ( k ) k! ,

( b ) G i s F-geodet ic w i t h F ( k ) = k ! ,

( c ) G i s F-geodet ic for some F.

l a t i o n graph o f G ' . The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t

A New Characterization of Hypercubes 139

Proof. Obv ious ly (b ) - ( a ) , ( c ) . We use t h e d e s c r i p t i o n o f G = G'xK = (V,E) as t h e

permuta t ion graph (G ' , idV, ) o f G ' = (V' ,E' ! ( 5 1) .

( a ) - ( b l : l e t x,y€ V. It i s enough t o suppose d ( x , y b 2 . I f x , y € V ' , then Y (x,y) =

=uG, 'x ,y ) = d G , ( x , y ) ! = dG(x ,y ! ! . S i m i l a r l y i f x , y ~ V " , then yG(x ,y ) = d G ( x , y ) ! .

I f x = x ' E V ' and y = y " ~ V " , l e t x " E V " and Y ' E V ' be t h e cop ies o f x ' and y"

r e s p e c t i v e l y . We have d ( x , y ) = d (x',y'') = d , ( x ' , y ' ) t d ( y ' , y " ) = d ( x ' , y ' ) t l . G G G G G

Moreover each geodesic o f G between x = x ' and y=y" i s o f t h e fo rm g ( x , y ) =

= ( x ' , ..., z',z" ,..., y " ) where g ' ( x ' , y ' ) = ( x ' , ..., z ' , ..., y ' ) i s a geodesic o f G '

between x ' and y ' and i t s copy g" (x " ,y " ) = (x " , ..., z", ...,y" ) i s a geodesic o f G"

between x " and y". (The case z'=x', i . e . z"=x", and t h e case z '=y ' , i . e . z"=y", a r e

no t exc luded) . I n o t h e r words, each geodesic g ( x , y ) i s ob ta ined e x a c t l y once f rom

a geodesic g ' ( x ' , y ' ) by choosing i n g ' ( x ' , y ' ) a ve r tex z ' (as t h e b a s i s o f t h e

b r i d g e (z',z") between V ' and V " ) . The number o f such z ' i s d G , ( x t , y ' ) t l . So

2

G

Y ~ ( x , Y ) = Y , ( x ' , y ' ) . (d , ( x ' , y ' ) + l ) = d G , ( x ' , y ' 1 . ( d G , ( X ' , Y ' ) + l ) = G G

= ( d G , ( x ' , y ' ) + l ) ! = dG(x,Y)!.

( c ) - ( b ) : a c t u a l l y F(0) = F ( 1 ) = 1. When x , y c V w i t h dG(x,y) = k > 2 , t h e r e e x i s t

X ' E V ' and y "EV" such t h a t d G ( x ' , y " ) = k . I f Y ' E V ' i s t h e copy o f y", we

d , ( x ' , y ' ) = d ( x ' , y O = k - 1 2 1 . With t h e same argument used f o r p r o v i n g

( a ) * ( b ) we o b t a i n f rom ( c ) :

have

G G

By us ing Theorem 1.1 we can now g i v e a new p r o o f o f t h e f o l l o w i n g w e l l known

r e s u l t .

LEMMA 2.2. Qntl = Q, x K p .

Proof . We can use t h e d e s c r i p t i o n o f Q x K = TQ as t h e permuta t ion graph n 2 n --

(Qn, idv , ) o f Q, = (V ' ,E ' ) ( 5 1) .

The graph Q, i s connected and b i p a r t i t e ( theorem 1.11, so t h a t TQ, i s a l s o

connected and b i p a r t i t e : i f A ' u B ' i s a p a r t i t i o n o f t h e ve r tex s e t V ' o f Q then

(A'u 6 " ) u ( A " u B ' ) i s a p a r t i t i o n o f t h e v e r t e x s e t V = V'U,V" o f TQ where A" n'

n

and 6" a r e t h e cop ies o f A ' and 6 ' r e s p e c t i v e l y .

140 P. V. Ceccherini and A. Sbppa

By theorem 1.1 t h e graph Q i s F ' -geodet ic w i t h F ' ( k ) = k ! , so t h a t (Lemma

2.1) Q xK i s F-geodet ic w i t h F ( k ) = k ! . The graph TQ i s consequent ly a hyper-

cube by theorem 1.1, and t h e lemma i s proved s ince diam (TQ,) = diamg t 1 = n t l .

n

n 2 n

n

COROLLARY 2.3. I f Tn denotes t h e n - t h power o f t h e opera tor T ( t r a n s l a t i o n n of a graph), we have Q = T K2.

n t l

We g i v e now t h e

Proof o f theorem 1.2. Le t G ' a hypercube. By theorem 1.1, c o n d i t i o n ( 1 ) i s

s a t i s f i e d . Moreover G = TG' = G'xK i s a hypercube by lemma 2.2, so t h a t (by the:

rem 1.1 again) G'xK2 i s F-geodet ic w i t h F ( k ) = k ! , ( 0 ~ kGd iam G), and c o n d i t i o n

( 2 ' ) i s s a t i s f i e d .

2

Conversely i f G ' i s a graph s a t i s f y i n g ( 1 ) and ( 2 ' 1 , then, by lemma 2.1, G '

s a t i s f i e s c o n d i t i o n ( 1 ) and (2 ) o f theorem 1.1, so t h a t G ' i s a hypercube.

3. A GENERALIZATION OF LEMMA 2.1.

I f G ' = ( V ' , E ' ) i s any graph and K i s t h e complete graph on m v e r t i c e s m

i s de f i ned by ( m 2 2 ) , t h e graph G = G'xK, = (V,E)

V = V'x11,2 ,..., m l = t ( x ' , h ) : x ' E V ' , h = 1,2,.. .,ml,

E = [ ( ( x ' , h ) , ( y ' , k )EVxV: [ ( x ' = y ' and ( h , k ) E E ( K 1 ) o r ( ( x ' , y ' ) E E ' and h=k)]l. in

An o t h e r d e s c r i p t i o n o f t h e same graph G = G'xK = ( V , E ) i s t h e f o l l o w i n g . L e t i i i im i

G = (V ,E 1 be a copy o f G ' = (V ' ,E ' ) and l e t x E V denote t h e copy o f x ' E V ' , 1

i=1 ,2 , . . . ,m, where we assume G1 = G I , x = x ' . Then

i ' i m . i m

v = u v , E = . u E' u i ( x ,~J).v x v j : X ' E V I , i , j=1,z ,..., m ; i + j l . i =1 1 =1

Lemma 2.1 i s i nc luded f o r m=2 i n t h e f o l l o w i n g

PROPOSITION 3.1. L e t G ' = (V',E'l be a connected graph and l e t be G = G'xK m

( m 2 2 ) . The f o l l o w i n g cond i t i ons are equ iva len t

( a ) G ' i s F ' -geode t i c w i t h F ' ( k ) = k !

( b ) G i s F-geodet ic w i t h F ( k ) = k !

( c ) G i s F-geodet ic f o r some F.

Proof. We can use t h e same argument as f o r Lemma 2.1. Indeed, i f x,yC V w i t h

x=x E V and y = y j ~ V j ( i# j ) , then t h e o n l y e s s e n t i a l p o i n t i s t h a t G x I i , j ) = i i i

i = TG " T G ' .

A New Characterization of Hypercubes 141

COROLLARY 3.2. A graph G ' i s a hypercube i f and o n l y i f

( 1 ) G ' i s connected and b i p a r t i t e ,

( 2 " ) G'xK i s F-geodet ic f o r some F and some m. m

REMARK 3.3. P r o p o s i t i o n 3.1 e a s i l y y i e l d s a f a m i l y o f connected graphs which

are F-geodet ic w i t h F ( k ) = k ! and which a re n o t hypercubes (because t h e y a r e no t

b i p a r t i t e ) .

ACKNOWLEDGEMENTS. Th is research was p a r t i a l l y supported by GNSAGA o f CNk

and by M P I .

BIBLIOGRAPHY

[ l ] L.R. Alvarez, Und i rec ted graphs r e a l i z a b l e as graphs o f modular l a t t i c e s , Can. J . Math. 17 (1965) 923-932.

[ 2 j S . Foldes, A c h a r a c t e r i z a t i o n o f hypercubes, D i s c r e t e Math. 17 (1977) 155-159.

131 J.M. Laborde, C h a r a c t e r i z a t i o n l o c a l e du graphe du n-cube, Journee Combina- t o i r e s , Grenoble (1978).

[ 41 J.D. McFal l , Hypercubes and t h e i r c h a r a c t e r i z a t i o n s , U n i v e r s i t y o f Water loo Dept. o f Combinator ics and Op t im iza t i on Research Report CORR 78-26 (1978) .

[ 51 J.D. McFal l , Charac te r i z ing hypercubes, Annals D i s c r e t e Math. 9 (1980) 237- 241.

[ 61 P . V . Ceccher in i , A q-analogous o f t h e c h a r a c t e r i z a t i o n o f hypercubes as graphs, J . Geometry 22 (1984) 57-74.

[ 7 1 A. Sappa, C a r a t t e r i z z a z i o n e d i g r a f i t r a m i t e geodet iche, Tesi , Univ. d i Roma, D ipa r t imen to d i Matematica (1984).

[81 P . V . Ceccher in i and A. Sappa, F-binomial c o e f f i c i e n t s and r e l a t e d combina- t o r i a l t o p i c s : p e r f e c t ma t ro id designs, p a r t i a l l y o rdered se ts o f f u l l b ino - m ia l t y p e and F-graphs, Annals D isc re te Math. ( t h i s volume).

Annals of Discrete Mathematics 30 (1986) 143-1 58 0 Elsevier Science Publishers B.V. (North-Holland) 143

F-BINOMIAL COEFFICIENTS AND RELATED COMBINATORIAL TOPICS: PERFECT MATROID DESIGNS,

POSETS OF FULL BINOMIAL TYPE AND F-GEODETIC GRAPHS

P i e r V i t t o r i o Ceccher in i and Anna Sappa D ipa r t imen to d i Matematica "G. Castelnuovo"

Uni v e r s i t l d i Roma "La Sapi enza" C i t t a U n i v e r s i t a r i a , 00100 Roma, I t a l y

We i n t r o d u c e F-binomial c o e f f i c i e n t s as a n a t u r a l genera l - i z a t i o n o f b inomia l and q-b inomia l c o e f f i c i e n t s . A genera l c a l c u l u s w i t h these numbers leads t o u n i f y t h e a r i t h m e t i c a l p r o p e r t i e s o f ( f i n i t e ) p r o j e c t i v e and a f f i n e spaces and o f S t e i n e r systems S(t ,k,v) i n t o those o f p e r f e c t m a t r o i d designs ( 5 2 ) . P a r t i a l l y o rdered s e t o f f u l l b inomia l t y p e ( 5 3 ) and graphs such t h a t t h e number o f geodesics between any two v e r t i c e s depends o n l y on t h e i r d i s tance ( 5 4 ) a re a l s o s t u d i e d by means o f t h i s fo rmal c a l c u l u s .

1. F-BINOMIAL COEFFICIENTS

L e t N ( resp . Q ) denote t h e se t o f non nega t i ve i n t e g e r s ( resp . r a t i o n a l

numbers) and l e t be N* = N \ ( O I , Q* = Q \ r O ) .

L e t F: N + Q* be any f u n c t i o n such t h a t F ( 0 ) = F (1 ) = 1 and l e t f : N + Q be

any f u n c t i o n such t h a t f ( 0 ) = 0, f ( 1 ) = 1 and f ( N * ) 5 Q*. I n what f o l l o w s , F and f w i l l be f u n c t i o n s s a t i s f y i n g t h e above cond i t i ons .

Given an F, t h e assoc ia ted f w i l l be de f i ned by:

f ( 0 ) 0, f ( n ) = F ( n ) / F ( n - l ) f o r n 1 1 .

Given an f, t h e assoc ia ted F w i l l be d e f i n e d by:

F ( 0 ) = 1, F ( n ) = f ( n ) F ( n - 1 ) i . e . F (n ) = f ( n ) f ( n - l ) . . , f ( l ) f o r n 3 1 .

Two such f u n c t i o n s F and f a r e then m u t u a l l y associated, and sometimes below t h e y

w i l l be used in te rchangeab ly .

DEFINITION 1.1. Given a p a i r (F , f ) o f m u t u a l l y assoc ia ted f u n c t i o n s and g i ven

any i n t e g e r s k,n w i t h O C k < n , we s h a l l d e f i n e t h e F-binomial c o e f f i c i e n t ( o r f- n

b inomia l c o e f f i c i e n t ) I 1 as t h e r a t i o n a l number k F

144 P. V. Ceccherini and A. Sappa

These numbers t u r n o u t t o be p o s i t i v e i n t e g e r s i n t h e f o l l o w i n g examples ( i n which

f and F a l so take i n t e g e r va lues ) .

EXAMPLE 1.2. Binomial c o e f f i c i e n t s . Consider t h e p a i r o f m u t u a l l y assoc i -

a ted f u n c t i o n s

and F ( t ) = t! f o r a l l t E N . 1

f l ( t ) = t

Then

n in) = = Ik} i s t h e usual b inomia l c o e f f i c i e n t ( O < k 6 n ) . k F1 k fl

EXAMPLE 1.3. Gaussian numbers. Given an i n t e g e r q > 1 , cons ider t h e p a i r o f

m u t u a l l y associated f u n c t i o n s

f ( t ) = [ t l and F ( t ) = [t],! 9 9 9

where [ ] and [ ] ! are de f i ned by 9 q

t - 1 [OI = 0, [ l l = 1, [ t l q = q t ... + q t l ,

[Ol,! = [ll ! = 1, [ t lql = [ t l [ t - 1 1 ... [ llq,

9 9

q q q t 2 2 .

Then

n - n - n IkIF - ikIf - [,Iq i s t h e q-b inomia l c o e f f i c i e n t (gauss ian number, c f . 141).

9 q

EXAMPLE 1.4. Constant c o e f f i c i e n t s . Given an i n t e g e r a z 1, cons ider t h e

p a i r o f mu tua l l y assoc ia ted f u n c t i o n s

f ( 0 ) = 0 , f(1) = 1, f ( t ) = a; F ( 0 ) = F ( 1 ) = 1, F ( t ) = a t - ’ ( t z 2 ) .

Then

n f o r O < k < n .

n n n F O F I 1 = i l = 1 , and I k j F = a

Usual b inomia l i d e n t i t i e s and q-binomial i d e n t i t i e s ( c f . [ 4 ] a re p a r t i c u -

F-Binomial Coefficients and Related Combinatorial Topics I45

l a r cases o f F-binomial i d e n t i t i e s (ob ta ined f o r F = F and f o r F = F r e s p . ) . We

g i v e now some o f them, w r i t t e n a s f - b inomia l i d e n t i t i e s . 1 q

P R O P O S I T I O N 1.5. The f o l l o w i n g f -b inomia l i d e n t i t i e s ho ld :

( O < k t n ) , n - f ( n ) n-1 - f ( n ) n-1 - f ( n - k + l ) n ‘ k ’ f - fo { k If - fo ‘ k - l ’ f - f ( k ) ‘ k - l ’ f

n - n-1 f ( n ) - f ( n - k ) n-1 f ( k ) ( k - l ’ f I k f f - I k If +

These fo rmulas suggest t h e f o l l o w i n g

DEFINITION 1.6. Given a f u n c t i o n f and any i n t e g e r s k,n w i t h 0 < k <n, d e f i n e

PROPOSIT IONS 1.7. Le t be as Def. 1 .6. The f o l l o w i n g c o n d i t i o n s a re n,k

equ iva len t :

( a ) i s independent o f n;

( b ) f = f

n,k

f o r some q E Q * (where f 9 q

i s f o r m a l l y de f i ned as i n Ex. 1 .3 ) .

( c ) df = qk f o r some q E Q*. n,k f

Proo f . Obv ious ly ( c ) = + ( a ) . We have ( b ) 9 ( c ) because gf = A n,k n,k

=

f ( n ) - f ( k ) n -1 k = qk. We prove now t h a t ( a ) ( b ) . L e t us p u t - 9 9 = 9 + . . . + q

f q (n -k ) qn-k-1,. . .+q+l

bf = q f o r a l l n, Then f ( 1 ) = 1 and f o r a l l i n t e g e r s n 3 2 we g e t

n-1

n , l

f ( n ) = A f f ( n - l ) + f ( l ) , so t h a t t h e e q u a l i t y f ( n ) = q + ... + q+ l f o l l o w s by n,l

i n d u c t i o n .

2 . PERFECT M A T R O I O DESIGNS

A comb ina to r ia l geometry ( o r s imp le m a t r o i d ) M on a f i n i t e se t S i s a ma t ro id

on S such t h a t 0, S andevery s i n g l e t o n o f S a re f l a t s o f M ( c f . [ 2 1 , [911.

A p e r f e c t ma t ro id des ign (PMD) i s a comb ina to r ia l geometry M such t h a t every

k - f l a t ( f l a t of rank k ) has t h e same number f ( k ) o f po in ts , k = O,l, ..., r k M

( c f . [ l o ] , [12 ] , [141) . We s h a l l say t h a t f i s t h e s i z e - f u n c t i o n o f M. Obviously

f ( 0 ) = 0 and f ( 1 ) = 1, so t h a t f -b inomia l c o e f f i c i e n t s t k l f can be cons idered

( O s k ( n 4 r - k M I . X

n

PMDs inc lude boolean se ts 2 , p r o j e c t i v e spaces, a f f i n e spaces, t - ( v , k , l )

designs ( c f . [ 1 4 ] ) .

PROPOSITION 2.1. L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f.

( a ) M i s t h e m a t r o i d o f a l l subsets o f S i f and o n l y i f f = f I n t h i s case 1' n n f

f ( n ) = n, 1 ~ 1 ~ = ( k ) and

i f f = f ( w i t h q a 2 ) . I n t h i s case f ( n ) = q

= 1.

( b ) M i s t h e ma t ro id o f f l a t s o f a p r o j e c t i v e space o f o rder q i f and o n l y n-1 n n

t . . . t q t l , 1 ~ 1 ~ = [ 1 and 9 k q

(Conversely, each o f these e q u a l i t i e s i m p l i e s f = f 1. f

9 n-1. , ",k = q *

( c ) If M i s thematroid o f f l a t s o f an a f f i n e space o f o rde r q, then f ( n ) = q n f k 2k-n

the converse i s t r u e when q a 4. I n t h i s case { k ] f = qk(n-kl and

(Conversely, each o f these e q u a l i t i e s imp l i es f ( n ) = q 1 .

= q - q . n-1

Proof . ( a ) i s obvious. For ( b ) ( resp . ( c ) ) , we have t o prove o n l y t h e " i f "

p a r t . A s imp le coun t ing argument shows t h a t every 3 - f l a t i s a p r o j e c t i v e ( resp .

an a f f i n e ) p lane o f o rde r q, so t h a t t h e r e s u l t f o l l o w s f rom a w e l l known charac-

t e r i z a t i o n o f p r o j e c t i v e ( resp . a f f i n e ) spaces by means o f p lanes, c f . [ Z ] ( resp .

[ l l ) . 0

With an argument which i s s tandard f o r f i n i t e p r o j e c t i v e spaces ( c f . [14]

one can prove t h e f o l l o w i n g

PROPOSITION 2.2. L e t M be a PMD w i t h s i z e - f u n c t i o n f.

( a ) The number o f k - f l a t s i nc luded i n a r - f l a t ( w i t h O < k 6 r ) i s g i ven by

f ( r ) ... f ( r - k + l ) A ~ , ~ . . . A ~ , ~ - ~ 'r, 1 " ' ' r , k - l

A k, 1 * ' k, k-1 f ( k ) . . . f ( l ) 'k, 1 " 'Ak,k- l

- - - ' k ' f '

( b ) When a k - f l a t i s i nc luded i n a r - f l a t , t h e number o f j - f l a t s between them

(01: k < j c r ) i s g i ven by

F-Binomial Coefficients and Related Cornbinatorial Topics 147

I n p a r t i c u l a r B(O,j,r) = c r ( j , r ) and

B(k ,k t l , r ) = - "" f ( r - k ) k+ l , k

A

( c ) When a k - f l a t

o f f l a t s between them

;(k,r) = B (k ,k t l

A i s i nc luded i n a r - f l a t 6, t h e number o f maximal cha ins

s g i ven by

Ar ,k . ' .Ar , r -2

A k t l , k ' * 'Ar- l , r - 2 r ) B ( k t l , k t 2 , r ) ... B( r -2 , r - l , r ) = F ( r - k )

where F i s assoc ia ted t o f .

PROPOSITIONS 2.3. L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f (and

assoc ia ted f u n c t i o n F ) and l e t a ( k , r ) , B ( k , j , r ) , ; (k , r ) be d e f i n e d as i n p ropos i -

t i o n 2.2. The f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t :

( 1 ) M i s t h e PMD o f a g raph ic space S o f o rder q ( i . e . a p r o j e c t i v e space o f

o rder q ( p o s s i b l y a " l i n e " ) , when 9 2 2 , o r t h e boolean s e t PS, when q = 1 ) ;

( l a ) a ( k , r ) = {;If f o r a l l D < k.S r c r k M;

r - k ( l b ) B ( k , j , r ) = ( j - k ~

( l b ' ) B(k ,k+ l , r ) = f

( l c ) ;(k,r) = F ( r - k )

Moreover, i f c o n d i t i o n ( 1

f o r a l l 06 k < ;i < r s r k M;

r - k ) f o r a l l O s k i r s r k M;

f o r a l l O < k < r < r k M.

i s s a t i s f i e d , t hen

r ( 2 ) f = f a (k , r ) = [ ] l3(k, j , r ) = [;][I,, ;(k,r) = [ r - k l !

q' k q'

Proo f . It i s we l l known ( c f . [ 3 ] , [ 1 6 ] ) t h a t ( 1 ) i m p l i e s ( l a ) - ( l c ) and ( 2 ) .

A r , l r - r s a ( 2 , r ) = A - * { 2 1 f - 121f ( l a ) - ( l ) . By Prop. 2.2, we ge t 2 3 1

'r k

' k t l , k ( I b ' ) =, (1 ) . By Prop. 2.2, we g e t R(k ,k t l , r ) = f ( r - k ) = f ( r - k )

. f o r k = l we o b t a i n A = A and ( 1 ) f o l l o w s as above. r,l 2,l 4 = r , k 'k+l ,k '

r - 0 - r ( l b ) =, ( l a ) . a ( k , r ) = R(O,k,rl = I k-O'f - 'k ' f '

( I c ) =) ( l b ) . I n each maximal cha in of f l a t s between a k - f l a t A and a r - f l a t

B t h e r e e x i s t s e x a c t l y one j - f l a t C ( 0 ~ k c j < r ( r k M ) ; g iven such a C, t h e j o i n -

i n g of a maximal cha in between A and C and o f a maximal cha in between C and El i s

a maximal cha in between A and B. The double coun t ing argument, a p p l i e d t o t h e s e t

{ ( C , $ ) : C i s a j - f l a t be long ing t o a cha in 0 between A and B), g ives :

F ( r - k ) = N k , j , r ) F ( j - k ) F ( r - j ) , i . e . B(k, j , r ) = F(r-jlF(j-kl j-kff. 0

REMARK 2.4. Le t M be t h e ma t ro id o f f l a t s o f an a f f i n e space o f o rde r q and

F ( r - k ) - { r - k

+ d e f i n e

f o l l o w s t h a t f ( 0 ) = 0, f ( t ) = qt-', F ( t ) = q t(t-1)'2 and f o r k > O

a ( k , r ) , R ( k , j , r ) and v ( k , r ) as i n Prop. 2.2. From ( 2 ) o f Prop. 2.3, i t

r - k r - 1 a ( k , r ) = q [k-llq, B(k , j , r ) = [;:;Iq, ;(k,r) = [ r - k l !. q

REMARK 2.5. We g i v e now an o t h e r example o f a PMD M with s i z e - f u n c t i o n f

such t h a t ( k , r ) # F ( r - k ) . L e t S be a S t e i n e r system S(Z,k,n) (e.g. t h e S t e i n e r

system whose b locks a re t h e l i n e s o f a p r o j e c t i v e ( resp . a f f i n e ) space S o f o rde r

q w i t h k = q t l ( resp . k = q ) ) and l e t M be t h e PMD ( o f rank 3) whose f l a t s a re the

empty se t , t h e s i n g l e po in ts , t h e b locks and t h e f u l l s e t o f p o i n t s . The s i z e

f u n c t i o n f i s such t h a t f ( 0 ) = 0, f ( 1 ) = 1, f ( 2 ) = k, f ( 3 ) = n; t he assoc ia ted

f u n c t i o n F i s such t h a t F(0) = F (1 ) = 1, F ( 2 ) = k, F (3 ) = nk; moreover =

= = 1, A = k-1, A = - '-' . There fore 2,l 3,l k

2 whenever n f k -k+ l ( i n t h e p rev ious examples, whenever t h e space S i s n o t a p r o j e c

t i v e p lane ) .

F-Binomial Coefficients and Related Combinatorial Topics 149

3. POSETS OF FULL BINOMIAL TYPE

These s t r u c t u r e s have been in t roduced i n [111, ( c f . a l s o [71). We s h a l l use

t h e n o t a t i o n o f $1.

L e t (P,*.) be a p a r t i a l l y o rdered s e t (pose t ) w i t h minimum 0, and l e t a,b be

elements o f P. I f a 4 b, t h e i n t e r v a l between them i s de f ined by I ( a , b ) =

= f c ~ P : a < c c b l . I f a< b a maximal cha in between a and b i s a cha in

a = a < a <...< an = b o f I (a ,b ) which i s n o t i n c l u d e d i n a l onger c h a i n o f

I ( a , b ) ; t h e number n i s c a l l e d t h e l e n g t h o f t h e cha in . We s h a l l denote by ;(a,b)

t h e number o f maximal cha ins o f I ( a , b ) , and we s h a l l assume ;(a,b) = 1 when a=b.

We s h a l l say t h a t P i s a JD-poset i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n

(JD) (Jordan-Dedeking cha in c o n d i t i o n ) : i f a,b E f w i t h a < b, a l l t h e maximal

cha ins o f I ( a , b ) have one and t h e same leng th , denoted by d(a,b) and c a l l e d t h e

d i s tance between a and b ( o r t h e l eng th o f t h e i n t e r v a l ( I ( a , b ) ) . If a=b, we pu t

d(a,b) = 0.

0 1

I f P i s a JD-poset w i t h minimum 0, then t h e rank o f an element a E P i s

P = max t r k a: a € $ , . de f ined by r k a=d(O,a), and t h e rank o f P i s de f i ned by r k

We no te t h a t i f a,bEP w i t h a < b, t hen d(a,b)< r k P.

We s h a l l say t h a t a JD-poset P ( w i t h minimum 0) has a cha in f u n c t i o n if

t h e r e e x i s t s a f u n c t i o n F : f O , l , ..., r k P I + N such t h a t , f o r a l l a,b E 2 w i t h

a < b, t h e number ;(a,b) o f maximal cha ins between them i s F(d(a ,b) ) , i .e . eve ry

i n t e r v a l o f l e n g t h n o f P has F ( n ) maximal cha ins . I n t h i s case we s h a l l say

t h a t p i s an F-chain poset .

DEFINITION 3.1. A poset

( 1 ) P has minimum 0,

( 2 ) P i s a JD-poset,

( 3 ) P i s a F-cha in poset, f o r some F.

P i s c a l l e d a poset o f f u l l b inomia l t ype i f

I f f i s a JD-poset w i t h minimum 0, f o r each i n t e r v a l I (a ,b ) o f P and f o r

each i n t e g e r k, w i t h O s k s d ( a , b ) , we s h a l l pu t

Ik (a ,b ) = I c E I ( a , b ) : d(a ,c ) = k )

We s h a l l say t h a t P has a s i z e func t ion_ i f t h e r e e x i s t s a f u c n t i o n f,

f : IO,1 , ..., r k P ) +. N such t h a t , f o r a l l i n t e r v a l s I (a ,b ) o f f ,

I50 P. V. Ceccherini and A . Sappa

I I l ( a , b ) I = f ( d ( a , b ) ) .

I n t h i s case we s h a l l say t h a t P i s an f - s i z e poset. The f o l l o w i n g p ropos i -

t i o n y i e l d s o t h e r equ iva len t d e f i n i t i o n s o f poset o f f u l l b inomia l t ype .

PROPOSITION 3.2. L e t P be a JD-poset w i t h minimum 0; t h e f o l l o w i n g cond i -

t i o n s a r e equ iva len t , where f and F a re m u t u a l l y assoc ia ted f u n c t i o n s :

(a )

( b l I I k (a ,b ) I = t d(a,b)]

( c ) P i s an f - s i z e poset.

Proo f . L e t a , b E P , a c b .

( a ) * ( b ) . Le t be O < k < d ( a , b ) . I n each maximal cha in o f I ( a , b ) t h e r e e x i s t s

e x a c t l y one c E P such t h a t d(a,c) = k. I f c i s such an element o f I ( a , b ) , t he

j o i n i n g o f a maximal cha in o f I ( a , c ) and o f a maximal cha in o f I ( c , b ) i s a maxi-

mal cha in o f I (a ,b ) . The double coun t ing argument app l i ed t o t h e se t I ( c ,o ) : C E ~ ,

CL i s a maximal cha in o f I (a ,b ) , d(a,c) = k ) g i ves :

P i s an F-chain poset ( i . e . P i s a poset o f f u l l b inomia l t ype ) .

(Oc k c d(a,b)) .

F ( d ( a , b ) ) = I I k (a ,b ) I F ( k ) F (d (a ,b ) -k ) ,

( b ) * ( c ) . For k = l we o b t a i n I1 (a ,b ) I = I d(;yb))f = f ( d ( a , b ) . 1

( c ) * ( a ) . A c t u a l l y I1 ( a , b ) I = f ( d ( a , b ) ) . We now app ly i n d u c t i o n on d(a ,b) ;

so t h a t ( b ) f o l l o w s .

1 i f d(a,b) = 0,1, then 1 = ;(a,b) = F (d (a ,b ) ) . I f d(a,b),2, app ly t h e double coun - t i n g argument t o t h e s e t { ( c , ~ ) : c E U , U i s a maximal cha in o f I (a ,b ) , d ( a , c ) = l l .

By t h e p rev ious argument and by t h e i n d u c t i o n hypothes is , we have

;(a,bl = ( I l ( a ,b ) I F (1 ) F (d (a ,b l - l ) = f ( d ( a , b ) ) F (d (a ,b ) - l ) = F(d(a ,b l ) . 0

EXAMPLE 3.3. I f X i s a f i n i t e s e t and i f P = 2 i s t h e s e t o f a l l subsets

o f X o rdered by i n c l u s i o n , then P i s a poset o f f u l l b inomia l type, w i t h s i z e

f u n c t i o n f ( t ) = t and cha in f u n c t i o n F ( t ) = t ! ( 0 ~ t c 1x1) ( c f . [131) . Conversely

any such f u n c t i o n s f and F ( w i t h dom f = dom F = t O , l , ..., n l ) , ( c f . example 1.21,

can be cons idered as s i z e f u n c t i o n and cha in f u n c t i o n o f a boolean se t p = 2 w i t h

1x1 = n. We no te t h a t , i f P = I X I U ( o ) U( , ) U...U ( k ) w i t h 2 4 k < 1x1 - 1, t hen t h e

poset P i s n o t of f u l l b inomia l t ype : i f Y E ( 2 ) and Z E(k r l ) l then d(0,Y) =

= d(Z,X) = 2, b u t t h e r e a re 2 cha ins from 0 t o Y and 1x1 - (k -1) > 2 cha ins f rom

z t o x.

X

X

x x X

X

F-Binomial Coefficients and Related Combinatorial Topics IS1

EXAMPLE 3.4. If X i s a f i n i t e p r o j e c t i v e space PG(n-1.q) o f o r d e r q and

dimension n-1

i s a poset o f f u l l b inomia l t ype w i t h s i z e f u n c t i o n f ( t ) = [ t ] and c h a i n f u n c t i o n

F ( t ) = [ t ] !. (c f . [ 3 ] , prop. 3.3.). Conversely, any such f u n c t i o n s f = [ ] and

F = [ I ! wi th q = p ( c f . exaE

p l e 1 .3 ) can be cons idered as t h e s i z e and t h e c h a i n f u n c t i o n s o f t h e poset o f

t h e subspaces of a p r o j e c t i v e space X = PG(n-l,q) o f o rde r q and dimension n-1.

We n o t e t h a t , if P i s t h e s e t of a l l t h e i -d imens iona l f l a t s o f X = PG(n-l,q),

w i t h i = -l,O,...,k,n-1 where 1 4 k < n - 2 , then t h e poset P i s n o t o f f u l l b inomia l

t ype : i f Y i s a l i n e and 2 i s a (k -1 ) -d imens iona l f l a t , then d(0,Y) = d(Z,X) = 2

b u t t h e r e a r e q+ l cha ins f r o m 0 t o Y and q wk-' + .. .+q+l > q+ l cha ins f rom Z t o X.

2 and i f ?' i s t h e s e t o f f l a t s o f X o rdered by i n c l u s i o n , t hen ?

q

q h 9

( h h l , p p r i m e 3 2 1 and dom f = d a n F = 10.1 ,..., n l f-l

EXAMPLE 3.5. I f $ = {xo,xl ,,..., xla ,..., xnl ,..., x

elements (a,n a l ) , ordered by assuming xo as minimum and x

2 i s a poset o f f u l l b inomia l t y p e w i t h s i z e f u n c t i o n f ( O ) = O , f ( l ) = l , f(t)a d with

cha in f u n c t i o n F ( O ) = F ( l ) = l , F ( t ) = a ( 2 6 t < n ) . Conversely, any such f u n c t i o n s f

and F ( c f . example 1.3.) can be cons idered as t h e s i z e and t h e c h a i n f u n c t i o n s o f

a poset o f f u l l b inomia l t y p e p as above.

1 i s any s e t w i t h an t1

< x i j hk

na i f f i < h, t hen

t - 1

REMARK 3.6. L e t M be t h e PMD o f t h e f l a t s o f an a f f i n e space X = AG(n-1,q)

o f d imension n - 1 3 2 , and l e t B be t h e poset o f t h e f l a t s o f X. The poset P i s

no t o f f u l l b inomia l t ype : i f Y i s a l i n e and Z i s an (n -3) -d imens iona l f l a t ,

t hen d(0,Y) = d(Z,X) = 2 , b u t t h e r e a re q cha ins f rom 0 t o Y and q+ l cha ins f rom

z t o x. Therefore, i f M i s a PMD on a f i n i t e s e t X and if P i s t h e s e t of t h e f l a t s

o f M ordered by i n c l u s i o n , then P i s n o t n e c e s s a r i l y a poset o f f u l l b inomia l

t ype : t h e number o f maximal cha ins i n an i n t e r v a l I (a ,b ) does n o t depend o n l y on

t h e l e n g t h o f t h e i n t e r v a l , b u t a l s o on t h e ranks o f a and b ( c f . a l s o Remark

2.5, and Examples 3.3, 3.4). The case when P i s a poset o f f u l l b inomia l t ype

i s cha rac te r i zed by t h e f o l l o w i n g p r o p o s i t i o n .

PROPOSITION 3.7. L e t M be a PMD on a f i n i t e s e t X and l e t P be t h e poset o f

t h e f l a t s o f M ordered by i n c l u s i o n . The poset P i s of f u l l b inomia l t ype i f and

o n l y i f M i s one o f t h e f o l l o w i n g PMD's:

( a ) M i s t h e t r i v i a l PMD o f rank 1 o r t h e t r i v i a l PMD o f rank 2 on X ( i .e .

M i s a t r i v i a l g raph ic space o f d imension 0 on 1 on X resp . ) ;

X (b ) M i s t h e m a t r o i d 2 o f a l l subsets o f X, i . e . M i s t h e graph ic space o f

o rde r 1 and dimension 1x1-1;

( c ) M i s t h e m a t r o i d Mn-l

P roo f . I f M i s a PMD as i n ( a ) - ( c ) , then t h e poset P o f i t s f l a t s i s a poset

( X I o f a l l f l a t s o f a p r o j e c t i v e space, o f dimen- Y q

s i o n n-1 and o rde r q * 2 , hav ing X as s e t o f p o i n t s .

o f f u l l b inomia l t y p e ( c f . Ex. 3.3 and Ex. 3.4).

L e t M be a PMD on X such t h a t t h e poset B o f i t s f l a t s i s a poset o f f u l l

b inomia l type. I f r k Mr 2, t hen we a r e i n t h e case (a ) . If r k M > 2, l e t f denote

t h e s i z e f u n c t i o n o f M.

Fo r any f l a t Y o f rank 2, we have t h a t l I l (O,Y) l = ( Y I = f ( 2 ) = f (d (0 ,Y) ) .

I n o the r words, t h e poset P has t h e same s i z e f u n c t i o n f than t h e PMD M. Thus

c o n d i t i o n ( b ) o f Prop. 3.2 holds; i t means t h a t c o n d i t i o n ( l b ) o f Prop. 2.3 ho lds .

So c o n d i t i o n ( 1 ) o f Prop. 2.3 ho lds too, and t h e r e s u l t ( b ) - ( c ) i s proved. 0

4. F-GEODETIC GRAPHS

I n what fo l l ows , a l l graphs w i l l be f i n i t e w i t h o u t loops o r m u l t i p l e edges,

an:i a l l d i r e c t e d graphs w i l l be w i t h o u t d i r e c t e d c i r c u i t s .

Any d i r e c t e d graph G' = (V,;) i s o b v i o u s l y t h e Hasse diagram o f a poset

p = p ( t ) = (V,<) where x < y i f and o n l y if t h e r e e x i s t s a d i r e c t e d pa th f rom x

t o y; converse ly t h e Hasse diagram o f a poset p = (V,<) i s a d i r e c t e d graph

6 =

t h a t E and

= (V,:), where (x ,y ) E E i f and o n l y i f x i s covered by y. We s h a l l say

P(E) are m u t u a l l y assoc ia ted .

I f G = (V,E) ( resp . 6 = (V,E)) i s a graph ( resp . d i r e c t e d graph) and i f two

v e r t i c e s x,yE V a re j o i n e d by a pa th ( resp . d i r e c t e d pa th) , t h e d i s tance d(x ,y )

( resp . i ( x , y ) ) i s de f i ned as t h e number o f edges i n a geodesic i . e . i n a s h o r t e s t

pa th (resp. d i r e c t e d s h o r t e s t pa th) between x and y. We denote by r ( x , y ) = r G ( x , y )

*

* + (resp. by r (x,y) = r G ( x , y ) ) t h e s e t o f

between x and y; we pu t :

v(x,y) = I r ( x , y ) l and ;(x,y) = I;

q (x ,x ) y (x ,x ) = 1 and d (x , x ) = d

diam G = max I d ( x , y ) : x,y E V j , d

+

We s h a l l say t h a t a connected graph G =

d i s t i n c t geodesics o f G (resp. of E l

x,y)I if x # y;

x,x) = 0;

am E = max ~+d(x,y) : x,y E W .

V,E) ( resp . a d i r e c t e d graph 5 = (V,:))

i s a graph w i th a geodetic func t i on i f there e x i s t s a map F : ( O , l , ..., diam G I + N

(resp. F : ( O , l , ..., diam G ) + N ) such t h a t u (x , y ) = F ( d ( x , y ) ) ( resp. u (x , y ) =

= F(J(x ,y) ) f o r a l l x,y E V . I n t h i s case we s h a l l a lso say t h a t G (resp. i) i s

F-geodetic. Note t h a t F(0) = 1 and t h a t F(n) # 0 f o r a l l n E dom F, so t h a t t he as - sociated func t i on f can be considered as i n $1.

* +

Let 2 = (V,;) be a d i rec ted graph and l e t x , y ~ V be such t h a t x-<y. Then the

i n t e r v a l I ( x , y ) i s defined by

+ I ( x , y ) : = I z E V : x < z - ( y l , i . e . I ( x , y ) i s defined as i n P ( G ) ,

and the geodetic i n t e r v a l Ig(x ,y) i s def ined by

P ( x , y ) := t z e V: Z E C;(x,y) f o r some C; E r ( x , y ) ~ . Note t h a t I 9 (x,y) 5 I ( x , y ) and t h a t I g (x,y) = t z e I ( x , y ) : i ( x , z ) = ;(x,y)-;(y,z)).

I;(x,y) : = I z e I g ( x , y ) : ~ ( x , z ) = k l = ( z E I 9 (x ,y) : J ( Z , Y ) = i ( x , y ) - k l .

For any 1 < k < i ( x , y ) , l e t

We say t h a t E = (V,:) has a source O E V i f f o r any X E V \ (01 there e x i s t s a d i r e c -

t e d path from 0 t o x.

A graph 6 = (V,i) w i l l be c a l l e d a d i rec ted graph o f f u l l binomial type ( w i t h

geodetic func t i on F ) i f :

( a ) E has a source 0,

(b ) f o r a l l x , y ~ V w i th x < y : I 9 (x,y) = I ( x , y ) ,

( c ) G i s F-geodetic f o r some F.

PROPOSITION 4.1. Le t E = (V,i) be a d i r e c t e d graph and l e t P = P ( 6 ) = ( V , < )

be the poset associated w i t h 6 (so t h a t = 6(P)). Then

( 1 ) P has a minimum 0 i f and on ly i f has source 0;

( 2 ) P i s a JO-poset i f and on ly i f I g ( x , y ) = I ( x , y ) , f o r a l l x , y ~ V

w i th x <y ;

(3 ) P i s a poset o f f u l l binomial type w i th chain func t i on F i f and on ly i f

6 i s a d i rec ted graph o f f u l l binomial type w i t h geodetic func t i on F.

Proof. (1 ) i s obvious. ( 2 ) : P i s a JD-poset 0 f o r a l l x , y ~ V w i t h x < y the -- set M(x,y) o f maximal chains o f P i n I ( x , y ) i s the set ? (x , y ) o f the geodesics

G(x,y;of E O f o r a l l x , y ~ V w i th x c y I ( x , y ) = I g (x , y ) . ( 3 ) : P i s a poset o f

154 P. V . Ceccherini and A . Sappa

f u l l b inomia l t ype w i t h cha in f u n c t i o n F O P has minimum 0, I' i s a JD-poset and

(M(x,y) I = F(+d(x ,y ) ) f o r a l l x < y i n V o "G has source 0, f o r a l l x < y i n V

Ig(x,y) = I ( x , y ) and [;(x,y)( = F (d (x , y ) )

t ype w i t h geodet ic f u n c t i o n F.

E i s a d i r e c t e d graph o f f u l l b inomia l

0

The poset p = (V,<), where V = IO,x,y,z,tl and O < x < z < y , O < t < y , i s no t a

JD-poset, b u t t h e graph 6 ( p ) i s F-geodet ic ( w i t h F = l ) ; no te t h a t Ig(O,y) =

= {O,t,Yl c I (0 , y ) = v. PROPOSITION 4.2. L e t "G = ( V , i ) be a d i r e c t e d graph. The f o l l o w i n g c o n d i t i o n s

a r e equ iva len t (where F and f are m u t u a l l y assoc ia ted ) :

( a )

( b ) f o r a l l x < y i n V and f o r a l l 1 g k c a ( x , y ) : ( I ( x , y ) I = i

( c ) f o r a l l x < y i n V : I I ~ I = f ( i ( x , y ) ) .

i s F-geodet ic f o r some F, 9 J(X,Y)

k I F ' k

Proof, L e t x,y be elements o f V w i t h x < y . ( a ) - ( b ) . The double coun t ing argument

app l i ed t o t h e se t Zk = ( z , < ( x $ y ) ) : z ~ < ( x , y ) , g e r ( x , y ) , z ( x , z ) = k l g ives :

F ( d ( x , y l ) = I I z ( x , y ) l F ( k ) F (a (x , y ) - k ) , so t h a t I k ( x , y ) = I

* +

+ 9 d ( x ,y

F.

( b ) * ( c ) f o r k = 1.

( c ) =) ( a ) . We app ly i n d u c t i o n on a(x ,y ) 31. I f a (x , y ) 1, then I;(x,y) I =

= 1 = F ( 1 ) . Suppose d ( x , y ) h 2 ; by t h e i n d u c t i o n hypothesis, when Z E Z we have

IT(z,y) 1 = F ( i ( z , y ) ) = F (G(x ,y ) - l ) . Therefore t h e double coun t ing argument a p p l i e d

t o t h e s e t Z1 g ives : I?(x,y) 1 = I I g (x , y ) l F ( l ) F ( i i ( ~ , y ) - l ) = f ( a ( x , y ) ) F ( a ( x , y ) - l ) =

= F ( a ( ~ , y ) ) . 0

1

1

I f 2 i s a d i r e c t e d graph, we s h a l l denote by G t h e und i rec ted graph assoc i -

a ted t o E .

PROPOSITION 4.3. L e t = (V,:) be a d i r e c t e d graph w i t h source O S V . Suppose

t h a t f o r each x E V t he p a r i t y o f t h e l e n g t h o f any d i r e c t e d path f rom 0 t o x

depends on ly on x; w r i t e p ( x ) =O i f i t i s even and p ( x ) = 1 i f i t i s odd. Then

t h e und i rec ted graph G assoc ia ted t o G i s connected and b i p a r t i t e . +

Proof . G i s connected s ince 0 i s a source. We v e r i f y t h a t G i s b i p a r t i t e by 0 1 i

assuming V = V U V where V = I x E V : p ( x ) = il, i = 0 , l . We have t o prove t h a t

i f ( x , ~ ) E E then p ( x ) # p ( y ) . We can suppose w i thou t loss o f g e n e r a l i t y t h a t

(X ,Y)E E. I f G(0,x) and G(0,y) a re geodesics f rom 0 t o x and t o y resp., then

g(0,y) and G(0,x) u (x,y) are both d i rec ted paths from 0 t o y, so t h a t t h e i r

lengths have the same p a r i t y p (y ) . It fo l l ows t h a t p ( x ) # p (y ) .

PROPOSITION 4.4. Let G = (V,E) be a connected b i p a r t i t e graph and l e t 0 be

any vertex o f G. Then by s t a r t i n g from 0, a na tu ra l o r i e n t a t i o n can be def ined

on E, i n such a way t h a t = (V,:) i s a d i r e c t e d graph w i t h source 0 (and wi thout

d i rec ted c i r c u i t s ) . It fo l l ows t h a t $ = p(G) = (V,C) i s a poset w i th minimum 0,

where x <y i f and only i f there i s i n E a d i rec ted path from x t o y .

-*

Proof. I f x,y E V w i th d(0,x) = d(O,y), then x and y cannot be adjacent be-

cause G i s b i p a r t i t e . Let (x ,y) be an edge o f G. We have e i t h e r d(0,y) = d(O,x)+l

o r d(O,x) = d ( 0 , y ) t l . Orient the edge from x t o y i n the f i r s t case, from y t o

x i n the second. It i s easy t o show tha t , whenever there i s an edge (x,y) EE,

then (x,y) i s t he on ly d i r e c t e d path from x t o y, and t h a t whenever the re e x i s t s

a d i rec ted path from x t o y then the re i s no d i rec ted path from y t o x; indeed

(by i nduc t i on on i ) , i f (xo,x l,...,xi) i s a d i r e c t e d path, then we have

+

(xo,xl) ,..., ( X ~ - ~ , X . ) E ~ 1 and d(O,xi) = d(O,xo) t i. 0

+ + THEOREM 4.5. L e t G = (V,E) be a connected b i p a r t i t e graph and l e t G = (V,E)

be the d i rec ted graph obtained by s t a r t i n g from a given vertex OEV as i n Prop.

4.4. Then, whenever x and y are elements o f V w i t h x < y , the f o l l o w i n g condi t ions

are equiva lent :

(a

(b

(C

Proof.

p(x,y) i s a d i rec ted path from x t o y i n t, p(x,y) i s a geodesic from x t o y i n 6,

p(x,y) i s a geodesic from x t o y i n G. *

a) * ( b ) . Assume x = 0 f i r s t . Let p(0,y) = (O=x ",,.,, x.=y) 1 be any d i r e c -

t e d path o f lenght i from x t o y i n 5. We have d(O,y)=i, by the i nduc t i on argu-

ment sketched a t the end o f the proof o f Prop. 4.4. Therefore p(0,y) = p(x ,y) i s

a geodesic i n G.

Assume now 0 < x < y . Let p(0,x) and p(x ,y) be any d i rec ted paths i n 8 from

0 t o x and from x t o y resp. By g lue ing p(0,x) and p(x,y) we get a d i rec ted path

p(0,y) i n 'G. For the previous case p(0,y) i s a geodesic i n G. Thus i t s subpath

p(x,y) must a l so be a geodesic i n 6.

( b ) =) ( a ) . Induct ion on i=d(x,y) . When i = l , we have p(x,y) = ( x , y ) ~ i since

x <y . so t h a t p(x,y) i s a d i rec ted path i n G. Assume now i Z 2 and suppose t h a t *

any geodesic o f G o f l eng th j w i t h 1 6 j < i i s a d i r e c t e d pa th i n 6 . The geodesic

p (x ,y ) = ( x =x,x ,..., x .

and ( X ~ - ~ , X ~ ) o f G. These are bo th d i r e c t e d pa ths i n 6 by t h e induc t ion hypothes is .

Then p(x ,y ) i s a d i r e c t e d pa th i n E .

,xi=y) i s ob ta ined by g l u e i n g t h e geodesics p(x.xi-,) 1 - 1

( c ) * ( a ) obv ious l y .

( b ) * ( c ) . If p(x,y) = ( x =x,x ..., x.=y) i s a geodesic i n G, then p(x,y) i s

a d i r e c t e d pa th i n (because ( b ) = . (a ) ) , and i t must be a geodesic i n 6 , because

any s h o r t e r d i r e c t e d pa th p ' ( x , y ) i n G would be a pa th o f G s h o r t e r t han p(x ,y ) ,

which i s imposs ib le s ince p(x ,y ) i s a geodesic i n G.

0 1 ' 1

+

0

COROLLARY 4.6. L e t G = ( V , E ) be a connected b i p a r t i t e graph and l e t be t h e

d i r e c t e d graph ob ta ined by s t a r t i n g f rom a g i ven ve r tex O E V as i n Prop. 4.4.

Then

( 1 ) whenever x , y ~ V w i t h x r y , we have d(x ,y ) = +d(x,y), r ( x , y ) = ? ( x , y ) so

t h a t r ( x , y ) = t ( x , y ) ;

( 2 ) G i s F-geodet ic i f and o n l y i f E i s F-geodet ic f o r a l l O E V .

Proof. (1 ) i s obvious. For ( 2 ) i t i s enough t o no te t h a t ;(x,y) = r ( x , y ) when we

assume 0 = x.

COROLLARY 4.7. L e t G = ( V , E ) be a connected b i p a r t i t e graph. Then t h e f o l -

l ow ing c o n d i t i o n s a re equ iva len t , where F and f a re mu tua l l y assoc ia ted f u n c t i o n s :

( a ) G i s F-geodet ic, d(x,y)

( b ) x,yEV, Otkcd(x,y)* I I z E V : d(x ,z ) = k, d(z,y) = d ( ~ , y l - k l ( = I

( c ) x,yEV * ~ [ Z E V : d(x,z) = 1, d (z ,y ) d ( x , y ) - l l I = f ( d ( x , y ) ) .

)f,

0

REMARK 4.8. The statement o f C o r o l l a r y 4.7 also ho lds i f G i s n o t b i p a r t i t e ;

i t can be proved by a d i r e c t argument ( c f . [161) . Th is can a l s o be ob ta ined f rom

t h e p r o o f o f Prop. 4.2, by r e p l a c i n g g(x,y) , r ( x , y ) , d(x,y) resp . w i t h g(x,y),

r ( x , y ) , d (x ,y ) (and by exchanging consequent ly t h e se t I!(x,y) and t h e s e t

-t + +

( z E V : d(x ,z ) = k, d (z , y ) = d ( x , y ) - k ) ) .

EXAMPLE 4.9. The complete graph Kn, a - t r e e G , t h e ( 2 k t l ) - c i r c u i t G a re F-geo -

d e t i c graphs w i t h F = l . An F-geodet ic graph w i t h F-1 i s c a l l e d geodet ic -g raph ( c f .

[161, [ l a ] ) . The i l k - c i r c u i t G i s F-geodet ic w i t h F ( t ) = l , f o r t <k , and w i t h F ( 2 ) = 2

o therw ise .

EXAMPLE 4.10. The complete b i p a r t i t e graph Kn,,=(V,E) w i t h V = V 'UV",

I V ' ( = IV" I = n i s F-geodet ic w i t h F (0 ) = F ( 1 ) = 1, F ( 2 ) = n.

Conversely any con-

131).

= t ( 0 < t s 2 cn,m)

EXAMPLE 4.11. A hypercube i s an F-graph w i t h F ( t ) = t ! .

nec ted b i p a r t i t e F-graph w i t h F ( t ) = t ! i s a hypercube ( c f .

EXAMPLE 4.12. The graph Kn x Km i s F -geodet ic w i t h F ( t

( c f . ( 5 1 , [151 1 -

m EXAMPLE 4.13. I f Qn i s t h e n-cube, t h e graph Qn x K

F ( t ) = t ! ( O s t < n t l ) ( c f . [51, Prop. 3.1).

s F -geodet ic w i t h

EXAMPLE 4.14. I f G i s a connected graph and i f GxK, i s F-geodet ic f o r some F

and some m x 2 , then F ( t ) = t!. Moreover, i f G i s b i p a r t i t e , then G i s a hypercube

(and GxK i s a l s o a hypercube i f and o n l y i f m = 2 ) ( c f . [ 5 ] , Prop. 3.1, Cor. 3.2) m EXAMPLE 4.15. L e t G be t h e d i r e c t e d graph whose v e r t i c e s a re t h e subspaces

i s an edge i f t h e

= [tlq! (c f . [31,

~- o f a g raph ic space o f d imension n and o f o rde r q 3 1 and (x , y

f l a t x i s covered b y t h e f l a t y. Then 6 i s F-geodesic w i t h F ( t

Prop. 3.3).

REMARK 4.16. L e t G be t h e und i rec ted graph assoc ia ted t o the d i r e c t e d graph

E cons idered i n t h e Ex. 4.15. I n o t h e r words, G i s t h e q-analogue o f Qn-,.

t h a t when q a 2, G i s n o t F-geodesic: i f x,y a r e two p o i n t s and z i s a p lane con-

t a i n i n g x, t hen d(x,y) = d (x ,z ) = 2, b u t 2 = v(x,y) # u ( x , z ) = q t l . Note a l s o

t h a t t h e d i r e c t e d graph 5 obta ined f rom G by s t a r t i n g f rom t h e empty f l a t i s t h e

graph o f Example 4.15; when we s t a r t f rom a f l a t O f 0 , then E i s F -geodet ic i f and

o n l y i f q = l .

Note

EXAMPLE 4.17. L e t 6 = 6(P) be t h e d i r e c t e d graph assoc ia ted t o t h e poset p

o f Ex, 3.5. Then 6 i s F-geodet ic acco rd ing l y t o Prop, 4.1 ( b u t G i s n o t F-geode-

t i c ) .

ACKNOWLEDGEMENT. Th is research was p a r t i a l l y supported by GNSAGA o f CNR and by

M P I .

BIBLIOGRAPHY

[ 1 1 F. Buekenhout, Une c h a r a c t e r i z a t i o n des espaces a f f i n s basee su r l a n o t i o n de d r o i t e , Math. Z. 111 (1969) 367-371.

[ 21 P . V . Ceccher in i , S u l l a nozione d i spaz io g r a f i c o , Rend. Mat. ( 5 ) 6 (1967) 78-98.

158 P. V, Ceccherini and A . Sappa

[ 3 ] P.V. Ceccher in i , A q-analogous o f t h e c h a r a c t e r i z a t i o n o f hypercubes as graphs, J. Geometry 22 (1984) 57-74.

[ 4 1 P.V. Ceccher in i , A. Dragomir, Combinazioni genera l i zza te , q - c o e f f i c i e n t i b i n o m i a l i e spazi g r a f i c i , A t t i Convegno Geometria Combinator ia e sue a p p l i - caz ion i (Peruaia. Settembre 1970) 137-158.

[ 5 1 P . V . Ceccher in i , A. Sappa, A new c h a r a c t e r i z a t i o n o f hypercubes,Annals -~ D i - s c r e t e Math. ( t h i s volume).

[ 6 1 L. Cer l ienco, F. P i ras , C o e f f i c i e n t i b i n o m i a l i g e n e r a l i z z a t i , Rend. Sem. Fac. Sc i . C a g l i a r i 52 (1982) 47-56.

[ 7 1 L. Cer l ienco, F. Piras, G-R-Sequences and inc idence coalgebras o f posets o f f u l l b inomia l t ype , ( t o appear) .

[ 8 ] R.J. Cook, D.G. Pryce, Un i fo rm ly geodet ic graphs, t o appear.

[ 9 ] H. Crapo, G.C. Rota, Combinator ia l geometries, MIT Press, Cambridge (1970).

[ 1 0 1 M. Deza, N.M. S ingh i , Some p r o p e r t i e s o f p e r f e c t m a t r o i d designs, Annals D i s c r e t e Math. 6 (1980) 57-76.

[ 11 ] P. Doub i le t , G.C. Rota, R.P. SRanley, On t h e foundat ions o f Combinator ia l t heo ry V I : t h e i dea o f genera t i ng f u n c t i o n , i n G . C . Rota (ed . ) , F i n i t e Ope- r a t o r Calculus, Academic Press, New York (1975) 83-134.

[ 12 1 J. Edmonds, U.S.R. Murt i , P. Young, Equ ica rd ina l ma t ro ids and ma t ro id d e s i - gns, i n "Combinator ia l Mathematics and i t s App l i ca t i ons " , Second Chapel H i 11 Conference (1970) .

[ 1 3 1 S. Foldes, A c h a r a c t e r i z a t i o n o f hypercubes, D i s c r e t e Math. 17 (1977) 155- 159.

[ 14 1 B.L. Rothsch i ld , N.M. S ingh i , Charac te r i z ing k - f l a t s i n geometr ic designs, J. Comb. Theory A 20 (19761, 398-403.

[ 15 1 A. Sappa, Cara t te r i zzaz ione d i g r a f i t r a m i t e geodet iche, Tesi , Univ. d i Ro- ma, D ipar t imento d i Matematica (1984).

[ 16 I R. Scape l la to , On geodet ic graphs o f d iameter two and some r e l a t e d s t r u c t u - r e s ( t o appear).

[ 171 B. Segre, Lec tures on modern geometry. With an Appendix by L. Lombardo-Ra- d i c e , Cremonese, Roma (1961).

[ 1 8 ] J.C. Stempe, Geodet ic graphs o f d iameter two, J. Comb. Theory B 17 (1974) 266-280.


POLYNOMIAL SEQUENCES ASSOCIATED +

WITH A CLASS OF INCIDENCE COALGEBRAS.

Luigi Cerlienco Giorgio Nicoletti Francesco Piras Dip. di Matematica Dip. di Matematica Dip. di Matematica Univ. di Cagliari Univ. di Bologna Univ. di Cagliari 09100 Cagliari 40127 Bologna 09100 Cagliari Italy Italy Italy

A few special sequences of polynomials associated with both automorphisms and hemimorphisms of a particular class of coalgebras as well as their links with locally finite posets of binomial type are analysed.

50 *

The purpose o f this paper is to briefly study a class of coalgebras, which we call coaZgebras of binorniaZ t y p e . These are the coalgebras C having a countable basis (bi) such that

i i 0

i i Abi = j ~ o hj bj@bi-jy ~ b . = 6 i 1

where h. are integers and hE=l.

Coalgebras of binomial type are the background for some recent work in combinatorics beginning with polynomials o f binomial type [17].We show that, under mild conditions, the structure constants hj behave much like binomial coefficients, and that the analog of sequence of polynomials of binomial type is obtained in any coalgebra of binomial type from coalgebra morphism.

The one new coalgebraic notion introduced in this work is the notion of k e m i m o r p h i s m , namely, a linear map f on C into itself such that

The image of the distinguished basis (bi) under a hemimorphism generalizes sequences of polynomials introduced by Goldman and Rota [ 1 4 ] .

We believe the general setting of coalgebras (and Hopf algebras) to be suited to a variety of combinatorial problems which we propose to study in future publications.

A*f = (1ef)oA.

§I * 1.1. Let C:=(V,A,E) be a coalgebra over a field K of characteristic zero. Here V is a K-vector space and e:V-K fcounitland A:VNVBV f c o m u Z t i p Z i c a t i o n o r diagonaZizationl are linear maps such that the following diagrams commute:

Research partially supported by "Fondi Ministeriali per la Ricerca 40% e 60%".

160 L. Cerlienro, G. Nicoletti and F. Rras

- V @ V h V -

/A@ I ( c o a s s o c i a t i v i t y )

v A ' ev I Q A V @ V @ V

€ @ I I @ E K@V - V @ V - V O K

($ i s t h e c a n o n i c a l isomorphism and I t h e i d e n t i c a l map).

I f , moreover, we have A = T o A , where T : V 0 V - V @ V y v@w"w@v ( t w i s t - - o p e r a t o r ) , C i s s a i d t o be c o c o m m u t a t i v e . A s u s u a l , we s h a l l deno te by C*=(V4,m,u) t h e d u a l K-algebra of C ; w e have U=E* and m=A*o j , where j:v'@v"-(V@V)" bedding. A l i n e a r map f:V-V i s an endomorphism o f C i f (3) Aof = ( f 8 f ) o A

i s t h e c a n o n i c a l em-

(4) € O f = E .

A l i n e a r map f : V - V i s s a i d t o be a r i g h t hemimorphism of C i f ( 5 ) A o f = ( I e f ) 0 A. L e f t hemimorphisms are d e f i n e d i n a s imi la r way. I f C i s cocommutati- v e , each r i g h t hemimorphism i s a l s o a l e f t hemimorphism, and conver- s e l y . L e t u s deno te by & m ( C ) t h e s e t of a l l r i g h t hemimorphisms of t h e c o a l g e b r a C . Hem(C) i s c l o s e d under l i n e a r combinat ion and func - t i o n a l composi t ion. Hence Hem(C) i s an a l g e b r a . Moreover, i f f" i s t h e d u a l map of feHem(C), t h e n f o r every arV w e have:

p ( a ) = f ' (m(a@l) ) = ( f ' o m ) ( a @ l ) = ( f 'O(A*o j ) ) ( a@Jl ) =

= ( ( A o f ) * o j ) ( a B l ) = ( ( ( I @ f ) o A ) * o j ) ( a @ l ) = ( A * o ( I @ f f o j ) ( a @ 1) =

(because o f ( f e g f o j = j o ( P @ g " ) )

= ( ( A * o j ) o ( f @ f " ) ) (a el) = (A* o j ) ( a @ f*(l)) = m(a@ f C ( l ) ) ,

i . e , , u s i n g aB i n s t e a d of m(a @ 6) : ( 6 ) F ( a ) = a * f " ( l ) w i t h l=u(lK)EV, which becomes f*(a) = ? ( l ) . a i n t h e c a s e of l e f t hemimorphisms. The element f ' ( l) i n d ( f ) : ( 7 ) i n d ( f ) := f'(1) . A s a consequence of ( 6 ) we o b t a i n :

Prop.1 . The map

1 7 ' ) i n d : & m ( C ) - C*

w i l l be c a l l e d t h e i n d i c a t o r o f f and denoted by

f c----c i n d l f )

i s a monomorphism o f aZgebras .

( N o t i c e , however, t h a t ( 7 ' ) i s an antimonbmorphism i n t h e c a s e o f

Polynomial Sequences and Incidence Coalgebras 161

l e f t hemimorphisms) . P r o o f , I t r ema ins t o p rove t h a t ind(f0:) = i n d ( f ) - i n d ( g ) ; i n f a c t , we have

( f o g f (1) = (g*of ' ) ( l ) = g ' ( f " (1 ) = P ( l ) * g * ( l ) . D

1 . 2 . I n t h e f o l l o w i n g w e s h a l l a lways assume t h a t C = ( V , A , E ) h a s a c o u n t a b l e b a s i s (b i ) iEN such t h a t

i i 0 Abi = jio h j b j @ b i - j , h 0 =1

I f t h i s i s t h e case, C i s s a i d t o be a c o a t g e b r a of b i n o m i a l t y p e . I t i s t h e n a s t r i c t l y g raded c o a l g e b r a . I n p a r t i c u l a r , bo i s t h e un ique g r o u p - l i k e e l emen t ( i . e . Abo=bo@3bo) and bl i s a p r i m i t i v e e l - ement ( i . e . Abl=bl@bo + b o @ b l ) .

The d u a l a l g e b r a C' i s n a t u r a l l y endowed w i t h a t o p o l o g i c a l s t r u c - t u r e ( t h e s o - c a l l e d f i n i t e t o p o t o g y ) assuming t h e f a m i l y

U i = I B E V 1 (Vj) ( j s i =$ B(b . )=Ol , i e N 1

a s a b a s e f o r a sys t em o f ne ighbourhoods o f z e r o . With t h i s t o p o l o g y , e a c h e l emen t a o f C y can be r e p r e s e n t e d a s f o l l o w s :

where

and

i n i a = 1 a . b := l i m .Z a . b i>o 1 n-m i=o 1

a := < a l b i > : = a ( b i ) E K i

b i : V - K i b - 6 .

j 3 T h i s i s sometimes e x p r e s s e d by s a y i n g t h a t ( b ) i E N i s a p s e u d o - b a s i s

With r e f e r e n c e t o a f i x e d b a s i s ( b i ) and i t s d u a l p s e u d o - b a s i s ( b i ) , we can r e p r e s e n t e a c h c o u p l e o f e l e m e n t s

i

of v".

i a = L a i b E C ' " i a = Z a b i E C , i = o i > o i i r e s p e c t i v e l y a s a co lumn-vec to r ( a ) o f e n t r i e s a and a s a row-vec-

t o r (a,) o f e n t r i e s a Thus , we may w r i t e 1 i ' i

. (a ) = : < a l a > = ( a i ) * ( a ) .

Moreover, f o r any g i v e n l i n e a r map f : V - V l e t u s d e f i n e t h e r e p r e - s e n t i n g m a t r i x M f f ) , i n t h e same way as i n t h e f i n i t e - d i m e n s i o n a l c a s e , t o be t h e (NrN)-mat r ix whose ( r , s ) - e n t r y i s g i v e n by

< r l f ( s > : = = . Of c o u r s e t h e same m a t r i x a l s o r e p r e s e n t s t h e d u a l map f*: M(f)=M(f*). N o t i c e t h a t t h e i - t h column f ( b i ) of M(f) h a s a f i n i t e s u p p o r t . Using t h e above n o t a t i o n a l c o n v e n i e n c e s , w e may e a s i l y deduce t h e f o l l o w i n g :

Prop. 2 . A l i n e a r m a p g:C*--tCu i s a c o n t i n u o u s ( r e l a t i v e t o t h e

162 L. Cerlienco. G. Nicoletti and F. Piras

f i n i t e t o p o l o g y ) i f and onZy i f t h e r e e x i s t s a l i n e a r map f :C-+C such t h a t g=f”. Proof. In fact g is continuous if and only if its representing ma-

If cr=~a.b , B = c ~ . b and y=zy.b :=m(a@B), we have

(10) j=o J J 1-j so that hi’s are also the structure constants of the algebra C* relative toJ the pseudo-basis (b’). imp1 ie s :

Prop. 3 . If C i s a coaZgebra of b i n o m i a l t y p e , t h e n map ( 7 ’ ) i s an i somorph i sm of a l g e b r a s . Proof. ear map m(-@g), because of (lo), has columns with finite support.

1.3. Commutativity of diagrams (1) and (2) gives.rise to the following identities relative to structure constants hl:

(11)

(12)

trix M(g) has columns with finite support. 0 i i i

1 1 i i

y i = 2 h. a. B .

This, together with Prop.1 and 2,

In fact, for every @EC* the representing matrix of the lin-

I i .

J r h. h’ =

Obviously, C is cocommutative if and only if i i J

h. = himj. (13)

The structure constants hi of a coalgebra of binomial type can all be expressed in terms &f h i alone, which will be more simply denoted by

Prop. 4 . ~E{z,z, ..., s-11 5e haue:

ni:=ht. In fact, we have the following:

If rl .#O f o r lrjss-1 b u t n s = O , t h e n f o r e v e r y i and e v e r y

a )

+1 + j - 1 b ) i f n =O f o r some p > s , t h e n h!=hP =. . .=h;

i i + l . h i + s - j e l h : h = 0 .

=O; P 3 j

ni! J j * . . . j i i

I f i n s t e a d q . # O f o r e v e r y j, t h e n h j = ti. = , (where 3 11 0 ‘ i’ lli-i*

q 0 ! : = 1 and n i ! : = r l i - l ! oi ) f o r e v e r y i and e v e r y j . I n such a c a s e ,

C i s cocommuta t i ve . Proof. For r-1, (11) gives hi = hj-l ni/nj. From this, we deduce by recurrence both a) and b). Let us first prove c) when j=1. From (11) and a) we get

i-1


i i+l i+s- l - i+s-l=h . hl . . . .hl -0. When j > l , n ini+b".n and t h e n , s ince n =O,

c) f o l l o w s from tge case J = and from b j . The second p a r t o f t h e s t a t e m e n t i s now t r i v i a l . 0

Thus, e a c h c o a l g e b r a o f binomial t ype C i s c h a r a c t e r i z e d , r e l a t i v e t o t h e b a s i s ( b i ) , by t h e sequence TI=(, ) n nc wr i te C,=(V,A,,E) i n s t e a d o f C=(V,A,c). Furrhermore, C, i s s a i d t o be o f full b inomia l t y p e i f q l = l and q i # O f o r e v e r y i > O .

P r o p . 5 . Any t w o c o a l g e b r a s o f f u l l b inomial t y p e , s a y C and CA, are i somorph ic a s coa lgebras : ( 1 4 )

. Accord ing ly , w e s h a l l

ll

c a n c----t qi ! / A i .f bi . bi

P r o o f . T r i v i a l .

i i i Here a r e some examples: 1) Coalgebra o f po lynomia l s : C,,=K[xl, b . = x , h . = ( . ) , nn=n. C; i s t h e a l g e b r a o f d i v i d e d power s e r i e s . In t h e l f o l l o w l n g J we s h a l l deno te t h i s c o a l g e b r a w i t h C N .

2 ) Coalgebra o f d i v i d e d powers: C =K[x], h . = n . = l . C* i s t h e a l g e b r a of formal power s e r i e s . 3 ) q - e u l e r i a n c o a l g e b r a : C =K[x], h . = ( . ) i i = (Gaussian

c o e f f i c i e n t s ) and n i = [ i J := l + q + q + . . .+q C* i s s a i d t o be t h e a l g e b r a o f fo rma l e u l e r i l n ser ies .

i 11 1 1 rl

[i] !

rl J J [ j l q ! [ i - j l q ! 2 i-1 .

n

Coalgebras l i k e t h e s e have a s i g n i f i c a n t c o m b i n a t o r i a l c o u n t e r - p a r t . Let 5' be a l o c a l l y f i n i t e p a r t i a l l y o r d e r e d s e t ( f o r s h o r t , 1 . f . p o s e t ) t h a t s a t i s f i e s t h e f o l l o w i n g f u r t h e r c o n d i t i o n s : a ) a l l maximal c h a i n s i n a g i v e n i n t e r v a l [x,y] o f 9 have t h e same c a r d i n a l i t y ( e q u a l t o " l + l e n g t h [ x , y ] " ) (Jordan-Dedekind c h a i n cond i - t ion ) ; b ) a l l i n t e r v a l s o f l e n g t h n i n 9 p o s s e s s t h e same number, s ay B n , o f maximal c h a i n s ; c ) t h e r e e x i s t s i n '7 o n l y one minimal e l emen t . A f t e r [ 1 2 ] , t h e s e p o s e t s are s a i d t o be 1.f. p o s e t s of full b inomia l t y p e . With e v e r y 1 . f . p o s e t o f f u l l b inomia l t y p e o f i n f i n i t e l e n g t h one can a s s o c i a t e a c o a l g e b r a of f u l l binomial t y p e C,=(K[X],A,,E) - t h e s o - c a l l e d maximally reduced i n c i d e n c e c o a l g e b r a o f ( 7 - by de - n o t i n g w i t h b i t h e r e s i d u a l c l a s s o f a l l i n t e r v a l s o f t h e same l e n g t h i i n 9 t h e number l e n g t h i. I n $ h i s way, t h e c o a l g e b r a s c o n s i d e r e d above co r re spond r e s p e c t i v e l y t o t h e f o l l o w i n g p o s e t s : a ) t h e l a t t i c e o f a l l f i n i t e s u b s e t s o f a c o u n t a b l e s e t ; b) t h e c o u n t a b l e c h a i n ; c ) t h e l a t t i c e o f a l l f i n i t e - d i m e n s i o n a l subspaces o f a v e c t o r space o f dimension w

over GF(q) .

and assuming n i = B i . Thus, each s t r u c t u r e c o n s t a n t hf g i v e s h l = Bi/B,Bi-j o f e l emen t s o f r ank j i n any i n t e r v a l o f

164 L. Cerlienco, G. Nicoletti and F. Bras

1 2 .

In t h i s s e c t i o n we s h a l l show how b o t h automorphisms and hemimorphisms o f a c o a l g e b r a o f f u l l b inomia l t y p e C, are a s s o c i a t e d w i t h s p e c i a l sequences o f p o l y n o m i a l s , whose g r e a t i n t e r e s t i s well-known ( a t l e a s t i n t h e p a r t i c u l a r c a s e o f t h e c o a l g e b r a o f p o l y n o m i a l s ) .

2 . 1 . Let u s beg in by g e n e r a l i z i n g t h e n o t i o n o f po lynomia l sequence o f b inomia l t y p e ( s e e (171 ) , I n o r d e r t o s t u d y a n a l i t i c a l l y a c o a l g e b r a C=(V,A,E) g iven i n some i n t r i n s i c way, i t i s c lear t h a t we may a r b i t r a r i l y choose any b a s i s ( v i ) o f V . Then, a l l we have t o know i s t h e v a l u e o f s t r u c t u r e con-

s t a n t s T~ Jr , c i o c c u r r i n g i n A V . = . L T J ~ v.@v and E ( ~ . ) = E ~ . How- e v e r , t h e chosen b a s i s (v i ) i s h f h i A g ' b u t a u s e f u f t o o l . Thus , it may happen t h a t t h e a n a l y s e s r e g a r d i n g C c a r r i e d o u t u s i n g two d i f f e r e n t b a s e s ( v i ) , (v;) canno t be compared t o each o t h e r by means o f t h e map v i -v i . Le t C,=(V,a , E ) be a c o a l g e b r a o f ( f u l l ) b inomia l t y p e and l e t ( b i ) be a b a s i s l i x e d on i t . A new b a s i s (b:) o f V i s s a i d t o be an q-bas i s of C i f t h e t h e map

1

T h i s remark j u s t i f i e s t h e f o l l o w i n g d e f i n i t i o n .

rl f : C -c 0 rl

b . - b: 1

i s an automorphism o f c o a l g e b r a s , t h a t i s

( 1 5 )

. . A,b! = [!'I b ! g b i - j .

1 j = O J q 3 ( 1 6 )

Cons ide r t h e isomorphism $I: c -cN

i n bi- n i ! / i ! x

from C n sequence p i ( x ) of po lynomia l s i s n-nomial i f t h e r e e x i s t s a n - b a s i s (b:) i n C,

P r o p . 6 . A po lynomia l sequence p i ( x j d ( [ x ] , ; E N , i s q-nornial i f and onZy i f t h e f o l l o w i n g s t a t e m e n t s h o l d :

t o t h e c o a l g e b r a o f po lynomia l s C N . We s h a l l s a y t h a t a

such t h a t p i ( x ) = $ ( b i ) . I t i s s imple t o p rove t h a t :

1 ) d e g f p i l = i;

3 ) p i l o ) = 0 f o r e v e r y i f 0 ; 2 1 p o ( x l = I;

0

The i n t e r e s t i n q-nomial s equences o f po lynomia l s i s due t o t h e f a c t t h a t t h e y e n a b l e u s t o c a r r y o u t n-ana log o f umbra1 c a l c u l u s a l o n g t h e l i n e s fo l lowed b y Rota and o t h e r s

The f o l l o w i n g p r o p o s i t i o n s p r o v i d e u s w i t h a u s e f u l t o o l i n o r d e t t o g e t 17-nomial s equences .

P r o p . 7 . f:C,,-C, be a morphism of coaZgebra8. Then t h e r e p r e s e n t a t i v e m a t r i x M ( f )

[17] , [18] ( s e e a l s o [ 8 ] , 191 ,[14).

Le t C, be a coa lgebra of fuZZ binomiaZ t y p e and Zet

i s c o m p l e t e l y de t e rmined by f"(bl):

where t h e i - t h power i s c a l c u l a t e d i n C , .

Proof. With a straightforward calculation, from (3) we get

0

t

t r + s <r+slflt>.[ 1 = j g o 151 <rlflj><slflt-j>; <0/f/t>=6

n rl which imply (18).

i i * i If a = . Z a.b , B = . Z Bib EC,, the element .Z (a./n. ! ) B E C ~ is said

to be the c o m p o s i t i o n of a and 13 and denoted by a 0 B .

l d 0 1 1) 1 130 1 1

Prop. 8 . The map A

( 1 9 1 Aut (C, l - C ;

f ' - f * ( b l ) i s an i somorph i sm o f t h e group Au t lC , ) o f - t h e automorphisms of t h e c o a l g e b r a C , on t h e c o m p o s i t i o n a l group (CG,o) of t h e e l e m e n t s a=ZaibicC; such t h a t ao=O#al.

Proof. Because of (18), map M(fog)=M(f)xM(g) it follows:

1 (fog)' (b ) = i$o <f"(bl)

19) is a bijection. Moreover, from

2.2. We come now to sequences of polynomials associated with hemimorphisms. For the sake of simplicity, in the remainder of this section we assume that the underlying vector space of the coalgebra of binomial type (not necessarily of full binomial type) C, is V=K[x] with its canonical basis bi=xi:

A x = Z 11. xj8x . Moreover, let,us identify the linear dual of K[x] with K[[x]] and denote also b1 by XI; thus, in general, the "series" xi is not the i-th power 6f x1 in C;. i i Consider in C: the element <=.I x (zeta-function) and let u=.Z LJX

i i i i-j rl j=o J

1+0 130 1

(Miibius function) be its multiplicative inverse: r,u =l. We have:

P r o p . 9. and l e t

L e t f: C,+C be a hemimorphism s u c h t h a t i n d ( f l = f * ( l ) = u rl

i D ( x i := f l x n i = .; h l ' I ~ - ~ x E C . ' n z=o n

Then t h e sequence ( P ix,yilneN of homogeneous p o l y n o m i a l s d e f i n e d by P ( x , l l = p ( x i s a t i s f z e s t h e i d e n t i t i e s :

( 2 0)

( 2 1 )

a

n P ( x , y J = Z hn P ( x , z l P I z , y l

P ( 1 , O ) = 1 .

n k=o k k n-k

n

More generally:

166 L. Cerlienco, G. Nicoletti and F. Piras

i P r o p . 1 0 . If u = . E u.x i s an element o f C;, g:C,-C i s t h e hemi-

morphism of i n d i c a t o r u, s 1xJ i s t h e polynomial g ( x I = L h a x

and S ( x , y l t h e homogeneous polynomial such t h a t S ( x , l ) = s ( x J , t h en ue have n (22 )

uhere Pk Ix ,y l i s as i n Prop. 9 .

I n o r d e r t o p rove t h e p r e v i o u s p r o p o s i t i o n s , w e need t h e f o l l o w i n g f o u r lemmas. Lemma 1 .

and S ( ~ , y ) = , ~ ~ sn x y

( 2 3 )

is e q u i v a l e n t t o ( 2 2 1 .

P r o o f . Conside6 t h t igcittity

Formula ( 2 2 ) becomes :

A n i 2 3 0 z n n i=o i n- i

n n

S ( x , y ) = c hn P ( x , z ) S f z , y ) n k=o k k n-k

n r r n-r

For a r b i t r a r i l y g i v e n polynomiaZs P ( ~ , y ) = ~ : ~ p n x y

4 z hn p r s q - k = 6 q s q , q w

n n r r n-r , t h e i d e n t i t y

n

k = r k k n - k r n

n n - t t + r Z E A ( k , r s t ) = tzo r f o k f r A ( k s r , r + t - k ) k=o r = o t = o ( 2 4 )

(because o f ( 2 4 ) ) r r n - r n n - t t+r r + t - k r n - r - t t - = r i o ~ n ~ ~ - E z z h n p r s X Y z - t = o r = o k = r k k n-k

n n n - t t+ r I: hn pr s r + t - k r n - r - t t

so IxrYn-' - t=l E r z o ' k = r k k n-k y z =o. = rso {Sn-hr P r n- r

0 P u t t i n g r + t = q , t h i s i s e q u i v a l e n t t o ( 2 3 ) .

Lemma 2 .

s a t i s f y i n g ( 2 0) and 1 2 1 ) .

& ( h n )

( 2 5 1

Thus, t h e sequence P (x,yJ i s compZetely determined by t h e c o e f f i - c i e n t s hn.

n r r n-r Let P I X , ~ ) = ~ $ ~ p n x y

( w i t h hn=o f o r n < k ) . Then

be a sequence o f polynomials n Le t u s c o n s i d e r t h e m a t r i c e s P = ( p r l and

n

t . k k

tl?.P = Pa h = I.

n k

P r o o f . From (ZO), when y=o and z = 1 , we g e t n

xn = hn P ( x , l ) . k i o k k T h i s , t o g e t h e r w i t h n

pn (x , l ) = kgo P: X k , e x p r e s s e s t h e change o f b a s e s xn-P ( x , l ) i n K[x], t h a t i s ( 2 5 ) ~ n


Lemma 3 . I n t h e h y p o t h e s e s o f Lemma 2 . , t h e i d e n t i t y

( 2 6 )

i s e q u i v a l e n t t o / 2 3 ) , and t h e n t o ( 2 2 ) .

P r o o f . sum w i t h r e s p e c t t o t h e index r , t h e n w e g e t :

r (23) i m p l i e s ( 2 6 ) . I n f a c t , i f we m u l t i p l y (23) by ht and

k ,q n hq t = r=t 2 k = r 9 hn k p r k ,qqk n-k hr t = k = t 2 hn k sq-k n-k r= t 1 p i hf =

(because o f ( 2 5 ) ) = 3 hn sq -k g k = hn sq-t

k = t k n-k t t n - t ' C o n v e r s e l y , (26) i m p l i e s (23) : 6: s: = ( b e c a u s e o f ( 2 5 ) ) = sn 9 9 k& h: pk r = 1 hn ,q-k r

k = r k n-k 'k' cr n

L e t P ( x , y ) = z Lemma 4 . p k ~ ~ y ~ - ~ be an a r b i t r a r y sequence of po lynomiaZs . Then , (20) a f 2 3 ) I ( ( 2 5 1 and (26)) w i t h s k k = p . P r o o f . A s i n Lemma 1. and Lemma 3 .

n k=o n

n n

0

n Proof o f P r o p . 9 . t h e h y p o t h e s i s c p = 1 i s e x p r e s s e d by ( 2 5 ) . T h e r e f f i r e , owing t o Lemma 4 . , i t i s s u f f i c i e n t t o p r o v e f o r m u l a ( 2 6 ) . I n f a c t we h a v e :

C o n s i d e r t h e c o e f f i c i e n t s pA=hi+ u ~ - ~ . . . Observe t h a t

= ( b e c a u s e o f ( 1 1 ) ) = hn hq - qwt = h: hn-t ~

ht P n - t q - t n-q q t n-q - ht P n * By a s i m i l a r a rgumen t , one can p r o v e P rop . 1 0

C o n v e r s e l y , i f we c a l l any sequence o f homogeneous p o l y n o m i a l s P (xd ( r e s p e c t i v e l y , S n ( x , y ) ) s a t i s f y i n g (20) and ( 2 1 ) ( r e s p e c t i v e l y , ( 9 2 ) ) r e l a t i v e t o s u i t a b l e c o e f f i c i e n t s hf a GoZdman-Rota-sequence ( r e spec - t i v e l y GoZdman-Rota- A e f f e r - s e q u e n c e ) , i t i s p o s s i b l e t o p r o v e t h a t :

Prop. 1 1 . The c o e f f i c i e n t s h: (hn#Ol a s s o c i a t e d w i t h any Goldman- -Rota-sequence can be assumed as s t g u c t u r e c o s t a n t s o f a coa lgebra o f b i n o m i a l t y p e . P r o o f . We have t o p r o v e t h a t (20) i m p l i e s b o t h (11) and ( 1 2 ) . We have

i h i h!-r = 1 A i hk hk-r = ( b e c a u s e o f Lemma 2 . ) r J - r k = r k r j - r

i i k i = z { z p hn}h: hk-r = ( b e c a u s e of ( 2 6 ) )

k = r n=k n j -r

i i k-r k- r = h: h: k z r n h . k - r k - r = = z c h i h .

( b e c a u s e o f Lemma 2 . ) k = r n=k n j - r hr 'n-r n = r J - r Pn-r

. . i . n n - r = c h: hr 6 j - r = h i h: , n = r t h a t i s f o r m u l a (11); Moreover , (12 ) i s a s t r a i g h t f o r w a r d c o n s e - quence o f (21) and Lemma 2 .

168 L. Cerlienco. G . Nicoletti and F. Piras

In conclusion, we give a combinatorial interpretation to both Gold- man- Ro ta- sequences and Goldman- Ro ta- Shef fer- sequence s.

Prop. 12. L e t C, be t h e maximal ly r e d u c e d i n c i d e n c e c o a l g e b r a o f a 1 . f . p o s e t of f u l l b i n o m i a l t y p e 9. In r e s p e c t o f t h i s c o a l g e b r a C q J l e t u s c o n s i d e r t h e Goldman-Rota-sequence Pnlx,yl and t h e Goldman- -Rota- S z e f f e r - s e q u e n c e a s s o c i a t e d w i t h t h e s e r i e s c=;.,~ x n . F u r t h e r - more, l e t pn(x)=Pn(x,l) and s n ( x 1 = ~ ( x , l ) . Then e a c h z n t e r v a l of l e n g t h n i n T has p n ( x l a s its c h a r a c t e r i s t i c p o l y n o m i a l and s , ( x l a s i t s l e v e l number i n d i c a t o r ( t h a t i s , t h e c o e f f i c i e n t s of S n ( x ) a r e t h e l e v e l numbers of s e c o n d k i n d o f t h e g i v e n i n t e r v a l ) .

?O

Proof. See [7].

REFERENCES:

Abe, E., Hopf Algebras (Cambridge Univ. Press, Cambridge, 1980).

Aigner, M., Combinatorial Theory (Springer-Verlag, New York, 1979).

Allaway, W.R., A Comparison o f two Umbral Algebras, J.Math. Anal.App1. 85 (1982) 197-235.

Andrews, G.E., On the Foundations of Combinatorial Theory V: Eulerian Differential Operators, Studies in Appl.Math. 50

Cerlienco, L . and Piras, F . , Coalgebre graduate e sequenze di Goldman-Rota, Actes du Sdminaire Lotharingien de Combinatoire, Publ. de l’IRMA, Strasbourg, 230/S-09 (1984) 113-125.

Cerlienco, L . and Piras, F. , Aspetti coalgebrici del calcolo umbrale, Atti del Convegno “Geometria combinatoria e d’incidenza: fondamenti e applicazioni”, Rend.Sem.Mat. Brescia 7

Cerlienco, L. and Piras, F., G-R-sequences and incidence coalgebras of posets o f full binomial type, J.Math.Anal.App1. (to appear).

Cerlienco, L., Nicoletti, G. and Piras, F., Automorphisms of graded coalgebras and analogs of the umbra1 calculus, Actes du Sdminaire Lotharingien de Combinatoire. Publ. de l’IRMA, Strasbourg, 230/S-09 (1984) 126-132.

Cerlienco, L., Nicoletti, G. and Piras, F., Coalgebre e Calco- l o Umbrale, Rend.Sem.Mat.Fis. Milano (to appear).

Cerlienco, L., Nicoletti, G. and Piras, F., Umbral Calculus, Actes du Sdminaire Lotharingien de Combinatoire, Publ. de l‘IRMA, Strasbourg, 266/S-11 (1985) 1-27.

(1971) 345-375,

(1984) 205-217.

1113 Comtet, L., Advanced Combinatorics (Reidel P.C., Boston, 1974).


I 191

Doubilet, P., Rota, G.-C. and Stanley, R . P . , On the Foundations o f Combinatorial Theory VI: The Idea of Generating Function, in: Rota, G.-C. (Ed.), Finite Operator Calculus (Academic Press, New York,1975) 83-134.

Garsia, A.M. and Joni, S.A., Composition Sequences, Comm. Al- gebra 8 (1980) 1195-1266.

Goldman, G.R. and Rota, G.-C., On the Foundations o f Combina- torial Theory IV:’ Finite Vector Spaces and Eulerian Generating Functions, Studies in Appl.Math. 49 (1970) 239-258.

Ihrig, E.C. and Ismail, M.E., A q-umbra1 Calculus, J.Math.Ana1.

Joni, S.A. and Rota, G.-C., Coalgebras and Bialgebras in Com- binatorics, Studies in Appl.Math. 61 (1979) 93-139.

Mullin, R. and Rota, G.-C., On the Foundations o f Combinato- rial Theory 111: Theory o f Binomial Enumeration, in: Harris (Ed), Graph Theory and its Applications (Academic Press, New York, 1970) 167-213.

Rota, G.-C., Kahaner, D. and Odlyzko, A., On the Foundations o f Combinatorial Theory V I I I : Finite Operator Calculus, J.Math. Anal.App1. 42 (1973) 684-760.

Taft, E.J., Non-cocommutative Sequences of Divided Powers, Lecture Notes in Math. 933 (Springer-Verlag, New York, 1980)

Appl. 84 (1981) 178-207.

203-209.

Annals of Discrete Mathematics 30 (1986) 171-184 0 Elavier Science Publishers B.V. (North-Holland) 171

R-REGULARITY AND CHARACTERIZATIONS OF THE GENERALIZED QUADRANGLE P(W(s), ( - ) )

M . De S o e t e a n d J . A . T h a s S e m i n a r of Geometry

S t a t e U n i v e r s i t y o f Ghent K r i j g s l a a n 2 8 1

B-9000 Gent -Belg ium

I n a g e n e r a l i z e d q u a d r a n g l e o f o r d e r ( s , s + 2 ) , s # 1, r e g u l a r p o i n t s c a n n o t o c c u r . T h e r e f o r e we i n t r o d u c e t h e n o t i o n o f R - r e g u l a r i t y f o r p o i n t s a n d l i n e s i n t h o s e g e n e r a l i z e d q u a d r a n - g l e s o f o r d e r , ( s , s t 2 ) w h i c h c o n t a i n a s p r e a d R U s i n g t h e s e new c o n c e p t s we g i v e t h r e e c h a r a c - t e r i z a t i o n t h e o r e m s f o r P ( W ( s ) , ( m ) ) .

I . I N T R O U U C T I O N

1. DEFINITIONS A f i n i t e generaZized quadrangle i s a n i n c i d e n c e s t r u c t u r e

S = ( P , B , I ) where P a n d B are s e t s o f e l e m e n t s c a l l e d points a n d Zines r e s p . , w i t h a s y m m e t r i c i n c i d e n c e r e l a t i o n I w h i c h s a t i s f i e s t h e f o l l o w i n g a x i o m s :

( i ) e a c h p o i n t i s i n c i d e n t w i t h l+t l i n e s ( t 1 ) a n d two d i s t i n c t

( i i ) e a c h l i n e i s i n c i d e n t w i t h l t s p o i n t s (s > 1) a n d t w o d i s t i n c t p o i n t s a r e i n c i d e n t w i t h a t most o n e l i n e ;

l i n e s are i n c i d e n t w i t h a t most one p o i n t ;

u n i q u e p a i r (y ,M) E P X B s u c h t h a t x I M I y I L . ( iii) f o r e a c h n o n - i n c i d e n t p o i n t - l i n e p a i r (x,L), t h e r e e x i s t s a

We c a l l s a n d t t h e parametersof t h e g e n e r a l i z e d q u a d r a n g l e , a n d ( s , t ) ( o r s i f s = t ) i s t h e o r d e r of S . T h e r e h o l d s IPI = v = ( l + s ) ( l + s t ) , IBI = b = ( l + t ) ( l + s t ) a n d s + t I s t ( s t l ) ( t + l ) 1 1 2 1 . M o r e o v e r t h e r e i s a point-line duality f o r g e n e r a l i z e d q u a d r a n g l e s o f o r d e r ( s , t ) ; t h i s means t h a t i n a n y d e f i n i t i o n o r t h e o r e m t h e w o r d s p o i n t a n d l i n e a n d t h e p a r a m e t e r s s a n d t may b e i n t e r c h a n g e d .

If t h e p o i n t s x , y ( r e s p . l i n e s L,M) a r e c o l l i n e a r ( r e s p . c o n c u r - r e n t ) w e w r i t e x - y ( r e s p . L - M ) . The l i n e d e f i n e d by d i s t i n c t c o l l i n e a r p o i n t s x , y i s d e n o t e d by x y ; t h e p o i n t d e f i n e d by d i s t i n c t c o n c u r r e n t l i n e s L,M i s d e n o t e d by LM o r L n M .

112 M. de Soete and J. A . Thas

Let S be a g e n e r a l i z e d q u a d r a n g l e of o r d e r ( s , t ) . For x E P , we 1

y l and i s d e f i n e t h e s t a r of x as x' {z E P II z - X I ; remark t h a t x E x . The t r a c e of two d i s t i n c t p o i n t s ( x , y ) i s t h e s e t x d e n o t e d by { x , y } . T h e r e h o l d s I { x , y I I = s + l i f x - y and I { x , y } 1 t + l i f x f y . More g e n e r a l l y , f o r A C P we d e f i n e

A' n{xl I1 x E A } . Fo r x # y we d e f i n e t h e span o f ( x , y ) as t h e s e t { x , y } l l { u E P II u E z', Vz E { x , y } I . If x f y , t h e s e t {x,y}" i s a l s o c a l l e d t h e h y p e r b o Z i c l i n e d e f i n e d by x and y . We have I [ x , y } I s + l i f x - y and I { x , y } I < t + l i f x f y . A p a i r of d i s t i n c t p o i n t s ( x , y ) i s r e g u l a r p r o v i d e d x - y o r x f y and l{x ,y} l11 = t + l . A p o i n t x i s r e g u Z a r p r o v i d e d ( x , y ) i s r e g u l a r f o r a l l y P , y # x . If S h a s a r e g u l a r p a i r o f r ion- .cn l l in , -a r p o i n t s , t h e n s 1 o r s t 1 9 1 . A triad of p o i n t s i s a t r i p l e o f p a i r w i s e n o n - c o l l i n e a r p o i n t s . A p o i n t u i s c a l l e d a c e n t e r of t h e t r i a d ( x , y , z ) i f f u E x

1

1 1

1

1

11 11

1 i n y l n z'. A s p r e u d o f S i s a s u b s e t R o f B s u c h

t h a t e a c h p o i n t o f S i s i n c h o l d s I R I s t + l .

2 . THE MODELS W ( q ) , T 2 ( 0 ' ) , ( a ) The p o i n t s of PG(3,q

d e n t w i t h e x a c t l y one l i n e o f R . T h e r e

T;(O) AND P ( S , x ) t o g e t h e r w i t h t h e t o t a l l y i s o t r o p i c

l i n e s w i t h r e s p e c t t o a s y m p l e c t i c p o l a r i t y , d e f i n e a g e n e r a l i z e d q u a d r a n g l e W(q) w i t h p a r a m e t e r s s = t = q , v = b = ( q + l ) ( q tl). We remark t h a t t h e l i n e s of W(q) a re t h e e l e m e n t s o f a l i n e a r l i n e complex i n PG(3 ,q ) [ 1 0 1 .

2

All p o i n t s o f W(q) a r e r e g u l a r ; t h e l i n e s of W(q) a r e r e g u l a r i f f q i s even [ 9 1 . We n o t e a l s o t h a t W(q) i s s e l f - d u a l i f f q i s even

I 9 1 . ( b ) L e t 0 ' b e a n o v a l [ 4 ] o f PG(2 ,q ) H , where H i s embedded i n

PG(3 ,q) P . D e f i n e p o i n t s as ( i ) t h e p o i n t s o f P \ H , ( i i ) t h e p l a n e s X o f P w i t h IX n 0'1 = 1, (iii) a new symbol (m). The l i n e s are ( a ) t h e l i n e s o f P which are n o t c o n t a i n e d i n H and meet O f ,

and ( b ) t h e p o i n t s of 0'. The i n c i d e n c e i s d e f i n e d as f o l l o w s . A

p o i n t o f t y p e ( i ) i s o n l y i n c i d e n t w i t h l i n e s of t y p e ( a ) and t h e i n c i d e n c e i s t h a t o f P . A p o i n t of t y p e ( i i) i s i n c i d e n t w i t h t h e l i n e s of t y p e ( a ) and ( b ) c o n t a i n e d i n i t . The p o i n t ( m ) i s i n c i d e n t w i t h no l i n e of t y p e ( a ) b u t w i t h a l l l i n e s of t y p e ( b ) . The o b t a i n e d s t r u c t u r e i s a g e n e r a l i z e d q u a d r a n g l e w i t h p a r a m e t e r s s = t q , v = b = ( q + l ) ( q t1) and i s d e n o t e d by T 2 ( O ' ) .

These q u a d r a n g l e s a r e due t o J . T i t s [4]. The q u a d r a n g l e T 2 ( 0 ' ) i s i s o m o r p h i c t o W(q) i f f q i s even and 0' i s a n i r r e d u c i b l e c o n i c 1 9 1 . F u r t h e r , a l l p o i n t s o f t y p e ( b ) a r e r e g u l a r , and t h e p o i n t ( m )

i s r e g u l a r i f f q i s even [ 9 l .

2

R-Reguiaritv of the Generalized Quadrangle P( W ( d . (-I) 173

( c ) L e t 0 b e a c o m p l e t e o v a l ( i . e . a ( q + 2 ) - a r c 1101) o f t h e p l a n e a t i n f i n i t y o f A G ( j , q ) , q e v e n . The p o i n t s o f t h e s t r u c t u r e T ; (O) a r e t h e p o i n t s o f AG(3,q) ; t h e l i n e s o f T;(O) a re t h e l i n e s of AG(3,q) i n t e r s e c t i n g t h e p l a n e a t i n f i n i t y i n t h e p o i n t s o f 0 ; t h e i n c i d e n c e i s t h a t o f AG(3,q). Then T;(O) i s a g e n e r a l i z e d q u a d r a n g l e o f o r d e r ( q - l , q + l ) . It w a s f i r s t d i s c o v e r e d by R.W. A h r e n s a n d G. S z e k e r e s [ 1 ] and i n d e p e n d e n t l y by P4. H a l l J r . I 5 1 .

A p a i r o f n o n - c o n c u r r e n t l i n e s (L,M) o f T;(O) is r e g u l a r i f f t h e

( d ) I n [ 7 1 S . E . Payne g i v e s a n i m p o r t a n t c o n s t r u c t i o n o f l i n e s a r e p a r a l l e l 1 9 1.

g e n e r a l i z e d q u a d r a n g l e s o f o r d e r (s-l,s+l). C o n s i d e r a g e n e r a l i z e d q u a d r a n g l e S = ( P , B , I ) o f o r d e r 5,s > 1, w i t h a r e g u l a r p o i n t x . D e f i n e PI as t h e s e t P \ x . I n B' t he re a r e two t y p e s o f e l e m e n t s :

t h e e l e m e n t s of t y p e ( a ) a r e t h e l i n e s o f B which a r e m t i n c i d e n t w i t h x , t h e e l e m e n t s o f t y p e ( b ) a r e t h e h y p e r b o l i c l i n e s { x , y } , y t. x. Now w e d e f i n e t h e i n c i d e n c e r e l a t i o n . If y E P ' , L E B ' w i t h

L a l i n e o f t y p e ( a ) , t h e n y 1' L i f f y I L ; i f y E I" a n d L E B' w i t h L a l i n e o f t y p e ( b ) t h e n y I ' L i f f y E L . Then t h e s t r u c t u r e S ' = ( P f , B f , I f ) i s a g e n e r a l i z e d q u a d r a n g l e o f o r d e r (s-l,s+l) a n d i s d e n o t e d by P(S,x).

o f W(q), i s i s o m o r p h i c t o a T;(O) ( h e r e 0 i s a n i r r e d u c i b l e c o n i c t o g e t h e r w i t h i t s n u c l e u s ) [ 9 1. The g e n e r a l i z e d quadrang!e P(T2(01),(m)), w i t h T 2 ( 0 ' ) as i n ( b ) a n d q e v e n , i s i s o m o r p h i c t o T;(O) where 0 = 0' U { n } w i t h n t h e n u c l e u s o f 0' [9].

I n P ( W ( q ) , x ) , q o d d , a p a i r of n o n - c o n c u r r e n t l i n e s (L ,M) i s r e g u l a r i f f o n e o f t h e f o l l o w i n g c a s e s o c c u r : (i) L a n d M a r e l i n e s o f t y p e ( b ) , ( i t ) L a n d M a r e c o n c u r r e n t l i n e s o f W(q) ( b u t a r e n o t c o n c u r r e n t i n P ( w ( q ) , x ) ) ; i n P ( w ( q ) , x ) , q e v e n , a p a i r of non-con- c u r r e n t l i n e s ( L , M ) i s r e g u l a r i f f o n e o f t h e f o l l o w i n g c a s e s o c c u r s : ( i ) L a n d M are l i n e s o f t y p e ( b ) , ( i i ) i n W(q) some l i n e o f {L,M)' i s i n c i d e n t w i t h x.

1

11

I n t h e e v e n c a s e t h e g e n e r a l i z e d q u a d r a n g l e P ( W ( q ) , x ) , x a p o i n t

1 7 . R-REGULARITY O F POINTS AND LINES

1. DEFINITIONS C o n s i d e r a g e n e r a l i z e d q u a d r a n g l e S ( P , B , I ) o f o r d e r ( s , s + 2 ) ,

s > 1. S i n c e 1 .: s .< t r e g u l a r p o i n t s c a n n o t o c c u r [ 9 1 . M o r e o v e r , i n t h e known e x a r z p l e s a l s o r e g u l a r l i n e s d o n o t o c c u r . T h e r e f o r e we i n t r o d u c e t h e c o n c e p t o f R - r e g u l a r i t y .

174 M. de Soete and J.A. Thas

I n what fo l lows w e a l w a y s assume t h a t t h e g e n e r a l S = (P,B,I) of o r d e r (s,s+2) c o n t a i n s a sp read R ( I R For x E P, we d e f i n e x = { z E P I1 z -. x , z # x, zx 1.

quadrangle 2

u { X I . s t l ) ) .

I * For a p a i r of d i s t i n c t p o i n t s x,y we deno te t h e set x I x , y } ' * . I f x f y o r x - y but xy 4 R , t h e r e h o l d s I { x , y ) More g e n e r a l l y , f o r A C P we d e f i n e A'* 9 Ix

uz 4 R , Vz E ( x , y I 1 * } . So we o b t a i n I I x , y ) ' * l * I < s t l . I f x - y = xy and s o ~ { ~ , y } ' * ~ * ~ = s t l . 1.1. but xy 4 R , t h e n c l e a r l y I x , y l

and xy 4 R , o r x f y and I Ix ,y} I = s t l . A p o i n t x i s R-regular

provided (x ,y ) i s R-regular for a l l y E P, y f x. A R-grid i n S i s a s u b s t r u c t u r e S ' = ( P f y B 1 , I 1 ) o f S d e f i n e d as

P' I x i j E P II i = l Y . . . , s t 2 , j = l , . . . ,s t2 , and i # j } ,

B' = I L 1 , . . . , L s t 2 , M 1 , . . . , M s t 2 } c B \ R , and

n yl* as 1. I s t l .

1. II x E A } . So f o r a l * l * - p a i r of n o n - c o l l i n e a r p o i n t s x ,y we have { x , y l - I u E P II u - z,

A p a i r of d i s t i n c t p o i n t s x,y i s c a l l e d R-regular provided x -. y 1.1,

fo l lows :

I' I n ( ( P I X B ' ) U ( € 3 ' X P I ) ) , w i th Li f L - x !j j i y

Mi -f- M j y j y Li 17 M j x i j i f i f j , Li j . M i , x X i j x j i = R i j R j i E R f o r 1 6 i, J G s t 2 .

and

11. We deno te t h e set {L 1y.. . ,Lst2! ( r e s p . !MI,.. * ,Mst2)) by { L i y L j 1 o r { M i , r C l 1 * ( r e s p . ( M i y M . I l l o r { L i , L . l l * ) f o r any i # j .

J J J I f L1,L2 E B \ R, L1 j . L 2 , t hen by d e f i n i t i o n t h e p a i r (L1 ,L2) i s

R-regular i f f ( L l a L 2 ) belongs t o a R-grid. I n such a case t h e r e e x i s t s a unique R E R for which L1 - R - L 2 . A l i n e L E B \ R i s weak R-reguZar i f f (L ,M) i s R-regular f o r a l l M E B \ R w i th L .f- M and IIL,M} RI 1. A l i n e L E B \ R i s R-reguZar i f f L i s weak R-regular and f o r a l l M E B \ R , L -f- M y w i t h I{L,Mll p a i r (L,M) i s r e g u l a r .

R - r e g u l a r i t y f o r p o i n t s .

1

R i f 1, t h e

F i n a l l y , n o t i c e t h a t R - r e g u l a r i t y f o r l i n e s i s n o t t h e d u a l of

2 . EXAMPLES

2 . 1 . Theorem. Cons ider P(W(q),x) ( P ' , B 1 , I ' ) and l e t R be t h e s e t

of a22 l i n e s of t y p e ( b ) in B ' ( s e e 1.2.(d)) . Then each p o i n t

i s R-reguZar. Each l i n e of B ' \ R i s R-regular iff q is e v e n .

P r o o f . L e t W(q) = ( P , B , I ) . Choose a p o i n t x i n W(q). It i s obvious t h a t t h e set ( ( x , y l I1 y E P , x f y l d e f i n e s a s p r e a d R i n P(W(q),x). 11

zed - -

4 R

R-Regularit). of the Generalized Quadrangle P(Wls1. 1-11 175

L e t y,z E I", y % ' z . Then i t f o l l o w s t h a t y 4 z. We know t h a t (y,z) i s re&ular i l l ' d ( g ) . L e t zO, 2 ' E x1 b e d e f i n e d by { y , z } l n x = { z g }

1 0

i [ y , z ) " '-I x1 = {?<,!,I. Prom 11.1. a n d 1;he d e f i n i t i o n o f R we 1 1 . 1 I * 1 ' * 1

s ~ t 3 i i ! ! tha t I{y,z,} 1 = I{y,z) \ {zojl q a n d l{y,z} I = 11

: { y , z I \ [ z i t i g. Helice e a c h p o i n t o f F ( W ( q ) , x ) i s R - m g u l a r .

L e t L l , L 2 6 D' \ R , L1 % ' L-,. if L1 - L2, (L1,L2) i s a r e g u l a r 1 '<

I- i n F ( W ( ~ l ) , x ) a n d {L1,L21 c R . I f L1 4 L, but L1,L2 a re b o t h k ,qncur r , c r t t w i t h a same l i n e or W(q) through x, t h e n (L1,L2) i s

L t 1 m (bot ,k i n W(?) a n d i n P(W(q) ,x)) i f f q i s even. IIere we h a v e { I , l , L q l coricuri7.ent w i t h a same l i n e t h r o u g h x i n l d ( q ) . T h e n I {L1,L2}" i? R I

= 1. If' q is e v e n , (L1,L2) i s r e g u l a r i n k l ( q ) . I n t h i s c a s e , l e t

L . I xi I !<Ii, x E x l . a R - g r i d i.n ? ( W ( q ) , x ) . I n d e e d , Li - I M . i f f i # j, a n d s i n c e f o r

J xi, x.. '- x . ( w i t h x - a n y i # j t x i , x . ) J c t x i , x i j l , x . . J 1 - J 1 J 11 i j -

L i (1 X. and x j i = I,. 9 M . ) a n d h e n c e x.. E I x , x . . I , w e h a v e x . . - ' x.. w i t h x . . x . 6 R . We c o n c l u d e t h a t e a c h l i n e of B' \ R i s R - r e g u l a r iff' q i s e v e n . n

i

I' 8 ) R = 0. P i n a l l y , suppose tha t L1 # L2 a n d L1,L2 a r e n o t

1 and l e t 1 1 1 - tL,;,Lql - L = tM1,... Y M(-lt 1 1 Y L1 Y L2 1 - [ L1, . . - , Lq+ 1

Then t h e l i n e s LiyMi, 1 S i S q+1, d e f i n e i

1

J J 1 J 1 1J

1 J J 1 1~ ~i

2 . 2 . Ttieoi~eni. L e t R be t h e s e t oj- a 7 1 l i n e s t h r o u g h ii p o i n t x of 0 i n 1 1 ' T ( 3 ) . A point y, r e s p . a 7ine L @ R , is R - r e g u l a r if a n d

L

IJ ;,+' O \ { X I < s n coizi,-.

, J P ~ ? : . It i s i m m e d i a t e t h a t R i s 3 ?? read i n T;(O). If 0 \ { X I i s a c'onic tilerc' h o l d s T: (3) 2 P ( W ( \ q ) , y ) , q e v e n . A p p l y i n g t h e f o r e g o i n g theorem therc. f o l l o w s t h e R-regulcrity.

L

Ti"0 p r o v e tkie c o n v e r s e , w? remark t h a t T;(O) P(T2(0'),(m)),

1 ' 0 \ t x } ( 1 . 2 , ( d ) ) , Assume t h a t T;(O) = ( P , B , I ) a n d T2(0') = ( P 1 , B 1 , I 1 ) . C o n s i d e r a p o i n t y1 E P s u c h t h a t yl i s R - r e g u l a r . T h e n yl i s I p o i n t o f type ( i ) i n T,(O'). We p r o v e t h a t y1 i s r e g u l a r i n T2(O').

tlie p a i r (yl,y,j i s R - r e g u l a r i n T;(O). L e t [y1,y2) {y,,y ?, . . . , yp). S i n c e {y1,y21

l l*l*

iY 1 ,Y?} l * l * C {y1,y21 ( v , , y , ) is r e g u l a r i n T,(O'). Now r u p p o s e t h a t y2 - y l y i . e . t h e l i n t - y y E R . { X I , a n d by J 1 , . . . , N t he p l a n e s d e f i n e d by y1 ,y2 ,x i , i = 1, . . . , q + 1, w h i c h meet 0 \ { x i = 0' i n a u n i q u e p o i n t . T h e n {ylYy21" = t W 1 , . . . , W q t l }

L e t y2 E P ' , y2 f' y l Y w i t h y 2 a p o i n t of t y p e (i). If y1 i' y2, 181' - -

j ' Z

I " C tyl,y211' t h e r e r e s u l t s y i --I

I' E {y,,y,] ( see 1 . 3 . 4 . i n 1 9 1 ) . C o n s e q u e n t l y ""J', I VYi E [YpY,]

. U s i n g a g a i n 1.3.4. i n [9 I w e o b t a i n t h a t

D e n o t e by X ~ , . . . , X ~ + ~ t h e p o i n t s o f 0 \ 1 2

q + 1

116 M. de Soete and J.A. Thas

1 'I ' = { z II z 1 y,y,l u I ( - ) ) s ince y y a n d {Y,,Y21 1 2 'i'

V i l,...,qtl. Again ( y l , y 2 ) i s a r e g u l a r p a i r i n T2(0'). N e x t , l e t X E P' be a p o i n t of t y p e ( i i ) w i t h y1 -f.' X . The t r a c e I y , , X l c o n t a i n s a t l ea s t two p o i n t s x1,x2 of t y p e ( i ) w i t h x1 7 L ' x2. On t h e o t h e r h a n d y 2 o f t y p e ( i ) . C l e a r l y y2 7L' y l . From t h e f o r e g o i n g i t f o l l o w s t h a t ( y l , y , ) i s r e g u l a r i n T 2 ( 0 1 ) . S i n c e {xl,x21 C Iy1 ,y2I1 ' a n d Iy l ,Xl C Ixl,x2li', t h e p a i r ( y l , X ) i s a l s o r e g u l a r i n T2(O'). F i n a l l y , f rom t h e r e g u l a r i t y of ( - ) i n T 2 ( 0 ' ) , q e v e n , t h e r e r e s u l t s t h a t ( y l y ( - ) ) i s r e g u l a r . We c o n c l u d e t h a t y1 i s a r e g u l a r p o i n t i n T 2 ( 0 ' ) a n d h e n c e 0 ' i s a c o n i c 1 9 1 .

Next we s u p p o s e t h a t L1 E B \ R i s R - r e g u l a r a n d t h a t ( L 1 , L 2 ) i s a R - r e g u l a r p a i r i n T;(O). We u s e t h e n o t a t i o n s of 11.1. w i t h q s+l. F o r n a c h p a i r (LiyMi), 1 Q i G qtl, t h e r e holds {Li,Mill =

I R i l y . . . y R i ,st1 d e f i n e t h e same p o i n t x o f 0 ( 1 . 2 . ( c ) ) . Hence ( L . , M . ) i s a r e g u l a r p a i r o f T;(O), a n d so d e f i n e s a p o i n t xi of 0 , i = 1, ...,q tl ( 1 . 2 . ( ~ ) ) . T h e r e f o l l o w s t h a t t h e se t s {L1,L2} a n d {L1,L21 h y p e r b o l i c q u a d r i c Q i n PG(3,q). The o v a l 0 ' i s a p l a n e i n t e r s e c t i o n of Q. Hence 0' i s a c o n i c . fl

I'

con ta ins , b e s i d e s y l a n d X , a t least o n e p o i n t

1 C R . Each p a i r o f l i n e s of R i s r e g u l a r ( t h e y

1 1

I 1'1' - - IL1 '" . 'Lqt l

I' = I M ly...,Mqtll a re t h e t w o sets of g e n e r a t o r s o f a

3. R-REGULARITY AND AFFINE PLANES

3.1. Theorem. L e t x be a R-regular p o i n t of t h e g e n e r a l i z e d q u a d r a n -

g l e S o f o r d e r (s,s+2) w i t h spread R . Then t h e i n c i d e n c e I * s t r u c t u r e ( P ' , B * , I * ) where P* = x , B' {L E B II x I L and L 4 9 l U

( ~ y , z l ' * l * II y,z E XI*, z 7~ y l and I* t h e n a t u r a l i n c i d e n c e , i s a

2 - ( ( ~ + 1 ) ~ , s + l , l ) d e s i g n i . e . an a f f i n e p l a n e of o r d e r s t l . P r o o f . I m m e d i a t e . 0

3 . 2 , Theorem. L e t S be a g e n e r a l i z e d quadrang le of o r d e r (s,st2)

which c o n t a i n s a spread R and a R- regu lar p o i n t x w i t h

x I L E R . Then each p a i r (L,M), M E R, is r e g u l a r and IL,M}' l C R . Proof. L e t x E L E R a n d M E R \ { L l . Choose a p o i n t y E M such t h a t x 7L y . Then (x,y) i s a R - r e g u l a r p a i r . L e t { x , y } { z i y l G i G stll w i t h x zl, y = z stl. Each p o i n t z i i s i n c i d e n t w i t h a u n i q u e l i n e Ri E R w i t h R1 = L , R s + l = M , a n d w i t h a u n i q u e l i n e Li which d o e s n o t c o n t a i n a p o i n t of I x , y l * * , f o r a l l 1

l * l * -

i G s+l. S i n c e z i 1 Rj, i # j , 1 < i , j 4 stl, w e o b t a i n ,

R-Regularity of the Generalized Quadrangle P( W(s), (mil 177

a p p l y i n g axiom ( i i i ) of 1.1. w . r . t . z a n d R t h a t Li j # i, 1 G i , j Q s + l . So (L,M) i s r e g u l a r a n d m o r e o v e r ILiy 1 d i G s+ll a n d {L ,M}l ' = I R i , 1 < i Q s+l).

i j '

0

3.3. C o r o l l a r y . Le t 1 b e a s e t o f n o n - c o n c u r r e n t l i n e s l i z e d q u a d r a n g l e o f o r d e r ( s , t ) . Then 1 i s normal

- R f o r a l l {L,MI =

j y 1

o f a g e n e r a - ( s e e I 7 l )

p r o v i d e d e a c h p a i r o f l i n e s o f L i s r e g u l a r a n d t h e i r s p a n i s a s u b s e t of 1.

The f o r e g o i n g theorem shows t h a t i n a g e n e r a l i z e d q u a d r a n g l e S o f o r d e r (s,s+2) w i t h a s p r e a d R i n w h i c h a l l p o i n t s are R - r e g u l a r , t h e s p r e a d R i s a n o r m a l s e t . Then a n a f f i n e p l a n e o f o r d e r s + l c a n b e d e f i n e d i n t h e f o l l o w i n g way. L e t P' = R , B' = { { M , N l l l y M , N E R l a n d I* t h e n a t u r a l i n c i d e n c e . Then t h e s t r u c t u r e (P' , B ' , I * ) i s a n a f f i n e p l a n e A R o f o r d e r s t l .

3.4. Theorem. L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r (s,s+2)

i s r e g u l a r w i t h { R , R ' I 1 ' {R1 R , R 2 , . . . , R s + l = R ' I C R . If a l l

p a i r s of p o i n t s i n c i d e n t w i t h d i f f e r e n t l i n e s of { R , R ' I L 1 a r e R- r e g u l a r , we can d e f i n e an a f f i n e p l a n e (P',B',I') of o r d e r s+l w i t h

B' t R , R ' } ' U t R , R ' l t h e n a t u r a l i n c i d e n c e .

Proof. F i r s t we p r o v e t h a t for z,z' E P ' , z f z ' , t h e set { z , ~ ' l ' * ~ * C P ' . L e t z I R i y z ' I R j y R i , R . E {R,R') p r o o f of 11.3.2. i t f o l l o w s t h a t for a n y zk E { z , z ' l Rk E R , zk E R k , b e l o n g s t o { R i , R . ) i m m e d i a t e t h a t ( P ' , B ' , I ' ) i s a 2-((~+1)~,stl,I) d e s i g n . Cl

w i t h a spread R . L e t R , R ' E R , R # R ' and assume t h a t ( R , R ' )

P I = t z E P II z I R ~ , R~ E ( R , R ~ I ~ ' , 1 < i Q s+i), 1.1'

II z,z' f PI, z + 2 ' ) and 1' 11 U { { z , z ' l

11 . From t h e t h e l i n e J l * l *

11 . Hence zk E P I . Now i t i s J

3.5. Theorem. L e t S be a g e n e r a l i z e d q u a d r a n g l e of o r d e r (s,s+2)

w i t h a spread R and a R- regu lar l i n e L. I f R , R ' E L1 R , R f R', 11

t h e n ( R , R ' ) is r e g u l a r and { R , R ' I Proof. L e t R , R ' E L n R , R # R'. C o n s i d e r a n a r b i t r a r y p o i n t x I R , x 1 L. The p o i n t x i s i n c i d e n t w i t h s + 2 l i n e s of B \ R . S i n c e IL1 n R ( = s+l t h e r e e x i s t s a t l e a s t o n e l i n e L ' E B \ R , x I L', s u c h t h a t { L , L ' l l n R = I R I . I n v i e w o f t h e R - r e g u l a r i t y o f L , ( L , L ' ) b e l o n g s t o a R - g r i d . U s i n g t h e n o t a t i o n s of 11.1. w e p u t L = L1, L' = L2. Then {L1,M1ll I R l i , i 2 ,..., s+2) L1 n R w i t h R R12, R f = Rlk, 3 G k Q s t 2 . A g a i n by t h e R - r e g u l a r i t y of L

11 (L1,M1) i s a r e g u l a r p a i r . Hence ( R , R ' ) i s r e g u l a r a n d { R , R ' ) L ' l n R . 0

L1 fi R . 1

=

178 hl. de Soete and J .A . Thas

3 . 6 . Theorem. L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r (s,s+2)

w i t h a s p r e a d R . If S c o n t a i n s a R - r e g u l a r l i n e L, t h e n t h e

i n c i d e n c e s t r u c t u r e (P',B~,E) w i t h P I = L* \

B' = { ~ K , K ' I ~ ' * n P I II K , K ' E P I , K Y K ' and I K , K ' I b e l o n g s t o a

R - g r i d } U I I K , K ' I II K , K ' E PI, K Z K ' and { K , K ' ) d o e s n o t beLong

t o a R - g r i d ; U { x B II i n c i d e n t w i t h x, i s ax a f f i n e pLune of o r d e r s + l . Proof. D e n o t e L o f t h i s s e t i s r e g u l a r (11.5.5.). L e t K , K ' E PI, K Y K ' a n d l e t L1 E I K , K ' I , L f L1. Prom t h e R - r e g u l a r i t y o f L t h e r e r e s u l t s if I { L , L ~ ~ ' n R I + I t h a t (L,L~) i s a r e g u l a r p a i r . I I ence ( K , K ' ) i s a l s o r e g u l a r w i t h l I K , K 1 l L 1 I {K,K' l ' ' i s t h e u n i q u e e l e m e n t of B' t h r o u g h K a n d K ' . Now s u p p o s e t h a t l {L1 ,L l l r- R / = 1. I r i v i e w o f t h e R - r e g u l a r i t y o f L t h e pair ( L , L 1 ) d e f i n e s a R - g r i d . T h i s R - g r i d c o n t a i n s K a n d K ' . Hence l t K , K ' I l l * n P'l = s+l. C l e a r l y { K , K ' I l l * i s t h e u n i q u e e l e m e n t o f B' t i i r o u g h K a n d K ' . F i n a l l y , i f K , K ' E P' w i t h K I x I K ' t h e n xu is t h e u n i q u e l i n e o f B' t h r o u g h K a n d K f . We h a v e a g a i n lxBl = s+l.

H e n c e , we c o n c l u d e t h a t (P',D',E) i s a 2 - ( ( s t 1 ) 2 , s t 1 , 1 ) d e s i g n . 0

( { L I u R ) ,

11

x I L l w i t h xB t h e s e t of a l l l i n e s J f PI

1 i- R by {R1, . . . , R s t l I . E a c h p a i r o f d i f f e r e n t l i n e s

I

= s+l. It i s o b v i o u s t h a t i n t h i s c a s e

3 . 7 . Theorem. L e t S b e a g e n e r a l i z e d q u a d r a n g l e o f o r d e r (s,s+2)

w h i c h c o n t a i n s a s p r e a d R . If x i s n R - r e g u l a r p o i n t and

(L1 ,L2) a R - r e g u l a r p a i r of l i n e s s u c h t h a t x i s i n c i d e n t w i t h no

l i n e x Proof. C l e a r l y x i s i n c i d e n t w i t h n o line o f [ L l , L 2 1 1 * . L e t {L1,L2I1'* = {L1 ,..., Ls+21 a n d l e t y i b e d e f i n e d b y x - y i , y i I L i , 1 G i 4 s + 2 . T h e p o i n t s x , y l , ... , Y , + ~ a r e s t 3 p o i n t s of t h e a f f i n e p l a n e o f o r d e r s t l d e f i n e d by x (11.3.1.). E a c h l i n e t h r o u g h x i n t h a t p l a n e h a s a u n i q u e p o i n t i n common w i t h t h e s e t { y l , . . . , Y , + ~ I . S u p p o s e t h a t y i , y j , y k , i f j # k f i , i , j , k E { l , . . . . . . , s t 2 1 , a r e c o l l i n e a r i n t h a t a f f i n e p l a n e . I f t h i s i s t h e c a s e ,

t h e r e h o l d s I I y i , y j , y k I y i , y . , y k is i n c i d e n t w i t h a l i n e M E {L1,L21

y i I M . L e t u ( r e s p . u,) b e t h e i n t e r s e c t i o n w i t h L i , L j , L k , e . g . o f M w i t h L j ( r e s p . L,). T h e n ' I , E {y i ,yk}" . H e n c e y - uk, w i t h u k y j @ R , a n d a t r i a n g l e u.u y

w i t h y i , y j o r y, a n d c o n c u r r e n t w i t h e a c t l y two of t h e l i n e s L i , L j or L,. S u p p o s e e . g . t h a t yi I M j , From t h e d e f i n i t i o n o f a R-grid i t f o l l o w s t h a t L I x

o f R ( w i t h t h e n o t a t i o n s of II.l.), t h e n s is o d d . i j x j i

1' I = s+l. S u p p o s e t h a t o n e of t h e p o i n t s I' w h i c h i s c o n c u r r e n t

J

j j

a r i s e s , a c o n t r a d i c t i o n . J k j

T h e r e f o l l o w s t h a t t h e r e i s a l i n e o f {L1,L211* w h i c h i s i n c i d e n t

I uk I Lk. M . E { L l , L 2 1 1 * , a n d M J j

j j k I Mk a n d

R-Regularity of the Generalized Quadrangle P(W(s), (-)I 179

I * E R . Hence y = x On t h e o t h e r h a n d uk E I y i , y k j a n d h e n c e 'j k'k j j k '

c o l l i n e a r i n t h e a f f i n e p l a n e of o r d e r stl. Hence I y l , . . . , y s t2} d e f i n e s a n o v a l i n t h e c o r r e s p o n d i n g p r o j e c t i v e p l a n e . The st2 t a n g e n t l i n e s o f t h e o v a l a r e c o n c u r r e n t a t x . So we c o n c l u d e t h a t t h e o r d e r s+ l of t h e p l a n e i s e v e n .

uk - y j , u k y j 4 R , a c o n t r a d i c t i o n . So t h e p o i n t s y i , y . , y k a r e n o t J

3.8. C o r o l l a r y . L e t S b e a g e n e r a l i z e d q u a d r a n g l e of o r d e r ( s , s t 2 )

R-regu lar l i n e s u c h t h a t x Z R, VR E L1 n R , t h e n s i s o d d .

a n d L n R (R1 , . . . , R s t l j . I f M E {Rl,R2)13 M # L , t h e n M - R i , for a l l 1 G i G s+l ( 1 1 . 3 . 5 . ) . Hence t h e r e e x i s t s a t l e a s t o n e l i n e L ' , L' # M , i n c i d e n t w i t h y , w h e r e y = P4 3 R j E {l, ..., stl}, s u c h t h a t L' f K i , 1 (L,L') i s a R - r e g u l a r p a i r . L e t I L . L ' ) = IL1, . . . ,Ls+2j. If R x i s o f t h e form x i j x j i ( w . r . t . t h e R - g r i d d e f i n e d by L a n d L ' ) , t h e n I L i , L . ) { L , L ' ) = $ a n d s o M . - L , Mi - L ' , M j - L , Mj - L ' . Hence o n e o f t h e l i n e s

which c o n t a i n s a spread R . If x i s a R - regu lar p o i n t and L a

P r o o f . L e t x I Rx , Rx E R . Then Rx 3 L . P u t IL ,Rxj l = I K 1 ,... J s t d 1

1 j ' i G s + l . Then { I , , L ' I n R {R.) a n d so

11' J

J 1

Ki i s c o n c u r r e n t w i t h L ' , a c o n t r a d i c t i o n . 0

1 1 1 . C H A R A C T E R I Z A T I O N S O F P ( W ( . b + l ) , ( m ) )

1. THEOREM

L e t S ( P , B , I ) be a g e n e r a l i z e d q u a d r a n g l e o f o r d e r ( s , s + 2 )

which c o n t a i n s a spread R . If aLL p o i n t s a r e R- regu lar , t h e n S i s

i s o m o r p h i c t o P ( W ( s + l ) ) , ( m ) ) w i t h ( m ) an a r b i t p a r y p o i n t of W ( s t 1 ) . P r o o f . W e d e f i n e an i n c i d e n c e s t r u c t u r e S' = (P',B',I') as f o l l o w s .

P' c o n t a i n s t h r e e t y p e s of p o i n t s : ( i ) t h e p o i n t s x E P ; ( i i ) t h e se t s IRl,R2} , R1,R2 E R ; (iii) a u n i q u e p o i n t ( - ) .

B' c o n t a i n s two t y p e s o f l i n e s :

1

( a ) t h e l i n e s L E B \ R ;

(b) s e t s , e a c h b e i n g t h e u n i o n of a l l t r a c e s o f p a i r s o f e l e m e n t s cf a same p a r a l l e l c l a s s i n A R ( c f . 1 1 . 3 . 3 . ) .

- a p o i n t o f t y p e ( i ) i s i n c i d e n t w i t h a l i n e o f t y p e ( a ) i f f t h e y a r e i n c i d e n t i n S ; i t i s i n c i d e n t w i t h n o l i n e o f t y p e ( b ) ;

I ' i s d e f i n e d as :

180 M . de Soete and J.A. Thas

1 - a p o i n t I R 1 , R 2 ) o f t y p e ( i i ) i s i n c i d e n t w i t h e a c h l i n e 1 o f t y p e ( a ) w h i c h b e l o n g s t o I R 1 , R 2 }

l i n e of t y p e ( b ) f r o m w h i c h i t i s a s u b s e t . and w i t h t h e u n i q u e

- t h e symbol ( m ) i s i n c i d e n t w i t h a l l l i n e s o f t y p e ( b ) b u t w i t h n o l i n e of t y p e ( a ) .

It i s e a s y t o c h e c k t h a t S ' i s a g e n e r a l i z e d q u a d r a n g l e o f o rder s t l ( s e e a l s o 1 7 I ) . We p r o v e now t h a t m o r e o v e r S' W ( s + l ) .

1st p r o o f . A p p l y i n g t h e t h e o r e m o f C.T. Benson 1 2 1 we o n l y h a v e I 'I ' t o c h e c k t h a t e a c h p o i n t o f S' i s r e g u l a r , i . e . I I x , y }

f o r a l l p a i r s ( x , y ) i n S ' w i t h x 7 L ' y . Suppose t h a t x a n d y a r e b o t h p o i n t s o f t y p e ( i ) w i t h x

Let x Y y . S i n c e a l l p o i n t s of S a r e R - r e g u l a r , we h a v e I{x s t l . L e t x L Rx E R , y I R y E R . Then I x , y } From t h e p r o o f o f 11.3.4. it , f o l l o w s t h a t e a c h p o i n t o f I x , y } l " l ' is

A p p l y i n g 1.3.4. o f [ 9 1 we o b t a i n I I x , y l I = s + 2 . I f x I R I y , R E R , t h e n I x , y ) l ' c o n s i s t s o f t h e s t 2 p o i n t s { R , R ' } ' , R ' E R a n d

3 I z II z I R I u { ( m ) } , a n d R # R ' , o f t y p e ( ii). C l e a r l y I x , y l s o I I x , y l 1 = s t 2 . Hence ( x , y ) i s r e g u l a r i n S'.

w i t h x 7 L 1 y . I f x1,x2 E { ~ , y } ~ ' , w i t h x1,x2 p o i n t s o f t y p e ( i ) , t h e

I' 1 = I x , y l l * U I R x , R y l 1 .

l * l * I 'I ' i n c i d e n t w i t h a l i n e of I R x , R I1. Hence { x , y l c I x , y ) . Y 1'1 1

1 'I 1

1'1 '

Let x b e a p o i n t o f t y p e ( i ) a n d y IR,R'I1 a p o i n t o f t y p e ( i i ) ,

p a i r ( x 1 , x 2 ) i s r e g u l a r i n S'. Hence t h e r e f o l l o w s t h e r e g u l a r i t y o f ( x , y ) i n S'.

Next we s u p p o s e t h a t x i s a p o i n t o f t y p e ( i ) a n d y = (m). I f x I R E R , t h e n Ix,(-)}l' c o n s i s t s o f t h e s t 2 p o i n t s { R , R ' ) , R' E R a n d R # R ' , i n S ' . C l e a r l y I x , ( - ) } "" 3 I y II y I R I u I ( m ) } ,

a n d s o / I x , ( - ) } I = s t 2 . Hence ( x , ( m ) ) i s r e g u l a r i n S'. 1 I F i n a l l y , l e t x = { R ,R;} , y = IR , R ' I , RxyR;, R , R t E R , b e

two p o i n t s of t y p e ( i i ) , { R x y R i I s u c h t h a t x1,x2 are two p o i n t s of t y p e (1). From t h e r e g u l a r i t y o f (x,,x,) i n S ' , t h e r e fol lows t h a t ( x , y ) is r e g u l a r in S ' .

S i n c e t h e s e are t h e o n l y p o s s i b l e c o m b i n a t i o n s for a p a i r of d i s t i n c t p o i n t s ( x , y ) , x f t y , we c o n c l u d e t h a t S ' --L W(st1). Hence t h e q u a d r a n g l e S i s i s o m o r p h i c t o P ( W ( s t l ) , ( - ) ) . The l i n e s o f t h e s p r e a d R i n S c a n b e i d e n t i f i e d w i t h t h e s e t s { ( = ) , X I Y

1 'I 1

1 y y Y Y I ' X 7L' { R R'}'. Choose x1,x2 E I x , y )

Y: Y

1 'I ' x Y ' (m).

2nd p r o o f . I n [ l l ] C . S o m a d e f i n e s t h e S t e i n e r s y s t e m l * l *

D = ( P ,B ,I ) w i t h P1 P , B1 = B U I I x , y } I1 x , y E P , x f y I and 1 1 1 I1 t h e n a t u r a l i n c i d e n c e . Now we p r o v e t h a t e a c h s u b s t r u c t u r e o f 0

R-Regularit). of the Geiieralized Quadrangle Pl W(sl, ( - I ) 181

1' g e n e r a t e d by a n a r b i t r a r y p o i n t x E P I , a n d a n a r b i t r a r y l i n e L E B w i t 1 1 x I, L,, i s ari a f f i n e p l a n e o f o r d e r s t l .

( i ) S u p p o s e x E Pi' L E U w i t h L = { y , z J '*I* and x Z1 1,. I * 1' 1'1. ( a ) I f x E ( y , z l t h e n xw E E \ R , Vw E I y , z l . The

s f f i n e p i a n e d e f i n e d by x ( c f . 1 1 . 3 . 1 . ) i s a s u b s t r u c t u r e o f D w h i c h c o n t a i n s x a n d L. Hence x and I, g e n e r a t e an a f f i n e p l a n e o f o r d e r s t l ,

1' ( b ) I f XI* n {y,zj = {ri, r # x, t h e n t h e n f f i n e p l a n e d e f i n e d by r ( c f . 1 1 . 3 . 1 . ) i s a s u b s t r u c t u r e o f 0 w h i c h c o n t a i n s x and L . So i n t h i s c a s e , x and L d e r i n e a g a i n an a f f i n e p l a n e o f 3 r d e r stl.

l * l * l *

R E R , t h e n w e c o n s i d e r t h e a f f i n e ( c ) S u p p o s e xl* a { y , z } = $ and x ~- w w i t h w E { y , z ~ ,

xw 4 R . If j i I R

p l a n e o f o r d e r s+l d e f i n e d by {R ,RZ) ( s e e 1 1 . 3 . 4 . ) . The l i n e XW E {Ry,HZI

r a t e d by x and L .

z I RZ, R Y ' y ' z

1 Y ( s e e 11.5.2.). I ience t h i s a f f i n e p l a n e i s a l s o g e n e -

I* l*l* (d) L e t x

xw E A . I f y I R p l a n e o f o r d e r s + l d e f i n e d by {R ,RZ1

f o l l o w s t h a t xw E {Zy,RZ}

o f o r d e r s + l .

C o u n t i n g t h e number o f p o i n t s u i n S s u c h t h a t u' n { y , z l p r u 17 t y , s l IIence t h i s c a s e c a n n o t o c c u r .

{ y , z l ' * = 8 a n d x .- w w i t h w E { y , z l , z I RZ, R ,R E R , t h e n w e c o n s i d e r t h e a f f i n e

( s e e 3.4.). From 1 1 . 3 . 2 . i t Y ' Y Z

11 Y . A g a i n x a n d L g e n e r a t e a n a f f i n e p l a n e

I*'* = 8 a n d x L * 5 { y , z l * * = 9 . ( 2 ) S u p p o s e t h a t x n { y , z l 1

f $ 1'1'

# 4 w e o b t a i n (stl)'(s-1)+2(~+1)~ = ( ~ + l ) ~ = v . 1' 1'

( i i ) S u p p c s e x E PI. , L E B w i t h L # R a n d x f, L . S i n c e S i s a g e n e r a l i z e d q u a d r a n & l e r ;here e x i s t , s a p a i r ( y , M ) E P X B s u c h t h a t x I M I y I L.

(a) I f M E R we c o n s i d e r t h e a f f i n e p l a n e d e f i n e d by ( M , P ! ' l ( c f . 3 . 4 . ) w i t h M - L , M' E R . T h i s p l a n e o f i ; r d e r s + l i s a l s o g e n e - r a t e d by x a n d L .

( b ) I f P4 R we c o n s t r u c t t h e a f f i n e p l a n e of r o d e r s t l

( i i i ) L e t x E PI, L E B w i t h L E R a n d x I1 L . S i n c e S i s a g e n e r a l ' z e d q u a d r a n g l e t h e r e e x i s t s a p a i r (y,.Nl) E P X B s u c h t h a t x I M I y I L w i t h :.I E R . L e t L' - 1.1, L' E R a n d L' # L. Then we c o n s i d e r t h e a f ' f i n e p l a n e d e f i n e d by {L,L'l ( c f . 11.3.4.). Again t h i s p l a n e i s g e n e r a t e d by x a n d L.

de i ' ined b y y ( c f . 11.3.1.). T h i s p l a n e i s g e n e r a t e d by x a n d L .

I n e a c h o f t h e c z s e s t h e p l a n e g e n e r a t e d by x a n d L i s an a f f i n e p l a n e o f o r d e r s+l. If s 2 3 w e c a n a p p l y t h e t h e o r e m o f F . Bueken-

182 M. de Soete arid J .A . Thas

h o u t 131 . Then D i s a t h r e e d i m e n s i o n a l a f f i n e s p a c e of o r d e r s + l . By t h e embedding t h e o r e m of J . A . T h a s 1131 a n d s i n c e s 2 3, we h a v e S f P(W(s+l),(-))y or S f T;(O) w i t h 0 a c o m p l e t e o v a l of PG(2,s+l) w i t h s o d d . Suppose we are i n t h e s e c o n d c a s e . The s p r e a d R o f T;(O! i s a n o r m a l s e t ( s e e 1 1 . 3 . 3 . ) . Hence a l l l i n e s o f R a r e c o n c u r r e n t i n a same p o i n t x o f 0 ( I I , l . ( c ) ) . S i n c e a l l p o i n t s a r e R - r e g u l a r , 0 \ { X I i s a c o n i c ( 1 1 . 2 . 2 . ) . So i n t h i s c a s e we also h a v e T ; (O) f

P ( W ( s t l ) , ( m ) ) ( c f . I I . l , ( d ) ) . I f s = 2 ( r e s p . s = 1) t h e r e i s , up t o i s o m o r p h i s m , o n l y o n e g e n e r a l i z e d q u a d r a n g l e of o r d e r ( 2 , 4 ) ( r e s p . ( 1 , 3 ) ) 1 9 I , a n d a g a i n t h e r e s u l t f o l l o w s . 0

2 . THEOREM L e t S = (P,B,I) b e a g e n e r a z i z e d q u a d r a n g l e of o r d e r (s,st2) c o n -

t a i n i n g a n o r m a l s p r e a d R . If e a c h l i n e o f B \ R i s weak R - r e g u Z a r ,

t h e n S 2 P ( W ( s + l ) , ( m ) ) , s o d d , w i t h ( m ) a n a r b i t r a r y p o i n t of w(st.1). Proof. A n a l o g o u s l y as i n 111.1. w e c o n s t r u c t t h e g e n e r a l i z e d q u a - d r a n g l e S' of o r d e r s t l . Now w e p r o v e t h a t S' = W ( s + l ) , s o d d , u s i n g t h e d u a l of Theorem 5 . 2 . 6 . i n 1 9 1. Assume t h a t t h e l i n e s L 1 , L 2 o f S d e f i n e a R-grid of S . From t h e d e f i n i t i o n of a R - g r i d i t f o l l o w s t h a t e a c h l i n e o f {Ll ,L211*, as w e l l as e a c h l i n e of IL1,L2)1L' i s c o n c u r r e n t ( i n S') w i t h a u n i q u e l i n e o f t y p e ( b ) , a n d t h a t d i f f e r e n t l i n e s f r o m I L ~ , L ~ I ( i n Sl) w i t h a common l i n e of t y p e ( b ) . Now we n o t i c e t h a t Li ,Mi w i t h Li E IL1,L21 , M i E {L1,L2} '* , 1 4 i < s + % , a r e c o n c u r r e n t w i t h a same l i n e of t y p e ( b ) and m o r e o v e r are i n c i d e n t w i t h a s a w

p o i n t of t h a t l i n e of t y p e ( b ) . I n d e e d , for t h e p o i n t s xki I M i , k f i , xik I L i , k # i , t h e r e h o l d s xikxki E R . T h e r e e a s i l y f o l l o w s t h a t (L1,L2) i s a r e g u l a r p a i r i n S'.

C o n s i d e r t h e p o i n t ( m ) . A n a l o g o u s l y a s i n t h e p r o o f o f 111.1. we show t h a t ( m ) i s r e g u l a r in 9 ' . Let x b e a n a r b i t r a r y p o i n t o f t y p e (i) 2nd p u t {x,(m)l = { ( R , R j . l , x I R E R , Ri E R } . L e t ( i l , L 2 , L , j b e a t r i a d o f l i n e s of S' w h i c h a r e i n c i d e n t w i t h p o i n t s of (x,(m) 1' . 'i'hen t h e following two c a s e s may occLir. The t h r e e l i n e s are of t y p e ( a ) , or t w o i i n e s a r e of t y p e ( a ) a n d o n e i s of t y p o ( b ) . Suppose t h a t L1,L2 a rc of t y p e ( a ) . I n S t h e l i n e s o f R

which are c o n c u r r e n t w i t h L1 ( r e s p . L 2 ) f o r m a l i n e L1 ( I -esp . L 2 ) sf t .he a f f i n e p l a n e A R , S i n c e i n 2' t h e l i n e s L1 ,L2 a r e c o n c u r r e n t w i t h d i f f e r e n t l i n e s o f t y p e ( t i ) , t h e l i n e s L 1 , l 2 of A R a r e n o t p a r a l l e l . I f R' E R i s t h e C O ~ T I C ~ ~ I e l e m e n t of L1 and L 2 , t h e n c l e a r - l y 3' i s t h e u:iique e l e m e n t of R !;~hicil i s c o n c u r r e n t (in s ) w i t h L1

I* o r { L ~ , L ~ 1"' a r e n e v e r c o n c u r r e n t

ll*

I' 1

J r

R-Regularit)' of the Generalized Quadrangle PI WIs) , ( - ) I I83

and L 2 . Hence t h e p a i r f L p r e v i o u s p a r a g r a o h i t i s a r e g u l a r p a i r i n S ' . By 1 . 3 . 6 . ( i i ) i n 1 9 1 t h e t r i a d ( L 1 , L 2 ' L ) h a s a t least o n e c e n t e r . U s i n g t h e d u a l o f 5 . 2 . 6 . i n I 9 1 , t h e r e r e s u l t s t h a t S' i s i s o m o r p h i c t o W ( s + l ) , s o d d . Hence t h e q u a d r a n g l e S i s i s o m o r p h i : t o P ( W ( s + l ) , ( m ) ) . The l i n e s o f t h e s p r e a d R i n S c a n b e i d e n t i f i e d w i t h t h e s e t s

L 1 b e l o n g s t o a R - g r i d i n S , a n d by t h e 1' 2

3

{ ( - ) , x H 1 ' l ' , x Y ' ( m ) . 0

3 . THEOREN L e t s b e 3 g e n e r a l i z e d q i l a d r a n g l e of o r d e r ( s , s + 2 ) c o n t a i n i n g a

s p r e a d R . If e a c h Z i n e of B \ R is R - r e g u l a r t h e n S P ( W ( s t l ) , ( = ) ) s ociLI, w i t h ( a) an a r 2 b i t r a r y poiqt o f W ( s + l ) .

P r o o f . T h i s i s i m m e d i a t e f r o m 1121.5. a n d Theorem 1 1 . 2 . 0

T a k i n g a c c o u n t o f I I I . l . , 1 1 1 . 2 . a n d III.3., w e c a n f o r m u l a t e t h e n e x t t h e o r e m .

4. THEOHEM L e t S b e a g e n e r a l i z e d q u a d r a n g l e o f o r d e r ( s , s + 2 ) w h i c h c o n t a i n s

a s p r e a d R . T h e n t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :

( i ) e a c h point i s R - r e g u l a r and s i s o d d ;

( i i ) e a c h line i s weak R - r e g u l a r and R i s a normal s e t ;

( i i i ) e a c h l i n e i s R - r e g u Z u r ;

( i v ) S 2 P ( W ( s + l ) , ( m ) ) , s o d d , w i t h ( m ) n n a r b i t m r y p o i n t of W ( s + l ) .

REFERENCES

I 1 1

1 2 ]

I 3 1

I 4 I

1 5 1

1 6 1

Phi.en; , R . l u . a n d S z e k e r e s , G., On a c o r n b i n a t o r i a l g e n e r a l i z a t i o n o f 27 l i n e s a s s o c i a t e d w i t h a c u b i c s u r f a c e , J . A u s t r . Math. S O C . 1 0 ( 1 9 6 9 ) 485-492.

B ( , t l s r - ,C.T. , O n the s t r u c t u r e ot' g e n e r a l i z e d q u a d r a n g l e s , J . A l g e b r a 15 ( 1 9 7 0 ) 443-4511.

B u e k e q h o u t , F . , Une c a r s c t ; r i s a t i o n d e s e s p a c e s a f f i n s b a s 6 e sur l a n o t i o n d e d r o i t e , I I P t h . - . 111 ( 1 9 6 9 ) 367-372.

Dembowski ,P . , F i n i t e g e o m e t r i e s ( S p r i n g e r - V e r l a g , l 9 6 8 ) .

kIa11,N. J r . , A f f i n e g e n e r a l i z e d q u a d r i l a t e r a l s , S t u d i e s i n P u r e [ l a t h . ( e d . L . M i r s k y ) , Academic P r e s s ( 1 9 7 1 ) 1 1 3 - 1 1 6 .

P a y n e , S . E . , The equivalent? o f c e r t a i n g e n e r a l i z e d q u a d r a n g l e s , J , Col,ib. Th. 1 0 ( 1 9 7 1 ) 7'34-289.

184 M . de Socte and J . A . Thas

I71 Payne,S.E., Quadrangles of order (s-l,s+l), J. Algebra 22

1 8 1 Payne,S.E. and Thas,J.A., Generalized quadrangles with symmetry,

(1972) 97-110.

Part 11, Simon Stevin 49 (1976) 8 1 - 1 0 3 .

L 9 1 Payne,S.E. and Thas,J.A., Finite generalized quadrangles, Research Notes in Mathei:idtics # 1 1 0 ( P i t m a r ? Publ. Inc. 1 1 8 4 ) .

I 10 ] Segre,R., Lectures on modern geometry (Ed. Cremonc.;? Rorn? l ' h 1 ) .

I111 Somma,C., Generalized quadrangles with parallelism, Annals of Discrete i""ath. 14 ( 1 9 8 2 ) 265-282.

117 1 Thas,J.A., Combinatorics of partial geometries and generalized quadrangles, in : Higher combinatorics (ed. M. Aigner), Nato Advanced Study Institute Series, Reidel Publ. Comp. (1976) 183-199.

[ 131 Thas,J.A., Partial geometries in finitc affine spaces, Math. Z. 1 5 8 ( 1 9 7 8 ) 1 - 1 3 .

Annals of Discrete Mathematics 30 (1986) 185-202 0 Elsevier Science Publishers B.V. (North.Holland) 185

ON PERMUTATION ARRAYS, TRANSVERSAL SEMINETS AND RELATED STRUCTURES

Miche l Deza and Thomas I h r i n g e r

Un ive rs i t i ! P a r i s V I I , U .E .R . de Math., P a r i s , France

Technische Hochschule, Fachbereich Mathematik, Darmstadt, Federal Repub l ic o f Germany

E x p l o i t i n g some ideas o f C41, t h i s paper i s focused on t h e equ i - valence between se ts o f m u t u a l l y o r thogona l pe rmuta t i on a r rays and a s p e c i a l c l a s s o f seminets ( t h e so -ca l l ed t r a n s v e r s a l seminets ) . Besides t h i s equ iva lence, Sec t i on 2 con ta ins a c o n s t r u c t i o n method f o r t r a n s v e r s a l seminets u s i n g groups. Nonsolvable pe rmuta t i on groups o f p r ime degree and t h e p r o j e c t i v e spec ia l l i n e a r groups PSL(2,Zm) bounds a r e proved f o r t h e number o f m u t u a l l y o r thogona l pe rmuta t i on ar rays , depending on t h e i n t e r s e c t i o n s t r u c t u r e o f these a r rays . With t h e r e s u l t s o f Sec t ions 4 i t i s shown t h a t a l l examples o f Sec t i on 2 a r e row-ex tend ib le . S e c t i o n 5 dea ls w i t h seve ra l i n c i - dence s t r u c t u r e s assoc ia ted t o t r a n s v e r s a l seminets. The consequences a r e i n v e s t i g a t e d when these inc idence s t r u c t u r e s have spec ia l p r o p e r t i e s ( f o r ins tance, when they a r e p a i r w i s e balanced des igns ) . Sec t i on 6 d iscusses b r i e f l y t h e r e l a t i o n s o f t r a n s v e r s a l seminets w i t h o t h e r mathematical s t r u c t u r e s , e.g. w i t h t r a n s v e r s a l pack ings , genera l i zed or thogona l a r rays , and se ts o f m u t u a l l y o r thogona l p a r t i a l quasigroups.

y i e l d examples f o r t h i s method. I n Sec t i on 3 some upper

1. INTRODUCTION

A v x r m a t r i x A = ( a . . ) w i t h e n t r i e s a . . f rom t h e s e t I1,2 ,..., r l i s 1J 1J

c a l l e d a p e r m t a t i o n amuy i f each row o f A

. . . , r e x a c t l y once, i . e . i f the rows o f A rep resen t permuta t ions o f { l , Z ,..., r l . The in tersec t ion s tructure o f A i s d e f i n e d as t h e vxv m a t r i x F(A) =

(Fi i , (A ) ) A = ( a . . ) and B = ( b . . ) a r e c a l l e d e i r n . L Z a i * i f F(A) = F(B) . C l e a r l y , A and B

a r e s i m i l a r i f and o n l y i f , f o r a l l i n d i c e s i , i ' , j ,

con ta ins each o f t h e elements 1,2,

w i t h F . . , ( A ) : = t j I l s j s r , a . . = a .

1J 1J

1 . The v x r pe rmuta t i on a r r a y s 1 1 IJ i ' j

The permuta t ion a r rays A and B a r e c a l l e d orthogonu2 i f they a r e s i m i l a r and

i f , f o r a l l i , i ' , j , j ' ,

aij = ai , j , and bij = bi ,j, ==3 j = j ' .

This concept o f o r t h o g o n a l i t y genera l i zes t h e wel l -known idea o f o r thogona l l a t i n

rec tang les . It has been de f i ned and i n v e s t i g a t e d by B o n i s o l i and Deza i n C41.

186 M. Deza arzd T. lhringer

(See a l s o 1 7 3 f o r c l o s e l y r e l a t e d cons ide ra t i ons . )

S i m i l a r l y as f o r l a t i n rec tang les and l a t i n squares, one w i l l be i n t e r e s t e d i n

se ts 4 = {A1,A 2,...,At} o f t m u t u a l l y o r thogona l permuta t ion ar rays , w i t h t

p o s s i b l y g r e a t e r than 2 . I n t h i s case t h e in tersec t ion s tructure

(Fi i,(fi) ) i . e .

F(A) =

o f A i s de f i ned as t h e common i n t e r s e c t i o n s t r u c t u r e o f t h e A k ' F(&):= F(A1) = F(A2) = ... = F(At) .

A permuta t ion a r r a y A = ( a . .) i s c a l l e d standardized i f alj = j f o r a l l j . 1 J

Without loss of generality, i t w i l l be assumed i n t h i s paper tha t each permu-

ta t ion array is standardized, and tha t i t s a t i s f i e s the following n o n t r i v i a l i t y

conditions (CII and (C2) .

(C,) The pe rmuta t i on a r r a y A has no cons tan t column, i . e . each column o f A

con ta in> a t l e a s t two d i s t i n c t values,

(C,) any two rows o f A a r e d i s t i n c t .

L e t X be a nonempty se t , and l e t Lo,L1, ..., Lt ( w i t h t z l ) be m u t u a l l y

d i s j o i n t s e t s o f nonempty subsets o f X. The elements o f X w i l l be c a l l e d

points and t h e elements o f u,,,,, ,..,, Lk Zines. Then 3 :=(X;Lo,L1 ,..., Lt )

i s c a l l e d a seminet o r (more p r e c i s e l y ) a ( t t 1 ) - s e m i n e t i f

(S1)

( S , ) each c lass Li p a r t i t i o n s t h e p o i n t s e t X.

Cond i t i on (S2) j u s t i f i e s t h e te rm paral le l c lass f o r each o f t h e l i n e s

n o t i o n o f a seminet genera l i zes such well-known s t r u c t u r e s l i k e a f f i n e p lanes ,

ne ts and (more g e n e r a l l y ) t he p a r a l l e l s t r u c t u r e s o f Andre 121. A subset o f X i s

c a l l e d a transi.arsa2 o f t he seminet 3 i f i t i n t e r s e c t s each l i n e o f 3 i n

e x a c t l y one p o i n t . I f 5 has a t ransve rsa l c o n s i s t i n g o f r p o i n t s , then each

p a r a l l e l c l a s s Li con ta ins e x a c t l y r l i n e s (and hence each f u r t h e r t ransve rsa l

o f t h e seminet c o n s i s t s a l s o o f e x a c t l y r p o i n t s ) . I f T1,T *,..., TV a r e t r a n s -

ve rsa l s o f 5 i s c a l l e d a transversa2

seminet ( o r transversal ( t t 1 , r ) - s e m i n e t i f each t ransve rsa l cons i s t s o f r

p o i n t s ) . B o n i s o l i and Deza r41 po in ted o u t t h a t t he re i s a c l o s e r e l a t i o n s h i p b e t -

ween set.s o f m u t u a l l y o r thogona l permuta t ion a r rays and o t h e r mathematical s t r u c -

t u r e s . For ins tance, they proved t h a t each s e t o f t m u t u a l l y o r thogona l v x r

permuta t ion a r rays i s e q u i v a l e n t t o a 1-design w i t h v t rea tments , r e p l i c a t i o n

number r and t t l mutua l l y o r thogona l r e s o l u t i o n s (see Sec t ion 5 ) . Moreover,

i t was shown t h a t any o f these a re e q u i v a l e n t t o a t ransve rsa l ( t t 1 , r ) - s c m i n e t

w i t h t r a n s v e r s a l s . There fore many o f t he examples and r e s u l t s i n t h i s paper

can be t r a n s l a t e d i n t o analogous statements on comb ina to r ia l designs w i t h m u t u a l l y

o r togona l r e s o l u t i o n s .

any two d i s t i n c t l i n e s i n t e r s e c t i n a t most one p o i n t ,

Li. The

then r:= (X;Lo,L1 ,..., Lt;T1,T2 ,..., Tv)

v

Oil Pennutatioii Arrays 187

2. AN EQUIVALENCE AND A CONSTRUCTION METHOD

L e t J := (X;Lo,L1,.. . ,Lt;T1,T2,. . . ,Tv) be a t r a n s v e r s a l seminet w i t h Lo:= 1 2 {lo,l o,...,lL). ( I n f a c t , th roughout t h i s paper t h e s e t o f p a r a l l e l c lasses , t h e

s e t o f t r a n s v e r s a l s and t h e s e t o f l i n e s of Lo o f any t r a n s v e r s a l seminet a r e

assumed t o be l i n e a r l y ordered, by t h e numbering o f t h e i r elements.) One can now

d e f i n e t m u t u a l l y o r thogona l v x r pe rmuta t i on a r rays Ak = ( a . . ) , k=1,2, ..., t, i n the f o l l o w i n g way: For

x be t h e unique po in t . con ta ined i n Ti n l;, and l e t 1 be t h e L k - l i n e th rough

x . L e t y be the un ique p o i n t w i t h y c T 1 n 1, and l e t 1; be t h e L o - l i n e

th rough y. F i n a l l y , d e f i n e a:j:= c . From t h e p r o p e r t i e s o f t r a n s v e r s a l seminets

one can conclude t h a t &J):= IA1,A2, . . . , A t )

o r thogona l pe rmuta t i on a r r a y s .

t r a n s v e r s a l seminets as f o l l o w s :

k 1 J

i E I1 ,2 ,... ?v1, j E I1 ,2 ,..., rl, k < I1,2 ,..., t) l e t

i s , i n f a c t , a s e t o f m u t u a l l y

The c o n d i t i o n s (C,) and (C,) o f Sec t i on 1 can be paraphrased i n terms o f

(D1) There i s no p o i n t o f t h e seminet wh ich i s conta ined i n a l l t r ansve rsa l s ,

1.‘. n i = 1 , 2 , ..., v i l i n e 1 , a n d v .2 . )

T . = 0. ( T h i s i m p l i e s , i n p a r t i c u l a r , I 1 I t 2 f o r each

(D2) Any two t r a n s v e r s a l s a r e d i s t i n c t , i . e . Ti i T f o r i z j . j

I n t h e c o n s t r u c t i o n procedure f o r J?-(r) o n l y those p o i n t s o f X have been

used which a re conta ined i n one o f t he t r a n s v e r s a l s

r e s t r i c t onese l f t o the reduced t r a n s v e r s a l seminet

Tv) de f ined by X I : = u i = 1 , 2 , . . . , v T i ’ L ’ : = k (1 n X ’ j l c L k I .

Ti. There fore one can always

( X ’ ;LA,Li,. , . ,LC;T1,T2,. . . ,

I n the r e s t of this paper a l l transoersa2 seminets are assumed t o be reduced

arid t o s u t i s f g the ron,Zitions (0,) m d (DJ. Y

The process of c o n s t r u c t i n g m u t u a l l y o r thogona l pe rmuta t i on a r rays f rom t r a n s -

ve rsa l seminets can be reversed: L e t & be a s e t o f t m u t u a l l y o r thogona l v x r

permuta t ion a r rays Y : = {1,2. ..., v ) x { l , Z , ..., r ; , and l e t + be t h e equ iva lence r e l a t i o n on Y w i t h ( i , j ) $ ( i ’ , j ’ ) i f and

o n l y if j = j ’ and j i Fi i , (A). Def ine the p o i n t s e t X o f t h e seminet as t h e

s e t o f equ iva lence c lasses o f 9 , i . e . X : = Y / $ = “ ( i , j ) l l b 1 ( i , j ) t : Y I . For c =

1 , 2 , . . . . r and k = 1,2,, . .,t l e t lo:= ( I ( i , j ) l $ I j = c , i=1,2, . . . ,v) and lk:=

{ I ( i , j ) l d j ~ a . . = c l , and d e f i n e Lo := {lo,lo ,..., lor},

F i n a l l y , l e t T . : = ’,r ( i , j ) l $ 1 j =1 ,2 , . ..,rI, i=1 ,2 , . . . , v . Then J(R):= (X;Lo,L1,.. .,Lt;Tl,T2,, . . ,Tv)

=A. Summarizing, one ob ta ins

k 1J

Ak = ( a . . ) , k=1,2, ..., t . Def ine

C C

k 1 2 1 2 1 J k L : = Ilk,lk ,..., 1 L I .

1 i s a (reduced) t ransve rsa l seminet w i t h &(T(R))

188 M. Deza arid T. Ihririger

2.1. PROPOSIT ION. The existence of a se t 01 t mutually orthogonal v x r perm-

tation arrays i s equivaZent t o the existence of a transversal (t+l,r)-seminet with

v transversals.

Th is equ iva lence was a l ready observed i n 141. One can show even a l i t t l e more.

L e t = (X;Lo,L1.. . . ,Lt;T1,T2,.. .,Tv) and ‘U = (Y;Mo,M1,. ..,Mt;U1,U2,. . .,Uv)

be reduced t ransve rsa l seminets w i t h Lo = ilo,lo ,..., 1 L I and Mo Imo,mo , . . . , m i l , and assume R(’3’) = A(%). Def ine a mapping $: X 4 Y as f o l l o w s . For

X E Ti n 1: l e t $ ( x ) be the unique element conta ined i n Ui nm:. Then 0 i s an

isomorphism o f and U , i . e . 0 i s a b i j e c t i o n which maps p a r a l l e l l i n e s i i onto p a r a l l e l l i n e s ( i n f a c t , 0 s a t i s f i e s $Li = Mi, $Ti = Ui and $lo = mo

f o r a l l i ) . Th is y i e l d s

1 2 1 2

2.2. PROPOSITION. I f y and are reduced transversa2 seminets with

J,(J) = L ( U ) then and u are isomorphic.

The t ransve rsa l seminet

s e t &(J) = IA1,A 2,.. . ,At l

T . n T . = 0 f o r a l l i,j, i z j . The a r rays A1,A 2,...,At form a s e t o f m u t u a l l y

o r thogona l Zatin squares i f and o n l y i f

tT1,T2,,..,Tv\, i s a ne t .

s p e c i a l i z e s t o the c l a s s i c a l correspondence o f m u t u a l l y o r thogona l l a t i n squares

w i t h ne ts .

gonal permuta t ion a r rays v i a seminets, us ing groups.

= ( X;Lo,Ll,. . . ,Lt;T1,T2,. . . ,Tv) corresponds t o a

o f m u t u a l l y o r thogona l Zatin rectanglos e x a c t l y i f

1 J (X;Lo,L1,, ..,Lt,Lt+l), w i t h Lt+l:=

I n o t h e r words, t he equ iva lence o f P r o p o s i t i o n 2.1

The f o l l o w i n g theorem prov ides a c o n s t r u c t i o n method o f s e t s o f m u t u a l l y o r tho -

2.3. THEOREM. Let G he a f i n i t e group with neutral eZement e . Let t and

S be pos i t ive integers, and Zet So,S1,.. .,St and F1,F2,.. .,F be nontrivial

subgroups of G such that the foZZowing conditions are sa t i s f ied fcr aZl i , j E

10,1, ..., tl, k , l E 11,2 ,..., ~ 1 :

S

( 2 ) i j 4 Si n S j = (e l ,

(2) S . O F = {e l , i k

(3) k * 1 =$ Fk z F,,

(41 l Fk I = CG:S i l .

Then there ex i s t s a s e t o f t mutualZy orthogonal v x r permtat ion arrays, with

r : = I F I and v:= -./GI . 1 r S

- Proof . For each i E (O,l, ..., t l l e t Li c o n s i s t o f t h e r i g h t cosets o f Si,

i . e . Li = {Sig 1 geG1. Then (G;Lo,L l,...,Lt) i s a ( t t 1 ) - semine t : Cond i t i on (S1)

On Permu tutiori Arru-vs 189

i s t r i v i a l l y s a t i s f i e d w h i l e ( S 2 ) i s a consequence o f ( 1 ) . Each r i g h t coset

o f one o f the subgroups

s h o w t h a t

1S.g n Fkh l i 1. Assumption ( 4 ) then i m p l i e s u f C F k S i f = G, and hence

lSig n Fkh l 2 1. Each t r a n s v e r s a l has r = lFll elements, and each Fk has

d i s t i n c t r i g h t cosets . Thus t h e r e a r e v = :*IGI form Fkh, w i t h k t t1,2, . .., s l and h e G . F i n a l l y , t h e n o n t r i v i a l i t y c o n d i t i o n s

(D1) and ( D p ) a r e a consequence o f t h e n o n t r i v i a l i t y o f t h e subGroups Fk and o f

(3), r e s p e c t i v e l y . By P r o p o s i t i o n 2.1, t h e p r o o f i s complete.

Fkh (G;Lo,L1, ..., L t ) : I t has t o be Fk

lSig n Fkh l = 1

i s a t ransve rsa l o f

f o r a l l g,hsG. As a consequence o f ( 2 ) one o b t a i n s

I G I 1

d i s t i n c t t r a n s v e r s a l s o f t h e

n

The seminet (G;Lo.L ,,..., L t ) o f t he above p r o o f i s , i n f a c t , a transZation

seminet, i . e . i t has a t r a n s l a t i o n group o p e r a t i n g r e g u l a r l y on i t s p o i n t s : I n t h e

r i g h t r e g u l a r rep rese r i t a t i on o f G each maoping XH xg, grG, maps eve ry l i n e

on to a p a r a l l e l l i n e . On the o t h e r hand, each t r a n s l a t i o n seminet can be ob ta ined

i n t h i s way f rom a group G and subgroups So,S l....,St s a t i s f y i n g c o n d i t i o n

(1). Analogous group t h e o r e t i c c h a r a c t e r i z a t i o n s have been given, f o r ins tance,

f o r t r a n s l a t i o n planes, t r a n s l a t i o n ne ts , t r a n s l a t i o n s t r u c t u r e s and t r a n s l a t i o n

group d i v i s i b l e designs (see e.g. r l l , 1151. C221, [31 and 1201). Marchi r181 uses

s i m i l a r ideas f o r his c h a r a c t e r i z a t i o n o f r e g u l a r a f f i n e p a r a l l e l s t r u c t u r e s by

p a r t i t i o n loops . Probab ly one can fo rmu la te an analogue o f Theorem 2.3 u s i n g loops

i n s t a e d of groups. The problem would be t o f i n d examples f o r such a genera l i za -

t i o n . The r e s t o f t h i s s e c t i o n y i e l d s two c lasses o f examples f o r Theorem 2.3.

C f . Huppert 1141 and Wie landt 1241 f o r t h e group t h e o r e t i c n o t a t i o n s .

2.4. EXAMPLE. L e t G be a nonso lvab le t r a n s i t i v e pe rmuta t i on group o f p r ime

degree p . L e t v:= p , r:= -, and l e t d be the p o s i t i v e i n t e g e r w i t h d < p - 1

and d = r (mod p ) . Then one can c o n s t r u c t a s e t o f t : = i- 1 m u t u a l l y o r thogona l

v * r permuta t ion a r rays : Assume G t o opera te on ial,a *,.... a I . For each k 6

t l ,Z , . . . , p 1 d e f i n e Fk t o be the s t a b i l i z e r o f ak i n G, i . e . Fk:= Ga Then

F k z F , f o r k z 1 s i n c e 6 i s doub ly t r a n s i t i v e ( c f . Theorem 11.7 o f r241 ) . L e t

So,S l , . . . ,St , be t h e Sylow p-subgroups o f G ( w i t h t ' 1 1 because G i s non-

s o l v a b l e ) . Obv ious ly , these subgroups s a t i s f y the assumptions o f Theorem 2.3.

Hence t h e r e a r e t ' m u t u a l l y o r thogona l v x r pe rmuta t i on a r rays . I t remains t o

show t ' = t o r , e q u i v a l e n t l y , t h a t G has e x a c t l y a Sylow p-subgroups. L e t

P be a Sylow p-siibgroup o f G. The o n l y Sylow p-subgroup o f t he no rma l i se r NG(P)

o f P i s P i t s e l f . Hence NG(P) i s s o l v a b l e and thus o f o r d e r pad' w i t h

d " p - 1 ( c f . r141, Satz 1 1 . 3 . 6 ) . There fore t h e number n o f Sylow p-subgroups

s a t i s f i e s n = [ G : N ( P ) l = p . d ' = a T . From n = l (mod p ) one ob ta ins

i . e . d = d ' and n =

2 I G I P

P

k '

r

G r d ' = r (mod p ) , G r a .

The nonso lvab le t r a n s i t i v e pe rmuta t i on groups o f p r ime degree have been com-

p l e t e l y determined, due t o the c l a s s i f i c a t i o n o f f i n i t e s imp le qroups (see

190 M. Deza Q I I ~ T. Ihringer

C o r o l l a r y 4.2 o f F e i t C111).

used i n the above c o n s t r u c t i o n : These groups have e x a c t l y one Sylow p-subgroup,

which would imp ly t = 0.

No t i ce t h a t solvabZe t r a n s i t i v e permuta t ion groups o f p r ime degree o cannot be

2.5. EXAMPLE. Fo r each i n t e g e r m r 2 one can c o n s t r u c t a s e t o f t m u t u a l l y m - 1 rn or thogona l v x r permuta t ion ar rays , w i t h t:= 2 ( 2 -1 ) -19 v : = (2m+1)2 and r : =

2m(2m-l) : Regard t h e p r o j e c t i v e s p e c i a l l i n e a r group

as a permuta t ion group o p e r a t i n g c a n o n i c a l l y on t h e q + l p o i n t s {al,a2,. . .,aq+l)

o f t he p r o j e c t i v e l i n e over the q-element f i e l d . For each

de f i ne Fk t o be t h e s t a b i l i z e r o f ak i n G, i . e . Fk:= G L e t So,S l , . . . ,St l

be t h e (mu tua l l y con jugate) c y c l i c subgroups o f G o f o rde r q + l . By the r e s u l t s

i n 1141, pp. 191-193, these subgroups s a t i s f y t h e assumptions o f Theorem 2.3. For

t h e number t ' o f conjugates o f So one o b t a i n s t '+ l = C G : N G ( S o ) l = w

= ,-= ttl, i .e. t ' = t.

m G = PSL(2,q), w i t h q = 2 ,

k t {1,2,. . . ,q+11

a k '

N o t i c e t h a t Hartman [121 used some o f t h e groups PSL(2,q) i n o r d e r t o con-

s t r u c t designs w i t h m u t u a l l y o r thogona l r e s o l u t i o n s . For ins tance, f o r each q E

{19,31,431 the re e x i s t s a des ign w i t h v = q t l t rea tments , r =? r e p l i c a -

t i o n s and t+l=q mutua l l y o r thogona l r e s o l u t i o n s .

3. BOUNDS FOR THE NUMBER OF MUTUALLY ORTHOGONAL PERMUTATION ARRAYS

A s e t {A1,A2,. . . ,At) o f m u t u a l l y orthogonal permuta t ion a r rays i s c a l l e d

maxima2 i f t h e r e e x i s t s no permuta t ion a r r a y

Ak, k = 1,2 ,..., t. A t ransve rsa l seminet

L-mo.ximaZ i f the re e x i s t s no a d d i t i o n a l p a r a l l e l c lass Lt+l such t h a t (X;Lo,L1,

.. .,Lt,Lttl;T1,T2,. . .,Tv) obv ious .

Attl which i s o r thogona l t o a l l

(X;L0,L1, ..., Lt;T1,T2 ,... ,Tv) i s c a l l e d

i s aga in a t ransve rsa l seminet. The f o l l o w i n g lemma i s

3.1. LEMMA. A s e t of mutually orthogond permutation arraps is maximal

i f mid only if the associated transversal seminet J (A) is L-maximal.

3.2. PROPOSIT IOM. Eaeh s e t of rnintualZy orthogoxu2 permutation arrays of

Example 2 . 5 is maximal.

Proof. L e t G = PSL(2,2m) be t h e group used f o r t he c o n s t r u c t i o n of A. The

assoc ia ted t ransve rsa l seminet 7 has G as p o i n t s e t . The subgroups SO'S1,

..., St a r e e x a c t l y t h e l i n e s th rough t h e n e u t r a l element e o f G, and t h e sub-

a r e e x a c t l y t h e t r a n s v e r s a l s th rough e ( c f . t h e p r o o f groups F1,F2,. . . , Fq+l

Oir Permu tation Arrays 191

o f Theorem 2 . 3 ) . By Satz 11.8.5 o f Huppert L141, these subgroups cover G. Hence

t h e r e cannot be any a d d i t i o n a l l i n e through e, and i s t h e r e f o r e L-maximal.

By Lemma 3 .1 the p r o o f i s complete, s ince P r o p o s i t i o n 2.2 y i e l d s J 2 J(&). 0

A c t u a l l y , t h e a s s e r t i o n o f Propos

s t r u c t u r e F(&) o f a: L e t 3 gonal pe rmuta t i on a r r a y s w i t h F(JJ )

Sec t i on 2 shows t h a t t h e t r a n s v e r s a l

p o i n t se ts and t h e same t r a n s v e r s a l s

t i o n 3 .2 depends o n l y on t h e i n t e r s e c t i o n

B1,B2 ,..., B t , I

= F (& ) . The c o n s t r u c t i o n procedure o f

seminets ')'(a) and 7113) have t h e same

The t r a n s v e r s a l seminet 7 o f t h e p r o o f

be a s e t o f m u t u a l l y o r t h o -

o f P r o p o s i t i o n 3 . 2 con ta ins

F1s, S E S ~ , w i t h u s e s o F1s = G. The re fo re each l i n e o f con ta ins e x a c t l y

n p o i n t s , and 7 7 (A) i m p l i e s t h a t t h e same i s t r u e f o r J(&) and a l s o

f o r y@). As a consequence, t h e r e i s a p o i n t x o f J(3) such t h a t t h e

number t '+ l o f l i n e s o f T(&) th rough x cannot exceed t h e number t+l o f

l i n e s o f Y th rough e ( i n f a c t , t h i s i s t r u e f o r each p o i n t x o f y(2) ) .

There fore P r o p o s i t i o n 3.2 can be improved as f o l l o w s .

n:= I S I = CG:F1l p a i r w i s e d i s j o i n t t r a n s v e r s a l s 0

3.3. PROPOSITION. Get a= {A1,A 2,...,Atl be one of the s e t s ofmutuaZZy

orthogonal permutation arrays of Exnmple 2 . 5 . Let

of mutually orthogonal permutation arrays with

3 = IB1,B2,. . . ,B t , 1 be a s e t

F ( B ) = F(&). Then t ' 5 C..

The nex t lemma g i ves an upper bound f o r t h e number o f m u t u a l l y o r thogona l p e r -

mu ta t i on a r rays depending on t h e i n t e r s e c t i o n s t r u c t u r e of t h e a r rays . The p r o o f

o f t h i s lemma i s a u n i f i e d v e r s i o n o f t h e p r o o f s o f severa l r e l a t e d r e s u l t s i n

'41 and 171.

3.4. LEMMA. Let A= IA1,A2 ,..., At ] be a s e t of mutualZy orthogonu2 v x r

pemutatioil arrays, and l e t I { l , Z ,..., v}, i o c 11,2 ,..., vl, J 5 {1 ,2 , . . . , r I ,

j o < { l , Z , . . . , r l s a t i s f g

al I f II,

bi

ai 'Ji E I : j o d F . 1 i, (A), d l V j c J 3 i . I : j + F i i (a.

t"il,i2t I : j O c Fil12 ' (A),

0

I"he1i t ' r - ! J I - 1 .

- Proof . L e t A = ( a . . ) be a v r pe rmuta t i on a r r a y w i t h F(A) = F (& . By a ) ' J

"jd the re e x i s t s an element ilk I . Def ine c := a i . From b ) one ob ta ins

f o r a l l

w i t n a i = a i j , by d ) . Thus j L j 0 and a i j z a . . = c . Hence a i o j c . There-

1" 0 i I , and c ) i m p l i e s aio,of c. F o r each j c J t h e r e e x i s t s an i c I

OJ 'JO

192 M . Deza arid T. Iliringer

f o r e a i o j = c f o r e x a c t l y one element j o f t h e ( r - i J l - l ) - e l e m e n t s e t { 1 , 2 , . . . ,rl \ ( J IJ { j , } ) . I n p a r t i c u l a r , t h i s i s t r u e f o r each o f t h e permuta t ion a r rays

A, t J?- w i t h c and j rep laced by ck and j, . By o r t h o g o n a l i t y , t h e

mapping k t + j, i s i n j e c t i v e . Th is i m p l i e s t s r - IJ I - 1. 0

The f o l l o w i n g c o r o l l a r y t r a n s l a t e s Lemma 3.4 i n t o t h e language o f t ransve rsa l

seminets. No t i ce t h a t t h i s c o r o l l a r y cou ld have been used i n o r d e r t o prove t h e

P ropos i t i ons 3.2 and 3.3.

3.5. COROLLARY. L a t = (X;Lo,L1,. . . ,Lt;T1,T2,. ..,Tv) b e a transversa2

seminet, and 7.et I c {1,2,. . . , v l , io c {1,2¶, . . ,v l and x E X s a t i s f y

al I f 6,

b ) mi,, T ~ ,

el x T i . 0

Then t s r - 6 - 1, w i t h r:= ITl] and 6 : : I T i n ( u i c I Ti) I . 0

3.6. COROLLARY. Let A= {A1 ,A21 . . . ,A 1 be a s e t of rnutuaZZy orthogonal v x r t uerrnutation arrays, and l e t

t i r - U - 1 . p : = max { I Fi , (A) I I i ,i '=1,2,. . . ,v, i *i ' I . Then

~ P roo f . Choose i , i o t . i l , 2 ,..., v l and j o t { 1 1 2 ] ..., r I1 w i t h I F i i o ( s Z ) I = IJ

and

s a t i s f y the assumptions of Lemma 3.4. 1-1 j o & F i i (A) . De f ine I : = t i 1 and J : = F i l o ( & ) . Then I , io, J , jo

0

3.7. COROLLARY. Let A= IAl,A2, . . . , A 1 be a s e t of mutzia1Zy orthogonal v x r t

pervnitution a r r a y s . Let h : = min { I F . #(.+?-)I 1 i , i ' =1 ,2 , . . , , v ) , and assume X z l .

T'hen l i

t s r - A - 2 .

Proo f . Choose io,i1,iz8 i1,2 ,..., v l , il*i2$ and joF11 ,2 ,..., r l w i t h j o '

Fili2(&), j o ~ F i l i o ( ~ ) and j o & F . '2'0 . (A). Def ine I : = {ilyi21, J : = F. '1'0 . (A) IJ F. (f-) .Then

proof i s complete i n t h e case IJI -Xtl. Assume now IJ I = A . Then X = IF. (&)I

= IF . . (&) I , and thus IJI =

A i s s e t t l e d by C o r o l l a r y 3.6. U

I , io, J , jo s a t i s f y t h e assumptions o f Lemma 3.4. Hence t h e 12iO

1210 1 2

1 l i O I F i i ( & ) I = IJ u tjo)l = X t l . There fore t h e case

The C o r o l l a r i e s 3.5, 3.6 and 3.7 y i e l d s l i g h t g e n e r a l i z a t i o n s for some o f t h e

r e s u l t s i n L41, 171 and r171 (which a r e fo rmula ted i n terms o f designs w i t h

m u t u a l l y o r thogona l r e s o l u t i o n s ) .

s e t o f permuta t ions opera t i ng t r a n s i t i v e l y on t h e s e t

A v x r permuta t ion a r r a y A i s c a l l e d row-transit ive i f t h e rows o f A form a

{ l Y 2 , . ..,rl.

Oil Permutatioii Arraj's 193

3.8. PROPOSITION. L e t A= IA1,A2, ..., A t ] be

v' r p e r m t a t i o n arrays. Assume one of these arrays

row-transitive. Then t c min Ir-1 , m t l } , with m orti?ogonal Latin squares of order r .

a s e t of mutuully orthogonal

(and hence a l l of them) t o be

the largest number of rm*tualZy

- Proof . R o w - t r a n s i t i v i t y i m p l i e s each l i n e o f t h e assoc ia ted t r a n s v e r s a l semi-

n e t T(&) = (X;Lo,L1 , . . . , Lt;T1,T2 ,..., Tv) t o have e x a c t l y r p o i n t s , s ince any

l i n e 1 cLi, i > l , i n t e r s e c t s each l i n e o f Lo e x a c t l y once. Thus (X;Lo,L l,..., Lt ) ex i s tence o f t -1 m u t u a l l y o r thogona l l a t i n squares o f o r d e r r . Hence t - l s m .

A complete ( t t 1 ) - n e t o f o r d e r r has e x a c t l y r t l p a r a l l e l c lasses . B u t a n e t

w i t h a t ransve rsa l cannot be complete. There fore t + l c r t l . U

i s a ( t - 1 ) - n e t o f o r d e r r . The e x i s t e n c e o f such a n e t i s e q u i v a l e n t t o t h e

4. EXTENSION BY ROWS

A s e t J1= IA1,A2,. . .,At} o f m u t u a l l y o r thogona l v x r pe rmuta t i on a r r a y s i s

c a l l e d row-ertercdible i f i t i s p o s s i b l e t o a d j o i n a new row t o each o f t h e a r rays

such t h a t t he r e s u l t i n g ( v t 1 ) x r pe rmuta t i on a r rays a r e aga in m u t u a l l y o r thogona l .

(X;Lo,L1, ..., Lt;TI,T2, ..., T ) i s c a l l e d zrwsversa l -

estendibZe i f t h e r e e x i s t s a t r a n s v e r s a l seminet (Y;Mo,M 1,...,Mt;T1,T2,...,TV,

Tvtl) w i t h Y 2 X and Li = I m n X 1 m c M i l . As transversal-extendibility i s t h e

obvious t r a n s l a t i o n o f r o w - e x t e n d a b i l i t y i n t o t h e language o f t r a n s v e r s a l semi-

ne ts , one ob ta ins

A t r a n s v e r s a l seminet V

4.1. LEMMA. A s e t o f inutun1L;j orthogonaz permutation arrays i s TOW-

extendible i j' and only i f the associated transversal seminet

sa 1 -e.-cteadib l e .

J (&) i s transver-

The above d e f i n i t i o n o f r o w - e x t e n d a b i l i t y i s s t r i c t l y s t r o n g e r than t h e one

g i ven i n r 4 1 where t h e r e s u l t i n g a r rays were o n l y assumed t o be s i m i l a r . Both de-

f i n i t i o n s c o i n c i d e i f Y = X i n t h e t r a n s v e r s a l seminets used i n the d e f i n i t i o n o f

transversal-extendibility. I n p a r t i c u l a r , t he d e f i n i t i o n s c o i n c i d e i f

..., L t ) l a t i n squares, r o w - e x t e n d i b i l i t y i s e q u i v a l e n t t o t h e ex i s tence o f a common

colLumz-tj.ans?~ersaZ ( i .e. a usua l t r a n s v e r s a l o f l a t i n squares w i t h t h e c o n d i t i o n

"no two c e l l s a r e on the same row" rep laced by t h e weaker c o n d i t i o n " n o t a l l c e l l s

a re on t h e same row" ) ; see 141, P r o p o s i t i o n 4.2.

Tv)

(X;Lo,L1,

i s a n e t ( c f . P r o p o s i t i o n 4.4 o f r 4 1 ) . I n t h e case o f m u t u a l l y o r thogona l

L e t (X;Lo,L1,...,Lt;T1,T2 ,..., be t h e t r a n s v e r s a l seminet assoc ia ted t o the l a t i n squares A 1 , A 2 , . . . ,A t .

194 M . Dezu urid T. Ilrringer

C l e a r l y , t h e l a t i n squares have a common column-transversal e x a c t l y i f t h e r e

e x i s t s a t ransve rsa l Tvtl o f (X;Lo,L1,. .. ,Lt) w i t h Tv+l a Lt+l:= tT1,T2,. .., T v I . There i s no such Tv+l i f the n e t (X;Lo,L1,...,Lt,Lt+l) i s an a f f i n e p lane

( i . e . i f A1,A 2,...,At fo rm a complete s e t o f m u t u a l l y o r thogona l l a t i n squares) .

I n genera l , i t i s an open ques t i on whether such a t ransve rsa l e x i s t s i f (X;Lo,L1,

..., Lttl) A l though t h e nex t p r o p o s i t i o n i s easy t o prove, t h e r e s u l t i s q u i t e s u r p r i s i n q .

i s a transversal-free ne t i n t h e sense o f Dow C101.

4.2. PROPOSITION. 411 se t s a of muti*aZZy o?thogonaZ permutation arrays 0.7 the Examples 2.4 and 2.5 are row-extendible.

- Proof . L e t G be the group used i n Example 2.4 o r 2.5 f o r t he c o n s t r u c t i o n

be t h e t ransve rsa l seminet o f JL, and l e t

assoc ia ted t o G and t h e subgroups So,S1 ,..., S and F1,F2 ,..., F, o f G.

Reca l l f rom t h e p r o o f o f Theorem 2.3 t h a t X = G, Li = ISig I gcG1 = Is i f ( f tF1 }

f o r a l l i E {O,l, ..., tl, and IT1,T2 ,..., T v } = IFkg 1 gcG, k=1,2 ,..., s } . By Lemma

4 .1 and because o f J r ( R ) , i t i s s u f f i c i e n t t o show t h a t t he t ransve rsa l

seminet i s t r ansve rsa l -ex tend ib le . L e t Fl = {fl fcF1) be a copy o f F1 w i t h F1OX=O, and d e f i n e Y : = XuF1. A s a l l subgroups Si, i=O, l , . . . , t , a r e

con jugate t o each o the r , t he re e x i s t ao,al, ..., at E G w i t h S. = a . S a , L e t

Mi:= {S ia i fu t - f } I feF1}, i = O , l , ..., t, and d e f i n e Tv+l:= F1. T r i v i a l l y then Y g X

and Li 2 tmnX I mcMi}. I n o rde r t o show Li = ImnX I mcMi}, assume fl,f2 E F1 and

S o n F1 l e } fl = f2. Th is i m p l i e s lLi I = ltmnX I mcMi} I , and t h e r e -

fo re Li = I m n X i mcMil. The same argument i m p l i e s c o n d i t i o n (S2) f o r

..., Mt). It remains t o show (S1). L e t i , j E { O , l , ..., t} , fl,f2 E F1 and x , y ~ X

s a t i s f y i z j , x r y and x,y 6 (Siaifl u I f l } ) n ( S . a . f IJ IT2}). A s (X;Lo,Ll,~..,

L t ) i s a seminet, e i t h e r x o r y cannot be conta ined i n X. Assume y k X. Then

y=Tf l=f2 and x c S . a . f n S . a . f = (a.S aT1)a.f n (a.S aT1)a. f = aiSoflnajSofl.

Th is i m p l i e s xf;' E a.S n a.S Hence aiSo = a.S

d i c t i n g i * j . 17

r = (X;Lo,L1 ,..., Lt;TI,T2 ,..., Tv)

t

-1 - 1 i o i

iaifl=Siaif2. Then (a.S 1 0 1 ay1)aifl = (a.S 1 0 1 ar1)aif2 and thus flf;'cSo. From

one ob ta ins

(Y;Mo,M1,

3 5 2

1 1 1 J J 2 1 0 1 1 1 J O J J 1 and thus Si = S con t ra -

1 0 J O ' J O j'

The a l t e r n a t i n g group A 5 y i e l d s an example f o r some o f t h e r e s u l t s o f t h e 2 preced ing sec t i ons . Since A 5 and PSL(2,2 ) a r e isomorphic as permuta t ion

groups, b o t h o f t h e Examples 2.4 and 2.5 imp ly t h e ex i s tence o f a s e t o f 5

mu tua l l y o r thogona l 25x12 permuta t ion a r rays . By P r o p o s i t i o n 3.2, t he s e t 59, i s

maximal, and by P r o p o s i t i o n 4.2 i s row-ex tend ib le , i . e . t h e r e e x i s t s a s e t

-9: o f 5 m u t u a l l y o r thogona l 2 6 x 1 2 permuta t ion a r rays . I t i s unknown whether

i s row-ex tend ib le o r n o t .

Oit Pcrniutatiori Arra),s 195

5. TRANSVERSAL SEMINETS CARRYING DESIGNS

The concept o f t r a n s v e r s a l seminets comprises a l a r g e v a r i e t y o f d i f f e r e n t

mathematical s t r u c t u r e s . For d e t a i l e d i n v e s t i g a t i o n s one has t h e r e f o r e t o add

f u r t h e r r e s t r i c t i o n s . Sec t ions 2 and 4 and p a r t s o f Sec t i on 3 t r e a t e d t r a n s v e r s a l

seminets w i t h a group o f t r a n s l a t i o n s o p e r a t i n g r e g u l a r l y on t h e p o i n t s . I n t h i s

s e c t i o n a d d i t i o n a l assumptions w i l l be imposed on t h e f o l l o w i n g i nc idence s t r u c -

t u r e s which a r e assoc ia ted t o each t r a n s v e r s a l seminet t ' 1' T2,...,TV) ( l e t L := LouL1u . . . uLt and T:= IT1,T2, . . . , T V l , i . e . L i s t h e s e t

o f l i n e s and T the s e t o f t r a n s v e r s a l s o f J ) :

= (X;Lo,L1,. . . ,L -T

I(X,L) - t h e t rea tments o f t h i s i nc idence s t r u c t u r e a re t h e p o i n t s and t h e

b locks a r e t h e l i n e s o f 'j',

t h e t rea tmen ts a r e t h e t r a n s v e r s a l s and t h e b locks a r e t h e p o i n t s I (T,X) - of J,

I(X,LUT) - t h e t rea tments

t h e t ransve rsa

The inc idence s t r u c t u r e I (X,L

cases t h e i n c i dence i s d e f i ned

c i d e n t i n I (T,X) e x a c t l y i f

are the p o i n t s and the b locks a re the l i n e s arid

s o f J.

i s e s s e n t i a l l y t h e seminet o f J. I n a l l t h r e e

n a t u r a l l y . For example, x E X and Ti ,- T a r e i n -

x r T . . I(T,X) i s t h e des ign w i t h m u t u a l l y o r tho - 1

gonal r e s o l u t i o n s mentioned i n Sec t i on 1: The b locks x , y t X a r e de f i ned t o be

p a r a l l e l i n t he i t h p a r a l l e l c l a s s i f and o n l y i f t h e r e e x i s t s a l i n e

w i t h x,y E 1 . Two i n t e r e s t i n g cases d iscussed l a t e r i n t h i s s e c t i o n occur when

I(T,X) o r I(X,LuT) a r e PBD's ( p a i r w i s e balanced des igns ) . Fo r i ns tance ,

I(X,LuT) has t h i s p r o p e r t y i f t h e assoc ia ted s e t o f pe rmuta t i on a r r a y s i s a com-

p l e t e s e t of m u t u a l l y o r thogona l l a t i n rec ta f ig les ( t h e examples a f t e r P r o p o s i t i o n

5 .3 show t h a t t h e converse o f t h i s s ta tement i s n o t t r u e ) . Before go ing i n t o de-

t a i l , some d e f i n i t i o n s a re necessary.

3 = (X;Lo,L1, . . . , L t ) i s c a l l e d n - r e p Z a r i f each l i n e con ta ins

e x a c t l y n p o i n t s . I n t h i s case 's i s a ( t t 1 , r ) - sen i i ne t w i t h r = I X l / n ( i . e .

ILi I = r f o r a l l i ), and 5 i s a l s o c a l l e d an ( r , n ) - h n o c o z f i p i m t i o z . One

has n s r, w i t h e q u a l i t y i f and o n l y i f 3 i s a n e t o r , e q u i v a l e n t l y , i f t h e

assoc ia ted pe rmuta t i on a r rays a r e r o w - t r a n s i t i v e (see Sec t ion 3 ) . An n - r e g u l a r

t ransve rsa l seminet s a t i s f i e s n 5 v (because r v = C . ITiI 2 1x1 = r n ) ,

w i t h e q u a l i t y i f and o n l y i f t h e T i ' s a re p a i r w i s e d i s j o i n t . I n t h i s s i t u a t i o n

the s e t o f t r a n s v e r s a l s can be cons idered as a new p a r a l l e l c l a s s Lt+l, and the

assoc ia ted pe rmuta t i on a r r a y s a r e l a t i n rec tang les . There fore , i f n = r = v , t hen

I(X,LIJT) i s a ( t+2 , r ) -ne t , and t h e assoc ia ted pe rmuta t i on a r r a y s a r e l a t i n

squares,

1 t Li

The seminet

1=1,2, . . . , v

The t r a n s v e r s a l seminet r= (X;Lo,L1 ,..., Lt;T1,T *,.. .,Tv) i s c a l l e d

196 M . Deza and T, lhririger

q- uniform i f each p o i n t i s con ta ined i n e x a c t l y q

t ransve rsa l seminet i s n - regu la r w i t h

c i a t e d permuta t ion a r rays conta ins each element i e {1,2, . .., r l e i t h e r q o r 0

t imes. An example o f an n - regu la r ( w i t h n=4) b u t not q-un i fo rm t ransve rsa l seminet

i s p rov ided by the or thogona l permuta t ion a r rays

t r a n s v e r s a l s . A q -un i fo rm

n = v/q. Each column o f each o f t he asso-

(The s e t

t o check t h a t A1,A2 have no or thogona l mate.)

IA1,A21 i s a row-extension o f two or thogona l l a t i n squares. I t i s easy

For each p o i n t x o f a t ransve rsa l seminet, l e t Y denote the number o f

t ransve rsa l s through

the i nc idence s t r u c t u r e I (T,X) i s a 1-design S r ( l , i , v ) , i . e . each o f t he v

t rea tments T i c T i s i n c i d e n t w i t h e x a c t l y r b locks , and each b lock x i s

i n c i d e n t w i t h E ; t rea tments . N o t i c e t h a t I (T ,X) may have repeated b locks .

The b locks o f I (T,X) have t t l m u t u a l l y o r thogona l r e s o l u t i o n s ( i . e . any two

p a r a l l e l c lasses o f d i s t i n c t r e s o l u t i o n s c o i n c i d e i n a t most one b l o c k ) . The reso-

l u t i o n s a r e g iven by

I f t h e r e i s a nonnegat ive i n t e g e r A w i t h ITinTT.l = A f o r a l l i,j w i t h

i f j ( i . e . i f t h e t r a n s v e r s a l s fo rm an ( r , h ) - e q u i d i s t a n t code), then I (T ,X)

becomes a PBD w i t h any two d i s t i n c t t rea tments conta ined i n e x a c t l y A b locks .

E q u i v a l e n t l y , t he assoc ia ted permuta t ion a r rays a re equidis tant w i t h Hamming-

d i s tance r-A, 1.e . any two rows o f each o f t h e permuta t ion a r rays c o i n c i d e i n

e x a c t l y A p o s i t i o n s . I f , moreover, t h e t ransve rsa l seminet i s q-uni form, then

I(T,X) i s a 2-design S,(2,q,v). As a consequence, r ( q - 1 ) = A ( V - 1 ) . Analogously,

if I(T ,X) i s an S A , ( t ' , q , v ) , w i t h t ' 2 2 , then r (?,--ll) = h ' (:,--\) . I n t h i s

case, any t ' d i s t i n c t rows o f each o f t h e permuta t ion ar rays co inc ide i n e x a c t l y

A ' p o s i t i o n s .

The t ransve rsa l seminets cons t ruc ted i n the p r o o f o f Theorem 2.3 a re n - r e g u l a r

and q-uni form, w i t h n = - and q = s . For none o f these examples I (T ,X) i s a

PBD.

x, and d e f i n e i := {: I X E X I . For each t ransve rsa l seminet

Lo,L1 ,..., Lt.

J

I G I r

Many examples o f t ransve rsa l seminets a re g i ven i n r171, w i t h I (T ,X) an

S A , ( t ' , q , v ) o r a g r o u p - d i v i s i b l e des ign ( i n t h i s case IT i n T . 1 < 1 f o r a l l d i s -

t i n c t i, j ) . For i ns tance , by Theorem 1 . 5 o f r171 the re i s a p o s i t i v e i n t e g e r v1 such t h a t , f o r

tence o f a t ransve rsa l seminet w i t h t 2 and t h e p r o p e r t y t h a t I (T ,X) i s an

S(2,3,v). There a re s i m i l a r r e s u l t s o f o t h e r au thors ; some o f them a re l i s t e d here:

J

v z vl, t he c o n d i t i o n v = 3 (mod 12) i s e q u i v a l e n t t o t h e e x i s -

On Pwmictation Arrays I97

k 1) D i n i t z l81. For each p r ime power o f t h e fo rm q = 2 c t 1, w i t h k a p o s i t i v e

i n t e g e r and c > l an odd i n t e g e r , t h e r e e x i s t s a t r a n s v e r s a l seminet such t h a t

t = c - 1 and I (T ,X) i s an S (2 ,2 ,q t l ) .

2 ) Kramer e t a l . C161. There e x i s t s a t r a n s v e r s a l semient such t h a t t = 12 and

I(T,X) i s an 5(5,8,24).

3 ) Hartman C121. There e x i s t t r a n s v e r s a l seminets such t h a t I (T ,X ) i s an

S(3,4,v) f o r ( v , t ) = (20,3),(32,5),(44,7).

T ransversa l seminets can be represented by t h e f o l l o w i n g ( t t 1 ) - d i m e n s i o n a l

a r r a y . o f s i d e r : t h e c e l l (io,il,. , , , i t ) con ta ins t h e s e t I T i c T 1 xcTi} i f

l ~ o n l ~ l n . . . n l ~ t = {XI, and i t i s empty o the rw ise ( t h e l i n e s o f each p a r a l l e l

c l a s s a r e assumed t o be l i n e a r l y ordered, i . e . Lk = {lk,l k y . . . , l k } ) . I f I(T,X)

i s an S(2,2,v), then t h i s a r r a y i s c a l l e d a Room ( t t 1 ) - c u b e o f s i d e r = v - 1 . Each

such a r r a y i s e q u i v a l e n t t o t t l p a i r w i s e or thogona l symmetric l a t i n squares o f

s i z e ( v - l ) x ( v - l ) (see 191). I f I(T,X) i s an S,,(t ' ,q,v), then t h e above

( t+ l ) -d imens iona l a r r a y i s a ( t + l ) - d i m e n s i o n a l Room design; f o r i ns tance ,

S(5,8,24) y i e l d s a 13-dimensional Room des ign o f s i d e 253, see [161.

1 2 r

The inc idence s t r u c t u r e I (X,L) o f t h e seminet 3 = (X;Lo,L l,...,Lt) i s a

1-design Sttl(l,; I 1 I I l c L } ,I X I )

then I (X ,L ) becomes an Sttl( l,n,rn). 5 i s c a l l e d an An&& seminet ( o r a f f i n e

purullel structure) i f I (X ,L ) i s a l i n e a r space, i . e . i f any two d i s t i n c t t r e a t -

ments o f I (X ,L) a r e i n c i d e n t w i t h e x a c t l y one b l o c k . Obv ious ly , AndrG seminets

do no t admi t t r a n s v e r s a l s . A t ransve rsa l seminet i s c a l l e d aZmost-Andr6 ( o r com-

p l e t e ) i f any two d i s t i n c t p o i n t s a r e e i t h e r connected by a l i n e o r by a t r a n s v e r -

s a l . As each almost-Andre t r a n s v e r s a l seminet i s L-maximal, Lemma 3.1 y i e l d s

w i t h o u t repeated b locks . I f 3 i s n - regu la r ,

5.1, REMARK. Let t h e warisverstil seniinet be aZmost-An&&. Then. the asso-

raiatecl s e t fL( r) 0.f m : i t i t a 2Zy mtho<gontrl perrrmtation arrays is mcLcima2.

A c t u a l l y , P r o p o s i t i o n 3.2 was proved e s s e n t i a l l y hy showing t h a t t h e i n v o l v e d

Reca l l t h a t a s e t o f t m u t u a l l y o r thogona l l a t i n rec tang les o f s i z e v x r i s

t r a n s v e r s a l seminets a re almost-AndrG.

c a l l e d .?ompZete i f t = r-1.

5.2. PROPOSITISN. Let be a trwi?suersiiZ ( t t 1 , r ) - semins t w i t h v trcrnsver-

sols. Then

pemnutation arrags is a z o ! p i e t e se% of ni!tunllLg orthogonal L i t i n rec tang les . I n

this i?me I(X,LuT) is dr: S ( P , j r , v } , r v ) .

t .. r-1, wit;z e p n l i t y if U ~ L ! oxLy if t he associated set A( 3') of

198 M. Dezu and T. Iliringiv

- Proof. The inequality t 5 r-1 i s a t r iv ia l consequence of Corollary 3.5. Assume now t = r-I . By Corollary 3.6 then u = O for Jt(T) or , equivalently, & ( T ) i s a se t o f l a t in rectangles. The completeness of fL(J) implies 7 t o be complete, i . e . any two elements of X are connected by a b l o c k from L u T . As T i r i T . = O for a l l i , j with i t j , one obtains t h a t I ( X , L u T ) i s a n

J S(Z,{r ,v) , rv) . n

Notice that complete sets of mutually orthogonal la t in rectangles have been constructed in Quattrocchi, Pellegrino C191 for a l l prime divisor of r .

v n o t exceeding the smallest

For a transversal seminet ( X ; L o , L 1 , . . .,Lt;T1,T2 ,..., T v ) and M c {1,2 , ..., v l ,

, . . . , v } ,Mz0 M Z ~ , l e t tM:= inicM T ~ I , and define d : = E ~ ~ ~ ~ , ~

Then -

with equality if the transversal seminet i s almost-Andr6.

5 .3 . PROPOSITION. Let J be a transversaZ (ttl,r)-semiMet i i ) i th v transuer-

s a k . L e t 7 be n-regular, and asswne I(X,LuT) to bc an S(2 ,{ r ,n l , rn ) . Thev t = w- 1 , anti there are eractly - transversals through. each x t X , ' V

n n- n

Proof. As the transversal seminet i s almost-Andr6, (1 ) i s valid with equality. t l i ' , ) = v ( ; ) . With l l l = n

- Since I T i nT.1 5 1 for i z j , one o b t a i n s o = El:.

for a l l 1 - L and 1x1 = rn, (1) turns into J '-1 I V

This can easily be transformed into the claimed equality for t . Let Q. be the number o f transversals through some x c X. Then v - l = r n - l=a ( r -1 )+ ( t + l ) ( n - 1 ) together with the equality ju s t proved yields Q. =:. 3

A class of examples satisfying the assumptions of Proposition 5 . 3 can be ob-

tained as follows. Let kz2, consider the transversal seminet where T1 , T 2 , . . . ,Trk are the lines contained in Ur-k<iir L i . Then sa t i s - fies the assumptions of Proposition 5 .3 , with n = r and v = rk . I n th i s s i tua- t ion, Corollary 3.6 yields the bound t s r-2 (since ~ = 1 ) while the exact value i s t = r-k .

( X ; L o . L 1 , ..., L r ) be an affine plane of order r . For J = ( X ; L o , L 1 , . . . ,Lr-k;T1,T2,. . . ,Trk)

However, there also exist transversal seminets for which I ( X , L u T ) i s a linear space S(Z,{r ,n} , rn) with r z n . I n terms of 1131 these transversal seminets correspond exactly t o the partiully resolvable 2-partitions P R P 2 - ( n , r , v ; t t l )

On Permurariorr A F T U ~ 199

w i t h t h e a d d i t i o n a l p r o p e r t y t h a t

r e s u l t s o f [131 and [51 on these designs imp ly

v = n r . Together w i t h P r o p o s i t i o n 5 .3 , t he

5.4. PROPOSITION. A n n- regu la r transversa2 ( t+ l , r ) - semine t w i t h I ( X,LuT) an

S (2, I r,nl, r n )

a ) i n the case n = 2, r ? 3 if and onZy if e i h t e r t = 0, r = 3 o r t = r-1,

bi in the case n = 3, r = 2 if and onZy if t = 0,

c i in t he case n = 3 , r = 4 if and onZy if t = O or t = 3 .

ez is ts

N o t i c e t h a t a l l seminets i n t h e above p r o p o s i t i o n a r e e i t h e r complete s e t s o f

m u t u a l l y o r thogona l ? a t i n rec tang les ( t = r-1) or t r a n s v e r s a l des igns ( t=O).

6 . SOME STRUCTURES RELATED TO TRANSVERSAL SEMINETS

It i s well-known t h a t ne ts a r e e q u i v a l e n t t o s e t s o f m u t u a l l y o r thogona l l a t i n

squares, t o t r a n s v e r s a l des igns ( v i a d u a l i t y ) , t o or thogona l a r rays , t o op t ima l

codes, e t c . P r o p o s i t i o n 2.1 i s a g e n e r a l i z a t i o n o f t h e f i r s t o f these equ iva len-

ces. Next, t h e second equ iva lence i s genera l i zed t o t r a n s v e r s a l seminets.

= (X;Lo,L1,. . . ,Lt;T1,T2,. . . ,Tv) i s e q u i v a l e n t t o

a ~ P ~ ~ ~ ~ J C I W J L pack ing , v i a t h e i nc idence s t r u c t u r e I (L ,X) . Each b l o c k X C X h i t s

each o f t h e gyi"7u:s L . i n e x a c t l y one t rea tmen t l cL i , two t rea tments f rom d i s -

t i n c t groups Li,L. a r e j o i n e d by a t most one b l o c k , and t h e r e i s no such b l o c k J

i f L . =L.. Moreover, t h e r e a r e a d d i t i o n a l mzin tr.eatmeurts Ti; each Ti p a r t i -

t i o n s t h e t rea tments o f I (L ,X) i n t o d i s j o i n t b locks .

Supoose now X and each Li t o be l i n e a r l y o rdered, i . e . X = Ix1,x2, . . . I and Li = { l i 7 1 .,..., l j 1 . L e t t h e m a t r i x B = ( b . . ) o f s i z e ( t t 1 ) X I X I be de f i ned

by b . . = k : 0 x . i . Th is m a t r i x i s an OA (orthogomZ a r r a y ) i f t h e seminet

i s a n e t (see e.g. 191). 11.1 t h e general case, t h e s e t o f a l l 1x1 columns o f B

forms a code o f ZeilgLh t+l over t h e a lphabet {1,2, ..., r l w i t h 1x1 wrds and

mi r i imaZ distance t , s i n c e any two d i s t i n c t columns c o i n c i d e i n a t most one p o s i -

t i o n . Each t r a n s v e r s a l T. corresponds t o a f a m i l y o f codewords (columns o f 6 )

such t h a t , f o r a l l k ' ~ 11,2 , ..., r I and a l l i c [0,1, . . . , tl, t h e r e i s e x a c t l y one

codeword i n t h i s f a m i l y w i t h va lue k i n row i , The t r a n s v e r s a l T . can a l s o

be regarded as an i i ! je,: t ioe , i i a g o n r l subset o f ( i . e . as a s e t o f

r words o f l e n g t h t + l over { l , Z , ..., r ? which d i f f e r i n an-y coo rd ina te ; see

161). now be l a t i n rec tang les

o r , e q u i v a l e n t l y , l e t Ti i . , T . = 0 f o r a l l i , j , i z j . Assume t h e numbering o f t he

Do in ts x c X and o f t he l i n e s li.Li ( i 2 1 ) now be more spec ia l than above: L e t

Each t r a n s v e r s a l s e v i n e t

1

1 $1

1 2 r 1 1 J

1J J

J

J {1,2, . . . , rIttl

L e t t h e assoc ia ted pe rmuta t i on a r r a y s A1,A2, ..., At

J

2 00 M . Deza and T. Ihririger

k k l = l k i f l n T 1 z l o , a n d l e t x = x

these assumptions t h e f i r s t row o f B s a t i s f i e s b = k i f j = k (mod r ) and,

f o r m.1, t he m t h row o f B becomes a " l i n e a r i z a t i o n " o f t he l a t i n r e c t a n g l e

'm-19 i.e. ( b m , ( i - l ) r + l 9 bm,( i -1)r+2 3.. 3 bm,ir) i s t h e i t h row o f Am-l.

N o t i c e t h a t t h e t r a n s v e r s a l Ti corresponds now t o t h e s e t o f columns ( i - l ) r + l , ( i - l ) r t 2 , . . . , ir o f B . For example, c o n s i d e r i n g t h e s imp le complete s e t

2) o f m u t u a l l y o r thogona l l a t i n rec tang les , one o b t a i n s

w i t h j = ( i - 1 ) r t k i f x c T i n l o . Under i j

U

A1 = 1 2 3 (i : ;) , A2 = (3

The permuta t ion a r r a y s A and A' a r e s i m i l a r e x a c t l y i f , f o r a l l j , column

j o f A ' can be ob ta ined f rom column j o f A by renaming t h e symbols. L e t

Q(A,A ' ) be t h e r x r m a t r i x ( q . . ) de f i ned by q i j = k i f the i ' s i n column j

o f A become k ' s i n column J o f A ' , and q i j = * o the rw ise . For example, 'J

Q ( [; : ;) ' (; : 23) = (; ; :) . * 1 3

As a l l permuta t ion a r rays a r e assumed t o be s tandard ized, qii = i f o r a l l i i n

Q(A ,A ' ) . I n no column o f t h i s m a t r i x a symbol a p e a r s tw ice . There fore Q(A,A ' )

can be cons idered as ( t h e m u l t i p l i c a t i o n t a b l e o f ) a p a r t i a l l e f t - c a n c e l l a t i v e

g roupo id de f i ned on {1,2, ..., r l . The pe rmuta t i on a r rays A , A ' a re o r thogona l i f

and o n l y i f t h i s g roupo id d l s o i s r i g h t - c a n c e l l a t i v e , and

a p a r t i a l quasigroup. Moreover, Q(A ,A ' ) i s a complete quasigroup i f and o n l y i f

t h e permuta t ion a r rays a re r o w - t r a n s i t i v e . L e t A1,A2¶. . . ,At be s i m i l a r permuta-

t i o n a r rays . By P r o p o s i t i o n 1.2 o f 141, these a r r a y s a r e m u t u a l l y o r thogona l

e x a c t l y i f Q(A1,A2j, Q(A1,A3), . .. , Q(A1,At) a r e m u t u a l l y o r thogona l p a r t i a l

quasigroups. N o t i c e t h a t t he assoc ia ted t r a n s v e r s a l seminet i s n - regu la r i f and

o n l y i f each Q(Ai,A.), i z j , has t h e f o l l o w i n g p r o p e r t i e s : Each i E (1,2, ..., r }

appears e x a c t l y n t imes, and t h e r e a r e e x a c t l y n d i s t i n c t symbols i n each rcw

and i n each column.

Q(A ,A ' ) then becomes

J

There a re many inc idence s t r u c t u r e s b u i l t f rom ne ts . Examples a r e t r a n s v e r s a l

geometries 161, d-dimensional ne ts 1211, and ex tens ions o f dua l a f f i n e planes

C231. I t w i l l be i n t e r e s t i n g t o s tudy s i m i l a r s t r u c t u r e s w i t h ne ts rep laced by

t ransve rsa l seminets.

On Permutation Arrays 101

REFERENCES

[ I1

r 2 1

131

141

151

I 6 1

I 7 1

181

191

r 101

I 1 1 1

r 122

r 131

1141

r 151

r 161

l 1 7 J

r 181

119J

1201

c211

c 221

[231

C 241

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J . D i n i t z and D.R. S t inson, The spectrum o f Room cubes. E d r o p . J . Combina- torics 2 (1981) 221-230.

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S . Dow, T ransve rsa l - f ree ne ts o f smal l d e f i c i e n c y . A w h . Math. 41 (1983) 472-474.

W . F e i t , Some consequences o f t h e c l a s s i f i c a t i o n of f i n i t e s imp le groups. Pruiteedin2s of i'ymposic; ix Fur-e kfu.athematics 37 ( 1980) 175-181.

A. Hartman, Doubly and o r t h o g o n a l l y r e s o l v a b l e quadrup le systems. I n : R.W. Robinson e t a l . (eds . ) , i,'~~!t,i,zat.jr?:uZ M~ithe.oiatics VII. Lec tu re Notes i n Mathematics 829 (Spr inge r , B e r l i n He ide lbe rg New York, 1980).

Ch. Huang, E . Mendelsohn and A . Rosa, On p a r t i a l l y r e s o l v a b l e L - p a r t i t i o n s . AvnuZs of iiiscrele k themc t i c s 12 (1982) 169-183.

B. Huppert , Erd l i zhe CY14pp?7! T (Sp r inge r , B e r l i n He ide lbe rg New York, 1967).

D. Jungn icke l , Ex i s tence r e s u l t s f o r t r a n s l a t i o n ne ts . I n : P.J. Cameron e t a1 , (eds . ) , Ipinitc geonetries uid desigi is. L e c t u r e Notes London Math. SOC. 49 (Cambridge U n i v e r s i t i y Press, Cambridge New York, 1981).

E . S . Kramer, S.S. Mag l iveras and D.M. Mesner. Some r e s o l u t i o n s o f S(5,8,24). J . Comb, Theory ( A ) 29 (1980) 166-173.

E.R. Lamken and S.A. Vanstone, Designs w i t h m u t u a l l y o r thogona l r e s o l u t i o n s . Ezirop. J . Curnbimtorics ( t o appear ) .

M. Marchi , P a r t i t i o n loops and a f f i n e geometr ies. I n : P.J. Cameron e t a l . (eds . ) , Finite 2eometr.ies m i designs. Lec tu re Notes London Math. SOC , 49 (Cambridge U n i v e r s i t y Press, Cambridge New Y G r k , 1981).

P. Q u a t t r o c c h i and C. P e l l e g r i n o , R e t t a n g o l i l a t i n i e " t r a n s v e r s a l des igns" con p a r a l l e l i s m o . A t l i i'e~~. ~ 2 t . pis. G .io. :.;s,?e?~: 28 (1980), 441-449.

R. H. Schulz, On t h e c l a s s i f i c a t i o n o f t r a n s l a t i o n group d i v i s i b l e designs. .'~!WO!J. J . Co/?iZ;lii;LZt,s.&?s ( t o appear) .

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A.P. Sprague, T r a n s l a t i o n ne ts . ~ i t c . ,..,th. Sem. Giei3etz 157( 1982) 46-68.

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t ~ P i i ? ~ 2 (1981) 193-204.

Annals of Discrete Matliematics 30 (1986) 203-216 0 Elsevier Science Publishers B.V. (Nortlr.Hollatid)

PASCALIAN CONFIGURATIONS I N PROJECTIVE PLANES

G i o r g i o F a i n a

D i p a r t i m e n t o d i Matematica U n i v e r s i t a d i P e r u g i a

06100 P e r u g i a I T A L I A

INTRODUCTION

Fo l lowing T i t s [ 19 ] , l e t P he a p r o j e c t i v e Oval of a p r o j e c t i v e

p l a n e TI and l e t P ( d ) be t h e f i g u r e formed by a i l o f t h e s e c a n t s or

t a n g e n t s t o 5: which are p a s c a l i a n l i n e s w i t h r e s p e c t t o R (see

[ 4 , p . 3 7 0 1 ) . The f i g u r e s P ( 2 ) a r e c a l l e d ? -pasCa l i an c o n f i g u r a t i o n s

of 2.

The problem o f d e t e r m i n i n g t h e c o n f i g u r a t i o n P(G) was i n t r o d u c e d

by F. Euekenhout i n [ 4 ] and , by u s i n g t h e s e c o n f i g u r a t i o n s , it is

p o s s i b l e t o p roduce an i n t e r e s t i n g c l a s s i f i c a t i o n f o r t h e p r o j e c t i v e

o v a l s (see, a l so , [ 7 ] ) . The p u r p o s e of t h i s p a p e r i s t c p rove t h e

f o l l o w i n g r e s u l t :

I t seems e x t r e m e l y i n t e r e s t i n g t o m e n t i o n t h a t i n a G a l o i s p l a n e

P G ( 2 , q ) , q=p , t h e problem of d e t e r m i n i n g t h e non-empty R - p a s c a l i a n

c o n f i g u r a t i o n s i s c o m p l e t e l y r e s o l v e d . I f q i s odd , by t h e theorem

o f S e g r e [18], t h e o v a l s are c o n i c s , f o r which e a c h l i n e is p a s c a l i a n

(see [4, p. 3 7 2 1 ) . I f i n s t e a d q i s even t h e n t h e r e is, o t h e r t h a n

h

204 G. Fairia

the class of conics, another class of ovals (called L k a n . & U v n ow&

[lo]) which have a single non-exterior pascalian line and such a

line is a tangent (see [lo]).

We also note that there are projective ovals having non-exterior

pascalian lines in non-desarguesian planes, but in this setting the

problem is very far from being resolved (see [4, p. 3801 , [21] 1 . At

present two other R-pascalian configurations were found:

a) If G is the Wagner's oval [21] then the R-pascalian configuration

contains a unique line (see [7] : Buekenhout also discovered that

this line is a tangent to S7 (see 1 4 1 ) .

b ) If R is the Tits ovoide 2i &abt&ldon [19], then P(Q) coincides with the

set of tangents to R (see [4, p. 3821 and 171).

So we see that in 2 ) and 3) of our Theoran we have exhibited

2-pascalian configurations.

new types of

1. DEFINITIONS PRELlMINARy RESllLTs

For definitions of the terms projectibe plane, aollineation, translation plare,

desarguesian plane and near-field see, for exanple, Segre [MI.

A phojective u u d is defined (see [19]) as a set of pints R of a projective

plane I such that M) three are collinear and through each there passes one and

only ore line (the tangent) that contains IY) other pints of R.

A hexagon of c) is a 6-ple (a a a b b b ) of p i n t s of R , m t recessarily

distinct, such that :

(i,j=lr2,3), where a ,h is a w e n t of R when a.=b Hexagons have three 1 1 - i j'

distinct hiagorule p o i a t h a. ,b, n c,% (ifj) and are called pcrlc&aii when

these three points are collinear. The €amus theorem of Euekenhout 143 m y be

stated as:

1' 2' 3' 1' 2' 3 _I _I.

ai,bj # aj,bi, ai#aj and b #b for i#j and ai#bi - i j

1 3 J i

Id each iach ibed lwxagot1 in a pm jcc..Citive v u d R 0 p a c d i a n -then R in a t u n i c

i n a p o jeotiwe pappian ptane.

Let R be a projective oval ii1 a projective plane TI. A line R of R

Pusculiatr Corij~gurutiorzs 111 Projective Planes 205

is c a l l e d R-puncae-Lun i f each hexagon i n s c r i b e d i n R which h a s t w o

d i a g o n a l p o i n t s on L a l s o has t h e t h i r d d i a g o n a l p o i n t on 1. The

f i g u r e P ( Q ) formed by a l l of t h e p a s c a l i a n l i n e s of II which a r e

s e c a n t or t a n g e n t to R i s c a l l e d R - p a d c a L h n c u n d i g u f i a t i o n of II.

1.1. Oval double loops

I t has been observed by many a u t h o r s ( [4] , [ 6 ] , [7] 1 , t h a t a p r o j e c t i -

ve o v a l may be used t o d e f i n e o p e r a t i o n s of a d d i t i o n @ and m u l t i -

p l i c a t i o n on i t s p o i n t s . W e w i l l d e s c r i b e a method f o r doing t h i s

which i s a s l i g h t m o d i f i c a t i o n of Buekenhout 's procedure [ 4 , p. 3731 . The r e s u l t i n g a l g e b r a i c system (Q ; @ , O ) w i l l c a l l e d an o v a l dou6Le T L o o p .

L e t R be a p r o j e c t i v e oval i n a p r o j e c t i v e p l a n e ll. A r b i t r a r i l y

select t h r e e p o i n t s on R and l a b e l them y l o , i and then l a b e l t h e

p o i n t of i n t e r s e c t i o n of t h e t a n g e n t s a t y and o w i t h x. The

p o i n t s of R, o t h e r than y , a r e then a r b i t r a r i l y a s s i g n e d symbols

w i t h t h e r e s t r i c t i o n t h a t o is a s s i g n e d a and i i s a s s i g n e d 1 .

I f a and b a r e t h e symbols a s s i g n e d t o t h e p o i n t s a and b of

Q\& }, we d e f i n e t h e sum a@6 i n t h e fo l lowing way:

- i f a#o t h e n t h e l i n e a , x w i l l m e e t R i n a p o i n t a ' o t h e r t h a n

0 ; i f a=o then l e t a ' = o . I f bfo then t h e l i n e o , b meets x,y i n a p o i n t z o t h e r than x; i f b=o t h e n l e t z=x. L e t C be t h e

p o i n t of i n t e r s e c t i o n of t h e l i n e a ' , z w i t h QN{a'j; i f

-

~

- a ' , z n R } \ { a ? =pl then c=a ' . L e t t i n g c be t h e symbol a s s i g n e d t o

t h i s p o i n t we d e f i n e u616=c.

I f a rbER\{oly] , w e d e f i n e t h e product a o b as fo l lows:

t h e t a n g e n t a t i meets t h e s e c a n t o , y i n a p o i n t j; i f b# i

t h e n t h e l i n e b , j w i l l m e e t R i n a n o t h e r p o i n t b ' # i ; i f b = i

then l e t b ' = i . L e t t h e i n t e r s e c t i o n of t h e l i n e a , i w i t h t h e

l i n e o , y be h. L e t c be t h e p o i n t of i n t e r s e c t i o n of t h e l i n e

__

- -

-

- - h , b ' wi th t h e s e t R\{b'] ; i f { h,b 'nR} N jb'} =$ then c = b ' . L e t t i n g

c be t h e symbol a s s i g n e d to t h i s p o i n t w e d e f i n e a 0 6 = c .

I f w e c a l l t h e set of symbols used Q , then i t i s e a s i l y seen t h a t

206 G. Fuiria

0 together with the operations defined above is a double loop which

is denoted by

where S:= {o,y,i}. This leads us to

(Q,;@,') which is also called an ova l double l o o p ,

LEMMA 1.[6]- The loop (Q ,@) is an abelian group if, and only if, S

the line

group if, and only if, the line c y is a R-pascalian line, where Qs:=Qx{o}.

i- - x , y is a R-pascalian line; the loop (Q,,') is an abelian

+

It is not difficult to verify that every point p€II\R can be identi-

fied with a involutorial permutation I(p) of the points of R as

follows:

if p~u\R, two points of R are a pair in the involutorial permutation

I(p) if they lie on the same line through p.

LEMMA 2 . [ 4 ] - For a line l of a projective plane containing a pro-

jective oval R , the following are equivalent:

1) .i? is R-pascalian, and

2) for each triple of involutions I(p) , I ( q ) , I (r) with centers on l, the composition I(p)I(q)I(r) is also an involution

with center on 1.

An a u t o m o 4 p h i n m of a projective oval R is a permutation $ of the

points of R which preserves the involutorial permutations I(p),

where p~ll\R, that is to say:

v PErI\R, 3! q&n\R : $ I(p)lp=I(q).

The automorphisms of R form a group. Denote this group by AutR.The

following is easily proven:

each c u L l k n e a t i o n 0 6 ll t h a t pekmuteh R i n t o i t b e t 6 i n d u c e s an a u t o -

mirkph44tn 0 6 0.

We also have the following result.

LEMMA 3.[4]- Let 52 be a projective oval of a projective plane Il and

let a be a collineation of TI that permutes R into itself. The line

Pascaliaii Coiifigirrarioiis iir Projwtive PIaiies 207

_. x,y , where x,y~:R, is a R-pascalian line if, and only if, the line

(x) ,w. (y) is a (1-pascalian line.

1.2. The near-field of order nine (Andrb [l])

Let x =-1 an irriducible quadratic over GF(3). Let K be the set

of all elements of the form a+bi as a and b vary over GF(3),

where we assume i =-1. We wish to define an addition and a product

on K in such a way that, using the field GF(3) addition, K will

be a near-field. We define the addition as follows

2

2

(a+bi) + (c+di) : = (a+c) + (b+d) if for all a, b, c, deGF (3) . We define the product in the following way:

ai=ia, for all a€GF(3)

a(btc)=ab+ac, for all a,b,c~K

ab+ba=O, for all a,bEK\GF(3) , where afb and a+b#O.

It is evident that (K,+) is an abelian group and that K\{O} is a

group.

G’ven a,b,cEK (a+b#O) , there is a unique XEK such that

ax+bx+c=O. 2

Finally, a =-1 for all acK\GF(3).

1.3. The non-desarguesian translation plane of order 9 (Andr6 [l] )

From the Rear-field of order 9 K we may now construct a transla-

tion affine plane of order nine, denoted by T , as follows (see, for example, [1] :

- points are the pairs (x,y) for all x,y~K;

- lines are defined as sets of points (x,y) whose coordinates x,y

0

satisfy an equation of one of the forms

(1) x=a (aEK) , (2) y=ax+b (a,bcK).

There is, up to isomorphism, a unique projective plane T such 0

that T = T \ { d } for a line d of T , where d is called the line at

infinity of To

same TO.

and its points are called points at infinity of the

208 C. Faitia

If p is the point at infinity of y=ax+b then it will be denoted

by (a). If p is the point at infinity of x=a then it will be

denoted by ( m ) .

It has been shown by Denniston [5] and Nizette

translation non-desarguesian plane of order nine T (and in its

dual T I ) the ovals fall into a single transitivity class under the

collineation group. The self-duality property make it unnecessary

to study T and T' separately; so the following example of pro-

jective oval in T will suffices:

[14] that in the

Rodriguez )6] discovered the oval R and Nizette [14] has studied

the group AutR of 32 collineations that leaves invariant an

oval of T and proved that AutR have generators

x ' =-x x'=x I x' =ix-iy

y' =ix+iy J Y"Y with io=-ir a € Aut K.

1.4. The Hughes plane of order nine (Zappa [22])

From the near-field of order 9 K we may now construct a projective

plane, denoted by H as follows:

- the points of H are the triplets ( x ,x ,x 1 , where x.EK, other

than ( O , O , O ) with the identification (x1,x2,x3)=(kxl,kx rkx3) for

all non-zero k in K;

- the lines of H will now be the sets of points (x,y,z) which

satisfy an equation of the form x+yt+z=O, teK, such that if cf is

any automorphism of K then the mapping

1 2 3

2

x'=a xa+b yo+c zu 1 1 1

z'=a xO+b yO+c z(J (VaitbircicGF3), i=1,2,3) 3 3 3

with det(a,,b,,c,)#O, is a collineation of H.

Pascaliun Configurations in Projective Planes 209

Denniston [5] and Nizette [14] have discovered that in the Hughes

plane H, ovals fall into two transitivity classes under the colli-

neation group of H. An oval, D, in one of these classes is inva-

riant under 4 8 collineations, as against 16 collineations for

the other class. So the following examples of non-isomorphic proje-

ctive ovals of H will suffice:

N={(l,i,O), (1,-i-l,O), (l,-l,i+l), (l,-l,-i-l) , (O,i,l), (O,i,-l), (l,~,-i-l),(l,~,i+l), (l,l,i), (l,l,-i)l.

In [14], Nizette proved also that AutN have generators

x'=x-y

y 1 =-y

a a x 1 =-x x'=x +y

y'=x+y y'=x -y a a

0 Z ' = Z Z ' = Z

(with i'=-i-l).

2. PASCALIAN CONFIGURATIONS JJ T AND IN T '

Let R be the Rodriguez oval of the non-desarguesian translation

plane of order nine T. First we show that each non-exterior line

(to R ) through the point ( 0 , O ) is a R-pascalian line. Let .t denote

the tangent y=ix and label the points of R in the following way:

(-i)=O, (i)=-, (-l,O)=l, ( 1 , 0 ) = 2 , (O,l)=i, (0,-i)=-i,

(i,-i) =2i+l, (-i, i) =2+i, (i, i) =2+2i, (-i, -i) =l+i . Letting Q be the symbol assigned to the set Rx{(i)l (i.e. Q is

coinciding with the set of elements of the near-field K ) , we select

the following triplet of points on R :

310 G. Fabra

By 1.1, the algebraic system (QS;@,O) is an oval double loop. Now,

with a straightforward proof which we omit for shortness, we can to

check that

a@b=a+b, for all a,bEQ (1.e. for all a,bEK).

Since (K,+) is an abelian group, we have that (Q ,@) is an abelian

group. Therefore, by Lemma 1, we have that the tangent at (i)=-

is a R-pascalian line.

S

Also, shce AutR is a transitive permutation group on the set

{ (i) , (-i) I C R (see [ 4 ] ) , by Lemma 3, we have that the tangent y=-ix is R-pascalian.

NOW, in order to show that the secant x=O is a R-pascalian line,

it is only necessary to prove that, for

loop ( Q s , ) , where long, but straightforward computation, shows it. It is well known

that (see [ 1 4 ] ) AutR fixes the point (0,O) and that it is transi-

tive on the points of R*{(i),(-i)}. Thus, from Lemma 3, we have that

all non-exterior lines through (0,O) are R-pascalian.

S=I (0,l) , (O,-l) , (i) 1 , the + o +

QS=R\{ (0,l) , (0,-1) 1 , is an abelian group. A very

Now we will prove the non-existence of non-exterior R-pascalian

lines not passing through the point (0,O). The points of R may, for

shortness, be denoted by digits from 0 to 9 as follows:

!i)=o, (-i) =1, (-1,0)=2, (1,O) =3 , (i, -i) =4 , (-i ,i) =5 , (1, i) =6 , (-i ,-I) =7 ,

(0,1)=8 and (0,-1)=9.

By [4, p. 3831 and [20, table 32/34], it follows that, if we denote

by G(8) the group of all elements in AutR which fix the point 8,

then I G ( 8 ) 1=4 and that G(8)={f f f f 1 , where 1’ 2 ’ 3’ 4

Finally, since AutR acts transitively on R\{ (i) , (-i) 1 , by Lemma 3,

the only thing remaining to be shown is that the lines

- - - - - 8 r 8 I 5 , 8 r O,8 i 2 1 8 r 0,l

x e not R-pascalian.

Pascalinn Configurations in Projective Planes 21 I

First of all, consider the following points of ll\R:

p =1,9r18,8, p =1,6n8,8, p =5,8nO,O, p =5,8nOI2, p =0,8fll,l, -~ -- - _ _ _ _ - -~

1 2 3 4 5

-- ~~ -- _ _ _ _ -- p =0,8ni,2, p =2,8no,o, p =2,8noI3, p =0,1n2,2, p =0,1n2,4, 6 7 8 9 10

-- pl1=0, 1fl2,6.

Now, without giving the proofs (which are straightforward but time-

consuming) we remark that p1,p2,p3~8,8 but I(p1)I(p2)I(p3) is

not a involutory permutation of R with center on 8,8; thus, by

Lemma 2, it follows that the line 8,8 is not R-pascalian.

-

-

__

Repeating this process, replacing

gives that the line 5'8 is not R-pascalian.

I (p,) I (p,) I (p,) by I (p,) I (p4)I (p,) , ~

Similarly: I (p5) I (P,) I (P,) , I (P,) I (P,) I (p7) I I (P,) I (pl0) I (pll)

not involutory permutations of R with center in 0,8, 2'8, 0,1,

are - ~ -

respectively. Hence these lines are not R-pascalian.

The R'-pascalian configuration of the dual T' of T is again of

the same type and we omit the analogous proof.

3. PASCALIAN CONFIGURATIONS IN H

In [8], Hughes reproduces the plane H in the useful following way:

- the points are the symbols

-seven of the lines are the following sets of points

A , B ,C.,D ,E.,FifGif i=O,1, . . . , 12; i i i i i

1) IAO,A1 'A3 I Ag, Bo tCo I Do I Eo I Fo 'Go 1

IAotB1rB8tD3 'Dl l tE2 tE5 ,E6,G7,Gg 1 2)

217 G. F U i f l O

G){A C C , D D D E E F , F ) 0' 7' 9 2 ' 5' 6' 3' 11' 1 8

7) {AOiB3rB11 r C 2 rC5rC6 r D7 i D g rG1 rG8);

- the remaining lines are found by successively adding one to the sub-scripts, reducing modulo 13.

In this notation, we remark that (see [5] and [ 1 4 ] ) the ovals D und

N of section 1.4 are the following sets of symbols:

P'{A4'A5'All'A12' B 0 rE 0 rC 6 r D 6 rC 7 rD 7 I

N={B c c ,G c D B F B 0' 0' 4 4' 6 ' 6' 7' 7' ll'E1l' '

We first show that the D-pascalian configuration is the empty set.

The suggestive term JLeaL is used for the points A of H I then in D

there are four real points and six i m a g i n a h y points. It is well

known that (see [5] , [14]) IAutPj=48 and we note further important

properties:

i

1) Auto is generated by: (A11A12) (BoEo) ( C 6 C 7 ) ( D 6 D 7 )

2) AutD is transitive on the set of real points of 0;

3 ) AutD is transitive on the set of imaginary points of D ;

4 ) I (AutD)xl=12 for all real point X E D ;

5 ) (AutD)x is transitive on the set of imaginary points of D for

all real point xrD;

6 ) if x is a real point of D , then (AutD)x is transitive on the

set of real points of D . { x l ;

7 ) if y is a imaginary point of D then I (Autp) 1=8; Y

8) (AutD) =(AutD) : BO EO

9) (AutD) is transitive on the set {C D C D I . 6 ' 6 ' 7 ' 7 BO

We omit the proof which is very long, but not difficult.

By the above properties of AutP and Lemma 3 , the only thing remaining

Pascalian Configurations in Projective Planes 213

to be shown is that no one of the lines

- - - - A4rA4r A12iA12 I A4iA5 r BOrEO I BOrC6 I A4rA12

is D-pascalian. As in the proof of section 2, it is sufficient to

exhibit some appropriate involutorial permutation of the points of

D . First of all, consider the following points of i l \P :

Now we remark that the following permutations

are not involutory permutations of type I(p) with the centers p - ~ - -

in A4,A4, A12,A12, A4,A5, A4,A12, BOIEO~ BO'C6

respective1.y. Hence, by Lemma 2, these lines are not D-pascalian.

Finally, we must show that the N-pascalian configuration is the

empty set too.

Also in this case, it is well known that (see [5] and [14])

1AutNI=16 and it is not difficult to check that:

1) AutN is generated by X=(C C F E G D B B and 4 6 7114 6 7 11

2) AutN fixes the set {Bo,Co};

3 ) AutN is transitive on the sets {B C 1 and I=N\$ ,C 1 respecti- 0' 0 0 0

vely;

4) (AutN) is transitive on 1; BO

5 ) (AutN) =(AutN) : BO cO

=(AutN) =(Id,ul . G4

Hence, in order to prove that P(N)=gf, it is sufficient to show that

no one of the following lines is N-pascalian:

214 G. Faina

A repetitionofthe arguments used in the earlier proof of this

C t i O n shows that the permutations

98-

are not involutory permutations of the points of N with center in

the above mentioned lines, respectively, wile we have that:

-- C41G4, E E EC C C G EC D E A EC 8' 0 4 ' 6' 5' 0 4' 6' 2' 1 4IB7'

Hence, by Lemma 2, no one of these lines is a N-pascalian line.

REFERENCES

1. J. ANDRE' , U b a nLckt-Dwqutbdche Ebenen miit .hanb.Ltiveh ThansLatLonghup-

2. A. BARLOTTI , Un'oodehvatione i & V h n O ad un ,teahema di B. Sqhe 6u.i q-mcki,

3. U. BARTOCCI, Condidekazioni d d h &O&a d&e ova& Tesi di Laurea,

4. F. BUEKENHOUT, Etude imkin6Eque d u o v a t u , Renl. Mat. (5) 25 (19661,

5. R.H.F. DENNISTON, On ~MCA i n po jec f i ve p h n w 06 o t d a 9, Manuscripts

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7. G. FAINA, Un m66inamento deRea oea6bi6icaziofle di BuetzenhoLLt peh g k 5 o v d i

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Le Matematiche 21 (1966) 23-29.

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Math. 4 (19711, 61-89.

(5) 15-A (19781, 440-443.

a b w , Boll. Un. Mat. Ital. (5) 16-B (19791, 813-825.

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okdeh 9 wkich podbe6 an invo.&Lion, J. C a b . Theory 33 (19821, 65-75.

Pascalian Configurations in Projective Planes 2 1 5

10. G. K O R C W R O S , SuRee ow& di .Ouu&zhne i n un piano di G d v D di ohdine p&, Rend. Accad. Naz. XL (5) 3 (1977-78), 55-65.

11. G. KORCHMAROS, U n a geneA&zzaLane d d teahema di Buehenhout A&e ow& p a s c a l h n e , Boll. Un. Mat. ItdL. (5) 18-B (19811, 673-687.

12. R. MAGARI, Le c o n d i g m a z i o n i p a h z i ~ ckiube contevwte net piano P 4ut quabicohpo a ~ n o c i a t i w o di ohdine 9 , Boll. Un. Mat. Ital. (3) 13 (1958), 128-140.

13. G. M E N I C H E T T I , Sapha i k-UJLCki [email protected] n& piano g h a d i c ~ di a h h ~ a z i o - ne di ohdine 9, Le Matematiche 21 (1966) , 150-156.

e,t du p&n de Hughen d'ahche mud, Bull. Soc. Math. Belg. 23 (1971), 436-44 6.

14. N. N I Z E T T E , Pe tehminaLbn den o w d e n du p h n de .i%anS.b.x%on nun ahguesien

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1-18.

M a t . Ital. (3) 14 (1959), 500-503.

13 (19641, 39-55.

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1980.

71 (19591, 113-123.

M a t . Ital. (3) 12 (1957), 507-516.


MONOMIAL CODE - ISOMORPHISMS

Pave1 F i l i p and Werner H e i s e (*)

Mathematisches I n s t i t u t der Technischen U n i v e r s i t a t Munchen

Germany

P.0.Box 20 2420, D -8000 Munchen 2

Le t C and D be two l i n e a r subspaces of GF(q)" and

v : C + D a Hamming weight preserving l i n e a r b i j e c t i o n . It

w i l l be proved t h a t cp i s t h e r e s t r i c t i o n o f a monmia l

t ransformat ion o f GF(q)" , i. e. an n x n - m a t r i x over

GF(q) , which i s t h e product o f a diagonal and a permutation

ma t r i x . An a p p l i c a t i o n shows t h a t t h e group o f a l l Hamming

weight preserving l i n e a r b i j e c t i o n s o f t h e q - a r y Hamming

code HAM(r,q) , r 2 3 , o f length 1 t q t q t ... t q i s

isomorphic t o t h e general l i n e a r group GLr(q) . 2 r-1

Let F be a f i n i t e se t cons i s t i ng o f q 2 2 elements and n z l be an in teger .

For i EZ" := { 1 , 2 . .. , n l we denote t h e ith project ion of t h e n - f o l d

Cartesian product F" by mi : F"+F ; ( X ~ , X ~ ~ . . . , X )+xi . With respect t o t h e

H m i n g dis tance p:F"xF"+lN0 ; ( ~ , $ ) j I { i € ~ " ; ~ ~ ( ~ ) * ~ ~ ( 3 ) } t , which

associates t o every p a i r (?$) E F" x F" t h e number p(?,g) o f pos i t i ons i n which

t h e words ? and 3 d i f f e r , t h e Cartesian product F" becanes a m e t r i c space.

A non empty m e t r i c subspace CcF" i s c a l l e d a (block) code o f length n over F .

I t s minimal dis tance d(C) i s def ined as d(C) : = m i n i p(?$);?,;€C, ?+;I i f

I C I 2 2 and as d(C) := n + 1 or d(C) :=a i f C cons is t s o f on ly one codeword.

Le t C and D be two codes o f length n over F . In con t ras t t o sane more

pragmatic d e f i n i t i o n s i n t h e coding t h e o r e t i c a l l i t e r a t u r e we take a sanewhat

mathematical ly p u r i s t i c a t t i t u d e . We def ine a code -isomorphism cp: C - r D as a

b i j e c t i o n cp frm C onto D which preserves t h e Hamming distance, i. e.

p(cp(?),cp($)) = p ( s f , $ ) f o r a l l x , y ~ C . Extending t h e range o f cp frm 0 t o F"

we a l so say t h a t c p : C+F" i s a code -rnonaorpMsm o f C , Clear ly , t h e

r e s t r i c t i o n OIc o f a code-autmorphism 0 from t h e whole space F" (which

fonns a t r i v i a l code) t o a code CcF" i s always a code-monanorphism of C .

+ - D

(*) The authors g r a t e f u l l y thank prof. L u c a - M a r i a A b a t a n g e l o f o r h e r exce l l en t organisat ion o f t he congress and S e r g i o P o v i a f o r h i s ext raord inary care dur ing our so journ a t t h e h o t e l Riva de l Sole i n Giovinazzo.

218 P. Filip and W. Heise

Every permuta t ion $

automorphism y : F"+F". We s e t "$(?)) IT$(^^(?) f o r a l l

Z E F " . Two codes

~1 o f Zn t h e group G : = { $ c S n ; y ( C ) = C }

Othert imes (e . g. i n [l] ) i t s f a c t o r group

phism group o f C . F r m our p u r i s t i c p o i n t o f v iew, we do n o t agree w i t h these

d e f i n i t i o n s (*) , because t h e r e a r e o the r types o f Hamming d i s t a n c e p rese rv ing

b i j e c t i o n s .

o f t h e symmetric group Sn o f Zn induces on F" a code-

i E n n and a l l

w i t h y ( C ) = D . Sometimes (e . g . i n [ 2 1 f o r t h e b i n a r y case q = 2 )

C,DCF" a r e c a l l e d equivalent, i f t h e r e e x i s t s a permuta t ion

i s c a l l e d t h e automorphism group o f t h e code C . := {y ; $ E G I i s c a l l e d t h e a u t m o r -

L e t x : = ( A ~ , A ~ , . . . , ~ , ) E S : b e a v e c t o r o f l e n g t h

i E Z n , a r e permutat ions o f t h e s e t F . We d e f i n e a code -au tanorph ism $! F" by s e t t i n g ni($!(?)) :=Ai(ni(?)) f o r a l l i EZn and a l l 2 E F " ,

F r m now on l e t q be a power o f a pr ime number and l e t F be t h e Ga lo i s f i e l d

o f o rde r q , F=GF(q ) . The Hamming m e t r i c on t h e vec to r space F"=V,(q) i s

induced by t h e k m i n g w e i g h t , i . e . t h e norm y : F " + N o ; x - , I ~ i E Z n ; n i ( ? ) + 0 I l ,

i n t h e usua l way. F o r ?,?EF" we have P(?$) =y(?-;) . I n case n = l , t h e

Hamming we igh t y : F+{O, l I i s t h e o r d i n a r y c h a r a c t e r i s t i c f u n c t i o n o f t h e s e t

F* : = F \ { O } . Using t h e Kronecker symbol 6i,j we have y(x) = 1 f o r a l l

x E F . For a subset C c F " we abbrev ia te y(C) : = F y ( t ) .

By a Zinear (n,k) -code C over F we understand a m e t r i c subspace C c F " , which i s a l s o a k - d i m e n s i o n a l l i n e a r subspace o f F" . L e t C c F " be a l i n e a r

(n,k) -code. For i EZn t h e r e s t r i c t i o n nilc : C+F of t h e ith p r o j e c t i o n

: F"+F fran F" t o t h e l i n e a r code C i s a l i n e a r form. Th is l i n e a r fo rm "i

nilc i s non - t r i v i a l i f and o n l y i f t h e derivat ion Ai(C) := ker(nilc) , i. e.

Ai(C) = { ~ E C ; mi(?) = 01 , of C i n the posi t ion i i s a ( k - 1) -d imens iona l

l i n e a r subspace o f C , A code-monomorphism (p:C+F" w i l l be c a l l e d Zinear, i f

i t i s a l i n e a r mapping. A l i n e a r i n j e c t i v e mapping c p : C+F" i s a code-monanor - phism, i f and on ly i f i t preserves t h e Hamming we igh t , i. e. i f f y(cp(-d))=y(?)

n , whose components hiesF , o f

p +

E C

f o r a l l ~ E C .

For i E Zn we denote by

aector of F" . Now l e t -# ei ._ .- (6i,l,6i,z,...,6 ) t h e ith canonical unit

i,n $€Sn b e aga in a permuta t ion o f Zn . Then

<K$ (i) (;).ti = e n i ( ? ) .t i = 1 k= 1 (i)

~

f o r a l l Z E F " . Hence i s a l i n e a r t r a n s f o r m a t i o n o f F" , $EGLn(F) , which can be represented

w i t h respec t t o t h e canon ica l

b a s i s t z i ; i ~ Z n } o f F" . Mow l e t h=(A1,A2,...,An)E(F*)" be a v e c t o r by t h e permuta t ion m a t r i x (6 i ,$-1 ( j ) 1 1 S i , j 9 n

(*) Remark by W. He ise : What do I c a r e about my rubb ish s a i d yes te rday !

Monornial Code-lsomorphisms 219

where a l l cmponents a re non -zero. For i E Z n we i d e n t i f y t h e element A ~ E F

w i t h the permutation XiES, def ined by Xi(x) :=Xiax f o r a l l x € F , Then

f(?) = f ( > ~ ~ ( ? ) - z ~ ) = >ni(z).xi.ti f o r a l l ?EF" . Thus, f i s a l i n e a r

t ransformat ion o f F" , '?EGLn(F) , which can be represented by t h e d iagona lma t r i x

n n

i = l i= 1

w i t h respect t o the canonical basis o f F" . ( A i o 6 i , j ) l < i , j < n

A l i n e a r t ransformat ion @EGLn(F) o f F" i s c a l l e d m o n a i a 2 , i f t h e r e e x i s t s a

vector X = ( X 1 , X 2 , . . . , A , ) E ( F * ) " and a permutation $ E S n such t h a t 0 = b y , i. e. i f n i ( O ( z ) ) = h i a n (?) f o r a l l i €Zn and a l l 2 E F " . Obviously,

monomial l i n e a r transformations o f F" preserve t h e Hamming weight o f every vec to r

?EF" . Le t C be a l i n e a r (n,k) -code over F . A l i n e a r code-monanorphism

c p : C-F" w i l l be c a l l e d monomia2, i f i t can be extended t o a monanial l i n e a r

t ransformat ion a : F"+F" ; i. e. i f the re e x i s t s a monanial transformation

OEGLn(F) , whose r e s t r i c t i o n t o C i s cp , (oJc = c p . A code -moncmorphism

c p : C+F" i s monmia l , i f and only i f the re e x i s t n elements h l y X 2y...,AnEF*

and a permutation $ E S , such t h a t mi(&) =Xi.n6ci.(t) f o r a l l i E Z n and

a l l ? E C .

$(i)

F. J. M a c W i l l i a m s and N . J. A . S l o a n e [2;p.2381 d e f i n e t h e autanorpnism

group o f a l i n e a r (n,k)-code C over F as t h e group o f a l l those monanial

l i n e a r transformations OEGLn(F) o f t he whole vector space F" , which leave t h e

code C i nva r ian t , i. e. f o r which we have a(?) E C f o r a l l Z E C . I n our te rm i - nology, t h e group o f l i n e a r code-autcmorphisms o f C cons is ts o f a l l l i n e a r

transformations cpEGLk(F) o f t he k -dimensional vector space C , which preserve

the Hamming weight, i. e. f o r which y(cp(2)) = y ( t ) ? E C . So t h e group o f

a l l l i n e a r code -automorphisms o f any l i n e a r equidistant

v ( z ) =d(C) f o r a l l ? € C \ l d l problem 33 o f [2;p.231fl from the binary, q = 2 , t o t h e general case we prove i n

t h i s paper - w i t h our d i f f e r e n t conception o f a code - isomorphism - t h a t every

l i n e a r code - ismorphism i s monomial. This then gives evidence f o r t h e f a c t , t h a t

our p u r i s t i c a t t i t u d e i s no t too f a r from F. J . M a c W i l l i a m s ' and

N. J. A . S l o a n e ' s pragmatic pos i t i on . For t h e proof we make sane a u x i l i a r y

propos i t i ons . L e t C be a l i n e a r (n,k)-code over F=GF(q) , cp:C+F" a l i n e a r code-mono-

morphism and D:=cp(C) ,

f o r a l l

(n,k) -code (i. e.

i s t r i v i a l l y t h e group GLJF) . Extending

The f i r s t proposi t ion, as an easy consequence of t he rank formula f o r matrices,

whose l i n e s form a bas is of t h e l i n e a r code C , does no t make use o f t h e m e t r i c

s t r u c t u r e of t h e code C .


PROPOSITION 1. L e t t be an i n t e g e r w i t h O i t s k and i ( 1 ) , i ( 2 ) , . . . , i ( t ) € Z n

p a i m i s e d i f f e r e n t i nd i ces . The t p r o j e c t i o n s TI^(^)^^ , h=1 ,2 , ..., t a r e

l i n e a r l y independent i f and o n l y i f f o r every cho ice o f t ( n o t necessa r i l y

d i s t i n c t ) elements a l y ~ 2 , . , . y a t E F t h e r e a r e e x a c t l y qk-t codewords t € C w i t h T T ~ ( ~ ) ( ? ) = ~ ~ , h=1 ,2 , ..., t .

Note, t h a t t h e case t = k prov ides us w i t h a c h a r a c t e r i s a t i o n of t h e l i n e a r so - c a l l e d MDS -codes, c f . [ l ;p . l64 f . I o r [2;p.317ff .I. I n t h i s paper p r o p o s i t i o n 1 i s o n l y used i n t h e case t = 1 . I n t h i s case p r o p o s i t i o n 1 i s a mere r e f o n n u l a - t i o n o f t h e "Satz ube r d i e G l e i c h v e r t e i l u n g d e r Zeichen i n l i n e a r e n Codes" f rom

r1;p.2103.

PROPOSITION 2 . L e t s be an i n t e g e r w i t h 02s I n - k . I f p r e c i s e l y s o f t h e

n p r o j e c t i o n s TI. i EZn , a r e t r i v i a l , t hen a l s o p r e c i s e l y s o f t h e n

p r o j e c t i o n s nj I D , j €Zn , a r e t r i v i a l .

Proof. F o r each non - t r i v i a l p r o j e c t i o n m i l c , i E Z n , by p r o p o s i t i o n 1 we have

F y ( n i ( z ) ) = qk-'- ( q - 1) . F o r each t r i v i a l p r o j e c t i o n nilc , i €Zn , we have E C

> y ( n i ( ? ) ) = 0 . There fore y(C) = > Y > y ( n i ( z ) ) = (n - s).qk-'.(q - 1) . Exac t l y i n t h e same manner we prove y(D) = (n -a) .qk- 'a (q -1) , where a

number o f t h e t r i v i a l p r o j e c t i o n s

serves t h e Hamming weight, so we have y(C) = y ( D ) , whence a = s .

1Ic

6 € C i = l ~ E c i s t h e

n j J D , j EZn . The b i j e c t i o n c p : C + D p r e -

The p roo f o f p r o p o s i t i o n 2 i s e s s e n t i a l l y t h e o n l y p lace i n which we make use o f

t h e f a c t , t h a t F=GF(q) i s a f i n i t e f i e l d . However H a n s K e l l e r e r (Math. I n s t .

d. TU Munchen, n o t Hans K e l l e r e r , Math. I n s t . d. LMU Munchen) showed t h a t p r o -

p o s i t i o n 2 ho lds a l s o f o r codes over an i n f i n i t e f i e l d F : Denote by

n ( h ) , h = 1,2,.. . , s , those s i nd i ces from Zn f o r which t h e p r o j e c t i o n s

n j I D , j €Zn . S ince F i s i n f i n i t e , t h e r e i s a codeword ? € C w i t h ni(?) * O

f o r each i E Z n \ I n ( l ) , n ( ( 2 ) , ..., n(s) 1 , hence y(cp(2)) = y ( t ) = n - s and t h u s

n - s 5 n - u . Since t h e r e i s a l s o a codeword d E D w i t h ~ ( d ) = n - U we g e t

y(cp-l(d))=y(d) = n - a and thus n - IS 5 n - s , whence a = s . Independent ly

L u d w i g S t a i g e r (ZKI d. Akad. d. Wiss. d. DDR) gave another p roo f o f t h e same

f a c t which w i l l be i nc luded i n h i s fo r thcoming paper "On cove r ing codewords" i n

t h e A t t i de l Seminar io Matematico e F i s i c o d e l l ' Un ive rs i t a ' d i Modena. As a

consequence t h e theorem o f t h i s paper does no t depend on t h e f i n i t e n e s s o f t h e

under l y ing f i e l d F .

a r e t r i v i a l . By a we denote t h e number o f t h e t r i v i a l p r o j e c t i o n s n(h) I C

TI

-P

+ - t

We denote by N = I n(l),n(2),...,n(s) 1 t h e s e t o f t hose i n d i c e s n ( h ) E Z n , h=l,2, . . .,s , f o r which t h e p r o j e c t i o n i s t r i v i a l . Then, by p r o p o -

nn(h) Ic

Monomial Code-Isomorphisms 22 I

s i t i o n 2 t h e r e i s a s e t M = I m ( l ) , m ( 2 ) , ..., m ( s ) l c Z n o f s i nd i ces , such t h a t

i E Z $ M t h e p r o - t h e p r o j e c t i o n s n

j e c t i o n nilD i s always non - t r i v i a l . F o r each index j EZn\N we i n d i c a t e by

K ( j ) := I j( l), j(2),...,. i(r.) 1 t h e s e t of a l l t hose r . i n d i c e s j ( h ) EZn\N , h = l , Z , . .. , r . such t h a t t h e r e e x i s t s an element ah EF* w i t h

3 n (h) Ic =ah-n j Ic . O f course, f o r a l l 1 -< r . 5 n - k t 1 . We

denote by R a system o f rep resen ta t i ves of t h e s e t s

IN1 U I K ( j ) ; j E R l i s a p a r t i t i o n o f 7Ln and we have n = c r t s .

PROPOSITION 3 . L e t j E R be an index . Then t h e r e i s a s e t

L ( j ) = { i ( l ) , i ( 2 ) ,..., i ( r j ) 1 cZn\M o f r j i n d i c e s and t h e r e a r e r ( n o t

n e c e s s a r i l y d i s t i n c t ) elements X i ( l ) , X i ( * ) ,... t h e r . p r o j e c t i o n s ni (h) ID , h = 1,2,. . . ,r. , a r e l i n e a r l y dependent and such

t h a t ni(,,(cp(Z)) = h i (h , . n j (h , ( ? ) f o r h = 1 , 2 ,... ,r. and a l l S'EC . If j l E R \ { j l i s another index, t h e n L ( j ) n L ( j ' ) = 0 .

, h=1,2, ..., s , a r e t r i v i a l . F o r m ( h ) ID

3 ' j E Zn\N we have

I K ( j ) , j E Z n \ N . Then

~ E R j

1 EF* such t h a t any two of "i (r )

1 1

3

- Proof . The d e r i v a t i o n A : = A . ( C ) = k e r ( n . I ) o f C i n t h e p o s i t i o n j i s a 3 I C

( k -1) -d imens iona l l i n e a r subspace o f t h e k -d imens iona l v e c t o r space C . It co inc ides w i t h t h e d e r i v a t i o n s A. ( C ) of C i n t h e p o s i t i o n s j ( h ) , h = 1 , 2 ,..., r . . The r . p r o j e c t i o n s n j ( h ) ( A , h =1,2 ,..., r a r e t r i v i a l as

3 7 j w e l l as t h e s p r o j e c t i o n s n n ( h ) I A , h = 1,2,...,s . The l i n e a r subspace

B :=cp(A) o f t h e k - d i m e n s i o n a l v e c t o r space D=(p(C) has d imens ion k - 1 . We

app ly p r o p o s i t i o n 2 t o t h e codes A and B and t o t h e l i n e a r code - isanorph ism

cpIA : A + B . There a r e r . t s t r i v i a l p r o j e c t i o n s nilB , i EZn among them t h e

p r o j e c t i o n s nm(hl J B , h = 1,2,. .. ,s . The o t h e r r . t r i v i a l p r o j e c t i o n s

n . 1(1) IB 'n i (2 ) IBI . . . ,~

1 (h)

7

3 a r e r e s t r i c t i o n s o f t h e non - t r i v i a l p r o j e c t i o n s

t o t h e subcode B c D . Thus t h e d e r i v a t i o n s i ( r j ) IB

i ( 1 ) ID"i(2) i ( r . ) ID

Ai (h) 3

n

(D) o f D i n t h e ' p o s i t i o n s i ( h ) , h=1 ,2y . . . y r . , a l l c o i n c i d e w i t h B . Thus t h e p r o j e c t i o n s I D , h=1,2 , ... ,r. , a r e p a i r w i s e l i n e a r l y dependent.

S ince C i s a l i n e a r code we can choose a codeword z E C \ A w i t h n . ( b ) = 1 . By -b I

s u p p o s i t i o n we have n j c h ) ( b ) = a h , h = l Y 2 , . . . , r , . It i s cp(z)Ecp(C\A) = D \ B , thus f o r h = 1,2,. . . , r . t h e element B ~ ( ~ ) := ni (h) (cp(b)) E F i s always non - z e r o .

Each codeword t E C = @ A can u n i q u e l y be w r i t t e n as c = B -b t ; w i t h

B E F and ~ E A . For h = l y 2 y . . , y r we s e t Xi h) : = B i ( h ) / ~ h E F * . Then f o r

h = l Y 2 , . . . , r . we have ~ ~ ~ ( ~ ) ( c p ( t ) ) ~ ~ ~ ( ~ ) ( B . c ~ ( i ) t v ( d ) ) = E*nich)(cp(lf)) =

1 +

1 +

L + +

j

1 -b + + = B.Bi(,)*ah/ah = B*Xi(h)*a h j * n (b ) =

= X i ( h ) . n j ( h ) ( 6 ' b + a ) = 'i(h)*"j(h)(')

j ( h ) ( b ) = X i ( h ) ' n j ( h ) ( B * b ) = + +

The s e t L ( j ) c o n s i s t s o f t h e r j ind i ces i ( 1 ) y i ( 2 ) y . . . , i ( r . ) .Analoguosly t h e

s e t L ( j ' ) c o n s i s t s o f rjl i n d i c e s i 1 ( l ) , i ' ( 2 ) , . . . , i ' ( r . , ) . Suppose t h e r e

e x i s t s a n index

3

3 i E L ( j ) n L ( j ' ) . Then, by what we have seen above, t h e r e e x i s t


two elements ni ,rl; E F*

j Ic a l l t E C . Hence n

supposi t ion t h a t j and

w i t h ni(w(t)) = ni-nj(?) and n,(cp(t)) = n;.n 1' (t) f o r

and IT j, Ic are 1 i n e a r l y dependent, con t rad i c to ry t o our j ' a re two d i s t i n c t ind ices from R .

THEOREM . The l i n e a r code -monmorphism cp i s monanial.

- Proof. The system I M 1 u { L ( j ) ; j ER 1 i s a p a r t i t i o n o f Zn .We define a pennu-

t a t i o n J I E S , o f Zn by past ing together t h e b i j e c t i o n s I)N : M + N ;m(h)+n(h)

and J, : L ( j ) + K ( j ) ; i ( h ) + j ( h ) , j E R . For h=1,2, ..., s w e s e t A m ( h ) : = I (we

could choose as w e l l any other element o f F* ) and get 1

(2) f o r h=1,2,.. . ,s and a l l t € C , by proposi - (t) f o r a l l j ER ,

'rn (h) ('(')) = = 'rn (h) " J , (m (h) ) + t i o n 2. By proposi t ion 3 we get

f o r h=1,2 ,..., r and f o r a l l ?EC . Thus @ = t o $ . ( ~ ( c ) ) = X i (h);nJ, ( i ( h ) )

j

Note t h a t t he permutation J, i s uniquely determined, i f r . = 1 f o r a l l j E R

and i f s = O o r s = l . I n c a s e r . = l f o r a l l j E R and s = O ; i . e . i f a l l

t h e n pro ject ions nj Ic , j E Z , , are pai rwise l i n e a r l y independent then t h e

vector A = (A1,AZ,.. . ,An) E (F*)" i f r . = 1 f o r a l l j E R , s = l and q = 2 .

I n F" we use the usual sca la r product F" x F"+F ; (2,;) :=>ni(x)*ni(y) . The dua2 (a more appropr ia te but unusual name would be "orthogonal")

C* := r?EF";?.t=O 1 of a l i n e a r (n,k) -code C over F i s a l i n e a r (n,n-k)-

code over F , which i s not necessar i ly a canplement t o C i n F" .

3

7

i s uniquely determined, a lso. The same i s t r u e

3

+ + n

i=1

Now l e t C be a l i n e a r (n,k) -code over F , such t h a t any two o f t h e n pro - jec t i ons nj Ic , j EZn , are l i n e a r l y independent. By the "Untere Abschatzung des

Minimalabstands 1 inearer Codes" [1;p.2271 t h i s cond i t i on i s equivalent t o the f a c t

t h a t t h e minimal d is tance o f t h e dual CL o f C i s a t l e a s t three, d(C*) 2 3 . (Other codes are i n many respects f a i r l y un in te res t i ng . ) Then the re i s f o r every

l i n e a r code -autmorphism cp o f C p rec i se l y one monomial t ransformat ion

Q, = 2 07 E GLn(F) o f F" w i t h X = ( A l , X *,..., An) E (F * n ) , J, €Sn and @Ic = Q . Therefore i n case d(C*) 2 3 the re i s no d i f f e rence between MacW i 11 iams I and

S l o a n e ' s concept o f the l i n e a r autanorphism group o f the code C and the author

authors' concept. These groups a re isomorphic. The t ransformat ion a' = t-' 07 as

a monomial transformation o f F" preserves the Hamming weight o f every codeword

d E C l . From O(C) =C we deduce O'(C1) = C * . Thus the map c p + Q ' IcL i s a hano - morphism fran t h e group o f a l l l i n e a r code-automorphisms o f C i n t o t h e group o f

a l l l i n e a r code-autmorphisms o f C* . If any two o f t h e n pro ject ions nilcI , i EZ, , are a l so l i n e a r l y independent, i. e. i f d(C) 23 , then the r a l e s o f

C = (CL)l can be interchanged i n our argunentation and t h e groups o f a l l l i n e a r code-autanorphisms o f C and C* are isanorphic.

Monomial Code-Isomorphisms 223

F o r r = 3 , 4 , ... t h e simplex code HAMl(r,q) i s d e f i n e d as a l i n e a r (n,r) - code

over F =GF(q) o f l e n g t h n = (qr - l)/(q - 1) , which as a l i n e a r subspace o f F"

i s generated by t h e rows o f a r x n - m a t r i x over F , whose columns fo rm a sys ten

o f r e p r e s e n t a t i v e s o f t h e one-d imens iona l l i n e a r subspaces o f Fr ( c f . e . g.

[ l ; p .232 ] ) . O f course, t h e r e a r e many code - i sanorph ic ve rs ions o f HAMl(r,q) . One can change t h e o r d e r o f t h e columns and g e t e q u i v a l e n t codes. One can a l s o

choose o t h e r systems o f rep resen ta t i ves . Note t h a t every l i n e a r code C w i t h

d (CI ) 2 3 can be ob ta ined by punc tu r ing ( i . e. d e l e t i n g t h e components i n sane

f i x e d p o s i t i o n s i n a l l codewords) a s u i t a b l e s imp lex code. The dua l o f HAMl(r,q)

i s t h e q - a r y Hannning code HAM(r,q) . The s imp lex code i s e q u i d i s t a n t w i t h

min ima l d i s t a n c e d(HAMl(r ,q)) = q r - l > 4 . Indeed, any l i n e of i t s genera to r

m a t r i x has (qr-' - l ) / ( q - 1) z e r o e n t r i e s and any non - z e r o codeword o f HAMl(r,q)

can be i n t e r p r e t e d as a l i n e i n a genera to r m a t r i x o f HAW(r ,q ) wh ich i s

o b t a i n a b l e f rom t h e o r i g i n a l genera to r m a t r i x by app ly ing o n l y elementary

ope ra t i ons on t h e l i n e s . The group o f a l l l i n e a r code -au tanorph isms o f tIAtP(r,q)

i s t h e genera l l i n e a r g roup GLr(F) . S ince by d e f i n i t i o n any two o f t h e l i n e a r

forms

d(HAM(r,q)) = 3 ) t h e group o f a l l l i n e a r code-autanorph isms o f t h e Hamming code

HAM(r,q) i s i sanorph ic t o GLr(F) .

F i n a l l y , we make a bow t o p r o j e c t i v e geanet ry and remark t h a t t h e theorem o f t h i s

paper a p p l i e s m u t a t i s mutandis t o "semi - l i n e a r code - i smorph isms" .

nilml(r,q) , i cZn , a r e l i n e a r l y independent ( i n f a c t i t i s

REFERENCES

[11 Heise, W . and Q u a t t r o c c h i , P. , In fo rma t ions - und Cod ierungs theor ie (Sp r inge r ,

B e r l i n -He ide lbe rg - N e w York -Tokyo, 1983).

121 MacWil l iams, F. J . and Sloane, N. J . A. , The theo ry o f e r r o r - c o r r e c t i n g

codes (Nor th -Ho l l and , Amsterdam - New York -Ox fo rd , 1977).


ONTHE CROSSING NUMBER OF GENERALIZED PETERSEN GRAPHS

S . Fiorini

Department of Mathematics, University of Malta

ABSTRACT The Generalized Pcte:-scfi Gmnn P ( n , k ) is defined to De the

la,, a*, ..., an,bl,b2 ,..., bn} and edges ta.b.,a.a. qraph on 2n vertices !abel led

1 1 1 i+1 'bibi+k:

i = 1,2,.. .,n; subscripts modulo nl. The crossing numbers

v(n,k)of P(n,k) are determined as follows: ~ ( 9 ~ 3 ) = 2, v(3h,3) = h, v(3h+2,3) = h+2, h+l&v(3h+l ,j)Lh+3,v(bh,4)=2h; various conjectures are formulated.

PRELIMINARlES All graphs C I (V(G),E(G)) considered will be simple, i.e. contain no loops or multiple edges. 'be Generalized Petersen Graph P(n,k) is defined to be

the graph of order 2n with vertices labelled ia,a2 ,..., an,bl,b2 ,..., bn} and edges (aibi,aiai+, ,bibi+k:i=l ,2,. . . ,n; subscripts modulo n , l6ki.n-I 1 'The derived Generalized Petersen Graph denoted Pt(n,k) is obtained from

P(n,k) by contracting all edges of form ai,bi, called spokes; edges of form bibi+k in P(n,k) are then called chords of the n-circuit al,a2, ..., an,a,. A drawing of a graph in a surface is a mapping of the graph into the surface in such a way tnat vertices are mapped to points of the surface and edges vw to arcs in the surface joining the image-points of v and w and the image of no edge ccntains that of any vertex.

surface we consider is the plane and all our drawings will be

sense that no two arcs which are images of adjacent edges have a common point

other than the image of the c o m n vertex, no two arcs have more than one

point in common, and no point other than the image of a vertex is c o m n to more than two arcs.

c m o n vertex is called crossing. A drawing is said to be optimal if it

minimizes the number of crossings; clearly, an optimal drawing is necessarily good.

by v(G); the number of crossings in a drawing U of C is denoted by vp(G).

In our case, the only

in the

A common point of two arcs other than the image of a

rhe number of crossings in an optimal drawing of a graph C is denoted

226 S. Fiorini

TECHNIQUES The technique of proving that t h e crossing number of some graph C is some p o s i t i v e i n t e g e r k is q u i t e s tandard.

is e x h i b i t e d whereby a n upper bound for k is e s t a b l i s h e d . method it is then shown that t h i s number i s also a lower bound. Embodied i n t h e theorems of t h i s s e c t i o n we p r e s e n t some conclusions of a general n a t u r e which hope fu l ly could be used also i n determining the lower bounds of c r o s s i n g numbers of o t h e r graphs.

Some g o d drawing By some ad hoe

THEORW 1 If two g raphs C and H are homeomorphic, t hen t h e i r crossing numbers are i d e n t i c a l . / I

COROLLARY1 ('Ihe Monotone l'heorem) If u = ( k , n ) , t h e greatest comnon d i v i s o r of k and n , and i f 2 6 u 6 k < i n ,

then

and 'n, k fvn-n f a , k-klo '

where w denotes v(P(n,k.) f . n,k

P B and l e t c a s e u 4

Let H be obtained from P ( n , k ) by d e l e t i n g k success ive spokes i( be obtained from P(n,k) by d e l e t i n g every k ' t h spoke i n t h e 2. Then H is homeomorphic to P(n - k,k) and i f u 5 2, then K is

homeomorphic t o P(n-n/o ,k-k/a). 'The r e s u l t fo l lows from 'Theorem 1. / /

If C is a graph and X 5 V(G)oE(G) t hen the subgraph induced by X is denoted by u(>.

THEORM 2

If v3= 0 , then

(The Decomposition Theorem) graph G and l e t E(C) = XWYJZ, XnY = YnZ = ZlrX 0 .

Let 0 be an opt imal drawing of a

v(G) = vo(uY> + vaCd,z>

PROOF v(G) E vD(C) = v O U ~ Y > + vyo[uz> - vVCD + k , where k

is t h e number of c r o s s i n g s of form Y x 2

5 VV<XUY> + w p z > - va*

= vyu(aY> + v aCuZ>, s i n c e v (x> 0 0 0

The followiw c o r o l l a r y r e a d i l y follows by induc t ion on k: COROLLARY 2 Let D be an opt imal drawing of a graph G i n which some s u b s e t

X of E(C) makes 0 c o n t r i b u t i o n t o vu(C).

, YinY. = 0 ( i d j ) is a d e c m p o s i t i o n J

of E(C) t hen v(G) f & v <XUYi>. / I

The Crossing Number of Generalized Petersen Graphs

THEOREM 3 graph G whose deletion fran G results in a planar subgraph H of C. u (GI b a.

PROOF edges being intersected results in a planar subgraph of GI contradicting the minimlity of a. / /

We often make use of this simple conclusion in conjunction with Euler's polyhedral formula as in the following:

(The Deletion 'heorem) Let u be the least number of edges of a Then

Assuming on the contrary that w < a, then deleting the (at most)v

THEOREM 4 w(9,3) I 2

PROOF M o s that it is also a lower bound we note that P(9,3), contains as subgraph a homeomorph of the graph C of Figure 1 (ii); obtained by deleting an edge from each of the three triangles of P(9,3). ) G has 12 vertices, 18 edges and girth 5, so that if u edges are deleted to obtain a planar subgraph HI Euler's formula for H implies that

The graph of Figure 1 (i) shows that 2 is an upper bound for v(9,3).

(the subgraph is

5 ( b - a) 6 2(18 - a). Thus, ~ ( 9 ~ 3 ) + a 5 r4/31 = 2. / /

(ii)

Fig.1.

227

228 S. Fiorini

THEOREM 5 le t e E E(G) make 0 contr ibut ion to vi) ( G ) . Let G be t h e graph obtained from C by c o n t r a c t i g t h e edge e = uv to a single vertex u = v and let 0' be the drawing of G induced by 0.

(lhe Contraction Theorem) Let 0 be a grawing of a graph C and

Then wv ( C ) p wDi

PROOF Let wv fEE(G) such that f is adjacent t o e uv,

( i ) If f d E(Ge) ( t .g uw E E ( G ) ) , then any crossing involving f i n 1) is missing i n 0' ;

If f E E(Ge) and f i s crossed by some edge t u i n 0, then t h i s crossing is a l s o missing i n 0'.

( ii)

Since a l l o ther crossings a r e unaffected, i n a l l cases

COROLLARY 1 t o v,(G), then v(G) 5 w(G 1.

I f v is an g p t i m l drawing i n which e E E ( G ) niakes 0 contr ibut ion

- Pi3OOF By the Contraction Theorem,

v(G) uD(C) a uu , (Ge) 9 "(Gel. / /

Repeated use of the Contraction Theorem yie lds the following:

-- COROLLARY 2 0 contr ibut ion to vD!h) i n iome drawing U . I f w e def ine recursively G = G, Oo = 0, Gi = (G1- ' Ie i , Vi the drawinp; of Gi induced by Ui-', then

0 1 t uOO(G ) 2 <,,,((?I 1 3 ... 2 wyt(G 1. / /

Let <e ..., e > be a sequence of edses of G each of whicb makes

CSROLLARY 3 C such that for Rach edge eHof H , e makes 0 contr ibut ion t o w U ( G ) . ' h e n w(G) 2 v(G ), where C is obtained from G by contract ing each edge e of' H.

-- PROOF We order t h e edges o f :I and apply Corol la r ies 1 and 2.

Let D be an optimal drawing of C and let H be a subsraph of

/ /

R M A H K S

(i) In t h e Contraction meorem and its c o r o l l a r i e s , t h e condition t h a t "e makes 0 contr ibut ion t o w l l is v i t a l . K by "expanding1' any vertex i n t o two adjacent ver t ices u , v , of valency 4. It is not d i f f i c u l t t o see t h a t 7 2 w ( C ) ) U ( C ~ ~ ) I v(K7) 9. 'The reverse inequal i ty w(Ce) 2 w ( C ) cannot i n general be proved, even i f o ther condi t ions (e.g; i f G conta ins no t r i a n g l e s ) a r e imposed.

Consider the graph G obtained from 7

(ii)

The Crossing Number of Generalized Petersen Graphs 229

THEOREM 6 there e x i s t s an opt imal drawing i n which the ( 3 k ) - c i r c u i t C does n o t i n t e r s e c t i t se l f .

If Ck deno tes t h e de r ived graph P1(3k ,3 ) , then v(Gk) = k and ---

PROOF That v(Gk) 4 k follows from t h e drawing of F igu re 2: ~

Fig . 2

To e s t a b l i s h the r e v e r s e inequal i . ty we n o t e t h a t t h e d e l e t i o n of any t h r e e success ive edyes of C y i e l d s a subgraph homeomorpnic t o G

is now proved by induc t ion on k. Theoren to G4, for which m = 24, n = 12 and g = 4 , so that

'The s t a t emen t K-1'

To start t h e induc t ion we apply t h e De le t ion

f = ( 2 4 - ~ ) - 1 2 + 2 = 1 4 - a => 4(14 - a) & 2(24 - a) => v 2 u 2 4,

and t h e s t a t emen t is v a l i d i n t h i s case.

We now cons ide r an opt imal drawing U of Gk and assume, for c o n t r a d i c t i o n , t h a t

v3 (C,) k-I. If' C does n o t i n t e r s e c t i t s e l f i n 3 , t hen by t h e Decomposition [heorem wi th CC, = C and Yi ( i = 1,2, ..., K), we conclude t h a t v(Gk) k , s i n c e t h a t i n t h i s c a s e v ( G , 1 = k and t h e r e e x i s t s a drawinq i n which C does n o t i n t e r s e c t i tself. I f , on t h e o t h e r hand, C i n t e r s e c t s i t s e l f i n some edge e , then by d e l e t i n g e and two success ive edges of C, we o b t a i n Ck-l f o r which Lhe

i n d u c t i v e hypo thes i s imp l i e s :

t h e i ' t h set of three success ive chords

( X U Y i ) = 1. It follows

x

a c o n t r a d i c t i o n . / I

The same arqcnen t , on ly s l i T h t l y modif ied, h o l d s for PV(3k+h,3) ( h = 1,2) and determines t h i s c r o s s i n g number as k + h. However, s i n c e the i n d u c t i v e argument fails i n its i n i t i a l s t e p for h = 1 ( t h e g i r t h of P 1 ( 7 , 3 ) = 31, we start wi th k = 3 for t h i s case.

230 S. Fiorini

THEOREM 7 and for h = 2 , k & 2 , v(G,) = k + h. i n which t h e ( jk+h) -c i r cu i t C does n o t i n t e r s e c t itself.

If Ck denotes t h e der ived graph P1(3k+h,3) , then f o r h = 1 , k 3 3 Fur the r , t h e r e exists an optimal drawing

PROOF That v(Gk) c k + h fol lows from t h e drawings of Fig. 3.

t h e r eve r se i n e q u a l i t y we proceed by induct ion and no te t h a t for ( h , k ) = ( 1 , 3 ) or (2 ,2 ) the g i r t h is 4 and (n,m) = (10,201 and (8,16) r e spec t ive ly . Delet ion Theorem, then y i e lds :

To e s t a b l i s h

'he

f = 12 - a and f 10 - a, r e s p e c t i v e l y , so that

4(12 - a ) L 2(20 - a) and 4(10 - a) c 2(16 - a), r e spec t ive ly ; i n e i t h e r case v 2 a .r 4 = k + h.

Now suppose t h a t C makes 0 con t r ibu t ion to vI) i n some drawing 0 .

is p lana r ly embedded and a l l chords e i t h e r l ie i n In t (C) or i n Ext(C). .men C

Case (i)

sub-cases a r i s e none of which i 3 opt imal ;

Case (ii) If some p a i r of ad.jacent chords ai,3ai, aiai+3 both l i e i n t h e same reg ion , then two f u r t h e r sub-cases, according as a i-2ai+, l ies i n the same or i n d i f f e r e n t r eg ions a s t h e s e , arise. ai,lai+2, a re-drawing is poss ib l e which both does not i n c r e a s e v and i n which some chord i n t e r s e c t s C.

if a l l ad jacen t chords l i e i n d i f f e r e n t r eg ions , then two d i s t i n c t

In a l l cases that l o c a t e

i4e conclude t h a t i n all. cases t h e r e e x i s t s an opt imal drawing i n which C is i n t e r s e c t e d i n some edge e. Assuming for c o n t r a d i c t i o n t h a t v(Ck) < k + h , d e l e t i n g t h e edge e and two success ive edges, we ob ta in a homeomorph of Ck-l for which t h e induc t ive hypothesis implies:

k + h - 1 v(Ck-.,) f v(Gk) - 1 5 k + h - 1 - 1,

a con t rad ic t ion .

The drawings of Figure 3 are then seen to be both opt imal and i n which C does no t i n t e r s e c t i t se l f . / I

The Crossing Number of Generalized Petersen Graphs 23 1

F ig .3 .

232 S. Fiorini

THEORkM 8 k + 3 w (3k + 1,3) 2 k + 1

- PROOF That v(3k + 1,3) 5 k + 3 follows from t h e drawing of Figure 4. 'lo show that the lower bound a l s o holds , we consider two cases for a minimal counterexample: Case (i) then t h e Contraction Theorem implies that

If there e x i s t s an optiml drawing i n which no spoke is in te rsec ted ,

v(3k + 1,3) 2 v'(3k + 1,3) =

That v(7,3) = 3 follows from t h e work of Exoo, Harary and Kabell.

Case (ii) successive spokes, w e ob ta in a homeormorph of P(3k - 2,3) whose crossing number is k, by the rninimality of k. But then,

k + 1 (By Theorem 71, for k ? 3.

If some spoke is in te rsec ted , then de le t ing t h a t spoke and two

~ ( 3 k + 1,3) 2 v(3k - 2, 3) + 1 = k + 1 ,

a contradict ion. / /

Fig .4 .

The Czossing Number of Generalized Petersen Graphs

The remaining two cases: exactly the same way once we prove that

u ( d , 3 ) 4 = u(12,3).

v(3k + h,3) = k + h (h = 0,2) are established in

4'

233

Fig .5

Proofs which are not case-by-case are elusive. 'To facilitate presentation we sketch the method of procedure. We assume, for contradiction, that the crossing number is at roost 3 and consider separately the cases where (i) no crossing is a spoke intersection, (ii) where a l l three crossings, (iii) two of the crossings, and (iv) exactly one crossing is a spoke intersection. 'The Contraction Theorem deals with (i) whereas 'Theorem 1 deals with (ii). 'Thereafter the armwent takes the following sequence: A large (usually Hamiltonian) circuit H is chosen in the grapn. If H is planarly embedded in some optimal drawing of the 2-spoke-deleted graph, then a contradiction is obtained by virtue of the Decomposition 'heorem with H = X. must intersect itself in exactly two of its edges to yield a 2-looped drawing of itself. heavy use is made of the following remarks. We define the planarization induced by a drawing of a graph C to be theplanar grapn obtalnL4 DY replacins eat?n Lrossins oj a new vertex with U incident edges, in the obvious way. aiso define a pair of parallel. ckol of a circuit C to be a pair of e&es (a,b),(c,d) in G\C such that s<o<cld*&, where "order" is defined clockwise along C. (In Lhis sense, parallelism is not a transitive relation).

Remark 1: If C intersects itself in edges (p,q) and (r,s), and if (a,b), (c,d) are parallel edges separating (p,q), ( r , s ) , ie; a 5 p < q 5 b < c 5 r < s 5 d < a , then one of (a,b),(c,d) must cross some edge.

Remark 2: of c.

If not, then H

A contradiction is obtained for each pair of edges. 'To this end

He

Each loop of C must contain at least two vertices in any good drawing

234 S. Fiorini

Remark 3: If C t oge the r with chords ( a , b ) , ( c , d ) , ( e , f ) is homeomorphic t o K then a 2-looped p l a n a r i z a t i o n o f H t oge the r w i th t h e s e edges a l so con ta ins a homeomorph of K con ta in ing a t rn0s3'~ one of t h e vertices { a , b , c , d , e , f ]

3 '3 i f t h e c ros sed edges of H bo th l i e i n a segment of H

-- PROOF: t h e reverse i n e q u a l i t y we no te t h a t s i n c e n 16, m = 24 and t h e g i r t h g I 6 , then t h e Delet ion Theorem impl i e s t h a t

That v(d,3) 5 4 foliows from t h e drawing of Fiqure 5(i). 'To establish

6(10 - CO = 2(24 - a) so t h a t

Lf t h e r e e x i s t s an opt imal drawing i n which no spokes i n t e r s e c t , then t h e Contract ion 'heorem t o g e t h e r w i th Theorem 7 imply that

v> u 2 3 .

w (8,3) 2 v'( t1 ,3) = 4.

Thus we can assume t h e r e e x i s t s an opt imal drawing i n which e x a c t l y t h r e e c r o s s i n g s occur one of wnich is a spoke i n t e r s e c t i o n :

Case (i) t h r e e crossed spokes should r e s u l t i n a p l ana r graph. t h r e e spokes ( i n a l l p o s s i b l e ways) t o g e t h e r with an a p p r o p r i a t e f o u r t h spoke we can always o b t a i n a honeormorph of t h e graph obtained from P(5 ,2 ) by d e l e t i n g a spoke, which is non-planar; a con t r ad ic t ion .

Case ( i i ) If e x a c t l y two spokes are i n t e r s e c t e d , tnen d e l e t i n g t h e s e two spokes should r e s u l t i n a grapn with c ros s ing number 1 and i n each optima.1 drawing of which no spoke is i n t e r s e c t e d . e i t h e r success ive ( a t d i s t a n c e 1 on t h e rim) or alternate (at d i s t a n c e 2 ) , then d e l e t i n g an a p p r o p r i a t e t n i r d spoke r e s u l t s i n a homeomorph of P ( 5 , 2 ) wnose c r o s s i n g number is 2. If tne d i s t a n c e is 3 ( r e sp . 4 ) t hen the r e s u l t i n g d e l e t e d graph c o n t a i n s a homeomorph of t h e graph of Figure 6 (i) ( r e sp . ( i i) ); both t h e s e cyaphs are seen t o posses s t h e Hamiltonian c i rcui t H < 1 , 2 , 3 , ..., 10,11,12,1>. If t h e r e e x i s t s s n o p t i r a l drawinq i n which H is p lana r ly embedded, t hen by t h e Decomposition 'Theorern wi th X = H , Y = t ( 1 , 5 ) , ( 2 , 7 ) , ( 4 , 9 ) 1 and Z = i ( 6 , 1 1 ) , ( 8 , 1 2 ) , ( 1 0 , 3 ) 1 , t h e f'irst graph is seen t o have crossing number 2 ; for t h e second graph we t a k e Y = ~ ( 1 , 5 ) , ( 2 , 1 0 ) , ( 3 , 1 2 ) } and z = t ( 4 , a ) , ( 6 , 9 ) , ( 7 , 1 1 ) } . I n each case ~ U Y > = K = GUD. Thus we m y assune that t h e only opt imal drawings are those i J ' d h i c h H intersects i tself exac t ly once i n non-spoke edges. segment <6,7 ,..., 12> and S2 = <1,2 ,..., 7>, and i f we assume that ( 6 , " ) = S nS is not c rossed , then some S2\ ( 6 , 7 ) , by Remark 3. e x i s t s a p a i r of parallel edges s e p a r a t i n g tnem excep t *or e is a spoke, anyway. 'Thus ( 6 , 7 ) must i n t e r s e c t some edge i n &rj, ..., S>, by Remark 2. But (6,7) is sepa ra t ed from each edge i n <10,11,. . . ,> by parallel edges (3,10),(4,9), so it cannot cross any of them, by Remark 1. O f t h e remaining edges, tne only non-spokes are ( d , g ) and ( 4 , 5 ) , both of which cases are d i s - pensed wi th by Remark 3. A s to t h e second graph, some edge i n segnent

If a l l t h r e e i n t e r s e c t i o n s are spoke c r o s s i n g s , then d e l e t i n g t h e But d e l e t i n g these

However, i f t h e two spokes are

For t h e first graph, i t S , , d e n o t e s t h e 1 2 edge i n S l \ ( 6 , 7 ) must cross some edge i n

Now, for sL1 p a i r s of edges (e ,e2)cSlxS, t h e r e ( 4 : 6 ) , which

S, = <6, 'T, . . . ,11> must c ross some edge i n segment S2 = <12,1 ,.. . ,5> by Remrk 3, s ince H together with chords (6,i2),(7111),(8,4) is K

same holds for chords (2 ,10) , (3 ,12) , (1 ,5) . Now for each p a i r of edges (el ,e2)e(S, x S2), there e x i s t s a pair of p a r a l l e l edges separa t ing them. Thus by Remark 1, d cannot i n t e r s e c t itself.

and t h e 393

2

Fig. 6

( i i )

B

Case (iii): If' exact ly one spoke is in te rsec ted , then d e l e t i n g t h i s spoke, 7 1 , l l ) say , should result i n a graph whose crossing number is 2 ; previous case we show t n a t t h i s l eads t o a contradict ion. 'The c i r c u i t (Fig 7 ( i ) ) C = <1,2,3,3~,6',11,4',4,5,6,7,'~',2~,5',8',8,1> is seen t o be a Hamiltonian c i r c u i t i n t h i s graph and has chords ( 2 , 2 ' ) , ( 5 , 5 ' ) , ( 7 , d ) which together w i t h

'Thus, i f C is planarly embedded i n some optimaL C a r e horneornorphic t o K

drawing, then one or other of' che spokes ( 2 , 2 ' ) , ( 5 , 5 ' ) is necessar i ly crossed, contrary to assumptions. We conclude that C must i n t e r s e c t itse1f;if eyact.Ly once, then t h i s i n t e r s e c t i o n m u s t occui* i n t h e segment <2,1,...,6,5> ; otner- wise e i t h e r ( 2 , 2 l ) or (5,5 ') is crossed. i n t h e segment <5r ,2 ' , . . . ,4 ' ,6 '>, s ince as before , C together wi th chords ( 6 , 0 * ) , ( 5 , 5 ' ) , ( 4 ' , 7 ) is hmemiorpnic to K We conclude that the c ross ing must occur i n the i n t e r s e c t i o n of these 3'3'seqnents, i e ; i n t h e segment <58,21 ,7f ,7 ,6 ,5>. two v e r t i c e s four cases a r i s e according as the crossing occurs i n : (a ) ( a ) (5,6) i( ( ' 7 f , 2 1 ) , ( b ) ( 5 , 6 ) s (2 , ,S ' ) , ( c ) ( 6 , 7 ) x ( 7 ' , 2 ' ) o r ( d ) ( t i , ' / ) x (2 ' ,5 ' ) .

i n the unique embeddin? of t h e p lanar iza t ion of' c spokes ( 2 , 2 ' ) , ( 5 , 5 ' ) , ( 6 , 6 ' ) can be drawn uncrossed i n only one way i n cases ( a ) , ( d ) , i n two ways i n case ( c ) and i n no way i n ( b ) . I n ( a ) ('7,d) and one (3,4) or ( 3 ' , 8 ! ) must each cont r ibu te 1 t o v ; i n ( d j , ( 7 ' , 4 ' ) and one of (3,4) o r ( 3 ' , d l ) each con- t r i b u t e 1 t o V ; and i n ( c ) each of ( 7 , 8 ) and ( 7 ' , 4 ' ) cont r ibu te 1 t o v i n e a c h embedding, Since a l l cases yi-eld a cont rad ic t ion we conclude t h a t C must be twisted twice, i n three Loops, i n such a way t h a t a l l o ther edges can be drawn i n without f u r t h e r crossinqs.

as i n t h e

313'

Tnis i n t e r s e c t i o n must also occur

Since (7,7') is a spoke and a loop nus t contain a t Least

235

236 S. Fiorini

F ip .7.

( i i )

'To ident i fy t h e two p a i r s of' i-ntersecting edges OP C we inake use of the foliowing remarks: Henark 1: m s i < j(b) crosses (h , k) (c$Kk<d), then unless a second in te rsec t ion occurs between some edge i n s e g e n t <a,..., i> wi th some edge i n segnent <k, ...,d> or between some edge i n <c,...,h> with an edge i n <j, ..., b>, tnen edges ( a ,b ) and ( c , d ) cannot be drawn without crossing each other . Furtnermor*e t h e planarizat ion of C and tne paraLJ.el edges must be drawn as i n Fip;ure / ( i i ) ie: with both end loops i n the e x t e r i o r (or equivalent ly , the i n t e r i o r ) of r;he middle loop.

Heinark 2: contain a t ieast 2 ver t ices . ocher than ver t ices of chords of t h a t loop, then there e x i s t s another drawing wi th a t most w crossings in which C does not i n t e r s e c t i t s e l f ' ; t h i s reduces t o 3 previous case.

i f ' both twists occur i n the segnent <2' ,'/ ',. . .6' ,3'> then the chords ( 7 , d ) , ( 2 , 2 ' ) , ( 3 ' , 8 ' ) necessar i ly i n t e r s e c t . so tha t one twist involves one of' the edr;;es ( 2 l , 5 ' ) , ( 5 I ~ l ) , ( ~ , 2 ) , ( 2 , 3 ) . We inves t iga te each of these separate1.y. I'he f i r s t ( i n a counter-ciockwise sense) candidate t o cross ( 2 ' , 5 ' ) is ( d , 2 ) which is separated from it by tne p a r a l l e l chords (5,5 ') ,(8 '- ,3 ') . Hence the second t w i s t must occur Detween one of (5',2'),(2','7'),('7,6),(6,5) and one of (d,2),(2,3) , by v i r t u e of Henark 1. O f the f i r s t se t , ( 7 , 6 ) and ( 6 , 5 ) are excluded s ince they a r e also separated by tne paralle.1 chords (8 ' ,3' 'The reminino; cases are disposed of by Hemark 2. c lude t h a t the set of edges t h a t can c ross ( 2 ' , 5 ' ) is empty. edges, the only crossings t h a t need discussing are (5' , d ' x ( ~ , 2 ) ,(5' ,a' x (2,3), (5 ' ,Y l ) x ( ' / l ,2l), (2 ,2) x (31 ,6v) . We assume witnout loss of Sener- a l i t y chat v e r t i c e s 8 and 8' l i e on r;he lef t loop and consider tn ree possible loca t ions of' 3 ' : on che r i g h t loop, on the lower branch of the middle loop, and on t h e upper branch. i n the last case, 2 must l i e on the P hr; 1 . 0 0 ~ ~ so t h a t ( 2 ' 2 ' ) is necessariLy crossed. I n the o ther cases , 7 must e on the upper branch of t h e middle loop, so that one of ( 2 , 2 ' ) or ( 5 , 5 ' ) is crossed. d e a l t with. / I

If (a ,b) and ( c , d ) (a<b<c<d) are a p a i r of p a r a l l e l chords and i f '

Ln a good drawing 7 h crossing riurnber V l each of the end-loops must t h e middle loop has a t most two vertices,

--..

, 4 % Arguing i n t h i s Way we con-

For the otner

In each cas ( 6 v , 3 c ) can be drawn i n uniquely.

The o ther cases are s imi la r ly

The Crossing Number of Generalized Peterseii Graphs

- Theorein 10 w(12,3) = 4.

- Proof: v ( 1 2 , 3 ) 5 4 fol lows from t h e drawing of F igu re 5 ( i i e s t a b l i s h t n e r e v e r s e i n e q u a i i t y we assume f o r c o n t r a d i c t i o n t h a t no te t h a t i f none of t h e spokes i n t e r s e c t i n some op t ima l drawinq app ly ing t h e k l e t i o n rneorem to t h e de r ived Sraph for wnich m = and g i r t h is 4 , we g e t

i'hat

4(14 - C X ) 5 2(24 - a) so t h a t

v > a > 4 .

237

. ro u 5 3 and

4 , n = 12 then

'Thus sone spoke is i n t e r s e c t e d and we proceed t o cons ide r three cases accord in5 as t h e number of spokes involved is e x a c t l y 3 , 2 or 1.

Case ( i ) : if^ a l l tnree i n t e r s e c t i o n s are spoke i n t e r s e c t i o n s , then delet in;< t n r e e spoKes should r e s u l t i n a p l ana r graph. If' two of the d e l e t e d spokes w e either consecu t ive ( a t d i scance 1 on t h e 12 -c i r cu i t , C ) or a l t e r n s t e ( a t d i s t a n c e 2 on C ) , then d e l e t i n 5 a f o u r t h a p p r o p r i a t e spoke r e s u l t s i n a homeomorph of P ( 9 , 3 ) less a spoke, which is non-planar. rhus we assume t h a t the d i s t a n c e on C between d e l e t e d spokes is a t least 3. spaced, then t h e resuLcin.7 p l ana r graph c o n t a i n s zi homenorph o f X a con t r ad ic t ion . There r e n a i n s t h e r e f o r e two s u b c a s e s acco rd ing 3 ' 3'as tne success ive d i scances on t h e r i m are ( 3 , 3 , 6 ) or (3 ,4 ,5 ) .

If they are e q u a l l y

( i i ) Fig. 8.

i n t he first i n s t a n c e , the r e s u l t i n g p l ana r graph is one or other of' t h e graphs ind ica t ed i n Figure d ( i ) , whereas t h e second case g i v e s rise t o one of t h e two g raphs implied i n F igu re d ( i i ) . n e c e s s a r i l y i n t e r s e c t s more than one edge.

Case (ii): if e x a c t l y tao spokes are i n t e r s e c t e d , then d e l e t i n g t h e s e two spoKes should r e s u l t i n a qraph whose c r o s s i n g number is one. As i n t h e p rev ious case, i f t h e d i s t a n c e between the d e l e t e d spokes is e i t h e r 1 or 2 a long t h e 12 -c i r cu i t C , tnen d e l e t i n g a f u r t h e r a d j a c e n t spoke a p p r o p r i a t e l y y i e l d s a hoioeomorph of P(9,3) whose c r o s s i n g number is 2. with s e p a r a t e l y . (5,5'),(6,6'),(7,7'), r e s p e c t i v e l y . posses s a t lamiltonian c i r c u i t H as follows: <1,2,2',5',5,4,3,3',6',6,7,7',4',1',

9',~',6,5,4,3,3',12',12,1>, < 1 , 2 , 2 ' , 1 1 ' , 1 1 , 1 0 , ~ , ~ , ~ ~ , 5 ' , 5 , 6 , ~ , 7 ' , 1 U ' , 1 ' , 4 ' , 4 , 3 , 3 ' , b8 ,9 ' , 12 ' , 12 ,1> . drawing.

Ln each i n s t a n c e one of the d e l e t e d edges

'The remaining cases are dealt I n each case we d e l e t e t h e spoke ( 1 , l ' ) and one of ( 4 , 4 ' ) ,

Each o f t h e F i r s t t h r e e graphs i s s e e n t o

10' , l o , 1 1 , l l ' , a ' , 6 , 9 , 9 ' , 12 ' , 12 ,1> , < I ,2,2' ,5' , d ' , 11 ' , 11 ,10 ,10 ' , 1 ' , 4 ' ,7 ' ,7,8,9,

We assume t h a t H is p l a n a r l y embedded is some op t ima l

238 S. Fiorini

Then, i n t h e first case the p a i r of chord t r i p l e s ( (L j1 ,8 l ) , (7 ,8 ) , (9 ,10 ) ) and ((31,121),(11,12),(21,111)) each c o n t r i b u t e 1 t o t h e c r o s s i n g number so t h a t by t h e Decomposition Theorem t h e c r o s s i n g number is a t least 2. case, t h e t r i p l e of chords ( ( 4 , 4 1 ) , ( d l ~ 1 ) l ( 9 , 1 0 ) ) n e c e s s a r i l y y i e l d s a c r o s s i n g involving one or o t n e r o f the spokes ( 4 , 4 ' ) o r ( 8 , 8 ' ) , reducing to Case ( i) ; s i m i l a r l y , i n t h e t h i r d case, a c r o s s i n g must arise among t h e t r i p l e of chords ( ( ~ , ~ ~ ~ , ( 1 0 , 1 0 ~ ) , ( ~ ~ , 1 1 ~ ~ ) , aga in reducing to Case (i). Since a l l cases imply a c o n t r a d i c t i o n we conclude t h a t H must i n t e r s e c t i t se l f , g i v i n g rise to e x a c t l y two loops , i n such a way that a l l remaining edges can be drawn i n without f u r t h e r c ros s ings . Remark: then t h e p l a n a r i z a t i o n of H obtained from i n t e r s e c t i n g i tself once c o n t a i n s a homeomorph of K con ta in ing a t most one Thus i n t h e f i r s t example, H t oge the r w i th chords ( 9 , 1 0 ) , ( 7 , 8 ) , ( 5 ' , 8 l ) is homeomorphic t o K so t h a t one of the crossed edges must l i e i n t h e segment < 7 , 7 ' , . . . ,8,0>, non-spoke edges i n tnis segment being: ( 9 , d ) ,(8', 11' ) , ( I 1,101 , ( l o ' , I ' 1 , ( 1 ' , 4 ' 1 , ( 4 ' , 7 * ) . Taking t h e s e i n t u r n , ( 9 , 8 ) is sepa ra t ed from each of the e d v s i n tne segment <91,12f , . . . ,61) by t h e parallel cnords ( g 8 , 6 * ) , ( 9 , 1 O ) and from those i n segment <101,1 ' ,4 ' ,71> by (9 ,10 ) , (101 ,7 ' ) so t h a t t h e only p o s s i b l e edges c r o s s i n g it are: ( d l , l l l ) , ( l O , l l ) and ( 6 , 7 ) . c r o s s i n s s are simi1arl .y found t o be: (81,111) x (11,101, ( 8 I , 1 l 1 ) x ( 6 , 7 ) , ( 1 0 , 1 1 ) x ( 6 , 7 ) and ( l O 1 , l f ) x (4 ' '7 ' ) . Of t h e s e the f irst and last are disposed of by v i r t u e of Remark (above). As f o r t h e remaining cases, i n (9,8) x (10,111 and ( 9 , d ) x ( 6 , 7 ) t h e r e s u l t i n g graphs are indeed p l a n a r , but i n the unique p l ane embedding either ( 1 , l ' ) or ( 4 , 4 * j crosses a t least two edges, i n a l l t h e rest, the p l a n a r i z a t i o n of H together with (6, ,9 ' ) ,('7,a), (8!,5' ) is uniquely embedded i n t h e p l ane , but t hen ( 1 l 1 , 2 " ) n e c e s s a r i l y i n t e r s e c t s some edge.

I n t h e second i n s t a n c e , H t oge the r with ( 6 , d t ) , ( 6 , 7 ) , ( Y , l 0 ) is homeomorphic with '<3,3 so t h a t some i n t e r s e c t i n g edge of H must l i e i n t h e segment < 6 , 6 ' , ..., lo>.

I n tne t h i r d i n s t a n c e , t h e i n t e r s e c t i n g edge must l i e i n segment (10' , 7 ' , ..., 1 1 ' ) s i n c e he re H and chords (10',10),(Y,01),(81,111) is K O f t h e s i x non-spokes i n t h e segment of the f o r n e r , it is r e a d i l y 3' 3 ' v e r i f i e d t h a t none q u a l i f y to i n t e r s e c t any other edge and of t h e seven i n t h e latter case on ly one, (d ,9 ) x ( 1 0 , l l ) . bu t the d e l e t e d spokes ( 1 , 1 1 ) , ( 6 , 6 q ) cross a t least twice each i n t h e unique embedding. ( 1 , l ' 1, (7 ,7 ' obtained by d e l e t i n g '7' and i n c i d e n t edges. d = <12' ,9 ' ,9, 1 0 , 1 1 , 1 1 ~ , ~ ~ , d , 7 , ~ , 6 8 1 3 ~ , ~ 1 4 , 5 , ~ ~ ,2 ' ,2 ,12> t h a t i nc ludes a l l v e r t i c e s except 4 1 1 1 f , 1 0 f , which l i e on a cha in j o i n i n g v e r t i c e s 4 and 10 on H. I f t n i s chain is no t i n t e r s e c t e d i n some opt imal drawing i n which H is p l a n a r l y embedded, so tnat i t l i e s i n .tnt(A) without loss of g e n e r a l i t y , then a l l cnords (9 ,d ) , (11 ,12 ) , (2,3),(5,6) must l i e i n Ext (H), y i e l d i n g a t least two c ross ings . If on the other nand, H i n t e r s e c t s i t se l f , then one of t h e crossed edges must l ie i n t n e segnent <12,12', ..., 8> and t h e other i n <6,6' , . . . ,2>. i n the second by t h e p a r a l l e l edges ( 2 ~ l l 1 1 ) , ( 5 1 , d ~ ) except for ( l l l , d t ) and ( 5 ' , 2 ' ) which are i n t u r n sepa ra t ed by (d,c)) and ( 3 * , 1 2 ' ) . We conclude that t h e cha in < 4 , 4 1 , 1 f , 1 0 1 , 1 0 > is i n t e r s e c t e d i n either ( 1 1 , 4 ' ) or ( l g , l O t ) . then any edqe i n t e r s e c t i n 4 one of t h e s e edges must also intersect one of ( ' / l l 4 ! ) , (71 ,101)

I n t h e second

We shall need the following:

3,3 If H t oge the r w i th chords ( a , b ) , ( c , d ) , ( e , f ) is homeomorphic t o I<

i f t h e crossed edges both l i e i n a segnent of H 3'3 ver t ex i n { a , b , c , d , e , Q .

3,3

The on ly o t h e r possib1.e

I n Lhis case, the r e s u l t i w graph is indeed p l a n a r

There r e n a i n s to cons ide r t h e f o u r t h graph obtained by d e l e t i n s which is n o t Haniiltonian. IJe cons ide r i n s t e a d its subgraph

Th i s graph i( has a c i r c u i t

But each non-spoke i n t h e first is sepa ra t ed from each non-spoke

But

i n t h e correspondinr: drawing of P (12 ,3 ) .

Case (iii): Lf e x a c t l y one spoke is i n t e r s e c t e d , tnen d e l e t i n g t n i s spoke, (12,12 ' ) s a y , ShOlJld r e s u l t i n a f r a p h whose c r o s s i n g number is 2. F i g r e 3 stiotas m%t t h i s .;rmli posses ses 3 xmi.Ltonian c i r c u i L kl.

239

dow ii t o3e the r w i t h chords (5,5'),(9,9'),(3,4J is K so t h a t i f H is p.lanarly emedded i n some op t imi drawinq, then one of t h e must be crossed. 'Thus, H crosses itself and we assume that i t i n t e r s e c t s itself e x a c t l y once i n two loops . i n t h e induced p l a n a r i z a t i o n , we show tnat two f u r t h e r crossiws cann0.i be avoided. dy Remark 3, one of the crossed edi:es of H , e s a y , must be i n segment S1 = <8',11', ..., ?I>; otherwise, one of t h e spokes (8,dt),(9,9') is i n t e r s e c t e d . 'The other, f , say,rnust be i n s e p e n t S2 = <4,j, ..., 5'>; otherwise, one or" (5,5'),(d,d') is crossed. Since p a r a i l e l spokes (5,5'),(6,df) s e p a r a t e e and f for a l l ( e , f ) E ( S \ (d,7)) x ( S 2 \ (4,5)), o t h e r than spokes, we need only cons ide r (4,5) E S1 and (8,'7) E S2. P i r s t case, ( 3 , 4 ) must CI'OSS i n each of t h e three d i s t i n c t Locat ions of' v e r t i c e s U and 9. I n the second case, (7';IO') must cross and a f u r t h e r c r o s s i n g arises from either (1 ' , 4 ' ) or ( 2 ' , 1 1 1 j . i n such a way t h a t a l l ocher edTes can be drawn i n without f u r t n e r c ros s inqs . As before, some crossed edge, e l i es i n S1 and some edTe f i n S2. f , then by the above reaso1-dq, ei ther e = (4,s) or f' (b,7). r f e = (4,5) and f E S = <d',Il', ..., 8>, then e and f are sepa ra t ed by one or o t h e r of the p a r a l l e l p a i r s (d,Y'),(6,7) and (8,8'),(3,4), so t h a t i f 5 and h are t h e crossed edges, then g (4,5) or (5,6), by Remark 1 of 'Theorem 9. ht 5 = ( 5 , b ) and h E S are also sepa ra t ed by (8,8'),(5,5') which do n o t s e p a r a t e e andh,so that 5 = e and f is a t d i s t a n c e a t least 3 from h a.long H by Remark 2 o f Theorem 9. and sepa ra t ed from (5,6) oy p r e c i s e l y t n e same sets of' p a r a l l e l edqes is seen to be empty. S i x i l a r l y , r' = ( d , ' ( ) cannot cross any e E < 5 , 6 , ..., S'> U ( 4 , s ) . 'Thus , e crosses y t f, 5 G '11 I (7' ,4', . . . , 8 ' > , and f c?osses n # e , n E U = < 5 ' , b 1 , . ..,4>. We n o t e cha t L fl U z (5',d1),,(4',7') and t h e spoke (4,4') which we ignore. Lf g ( t I ' , S ' ) , t nen g cannot cross any edge i n t h e sub-segnent <4,5,. . . ,3> , s i n c e they are separated by p a r a l l e l cho rds ( 2 ' , 11 ), ( 3 , 4 ) and no edge i n I' is bounded by (3,4) . i f ' g crosses e = (2,3), then h is either ( 5 ' , 6 ' ) or (8l,llf) and f E 0, so that # (dl,5f). If h ( 8 ' , 5 ' ) crosses some edge i n <7',7,.. .,ll>, these are seDarated by (2',11') and one of (4',l1),(10,l'l) which bound no ed<e i n <5' ,2 ' , ..., b; t h u s (8',5') crosses no edge.

3'3 'spokes (S,51),(Y,r1)

Ln t h e

A i W t h e r c r o s s i n g arises f'rom e i t h e r ( 6 , 7 ) or (1',4').

Vie conclude t n a t H i n t e r s e c t s itself e x a c t l y twice

If' e crosses

l'he set of' edge p a i r s i n S s a t i s f y i n g these c o n d i t i o n s

240 S, Fiorini

If q = (49,71), then g crosses no edge i n <4,5, ..., 5'> since these edges are separated by (11,4') and either (3,4) or (5,5'), which bound no edge in <7',7, ..., df>. Similarly, h z (4',7') cannot cross any edge in <8,9, ..., 8l> since they are separated If h I (4',7') and f = (7,8) then e and g cannot be separated by parallel edges since h and f are not. rhus (f,h) are either ((2,3),(5,6)) or ((2,3),(4,5)). In the unique planarization of each case (/',lot) is necessarily crossed. l'hus {e,gl !& <4,5 ,..., jr> and {f,h} C_ <7','7 ,..., If e, say, is (5I,2l), then g E <2,3 ,..., 4> and f = (b1,Il1), h E <l,l', ..., 7l>. But (5',2') is separar;ed from g E <3,3', ..., 4> by (21,111),(3,4) which do not separate f , h, so that g = (2,3). But each h in < 1 , 1 1 , ..., 7 l> is separated from (dl,llr) by (2l,1l1) and another parallel edge wnich do not separate e and g, so that neither e nor g is (5f,21). If e = (2,3), then g E <3',9', ..., 4> excluding <3', ..., 6'>, since these latter edqes are separated from (2,3) by (31,6v)l(3,4) which do not separate f,n. Thus g is either (6,5) or (5,4) both of which are separated from e by (6,7),(9,9'). But then no edge other than the spoke (7,7*) qualifies as either f or h, so that neither e nor g lies in (2,3) u <3', ..., 61>. Since these edges account for all distinct possibilities in <5',2', ..., 4>, the result is proved. I1

Comnents and Problems:

It is known [3] that (i)

(ii) P ( n , k ) '2 P(n,n-k).

It foliows from out' conclusions that

by (6,'/),(8,8') which bound no edge in <4,5, ..., 5'>.

if 1 5 k, in 5 n-1 and !un P 1 (mod n), then P(n,m) 5' P ( n , k ) ;

v (3h+1,3) = v (3h+2,3) =

~(3h+l,h) = v(3n+1,2h+l) = v(3h+193h-2) = h+l ~(3h+2,h+l) ~(3h+2,2h+l) = ~(3h+2,3h-1) = h+2

'The following table of knom values for u(n,k) can be drawn up:

1

2

3 4

5

6

7 8

9 10

11

12

13

1 2 3 4 5 6 7 d 3 10 1 1 12 13 14

0 0 0 0 0 0 0 0 0 0 0 0

0 0 2 0 3 0 3 0 3 0 3 0

0 2 1 3 4 2 4 5 4 5 6

0 0 3 1 2 ? 5 * 5 ? 0 3 4 3 1 2 ? ? 6

0 0 2 ? 2 1 3 ?

0 3 4 5 ? 3 1

0 0 5 ' ? >

0 3 4 5 6

0 0 5 ?

0 3 6 0 0

0

The Crossing Number of Generalized Petersen Graphs 24 1

Regarding e n t r i e s marked (*), t h e fo l lowing can be s a i d : P( k t , t 'The drawing of P ( k t , t ) i n which tne k t - c i r c u i t is p l a n a r l y drawn and t h e t"k-helms" are drawn success ive ly a l t e r n a t e l y in t h e i n t e r i o r and e x t e r i o r of' t h e k t - c i r c u i t g i v e s t h e following upper bound for t h e c r o s s i n g number ct: -

It is r e a d i l y v e r i f i e d that i f th i s estimate is v a l i d for a p a r t i c u l a r odd value of t , then it is also va1.i.d for t + l . 'The same cannot be s a i d for even t. b i p a r t i t e ~yaphs :c f . [2 ] 1.

(It is of i n t e r e s t t o n o t e t h a t a similar s i t u a t i o n o b t a i n s for t h e complete We conclude that v (4k ,4 ) 2k.

References:

1 . G. EXOO, F. Harary, J. Kabel l , Ine Cross ins numbers o f some General ized Pe te r sen Graphs, irlath. Scand. 2 (1981) 184-188.

R. Guy, the d e c i i n e and rali of Zarankiewicz 's Theorem, Proof rechniguz? .-- i n Graph l'heory.(F. r k r a r y , ed.)

Iy. Watkins, A 'Theorern on hit Colourings ..., J .C.T. (B) & (1969) 152-104.

2.

3.

Annab of Discrete Mathematics 30 (1986) 243-250 8 Elsevier Science Publishers B.V. (North-HoUand) 243

COMPLETE ARCS IN PLANES OF SQUARE ORDER

J.C. Fisher1, J.W.P. Hirschfeld2 and J . A . Thas3

'Department of Mathematics, University of Regina, Regina, Canada, S4S OA2.

2Mathematics Division, University of Sussex, Brighton, U.K. BN1 9QH.

3Seminar of Geometry and Combinatorics, University of Ghent, 9000 Gent, Belgium.

Large arcs in cyclic planes of square order are constructed as orbits of a subgroup of a group whose generator acts as a single cycle. order, this gives an example of an arc achieving the upper bound for complete arcs other than ovals.

In the Desarguesian plane of even square

1. INTRODUCTION

2 Our aim is to demonstrate the existence of complete (q - q + 1)-arcs in a

cyclic projective plane II(q ) of order q . The only such plane known is

PG(2,q2), the plane over the field GF(q2) . These arcs were found incidentally

by Kestenband [S], using different methods, as one of the possible types of inter-

section of two Hermitian curves in PG(2,q2) .

2 2

The importance of these arcs, not

observed in [S], is Segre's result that for q e

PG(2,q) with m < q + 2 satisfies m 5 q - Jq +

complete arc attains the upper bound f o r q even

gation, it is shown that a Hermitian curve in PG

q + l of these arcs.

2. NOTATION

en, a complete m-arc in

1 . Thus, this example of a

As a by-product of the investi-

2,q') is the disjoint union of

2 Let n = n(q ) be a cyclic projective plane of order 9'. One can

identify its points with the elements i of ZV , v = q4 + q2 + 1 ,

cyclic group is generated by the automorphism u with o(i) = i + 1 , i E Zv , [3] , 54 .2 .

lo = Ido,dl,. , . dq2} as the sets u J ( l o ) ,

so that the

The lines are obtained from a perfect difference set

j = O,l,.. ., v - 1 .

2 Let b = q2 + q + 1 and k = q - q + 1 ; then v = bk. Since b and k

are relatively prime, Zv = Zb x Zk . For i in Zv , we write

i = (1,s) where i r(mod b), i L s(mod k) .

In this notation u(i) = (r + 1, s + l), where the sum of the first component is

taken modulo b and the second modulo k . The notation extends in a natural way

to any arithmetical operation in Zv .

244 J.C. Fischer, J . W.P. Hirschfeld and J.A. Thar

By the multiplier theorem of Hall [2], q3 is a multiplier of II ; this 3

means that the mapping J, given by $(i) = q i is an automorphism of Il. Since

q6 F 1 (mod v) , so J, is an involution. Indeed, J1 is a Baer involution since

it fixes all b points of kZv = [(r,O) : r E Zb1; this is because q r - r =

r ( q 3 - 1) E 0 (mod b) . 3

If we define

B = I(r,s) : r E Zbl for s = O,l, ..., k - 1 ,

then o(Bs) = BS+l and the q2 - q + 1 Baer subplanes Bs partition Il.

l i n e of a Baer subplane Bs is a line of Il meeting Bs in q t 1 points.

Similarly define

A

K = {( r , s ) : s E iZ 1 for r = 0,1, . . . , b - 1 ,

whence o(Kr) = K and the Kr also partition Il. Thus i = ( r , s ) = B n K ,

It will turn out that Kr is a complete

k

r+l s r (q2 - q + 1)-arc.

3 . COMPLETE k-ARCS

LEMMA 3 . 1 : For each i = (r,s) i n izv = izb x Zk , we have that

$(i) = (1, k - s) .

Proof: It was noted in 52 that fixes the first component r of i

Now, for each s in iZk ,

3 q s + s = s ( q + l)k 3 0 (mod k) ,

q 3 s F - s s k - s (mod k ) . 0 whence

LEMMA 3.2: For any l ine .t of the Baer subpZane BS , wi th ( r ,s) = Bs n Kr,

odd if (r,s) E K

even if (r ,s) # p..

Kr 3 Proof: By lemma 3.1, the involution $J fixes exactly one point of

namely the point (r,O) where it meets B o ; the other points of Kr are inter-

changed in pairs. If K is a line of Bo it is fixed by $ , which implies

that the number of points of .t n Kr outside Bo is even. Thus the parity of

1.t n K r l varies as L n Kr n Bo is empty o r the point (r,O) . For a line .t

of Bs, apply the same argument to o- ’ ( . t ) , which is a line in Bo. 0

I K n ~~1 is

Complete Arcs in Planes of Square Order 245

LEMMA 3 .3 : Let S be an automorphism group that ac t s regularly on the

points of some project ive p h n e n(n) of order n , and suppose tha t

VO,V1, ..., Vt s . If .t i s a Zine of n(n) and A . = 1.9. n v .1 , then

are the orbits of the points under the act ion of a subgroup G of

1 3

A . ( A . - 1) = I G I - 1 . j=1 J J

Proof: To each of the n2 + n elements y of S \{ l} there corresponds

1 a unique pair of points P , Q of 9. for which y ( P ) = Q ; in fact,

P = y - ( 2 ) n 2 and Q = 2 n y(L) . If there was another such pair on 2 , then

S would not act regularly on the lines of n(n) . Now we count the set

in two ways. First, each y other than the identity gives a unique pair ( P % Q ) ,

whence I J I = I G I - 1 . Second, 9. is a disjoint union of the sets 9. n V. ,

and to each pair (P,Q) , P # Q , in 9. n V. there is a unique y in G such

that y ( P ) = Q ; hence IJI = 1 4 . (A. - 1) and so I J I = A . (A. - 1 ) . 0

3 J

Aj>l J J j = 1 J J

We are now ready to prove the main result. In 14, an alternative proof is

provided that makes use of the properties of perfect difference sets.

THEOREM 3.4: For q > 2 , each orb i t Kr is a complete k-arc with

k = q2 - q + 1 i n n(q2) . lie i n B are the q + 1 tangents t o Kr a t (r,s) .

Furthennore, the l i nes through Bs n Kr = (r ,s) tha t

Proof: Fix a Baer subplane B and let II be one of its lines. For

each orbit K r j ( j = 0 , 1 , . . . , q ) that meets .9. n Bs, set CI. + 1 = 12 n K I ; for the remaining orbits, set @ . = n K 1 , j = q + 1 , q + 2 , . . . , b - 1 .

By lemma 3.2 both a . and B . are even.

J rj

1 'j

I 3

By definition,

By lemma 3.3,

2 b-1 1 Bj (Bj - 1) + ( a . + 1) a . J = q - 9 ,

j =q+1 j=1 J

whence subtraction yields

b-1

j =O

246 J.C. Fischer, J. W.P. Hirschfeld and J.A. Thas

Consequently B . E {0,2} f o r j 2 q t 1 . I

Summarily, f o r any l i n e II o f t h e subplane Bs, e i t h e r

( i ) (r j ,s) E II n Kr so t h a t a j = 0, (!2 n K 1 = 1 and II is t angen t t o j r j

K r j a t t h e p o i n t i = (r.,s) 1

o r

( i i ) .t n K,. n Bs = 0 and II meets K r j i n 0 o r 2 p o i n t s . 1

Since each l i n e of il i s a l i n e of e x a c t l y one o f t h e subplanes Bs, it fo l lows t h a t no l i n e meets K r j i n more than two po in t s ; t h a t is , Krj is a (q2 - q + 1 ) - a r c . From ( i ) it is c l e a r t h a t , f o r each p o i n t t h e q + 1 l i n e s of Bs through ( r j , s ) a r e t h e q + 1 t angen t s o f K r j a t t h i s p o i n t .

( r j , s ) o f K,. , I

For q 2 4 , a s imple count ing argument suffices t o show t h a t t h e k - a r c Kr i s complete. Assume t h e c o n t r a r y . Then t h e r e is a p o i n t P through which pass q - q t 1 t angen t s o f some r ’ # r , it fo l lows t h a t through each p o i n t o f K r l t h e r e a r e q 2 - q + l t angen t s o f K (because Krl and K r are o r b i t s under t h e a c t i o n o f

t h e group genera ted by a ) . Since K r l is i t s e l f a k-arc , none of t h e s e t angen t l i n e s i s counted more than twice , whence K r ha s a t l e a s t ; ( q t a n g e n t s . But a s K r has e x a c t l y (q + l ) ( q - q t 1) t a n g e n t s , we have

i ( q 2 - q + 1)’ L (q t l ) ( q 2 - q + 1) ,

K r , one from each of i t s p o i n t s . S ince P E K r , f o r 2

br

2 - q + l ) 2

a c o n t r a d i c t i o n f o r q 2 4 .

When q = 3 , it must first be observed t h a t PG(2.9) is t h e unique c y c l i c p lane of o r d e r 9 , Bruck [l]. Then t h e only 7 -a rc o f PG(2 ,9) whose automorphism group c o n t a i n s an element o f o rde r 7 i s a complete arc, [ 3 ] , 514.7. The c a s e

q = 2 i s a genuine excep t ion : a 3 - a r c i s never complete. 0

Remark

A theorem o f Segre [ 3 ] , 510.3, s t a t e s t h a t a complete m-arc i n P G ( 2 , q ) ,

q even, i s e i t h e r an o v a l , t h a t i s a (q + 2 ) - a r c , o r m 5 q - Jq + 1 . So, f o r q

Complete Arcs ira Planes of Square Order 247

an even squa re , theorem 3 . 4 g i v e s an example of a complete (q - Jq + 1 ) - a r c and shows t h a t Segre ' s theorem cannot be improved i n t h i s ca se . For q odd, [ 3 ] ,

§10.4, t h e comparable theorem s t a t e s t h a t a complete m-arc i n PC(2 ,q ) , q odd, is e i t h e r a con ic , t h a t i s a (q + 1 ) - a r c , o r has been s l i g h t l y improved by t h e t h i r d au tho r . complete 8 -a rc i n PG(2,9) shows t h a t q - i q + 1 is n o t t h e b e s t bound f o r a l l odd q , [3] , 514.7.

m 5 q - Jq/4 + 7 / 4 . This r e s u l t However, t he e x i s t e n c e o f a

4 . LINES IN lI(q2)

THEOREM 4 . 1 : Any l i n e of Bo c o n s i s t s of

( i ) q + 1 po in t s of the form (d,O) , where d i s an element o f a p e r f e c t

d i f f e rence s e t D for Z b ;

d ( k - 1) = ('1 pairs of po in t s of the form ( r j , j ) and ( r j , k - j ) for

j = 1 , 2 , . . ., $ ( k - 1) , w i t h the r . d i s t i n c t elements of Zb\D .

Proof:

( i i ) 2

3

Bo is i t s e l f a c y c l i c p l ane of o r d e r q , s i n c e ok gene ra t e s a c y c l i c group f o r B o . Each of i t s l i n e s II t h e r e f o r e con ta ins q + 1

elements (d,O) where d i s an element of a p e r f e c t d i f f e r e n c e set D f o r Zb .

A l i n e L of Bo meets any o t h e r subplane Bs of t h e p a r t i t i o n i n e x a c t l y one p o i n t . Thus each element j of Zk\{O} occurs a s t h e second component of exac t ly one p o i n t of L . Lemma 3 . 1 shows t h a t (I i n t e rchanges ( r j , j ) and ( r j , k - j ) . Since L i s f i x e d by $ , it fol lows t h a t both j and k - j are pa i r ed with t h e same element r o f Zb .

j

I t remains t o show t h a t r . # D and t h a t no r . i n Zb\D can appear more This fo l lows from t h e fact t h a t t h e p o i n t s of

3 I L . t han twice among t h e p o i n t s o f

a c o n s t i t u t e a p e r f e c t d i f f e r e n c e set f o r Zv : each of t h e k - 1 d i f f e r e n c e s of t h e form i = (0,s) , s # 0 , must occur e x a c t l y once, and t h e s e are accounted f o r by t h e k - 1 d i f f e r e n c e s t ( ( r j , j ) - (rj, k - j ) ) . 0

The d e s c r i p t i o n of a l i n e of Bo given i n t h e theorem i s e s s e n t i a l l y t h e

a l t e r n a t i v e proof t h a t Kr is a k-arc whose t angen t s are t h e l i n e s through (r ,s) t h a t l i e i n B s . l i n e of Bo is a tangent t o those K t t h a t meet i t i n a p o i n t of B o ; it meets t h e o t h e r K t i n 0 o r 2 p o i n t s . The proof i s completed by no t ing t h a t

os(Bo) = Bs , common with i t .

The d e s c r i p t i o n of p o i n t s of t ype ( i ) and ( i i ) shows t h a t any

which e i t h e r co inc ides with Bo o r has no p o i n t s o r l i n e s i n

248 J.C. Fischer. J . W.P. Hirschfeld and J.A. Thas

5 . HERMITIAN CURVES

The only known cyclic planes are the Desarguesian ones and, in this section,

we restrict our attention to PG(2 ,q2) .

It is convenient to distinguish one line Lo of Bo and define

R . = o-’(Ro) . Then i and R . are incident exactly when i + j (mod v) is an element of L o . In particular, L n B = {i = (d,O) : d E DI as in (i) of 0 0 theorem 4.1; now, D is a distinguished, perfect difference set for

1 1

‘b .

THEOREM 5 . 2 : iil The s e t H = { (d/2, s) : d 6 D , s E Zk} i s a

Kd/2 ’

liii H n Bs is a conic or a l i n e of Bs according as q

is odd or even, whence tI is a disjoint union o f k subconics or k subl ines

accordingly.

Hermitian curve and is the d i s j o i n t union of the q + 1 compZete k-arcs

Define the correlations $ : i +-f a . and p : (r,s) c-f 2 (r,-s) . Pro0 f:

Then $ is an ordinary polarity for q odd and a pseudo polarity for q even,

[ 3 ] , g 8 . 3 . Thus, with J , as in 12, we have that p = $0 = $ 9 . In fact, p is

a Hermitian polarity since the self-conjugate points of p are the q3 + 1

points (r,s) satisfying ( r , s ) + (r,-s) = (d,O) for d in D , From this (i)

follows.

both with r (r,O) ’ In Bo the points are (r,O) while the lines are R

in Zb . So Bo is self-polar with respect to p and meets H in the q + 1

self-conjugate points of the polarity @ induced on B0 by p . These self-

conjugate points form a subconic when q is odd and a subline when q is even.

Given s , there exists s ‘ such that bs’ t s (mod k) since b and k are

coprime. Thus H n Bs = 0

of Bs according as q is odd o r even, and the last part of (ii) follows. 0

bs’ (13 n Bo) = I(d/2, s) : d 6 D} is a conic or a line

2 THEOREM 5 .2 : The tangents t o any complete ( q 2 - q + 1)-arc i n PG(2 ,q ) , q even, form a dual Hermitian arc.

Proof: See Thas [ 6 ] .

THEOREM 5.3: The tangents t o any o f the complete (q - q + 1)-arcs Kr i n 2

2

PG(2,q ) form a dual Hemi t ian curve i f and only i f q i s even.

Proof: Let q be even and consider the arc K O , where D has been

chosen so that 21) = D (which is always possible since 2 is a Hall multiplier

and each multiplier of Bo fixes at least one line of Bo) . Then the tangents

Complete Arcs in Planes of Square Order 249

t o K O a t (0,O) , namely t h e l i n e s o f Bo con ta in ing ( 0 , O ) , have t h e form

II

II

s E Zk} . l i n e s o f t h e p o l a r i t y p determined by H . Thus t h e s e t of t a n g e n t s t o K O

co inc ides with t h e s e t o f t angen t s t o H .

wi th d i n D . S ince obs ' , which t a k e s (0 ,O) t o ( 0 , s ) , t a k e s t h e set o f t a n g e n t s t o KO i s {I. : j = ( d , s ) , d 6 U , t o 9,

From t h e assumption t h a t D = 2 D t h e s e l i n e s a r e t h e s e l f - c o n j u g a t e

(d , 0 )

(d,O) (d , - s ) ' J

Now l e t q be odd. S ince t h e number o f t a n g e n t s from a p o i n t P no t i n

Kr t o Kr has t h e p a r i t y of q' - q + 1 and so is odd, t h i s number i s never

q + 1 . Hence t h e t a n g e n t s t o Kr do n o t form a dua l Hermi t ian arc. 0

Tb'EOREM 5 .4 : Each of t h e (q2 - q + 1 ) - a r c s K r i s t h e intersection of t u o

Wermitian curves.

Proof: F i r s t , l e t q be even. Then a s i n theorem 5 . 1 , t h e a r c K r i s

conta ined i n a Hermitian curve H , which de termines a p o l a r i t y p . Let

p(H) = H and l e t H* be t h e dua l Hermi t ian curve o f theorem 5.3 t h a t is formed

by t h e t a n g e n t s t o Kr . Then p(Kr) = H* n h , whence

Now, l e t q be even o r odd. Then, a s i n theorem 5 .

s c izk H = Ho = { ( d / 2 , s ) : d E D ,

is a Hermitian curve . Hence

Hr = { ( d / 2 + r,s) : d E D , s E iZk

K = p(H*) n H .

i s a l s o a Hermitian cu rve . In f a c t , s i n c e t h e r e e x i s t s r ' such t h a t

k r ' : r (mod b) , we have t h a t H r = o ( H o ) . S ince D i s a p e r f e c t

d i f f e r e n c e s e t i n Z b , so IHrl n Hr21 = k f o r any r1 # r 2 . A l s o H r l n Hr2 = Kt , where t = ad + r l = i d + r 2 , s i n c e t h e r e e x i s t un ique

and d2 i n D such t h a t d l - d2 E 2 ( r 2 - r l ) (nod b ) . 0

k r '

d l

THEOREM 5.5: Let H be a Hemitian curve in PG(2,q') , q even, and l e t

K be an m-izric contained in t i .

( i ) If there is no (m+1)-'zw in H , then

if q > 2 ; 2 (a) m = q - q + l

(b) m = 4 if q = 2 . 2 ( i i ) If m = q - q + 1 and q > 2 , then K is compLete.

Pro0 f: ( i ) Suppose m > q' - q + 1 . Then by S e g r e ' s theorem ( [ 3 ] ,

theorem 10 .3 .3 , c o r o l l a r y 2 ) , K is conta ined i n an ova l 0 , t h a t i s a

(q2 + 2 ) - a r c . Now, count t h e p a i r s ( P , Q ) such t h a t P E K , Q E 0 , P # 9 and

250 J. C. Fischer. J. W.P. Hirsch feld and J. A. Thas

PQ is tangent to H. There are at most two points P f o r a given Q , since

three would be collinear. So

m .C 2 ( q 2 + 2 - m) .

Hence 3m 5 2q2 + 4 , and 3q2 - 3q + 3 c 2q2 + 4 implies that q = 2 . This

gives the result.

(ii) Suppose K is not complete, then the same argument as (i)

gives

q' - q + 1 5 2(q + l),

whence q2 - 3q - 1 5 0 ; that is, q = 2 . U

Remark: For q odd, the points of Bo together with the q2 + q + 1

conics Cr = { (6d + r, 0) : d E D} , r E Zb , form a plane of order q . This

plane is isomorphic to PG(2,q) via the isomorphism 6 given by 8(x,O) =

(gx, 0) . For all q , this configuration of conics also appears as the section

by a plane TI of the q2 + q + 1 quadric surfaces through a twisted cubic T

in PG(3,q), where 71 is skew to T ; see [4], theorem 21.4.5.

REFERENCES

[l] Bruck, R.H., Quadratic extensions of cyclic planes, Proc. Sympos. AppZ.

Math. 10 (1960), 15-44.

[ 2 ] Hall, M., Cyclic projective planes, Duke Math. J. 14 (1947), 1079-1090.

[3] Hirschfeld, J.W.P. Projective Geometries over Finite Fields (Oxford

University Press, Oxford, 1979).

[4] Hirschfeld, J.W.P., Finite Projective Spaces of Three Dimensions (Oxford

University Press, Oxford, to appear).

[5] Kestenband, B., Unital intersections in finite projective planes, Gem.

Dedicata 11 (1981), 107-117.

[ 6 ] Thas, J.A., Elementary proofs of two fundamental theorems of B. Segre

without using the Hasse-Weil theorem, J . Combin. Theory Ser. A . 34 (1983),

381-384.

Annals of Discrete Mathematics 30 (1986) 251 -262 0 Elsevier Science Publishers B.V. (North.Holland) 25 1

ON THE MAXIMUM NUMBER OF SQS(o) H A V I N G A PRESCRIBED PQS I N COMMON"

M a r i o G i o n f r i d d o ' , A n g e l o L i z z i o ' , M a r i a C o r i n n a M a r i n o '

S u m m a r y . We d e t e r m i n e some r e s u l t s r e g c r d i n g t h e p a r a m e t e r D ( v , u l , where D ( v , u ) i s t h e maximum number of S Q S l v l s s u c h t h a t any two o f t h e m i n - t e r s e c t i n u q u a d r u p l e s , w h i c h o c c u r i n g i n e a c h of t h e S Q S l v l s .

1. I n t r o d u c t i o n

A p a r t i a l q u a d r u p l e s y s t e m ( P Q S ) i s a p a i r ( P , s ) , w h e r e P

i s a f i n i t e s e t h a v i n g v e l e m e n t s a n d s i s a f a m i l y o f 4-sub-

s e t s c f P s u c h t h a t e v e r y 3 - s u b s e t o f P i s c o n t a i n e d i n a t mos t

a n e l e m e n t o f s . I f ( P , s l l a n d ( P , s 2 ) a r e two PQSs , t h e y

a r e s a i d t o b e d i s j o i n t a n d m u t u a l l y b a l a n c e d ( D M B ) i f s n s = @

and any t r i p l e { x , y , z } c p i s c o n t a i n e d i n a n e l e m e n t o f s i f

a n d o n l y i f i t i s c o n t a i n e d i n a n e l e m e n t o f s I f ( P , s l l a n d

l P , s 2 ) a r e D M B , t h e n l s l l = 1s21 . I f ( P , s ) i s a PQS s u c h

t h a t e v e r y 3 - s u b s e t o f p i s c o n t a i n e d i n e x a c t l y o n e e l e m e n t o f

s , t h e n ( P , s ) i s s a i d a S t e i n e r q u a d r u p l e s y s t e m ( S Q S ) . The n u t

h e r / P I = v i s t h e o r d e r a n d i t i s w e l l - k n o w n t h a t a n SQS(vl t h e -

re e x i s t s i f and o n l y i f v :2 o r 4 (mod. 6 ) .

1 2

1

2 .

I n w h a t f o l l o w s a n S Q S l v ) w i l l b e d e n o t e d b y ( & , a ) . We h a v e

19 I = ( I , = v ( v - 1 ) (~-2)/24 . On o f t h e most i m y o r t a n t p r o b l e m i n t h e t h e o r y o f S Q S s i s t h e

d e t e r m i n a t i o n o f t h e p a r a m e t e r :

D l v , u ) =Max 112 : 1 h SQSlvl ( Q , q l ) ,..., l Q , q h l / q i n q j = A ,

ac i , j J i # j J IAl = u } . "Lavoro e s e g u i t o n e l l ' a m h i t o d e l GNSAGA e c o n c o n t r i b u t o d e l MPI

' D i p a r t i m e n t o d i M a t e m a t i c a , U n i v e r s i t B , V i a l e A . D o r i a 6 , 95125 Ca-

' D i p a r t i m e n t o d i M a t e m a t i c a , U n i v e r s i t g , V i a C . R a t t i s t i 9 0 , 98100

( 1 9 8 3 ) .

t a n i a , I t a l y .

M e s s i n a , I t a l y .

252 M. Gionfriddo. A . Lizzio and M.C. Marino

I n [ 2 ] J. Doyen has p o i n t e d o u t t h i s problem f o r S t e i n e r t r i p l e

sys t ems .

I n t h i s p a p e r we p r o v e some r e s u l t s r e g a r d i n g D(v,ul f o r

SQSS . 2 . K n o w n r e s u l t s

Let ( P , s ) b e a FQS . We w i l l s a y t h a t an e l emen t x,.:. P has

d e g r e e d ( x ) = r i f x b e l o n g s t o e x a c t l y r q u a d r u p l e s o f s . I f

X,YEP 9 X # Y , we w i l l i n d i c a t e by ( x , y l r a p a i r { x , y ~ c P con-

t a i n e d i n e x a c t l y k q u a d r u p l e s o f s . We have d l x ) = 4 1 s 1 . X E P

The d e g r e e - s e t of a PQS I P , s ) i s t h e s e t DS = [ d ( x ) , d l y ) , . . . ]

where x l y , . . . are t h e e l e m e n t s o f P . I f ri e l e m e n t s o f P ha-

ve deg ree hi , f o r i = 1 , 2 , . . . , p , we w i l l w r i t e DS = ( h )

( h e ) , ..., ( h I where r + . . . f r = IPI c 1 P I J

2 s .,I 1

b e t h e comple t e graph on X . 1x1 I f X i s a f i n i t e s e t , l e t K

An I-factorization (o f K ) on X is a f a m i l y F = [ F l , - - . , F h ~ , where Fi i s a factor [I] o f K

F . n F . = @ f o r e v e r y i , j = 1 , 2 , ... , i # j . I t i s h = 1x1-1 . I f

I - < h < IxI-1 , F i s c a l l e d a p a r t i a Z 1 - f a c t o r i z a t i o n (o f

x .

1x1 (on X) and , f u r t h e r , 1x1

1 3

On

A p a r t i a l 1 - f a c t o r i z a t i o n Fa\ = { F ; , F ; , ..., Fe} on a s e t Y i s

embedded i n an 1 - f a c t o r i z a t i o n F = {F . , F 1 on X , i f and on ly

i f Y s X , and e v e r y F9‘: E Fg: i s c o n t a i n e d i n a F . € F .

h

l’** k

z 3 Let X and Y b e two f i n i t e s e t s such t h a t 1x1 = I Y I = u and

X n Y = @ . I f F = { F l ,..., F 1 i s an I - f a c t o r i z a t i o n on X , G = I G I ,..., G u - l } an I - f a c t o r i z a t i o n on Y , a a p e r m u t a t i o n on

{1,Z,...,u-Z} , t h e n I F , G , c l ) i n d i c a t e s t h e s e t o f t h e q u a d r u p l e s

{ x l , x 2 , y l , y e } C X A Y such t h a t

V - 1

On the Maximum Number of SQS(vI 253

I t i s wel l -known t h a t , i f ( X , A ) a n d f Y , B ) are two S Q S l v ) , w i t h X n Y = 0 , t h e n I Q , q ) = [ X u Y ] I A , B , F , G , r x I , w h e r e Q = X u Y a n d

q = A u E w r ( F , G , c O , i s a n S Q S ( 2 v ) . I n 1-31, [4], [6] M . G i o n f r i d d o h a s c o n s t r u c t e d , t o w i t h i n i s o -

morphism, a l l DMB PQS h a v i n g m = 8 , 1 2 , 1 4 , 1 5 ( i . e . m ~ 1 5 ) q u a d r u -

pies. -

T h e s e r e s u l t s a r e t h e f o l l o w i n g :

97 92

1,2,:,4 1,2,3,5 1,2,5,6 1,2,4,6

1,4,6,7 1,5,6,7 2,3,5,8 2,3,4,8 2,4,6,6 2,5,6,8

5,6,7,9 4,6,7,9 3,4,8,0 3,4,9,0

1,3,5,7 1,3,4,7

3,4,7,9 3,5, 7,9

3,5,9,0 3,5,8,C 4,6,9,0 4,6,8,0 5,6,8,0 5,6,9,0

1,4,5,6 1,4,5,7 1,4,?,8 1,4,6,6 1,5,7,9 1,5,6,9 1,6,8,9 1,7,8,9 2,4,5,7 2,6,8,9 2,6,7,8 2,4,5,5 2,6,5,9 2,4,7,8 2,4,8,9 3,6,5,7 3,4,6,8 3,5,7,9

3,4,5,9 3,4,8,9 3,7,8,9 3,6,7,8

3,5,&,7 3,4,5,&

1,2,3,4 1,2,3,5 1,2,5,6 1,2,4,7 1,2,7,6 1,2,6,8 1,3,5,7 1,3,4,6 1,4,6,7 1,5,6,7 1,3,6,6 1,3,7,8

2,4,5,7 2,4,5,6 2,4,6,8 2,5,7,8 3,4,5,6 3,4,5,7 3,4,7,8 3,5,6,8 5,6,7,8 4,6,7,8

2,3,5,8 2,3,4,a

254 M . Gionfriddo. A . Lizrio and M.C. Marino

1 , 2 , 3 , 4 1 , 2 , 3 , 5 1 , 2 , 5 , 6 1 , 2 , 4 , 6 1 , 3 , 5 , 7 1 , 3 , 4 , 7 1 , 4 , 6 , 7 1 , 5 , 6 , 7 2 , 3 , 5 , 8 2 , 3 , 4 , 8 2 , 4 , 6 , 8 2 , 5 , 6 , 8 3 , 4 , 7 , 8 3 , 5 , 7 , 8 5 ,6 , 7 , 9 4,6, 7 , 9 5 , 6 , 8 , 0 4 , 7 , 8 , A 5 , 7 , 8 , A 4 , 6 , 8 , 0 5 ,9 ,O,A 5 , 6 , 9 , 0 4 , 7 , 9 , A 5 , 7 , 9 , A 4 , 8 , 0 , A 5,8 ,0 , A 4 , 6 , 9 , 0 4 , 9 , 0 , A

4 1 42

7 , 3 , 4 , 5 1 , 3 , 4 , 6 1 , 3 , 6 , 7 1 ,3 ,5 ,8 1 , 3 , 8 , 9 1 , 3 , 7 , 9 1 , 4 , 6 , 8 1 , 4 , 5 , 9 1 , 5 , 7 , 8 1 , 5 , 6 , 7 1 , 4 , 7 , 9 1 , 6 , 8 , 9 1 , 5 , 6 , 9 1 , 4 , 7 , 8 2 , 3 , 4 , 6 2 , 3 , 4 , 5 2 , 3 , 5 , 8 2 , 3 , 7 , 6

2 , 3 , 7 , 9 2 , 3 , 8 , 9 2 , 4 , 5 , 9 2 , 4 , 6 , 8 2 , 5 , 6 , 7 2 , 5 , 7 , 8 2 , 6 , 8 , 9 2 , 4 , 7 , 9 2 , 4 , 7 , 8 2 , 5 , 6 , 9

1 , 2 , 3 , 4 1 , 2 , 3 , 5 1 , 2 , 5 , 6 1 , 2 , 4 , 6 1 , 3 , 5 , 7 1 , 3 , 4 , 7 1 , 4 , 6 , 7 2 , 5 , 6 , 7 2 , 3 , 5 , 8 2 , 3 , 4 , 8 2 , 4 , 6 , 8 2 , 5 , 6 , 8

3 , 4 , 8 , 0 3 , 6 , 8 , 0 3 , 6 , 9 , 0 3 , 5 , 7 , 8

3 , 6 , 7 , 8 3 , 6 , 7 , 9 5 , 6 , 8 , 0 4 , 6 , 7 , 8 5 , 6 , 7 , 9 5 , 6 , 9 , 0 4 , 5 , 9 , 0 4 , 5 , 8 , 0

3 , 4 , ? , 9 3 , 4 , 9 , 0

4 , 5 , 7 , 8 4 , 5 , 7 , 9

4 , 9 "

1 , 2 , 3 , 4 1 , 2 , 3 , 5 1 , 2 , 5 , 6 1 , 2 , 4 , 7 1 , 2 , 7 , 8 1 , 2 , 6 , 8 1 , 3 , 5 , 7 2 , 4 , 5 , 8 I, 4 , 7 , 6 2 , 5 , 6 , 7 I, 3 , 6 , 8 2 , 3 , 7 , 8 1 , 4 , 5 , 8 2 , 3 , 4 , 6 2 , 3 , 5 , 8 1 , 3 , 4 , 8 2 , 4 , 5 , 7 1,4,5,6 2 , 4 , 6 , 8 1 , 5 , 7 , 8 2 , 3 , 6 , 7 1 , 3 , 6 , 7 3 , 4 , 5 , 6 3 , 4 , 5 , 7 3 , 4 , 7 , 8 4 , 6 , 7 , 8 5 , 6 , 7 , 8 3 , 5 , 6 , 8

3. The v a l u e o f ~ ( v , q ~ - m l f o r s o m e c l a s s e s o f S Q S f v l

We prove t h e f o l l o w i n g theo rems .

THEOREM 3 . 1 . L e t ( P , s I I ,..., ( P , s h ) be h DMB PQS . If t h e r e

On the Maximum Number of SQSlv) 255

e x i s t an i € { l , , , . , h l and a p a i r { r , y l ~ P such t h a t i t i s

I x , ~ ) ~ i n I P , s . ) , t h e n h ( 2 k - 1 . 2

P r o o f . It f o l l o w s i x , y l k in ( P , s . l , f o r every , j ~ I 1 , 2 , ..., h l .

If { x , ~ , a ~ ~ , a ~ ~ } a { ~ , ~ , a ~ ~ , a ~ ~ } a . . . , { x , y , a k l 3 a k 2 } ~ s i ' let F i

3

be the 1-factors IIallJa12}a{a21,a22},.. ., {akl,ak211 . It f o l l o w s

that F ={F , F 2 , . , .,Fkl is a partial 1-factorization of K Z k on

the set A = ~all,al2,aZlJaz2,.. . , is exactly an 1-factorization of

1 l (in the case h =2k-I F

ak1"ak2

K Z k on A).

can (at most) be an ]-fa5 Fi

Since the set of the I-factors

torization on 2k-1 elements, it follows h - < 2 k - 1 necessarily.

We have I F 1 = k i 2 k - l ,. THEOREM 3 . 2 . L e t X and Y be two s e t s s u c h t h a t 1x1 = I Y I = 2 k

and XnY = @ . F u r t h e r , l e t F and G b e t w o I - f a c t o r i z a t i o n s o f

K z k on X and Y r e s p e c t i v e l y , and l e t CY be a p e r m u t a t i o n on

{ 1 , 2 ,..., Zk-11 . If t h e r e e x i s t s an S Q S l v l c o n t a i n i n g I'(F,G,al , t h e n Dlv,qv-k'(2k-1)l - > 2k-1 .

P r o o f . Let (Q,ql be an S Q S ( v l containing the family lF,G,aI . It is

If

[a .+i E z 1 3 2k-I

1 2 ... 2k-1 1 2 ... 2k-1

a +i a ti ... a +i Zk-1 1 2 2k-1

for i = 1 , 2 , . . . , 2 k - 2 , then the quadruples of khe families TfF,G,a.l,

w h e r e i =0,1,2, ..., 2lk-1) , form 2k-1 DMB PQS fP,s ) , I P , s I ,... ' ' * ('9 2 (k-2 ) l , all embeddable in an S Q S ( v ) . Hence

1

2 D(v,q-k ( Z k - 1 ) ) L 2 k - 1 . ,

THEOREM 3.3. I f k € N i s s u c h t h a t 2 k i2 o r 4 (mod. 6 ) , t h e n

256 M. Cionfriddo. A. Lizzio and M.C. Marino

Proof. I f k € N i s s u c h t h a t 2 k - 2 o r 4 (mod. 6 ) , t h e n i t i s

p o s s i b l e t o c o n s t r u c t a n S Q S o f o r d e r Zk . Let l Q 1 , q l ) a n d

( ~ ~ , q ~ l b e two s ~ ~ l 2 k l w i t h Q ~ ~ Q ~ = @ , and l e t F and G b e

two I - f a c t o r i z a t i o n s o f K 2 k on Q , and Q2 r e s p e c t i v e l y . From

Theorem 3 . 2 we c a n c o n s t r u c t e x a c t l y 2 k - 1 S Q S ( 4 k . J h a v i n g

q 4 k 2

- k ( 2 k - I ) q u a d r u p l e s i n common. Hence

2 D ( 4 k , q q k - k ( 2 k - 1 ) ) Z 2 k - 1 ..

TI-IEOREM 3 . 4 . For e v e r y k € N , k ' 2 , l e t w =min { v E N : u , 4 k ,

v - 2 o r 4 (mod. 6'11 , I t f o l l o w s D ( 2 w , q Z M - k ' ! 2 k - l ) ) - > 2 k - l . Proof. Let X a n d Y b e two f i n i t e s e t s w i t h 1x1 = I Y I = 2 k - > d

and X n Y = @ . F u r t h e r , l e t F = I F 1 a n d

G = { G 1 b e two I - f a c t o r i z a t i o n s o f K e k on X a n d Y

r e s p e c t i v e l y . From Theorem 8 o f 18.1, t h e r e e x i s t s a n l - f a c t o r i z a -

t i o n s F ' = f F I I on a s e t X ' , s u c h t h a t XCX' , l X f 1 , 2 1 X I = 4 k > 8 , -

w i t h F embedded i n F ' . Let I X ' I = ~ = m i n C V E N : u - > 4 k , 8 , v ~2 o r 4 (mod. G ) } . I f Y' i s a s e t , c o n t a i n i n g Y , s u c h t h a t

X ' n Y ' = @ and lY'I = / X ' I ip a b i j e c t i o n X ' + Y f y

G ' = { G I } t h e I - f a c t o r i z a t i o n on Y' s u c h t h a t

i i=I,. . . , 2 k - 1

i i = I , . . . , 2 k - l

z i=l,...,w

-1 -1 { r , y l E G E { g (x),m ( y ) l E F I ,

t h e n G is embedded i n G' . F u r t h e r , i f f X ' , q l ) , I Y ' , q 2 ) a r e two

SQS(wl and a i s a p e r m u t a t i o n o f { l , Z , . . . , w - 1 1 , t h e n we c a n c o g

s t r u c t a n S Q S ( 2 w l = [ . x ' u Y ' ] I q 7 , q Z , F ' , G ' , a l c o n t a i n i n g T ( F ' , G ' , a l . 2

From Theorem 3 . 2 i t f o l l o w s D ( 2 w , q g M - k ( 2 k - I l l 2 2 k - 1 ,. C O R O L L A R Y , F r o m the s a m e hypotheses of T h e o r e m 3 . 4 it f o ~ l o w s

2 D ( 2 v , q w - k ( 2 k - 1 ) ) - > 2 k - l , f o r every v 'w , v : 2 o r 4 (mod. 6 ) .

P r o o f . The s t a t c r n e n t f o l l o w s f r o m p r o o f o f Theorem 3 . 4 a n d f r o m

Theorem 8 of 181 .. From p r e v i o u s t h e o r e m s w e h a v e t h e f o l l o w i n g scheme:

On the Maximum Number of SQS(v) 257

k -

2

3

4

5

6

7

8

w > 4k -

8

14

16

20

26

28

32

.. * .

2 q2u-k i2k-1)

q16 - 1 2

qs2 - 112

q4@ - 225

q s 2 - 396

q50' - 637

964 - 960

928 - 45

........

........ ... ...

V L W , v - 2 or 4 (mod. 6)

2 Z i m D ( v , q v - k ( 2 k - 1 ) ) = + m . I t i s e a s y t o s e e t h a t

v - t + m

4 . The v a l u e o f D(v,qv-rn) f o r m = 8 , 1 4 , 1 5 a n d t h e v a l u e o f

D ( 8 , q g - 1 2 )

I n t h i s s e c t i o n we d e t e r m i n e D ( v , q v - m ) f o r m = 8 , 1 4 , 1 5 a n d

D18,q8-121 .

THEOREM 4 . 1 . L e t ( P , s . . J be h DMB PQS ( f o r i = 1 , 2 ..... h ) . If

, > a s ( i ~ i

a q u a d r u p l e b

h < 2 . -

P r o o f . From

I,. . . , J i l ) t h e r e exist t h r e e e l e m e n t s x , y , z E P and

such t h a t ( ~ , ~ ) ~ , ( x , ~ ) ~ , { x , y , ~ } = b E s . , t h e n

Theorem 2 . 1 i t i s h - < 3 . I f

a n d h = 3 , t h e n { ~ , 1 4 , ~ , ~ } , { 2 , y , a , b } E s I t f o l l o w s { z , z , c , ~ ~ € s ~ , 3 .

h e n c e ( x , z ) > ~ . -

THEOREM 4 . 2 . I t i s n o t p o s s i b l e t o c o n s t r u c t t h r e e D M B P Q S w i t h

m = 8 , 1 4 , 1 5 q u a d r u p l e s .

258 M . Gionfriddo, A . Lizzio and M.C. Marino

P r o o f . I t i s easy t o s e e t h a t i n t h e unique p a i r s of D M B P Q S

w i t h rn = 8 and rn = 1 5 , and i n t h e p a i r s o f D M B PQS w i t h m = 14

and D S = [ ( 7 1 4 , ( 6 1 2 , ( 4 ) 4 1 , DS = [ ( 7 / 2 , ( 6 1 3 , ( 4 ) 6 ] ,

t h e r e e x i s t ( i n eve ry c a s e ) a t l e a s t t h r e e elements x , y , z E P such

t h a t ( x , y 1 2 , ( x , z J 2 and a quadruple b = { z , y , z l ( s e e § 2 ) . The-

r e f o r e , from Theorem 4 . 1 , i n t h e s e c a s e s i t i s h = 2 . Consider t h e

case m = 1 4 and D S = [ ( 7 ) 8 ] . I f i t i s h = 3 , s i n c e

D S = [ 1 7 / 2 , ( 6 1 7 ] ,

t hen { 1 , 2 , 3 , x } ~ s w i t h z ~ { 6 , 7 , 8 } . But, x = 6 [ r e s p . x = 7 ]

i m p l i e s { 2 , 2 , 4 , 8 1 , { 3 , 2 , 5 , 7 1 E s [ r e sp . { 1 , 2 , 4 , 6 ) , { 1 , 2 , 5 , 8 ) E s 3 ] , 3

w i t h { 1 , 4 , 6 , y } ~ s ~ [ { 1 , 3 , 8 , y l ~ s ~ ] and y @ { l , Z ,... ' 81 . From

x = 8 i t fo l lows { 1 , 3 , 5 , 6 ) , ( 2 , 3 , 4 , 7 ) ~ s w i t h { 1 , 5 , 8 , y l ~ s ~ and

y e { 1 , 2 , . , . , 8 1 . T h e r e f o r e , i t i s h = 2 .. 3

3 '

THEOREM 4 . 3 . T h e r e e x i s t t h r e e D M B PQS w i t h m = l 2 q u a d r u p l e s .

T h e i r d e g r e e - s e t is DS = [ ( 6 1 8 ] . P r o o f . In t h e p a i r s of DMB PQS w i t h m = 1 2 hav ing DS=[f6)6,(4)31

o r DS = [ ( 6 ) 4 , ( 4 ) & t h e r e e x i s t t h r e e elements z , y , z such t h a t :

( X , Y ) ~ , ( x , z ) and a quadrup le b = I x , y , z ) ( s e e 5 2 ) . T h e r e f o r e ,

f o r them i t i s h = 2 (Theor. 4 . 1 ) . Consider t h e l a s t p a i r o f D M B

PQS w i t h m = 1 2 . I t has d e g r e e - s e t DS = [ ( 6 j 8 ] . Let F = { F F F 1 , 1, 2' 3

G = {G ,G ,G 1 be t h e f o l l o w i n g 1 - f a c t o r i z a t i o n s of K 4 on

2

1 2 3 X = { 1 , 4 , 5 , 8 )

F =

and Y = { 2 , 3 , 6 , 7 ) r e s p e c t i v e l y :

F l I I F 3 I I I G 2 I G 3

On the Maximum Number of SQS(v) 259

F u r t h e r , l e t a b e t h e f o l l o w i n g p e r m u t a t i o n s on { 1 , 2 , 3 } i

1 2 3 1 2 3 1 2 3

1 2 3 3 1 2 ' 2 3 1 = ( ) 3 ) J ) *

I f P = { 1 , 2 ,..., 8 1 , w e can v e r i f y t h a t ( P , I ' ( F , G , a l) , 1

( P , r ( F , G , a 2 1 ) , ( P , T ( F , G , a I ) a r e t h r e e D M B PQS w i t h m = 1 2 and

DS = 1 ( 6 1 8 ] . 3

F u r t h e r , r ( F , G , a ) and r ( f , G , c i 2 ) are t h e two f a m i l i e s i n d i -

c a t e d i n 5 2 . S i n c e i t i s , i t f o l l o w s t h a t i t is n o t p o s s i -

b l e t o c o n s t r u c t 7 2 ~ 3 DMB PQS w i t h m = 1 2 and DS = [ ( 6 j 8 ] . Hen-

c e , i t f o l l o w s t h e s ta tement . .

1

THEOREM 4.4. D ( 8 , q 8 - 1 2 1 = 3 . Proof. Let ( X , A ) and ( Y , 8 ) b e two S Q S ( 4 ) , where X={1,4 ,5 ,8}

and Y = { 2 , 3 , 6 , 7 } . F u r t h e r , l e t F = f F ,F ,F 1 , G = { G ,G G 1 be

t h e two I - f a c t o r i z a t i o n s of K 4 on X and Y , and l e t a

( i = 1 , 2 , 3 ) b e t h e p e r m u t a t i o n s , d e f i n e d i n Theorem 4 . 3 . I t i s known

1 2 3 1 2' 3

i

191 t h a t t h e p a i r

i n an SQS(8) . We have :

260 M. Gionfriddo, A . Lizzio and M.C. Manno

We can see immediately that:

q 1 n q 2 = q l n q 3 = q 2 n q 3 , l q i n q . l = 2 3

for every i , j € { I , Z , 3 1 , i # j .

Hence D(8,q8-12) - > 3 . From Theorem 3.1, in the case k = 2 , we have Di8,q8-12) = 3 .

THEOREM 4 . 5 . We have D(v ,qv-8) = D ( v , q - 1 4 ) =D(u ,q - 1 5 ) = 2 , for v v

n+2 n n e v e r y v = 2 , u = 5 - 2 , u = 7 . 2 , and n L 2 .

n+2 n Proof . Since for v = 2 , v = 5 . 2 , v = 7 . e n it is possible

to construct at least two S Q S ( v l with qv-8 or q - 1 4 or 4,115

quadruples in common (see [ 6 ] , [ 7 ] , [13]), the statement follows

from Theorem 4.2, directly..

V

R E F E R E N C E S

111 C. Berge, Graphes e t hypergraphes , Dunod, Paris, 1970.

1-21 J. Doyen, C o n s t r u c t i o n s of d i s j o i n t S t e i n e r t r i p Z e s y s t e m s ,

[3] M. Gionfriddo, On some p a r t i c u l a r d i s j o i n t and mutuaZly b a l a n - ced p a r t i a l quadrup le s y s t e m s , Ars Combinatoria, 12 (1981),

Proc. Amer. Math. SOC., 32 (1972), 409-416.

123-134.

141 M. Gionfriddo, Some r e s u l t s on p a r t i a l S t e i n e r quadrup le sy- s t e m s , Combinatorics 8 1 , Annals o f Discrete Mathematics, 18 (1983), 401-408 .

1 5 1 M. Gionfriddo, On t h e b l o c k i n t e r s e c t i o n prob lem for S t e i n e r quadrup le s y s t e m s , Ars Combinatoria, 15 (1983), 301-314.

161 M. Gionfriddo, C o n s t r u c t i o n of a l l d i s j o i n t and m u t u a l l y ba lan - ced p a r t i a l quadrupZe s y s t e m s w i t h 1 2 , 1 4 o r 1 5 b l o c k s , Rendiconti del Seminario Matematico di Brescia, 7 (1984), 343- 354.

171 M. Gionfriddo and C.C. Lindner, C o n s t r u c t i o n of S t e i n e r quadru- p ? e s y s t e m s h a v i n g a p r e s c r i b e d number of b l o c k s in common, Di- screte Mathematics, 34 (1981), 31-42.

1-81 C . C . Lindner, E. Mendelsohn, and A. Rosa, On t h e number of I - f a c t o r i z a t i o n s of t h e comple t e graph , J. o f Combinatorial Theory, 20 (B) (1976), 265-282.

On the Maximum Number of SQSlvl 26 1

[ 9 ] C.C. Lindner and A. Rosa, S t e i n e r q u a d r u p l e s y s t e m s - A s u r v e y , Discrete Mathematics, 2 2 ( 1 9 7 8 ) , 1 4 7 - 1 8 1 .

[lo] A . Lizzio, M.C. Marino, F. Milazzo, E x i s t e n c e of S ( 3 , 4 , v l , v ~ 5 . 2 ~ and n z 3 , w i t h qv-Zl and qv-25 b l a c k s i n common,

Le Matematiche

1111 A . Lizzio, S. Milici, C o n s t r u c t i o n s of d i s j o i n t and m u t u a l l y baZanced p a r t i a l S t e i n e r t r i p l e s y s t e m s , B o l l . Un. Mat. Ital. ( 6 ) 2 - A ( 1 9 8 3 ) , 1 8 3 - 1 9 1 .

1121 A. Lizzio, S. Milici, On some p a i r s of D a r t i a l t r i p l e s y s t e m s ,

1131 G . L O Faro, On t h e s e t J l v l f o r S t e i n e r quadrup le s y s t e m s o f

Rendiconti 1st. Mat. Un. Trieste, (to a p p e a r ) .

o r d e r v = 7 . 2 n w i t h n 2 2 , Ars Combinatoria, 17 ( 1 9 8 4 ) , 39- 4 7 .

1141 A . Rosa, I n t e r s e c t i o n p r o p e r t i e s o f S t e i n e r q u a d r u p l e s y s t e m s , Annals of Discrete Mathematics, 7 ( 1 9 8 0 ) , 1 1 5 - 1 2 8 .


ON FINITE TRANSLATION STRUCTURES WITH PROPER DILATATIONS

Armin Herzer

Fachbereich Mathematik Johannes Gutenberg-Universitat

Mainz, Germany

Recently, Biliotti and the author obtained a certain number of results on translation structures with proper dilatations including structure- and characterisation-theorems, which here will be reformulated in a different manner, throwing a new light on some of the regarded questions.

1 . GROUPS OF EXPONENT p AND CLASS 5 2 .

Let K be a (commutative) field of characteristic p > 0 with automorphism Y and V a vector space over K. For a subspace W of V we consider mappings f: VxV + W with property ( * ) : namely f is alternating, vanishing on VxW and bisemilinear with automorphism y, i.e. f satisfies the following conditions:

( * I (i) f(UrV) = -f(v,u)

(ii) f (ul+uz ,v) = f (u1 ,v) +f (uz ,v)

(iii) f(uk,v) = f(u,v)ky

(iv) f(u,u) = 0 = f(u,w)

for all U , U ~ , U ~ ~ V E V, w E W, k E K. Clearly f is bilinear iff y=l.

A group G = (G,.) is called of exponent p, if xp = 1 for all x G holds, and G is called of (nilpotency) class 5 2 , if the commutator subgroup of G is contained in the center of G:

G' 5 Z ( G ) .

We define a multiplication a on V by

xoy := x+y+f(x,y) for all x,yEV.

We write (V,f) for the structure consisting of the set V and the multiplication a on it, where f has property ( * ) .

PROPOSITION 1 . G = (V,f) is a group of exponent p and class 5 2 .

Proof: The neutral element is 0 , the inverse of x is -xI and an easy computation shows

(xay)oz = x+y+z+f (x,y)+f (x,z)+f(y,z) = XO(~*Z).

Moreover Pacteristic p . At last for the commutator of x and y we have

xn = x+...+x (n times) holds and so xp=ol since K has cha-

[x,yl = x-I= y-l- x'y = 2f (x,y) , and so G' 5 W 5 Z(G) is valid.

264 A . Herzer

Conversely the following is true:

PROPOSITION 2 . Every group of prime exponent p and class 52 is isomorphic to a group (V,f) as defined before.

- Proof: Let G be such a group. We define an abelian group = (G,+) in the following manner. For p=2 let be x+y=xy; for pf2 we define

p-l x+y := xy[x,yl 2 for all x,yEG.

Then is a K-vector space for some field K of characteristic p (at

for p odd, least K=GF (p) ) . Defining p+l

f(x,y) := [x,yl 2

and f(x,y)=o for p=2, the mapping f: GxG + G' has property ( * ) with y = l , and G=(G,f) holds.

It is easy to construct such mappings f with property ( * I . Let V,W,K, y be as before and vlf...vh a base of a complement of W in V. We choose elements wijEW for l<i<j_<h. Given u,vEV we have x ,y.EK and w,w'EW, such that i i

h h u = w + t v x . i 1' i= 1

holds. Then we define f(u,v) = 1 Wij(X.Y -x.y.) Y I

l<i<jZh 1 ' and obtain f with property (*I.

For p an odd prime there is also a connection between such groups and the kinematic algebras: We look for A a local K-algebra with A= K @ M, where M is the maximal ideal of A satisfying x2=0 for all xEM. Then defining f on M by f(a,b)=ab for all a,bEM one can show that the bilinear map f has property ( * ) with y=l, since char K # 2 . On the other hand for A* the group of units of A the factor group A*/K*={(l+a)K*I aEM} is a two-sided affine linearly fibered incidence group1, and is isomorphic to (M, f . For, since (1 +a) (1 +b) =1 +a+b+f (a , b) holds , the mapping M --t A*/K*; a (l+a)K* is an isomorphism of groups, " 6 1 ) ~

Conversely from the preceeding results every group of prime exponent p and class 5 2 has (in at least one way) the structure of such an affine incidence group as cited before.

2 . GROUPS WITH A PARTITION TI AND A NON TRIVIAL TI-AUTOMORPHISM.

A (non-trivial) partition TI of a group G is a system of subgroups of G called components with the properties:

(1) G and 1 1 1 are no components (2) every element of G different from 1 is contained in

exactly one component.

An endomorphism a of G is called a n-endomorphism, if hold for every component U of n.

For an abelian group G with partition n we have the following RESULT 1 (And&, [l]). The n-endomorphisms form a ring K without zero- divisors. So in the finite case K is a field (called the kernel of n) and G is a K-vector space, whereas the components are K-subspaces.

More general we have

Ua -< U does

Translation Structures with Dilatations 265

PROPOSITION 3 (Biliotti/Scarselli [ 4 ] , Herzer [ 7 ] ) . Let G be a finite group with a partition n and a non-trivial n-automorphism a. Then G is of prime exponent p and class 22.

Now suppose we have a group G = (V,f) for some K-vector space V, and there is some aEK with O+a+l, and a defined by

va = va for every vEV

is n-automorphism of G for a partition n of G. This forces (x0y)a = (xa)o (ya) and so

2Y f(x,y)a = f(xa,ya) = f(x,y)a . Thus, if G is non-abelian and therefore f(x,y)#o for appropriate x,y,

we have a = a2' which is only possible for y # 1 . (For every odd number n one can construct such an a of period n for an infinite series of finite fields K and suitable automorphism y of K , see 171, pg.387.)

In a certain sense also the converse of the above statement is true: PROPOSITION 4 (Herzer 1 7 1 ) . Let G be a finite non-abelian group with partition n and non-trivial n-automorphism a. Then there is a field K, an automorphism y of K, a K-vector space V and a mapping f with property ( * ) , such that G is isomorphic to (V,f). Moreover there is some aEK with va=va for all vEV, and every component is a K-subspace of V.

3. THE GEOMETRIES BELONGING TO THOSE GROUPS

(Most of the following ideas can be found in Biliotti/Herzer [ 3 1 . ) Let P be a set the elements of which are called points, and 8 a set of subsets of P called blocks with the property, that any two points x and y are contained in exactly one block denoted by (x,y) and moreover every block contains at least two elements and is different from P. We call s = (Pl6,Il) a arallel structure, if /I is a parallelism of s , i.e. an equivalence re?ation on 8 the classes of which are partitions of P. A permutation u of P is called a dilatation, if for every block B also Bo and Bo-l are blocks and moreover BHBo holds. A dilatation is called a translation, if it has no fixed points or is the identity map, and is called a ro er dilatation, if it has exactly one fixed point. A parallel structp&= ( P , B , ' ) is called a translation structure, if there is a group of translations of s acting transitively on P.

Let G be a group with a partition n.Then S(G,n) = (G,6,//) is a translation structure, where 8 is defined to be the set of all right cosets of the components and any two blocks are parallel, if they are right cosets of the same component:

6 = {UxlUEn, xEGI; UyIlWz * u = w. A transitive translation group of S ( G , n ) is G acting on the pointset G by right multiplication.

PROPOSITION 5. S ( G , n ) possesses proper dilatations if and only if G possesses at least one non-trivial n-automorphism. (So far we have repeated basic concepts, which can be found in Andrg's paper [ I ] and as a short survey also e.g. at the beginning of Biliot- ti [ 2 1 . )

NOW, by the preceding results, if G is finite and possesses a non-tri-

266 A . Herzer

vial n-automorphism, then G is of prime exponent p and class 52: moreover G has the structure of a K-vector space such that the components are subspaces. Then the concept of parallel structure leads to the following definitions:

The parallel structure s = (P,B,I() is called affinely embedded, if there is an affine space A such that P is the set of points of A and B is a set of affine subspaces of A ( , whereas I1 in general is not the natural parallelism of A ) . For different points x,y of A the line of A joining x and y is expressed by xvy. Let R be the set of all lines of A and Ilr a parallelism on R. For BEB and xEP define C = n(x,B) by CEB and BllC and xEC. Similarly for RER and YEP define S = nr(y,R) by S E R and RllrS and yES.Let the parallel strucure A = (P,B,W) be affinely embedded in the affine space A as defined before. A parallelism Ilr on R is called compatible with s , if for every BEB and RER with IBnRl = 1 the following hold:

(i) In(x,B)tlnr(y,R) I = 1 for every xER and every yEB

(ii) Let xER and yiEB and define z by [ z , )=ll(x,B)nnr(y.,R) for i=l , 2,3. Then y ,y ,y

‘2 “3

are colfinear fn A if and on$y if z 1 , are collineir In A,

(If an affinely embedded parallel structure s = (P,B,/l) possesses a compatible parallelism /r on R for BE8 and RER with lBnRl = 1 , the set

U n(x,B) = U nr(y,R) xER YEB

is a generalized affine Segre variety, and the set { X E B I X I I B , X n R # @ } is the projection of an affine d-regulus, where d is the dimension of B in A, as defined in Herzer [ a ] . )

PROPOSITION 6. An affinely embedded parallel structure s possesses at most one parallelism Ur on R beeing compatible with s , The parallel structure s = (P,B, l l ) is called an affine microcentral translation structure (of odd order), if s in an embedding in an a€- fine space A (of odd order) is a translation structure with translation group T, such that the elements of T also are affinities of A and there is a parallelism Ilr on R which satisfies for every TET with t f l the condition

XVXT jr yvyr €or all x,yEP.

PROPOSITION 7. For an affine microcentral translation structure s the parallelism Ir on R as given by the definition is compatible with s . Let s be a finite affinely embedded parallelism f/r on R compatible with 5 . We define the following incidence proposition (D) ( - a kind of Desargues’ theorem - 1 :

(D)

arallel structure possessing a

For i=1,2,3 let xi,yi be points no three of which are collinear.

If then xlvyl Ilr x vy and j j

( X I J 1 I (Y1,Yj) hold for j=2,3, j

so also (X,IX3) u (YyY,). PROPOSITION 8. Under the conditions for s given in the preceding section s is an affine microcentral translation structure, if and only if (D) is valid for s .

Translation Structures with Dilatations 261

By omitting the more complicated case p=2 (which can be found in [3]) all this together gives the following characterisation:

THEOREM. For the finite parallel structure s the following statements are equivalent: a)There is an affine embedding of s in an affine space of odd order,

b)S is an affine microcentral translation structure of odd order c)There is a group G of prime exponent p > 2 and class 5 2 and a par-

such that s possesses a compatible parallelism Ir on R , for which (D) is valid

tition n of G, such that s = S ( G , n ) holds.

Concerning the proof the implications a)*b) and c)-a) follow from the preceding propositions. The implication b)-*c) follows from the fact, that s fulfilling b) can be considered as an affine two-sided linearly fibered incidence group1, which then is represented by an affine kinematic algebra A= K (3 M as defined at the end of chapter 1.

EXAMPLE. Let V be a K-vector space of dimension at least 2, and for a subspace W of V and automorphism y of K let f: vxv + W have property ( * ) . Define G = (V,f) and n = {vKlo#v€V}. Then the translation structure s = S(G,n) is even a microcentral affine translation Sperner space (i.e. all blocks have the same cardinality): For any subfield F of the fixed field of y we can choose A = AG(V/F) as an embedding affine space of s. (In the last chapter of 131 these special geometries of most simple shape in their class are characterized.) The properties of f imply f(v,u)=o for some fixed u#o and all vEV. So in s there is at least cne parallel class of blocks which is also a parallel class of subspaces of A w.r.to the natural parallelism in A , holds for the lines and their parallelism Ilr.)

REMARK. Since the parallel structures s of the theorem are the only candidates for finite translation structures with proper dilatations, one could ask how to characterize this additional property. But for a parallel structure s which is affinely embedded and possesses a parallelism //r on R compatible with s , for different points xlylz of the same block it is easy to give an incidence proposition (Dxyz), which guaranties the existence of a (proper) dilatation 6 with fixed point x and y6 = z (see [31, 3.8).

(The analogic thing

FOOTNOTE

1. G is an affine two-sided linearely fibered incidence group, if G is the set of all points of an affine space A, such that for every element a of G by left multiplication and right multiplication a on A are induced affinities and moreover all lines through 1 are subgroups of G. For A = AG(V/K) every such grou can be represented by means of a local K-algebra A = K @ M with x'=, VxEM, see [ 5 1 or [91.

REFERENCES

[I] Andrg, J . , Uber Parallelstrukturen I.II., Math.Z.76 (1961) 85-102

[ 2 ] Biliotti, M., Sulle strutture di traslazione, Boll.U.M.I.(5) 1 4 - A (1978) 667-677.

[3] Biliotti, M. and Herzer, A . , Zur Geometrie der Translationsstruk- turen mit eigentlichen Dilatationen, Abh.Math.Sem.Ham- burg (1984) 1-27.

and 155-163.

268 A . Iferzer

[ 4 ] Biliotti, M. and Scarselli, A., Sulle strutture di traslazione dotatedi dilatazioni proprie, Atti Acc.Naz.Lincei, Rend. C1.Sci.Fis.Mat.Nat. (8) 6 7 ( 1 9 7 9 ) 75-80.

[ 5 ] BriScker, L., Kinematische Raume, Geom.Ded. 1 (1973) 2 4 1 - 2 6 8 .

[6] Herzer, A., Endliche translationstransitive postaffine Riume, Abh.

171 Herzer, A., Endliche nichtkommutative Gruppen mit Partition ll und Math.Sem.Hamburg 48 ( 1 979) 25-33.

fixpunktfreiem WAutomorphismus, Arch.Math.34 (1980) 385-392,

181 Herzer, A., Varietg di Segre generalizzate, Rend.Mat.(Roma) to ap-

[ 9 1 Karzel, H., Kinematic Spaces, Symposia Mathematica XI ( 1 9 7 3 ) 4 1 3 -

pear 1986.

439 (Istituto Nazionale di Alta Matematica).


Sha rp ly 3 - t r a n s i t i v e g roups g e n e r a t e d by i n v o l u t i o n s

Monika H i l l e and He in r i ch Wefelscheid

The set J: = { y E G ) y 2 = i d , y + i d } of i n v o l u t i o n s of a g roup G which o p e r a t e s s h a r p l y 3 - t r a n s i t i v e l y on a set M, i n d u c e s , t o a l a r g e e x t e n t t h e s t r u c t u r e of t h i s group G . For example t h e c h a r a c t e r i s t i c

2 of G and t h e p l a n a r i t y of G a r e e x p r e s s i b l e as p r o p e r t i e s of J .

I n t h i s pape r w e a r e l o o k i n g f o r s h a r p l y 3 - t r a n s i t i v e permutat ion- groups G which a r e g e n e r a t e d by t h e i r i n v o l u t i o n s . I t i s shown t h a t : G = J 2 n G S P G L ( 2 , K ) fi K c Z 2 and r e s u l t s on groups G w i t h G = J3 a r e g iven . Also a c l a s s of examples of g roups w i t h G = J3 are p r e s e n t e d The q u e s t i o n , f o r what n E N t h e r e e x i s t g roups G such t h a t G = Jn

b u t G c Jn-l i s s t i l l open. On t h e o t h e r s i d e t h e r e do e x i s t s h a r p l y 3 - t r a n s i t i v e g roups G which a r e n o t g e n e r a t e d by t h e i r i n v o l u t i o n s ; e . g . a l l f i n i t e such g roups G which are n o t isomorphic t o a P G L ( 2 , K ) f o r some K have t h i s p r o p e r t y .

For unde r s t and ing t h e f o l l o w i n g w e need t h e n o t i o n of a KT-field and t h e b a s i c r e p r e s e n t a t i o n theorem:

D e f i n i t i o n 1 : F ( + , . , a ) is c a l l e d a KT-field i f t h e f o l l o w i n g axioms a r e v a l i d :

FB 1 ( F , + ) i s a l o o p ( w i t h i d e n t i t y 0) which h a s t h e p r o p e r t i e s : a + x = 0 x + a = 0. ( w e p u t x:= -a) Fo r each p a i r of e l emen t s a , b F F t h e r e e x i s t s a n element d a I b E F such t h a t a + ( b + x ) = ( a i b ) +d - x f o r each x E F. a , b

- FB 2 (F* , - ) i s a group ( w i t h i d e n t i t y 1; F*=F { O } )

FB 3 a . ( b + c ) = a - b t a - c and 0 . a = o f o r a l l a , b , c C F

- K T o i s an i n v o l u t a r y autornorphism of t h e m u l t i p l i c a t i v e group ( F * , * ) which s a t i s f i e s t h e f u n c t i o n a l e q u a t i o n :

-

270 M. Hille and H Wefehrheid

O(l + U(x) = 1 - a ( l + x ) f o r all x E F \ { 0 , 1 } .

Representation-Theorem 2: i n F and denote by F:= FUI-1.

The t r a n s f o r m a t i o n s of t y p e a and B of F o n t o F:

L e t F be a KT-field, - an element n o t

a : x --f a + m - x 8: x + a + c s ( b + m - x )

with a , b , m E F , m $ O and a ( - ) = - , o ( n ) = 0 , o ( 0 ) =-, form a group T~ (F) which o p e r a t e s s h a r p l y 3 - t r a n s i t i v e l y On F'. Conversely each s h a r p l y 3 - t r a n s i t i v e group i s isomorphic as a permutat ion group t o t h e groups T (F) of a uniquely determined KT-field. 3

In t h e fo l lowing l e t G be a sharp ly 3 - t r a n s i t i v e group a s be ing r e p r e s e n t e d i n t h e form T 3 ( F ) The i n v o l u t i o n s of G a r e :

a0: x + a - x a E F ( i f c h a r F = 2 , then a * O )

8,: x + -b + no(b+x) b E F and n E S := { s E F*lo(s).s=lI

I n case of c h a r F = 2 then a. has t h e only f i x e d p o i n t ; i n case of c h a r F * 2 then a. has t h e two f i x e d p o i n t s - and a - 2 - I , whereas n E R := { z . a ( z ) I z E F * ) . I f t h a t is t h e case with n = z .o (z ) .

p o s s e s s e s f i x e d p o i n t s x1 ,x2 i f and only i f = - b _ + ~

- ;o x1,x2 -1

2 - D e f i n i t i o n 3: A t r a n s f o r m a t i o n 6 E G wi th 6 * i d i s c a l l e d a pseudo-involut ion i f t h e r e e x i s t s two d i f f e r e n t p o i n t s wi th 6 ( a ) = b and 6 ( b ) = a

a , b E F

2 Lemma 4: If G c o n t a i n s a pseudo-involut ion 6 then 6 4 J and S * F*.

Proof: L e t be 6 E G a pseudo-involut ion w i t h S ( 0 ) = m and 6(-) = 0.

Then 6 has t h e form 6 (x ) = a(m-x) = o(m) - a ( x ) with m E F*\S. Suppose

t h e r e e x i s t two i n v o l u t i o n s 6 , , 8 , E J such t h a t 6 = B l 8,. Thenwehave:

m = 6 ( 0 ) = e102co, * 8, (-1 = B2(0)

0 = 6 ( m ) = B , B 2 ( - ) =b B, ( 0 ) = 0 2 ( 4

1 . Case: L e t be B , ( m ) = B(0) = z * 0,-.

Then 66(z) = 6 S B 2 ( 0 ) = 6R,B2B2(0) = 6 8 , ( 0 ) = 6 B 2 ( m ) = I ~ ~ B , B , ( - ) =

Sharply 3-Transitive Groups 27 1

= P I ( - ) = z. * u ( m ) * m - z = 6 6 ( z ) = z - a(m).m = 1 * m E S c o n t r a r y t o

o u r assumption m E F * \ S .

2 . Case: 6 , (a) =

Then w e g e t B 2 ( 0 ) = B, (-1 = QD. =. 0 = B2(m) = @ ( 0 ) * B1 h a s t h e f i x e d p o i n t s 0 and m =$ B, (x ) = - x . B 2 i n t e r c h a n g e s 0 and OJ.

There fo re (3, has t h e form @ , ( x ) = n.o[x) = B15,(x) = - n - a ( x ) . * o ( m ) = - n * m E S s i n c e n E S and -1 l i e s i n t h e c e n t e r of (F*,’) . C o n t r a d i c t i o n t o m E F* \ s.

1

wi th n E S. * u ( m ) -r?(x) = 6 ( x ) =

3 . Case: @ , ( m ) = 0

* B 2 ( 0 ) = B1 (m) = 0. T h i s l e a d s t o t h e same proof as i n case 2 , i f w e i n t e r c h a n g e B and B 2 . 1

Theorem 5: L e t G be a s h a r p l y 3 - t r a n s i t i v e group. Then G = J2 i f and on ly i f G S P G L ( 2 , K ) and I K I > 2 .

2 Proof: “4‘ PGL(2,K) S J i f I K I > 2 i s commonly known and c a n b e v e r i f i e d by c a l c u l a t i o n . w*11

pseudo-invo1ution.Therefore F * \ S = #. A KT-field w i t h F* = S i s a commutative f i e l d ( c f . [31 , S a t z 1 . 6 ) .

2 L e t b e G = J . Because of lemma 4 t h e g roup G c a n n o t c o n t a i n any

3 Now w e t u r n t o groups G w i th G = J .

D e f i n i t i o n 6: A KT-field (F,+,*u) is called p l a n a r , i f t h e e q u a t i o n ax + b x = c w i t h a + - b always h a s a s o l u t i o n x E F .

P l a n a r KT-fields a l r e a d y a r e n e a r f i e l d s ; i . e . (F,+) i s an a b e l i a n group. ( c f . [ I ] , 5 . 6 ) .

Theorem 7 : L e t be (F,+,.,u) a KT-field w i t h I F / > 2 and l e t be G t h e induced s h a r p l y 3 - t r a n s i t i v e group. Then

G ~ : = I y E G l y ( a ) = a ~ c ~ ~ f o r a n a E F

i s v a l i d i f and o n l y i f F i s p l a n a r and S - S = F* w i t h S:=

= C s E F * l s - U ( s ) = I } .

212 M . HiNe and ti Wefelscheid

P r o o f : W e c o n d u c t t h e d e m o n s t r a t i o n i n s e v e r a l s t e p s :

A : L e t b e F a p l a n a r KT-f ie ld and S - S = F*.

1 . Gm cJ '

P roof : have t h e form:

1 0 -

The e l e m e n t s of t h e g roup Gmlo:={yEGI y ( 0 ) = 0 , y ( m ) = m )

Gm10 = {LIEGI p : = - , k x w i t h k E F * ] . .

Because of S - S = F* t h e r e e x i s t n 1 , n 2 E S such t h a t nl' n2 = k. Then 'V ( x ) = n n ( x ) = n n (x) = n l * u a ( n x ) = B 1 B 2 ( x ) w i t h B ( x ) = n l . u ( x ) ,

p,(x) = u ( n 2 x ) and B 1 , B , - E J . 1 2 1 2 2 1

A

- f o r e a c h a E F .

P roof : L e t b e a : x-, a t x . Then G = ci G ci c c1 J - - J

because of ci J c1-I = J.

- 3. Gm= { y E G l y ( x ) = c + m-x, c , m E F , m t O } c J

2 * G m , a c J 2

-1 2 a- l 2 - 1 a = I 0

2

P r o o f : L e t b e T := { T E G I T ( x ) = ~ = , x = ~ } . Then w e have

S i n c e F i s p l a n a r , t h e se t T c o n s i s t s of t r a n s f o r m a t i o n s o f t h e form:

T = { T E G ~ I T ( X ) = b t x w i t h b E F * l

But f o r IF1 > 2 w e have T = a2a1 E J2 w i t h a i ( x ) = ai-x and a 2-al = b . Hence T c J . B . L e t be G a c J 2 f o r some a E F .

Then Gm = y G a y c y J y = J f o r some y E G w i t h y ( a ) = m.

L e t now b e c 1 E Gm w i t h a ( x ) = c +mx, m + 1 and c1 = B 2 B l E J

B , ( x ) = - b . + n * o ( b i + x ) , i = 1 , 2 .

W e show t h a t c1 p o s s e s s e s a n o t h e r f i x e d p o i n t d i f f e r e n t f r o m - . We have

2

-1 2 - 1 2

2 w i t h

Because of B i E J w e have n E S . i i i

m = a ( - ) = B 2 B 1 (-1 = - b 2 + n 2 u ( b 2 + ( - b +n o ( b l + - ) ) )

= -b +n. o ( b -b )

1 1

2 2 2 1 =, b2 = b l = :b . +. a ( x ) = -b t (n2n1- ' b t n2n1 -2 . x )

=) The second f i x e d p o i n t of ci i s -b .

Sharply 3-Transitive Groups 213

S i n c e each a E G m h a s two f i x e d p o i n t s , F i s p l a n a r whereas f o l l o w s t h a t (F,+) i s an a b e l i a n group. * a ( x ) = (-b+n2n1-lb)+n2n,- ' -x =) m = n n - 1 ES.S

2 1

Thus F* =S-s s i n c e m E F * was a r b i t r a r i l y chosen.

Remark: I f Ga c J2 t h e n G c J3. Wether t h e conve r se i s t r u e , w e do n o t know.

I n c o n c l u s i o n w e g i v e examples of s h a r p l y 3 - t r a n s i t i v e groups G w i t h

G = J

s h a r p l y 3 - t r a n s i t i v e g roups due t o Xerby ( c f . [ 2 ] , p . 6 0 ) :

L e t (F,+, - 1 be a commutative f i e l d and l e t a ( x ) = x- ' . Suppose A < (F*,.) such t h a t :

( 1 ) Q : = {a E F * l a E F * } c A

( 2 ) There e x i s t s a monomorphism TI: F*/A + A u t ( F , + , ' )

( 3 ) i ( x ) E xA f o r all T E TI (F*/A) and a l l x E F*.

Define Oob = 0, and f o r a + O , aob = a - a (b) where a = a ( a . A ) . Then ( F , + , o , o ) i s a KT-field. (To be e x a c t : ( F ! + # o ) i s a s t r o n g l y coupled Dickson-nea r f i e ld , which i n a d d i t i o n i s p l a n a r ( c f . [ 5 ] ) )

3 b u t G t. J2. We u s e t h e f o l l o w i n g g e n e r a l method f o r c o n s t r u c t i n g

2

cp cp

( 1 ) i nduces [F*:A] = 2111 where I d e n o t e s an a p p r o p r i a t e index s e t . One can c o n s t r u c t f o r a r b i t r a r y index sets such KT-field f o r which t h e induced group G s a t i s f i e s G = J3 b u t G t. J2, W e i l l u s t r a t e t h i s method of c o n s t r u c t i o n f o r I ={1,2}: L e t be K a commutative f i e l d and L : = K ( x 1 , x 2 ) a t r a n s c e n d e n t a l e x t e n s i o n of K . Consider t h e f o l l o w i n g automorphisms a1,a2 E AutKL :

+ 1-x2

1 a2: x + x r1 k E K --* k

X + 1-Xl

al: { x2 + x2

k E K + k u1 and a2 are i n v o l u t a r y and s a t i s f y u 1 u 2 = a2a1.

Take F:= L ( t l f t 2 ) where t h e ti are t r a n s c e n d e n t a l o v e r L. L e t be 7 i EAut F t h e f o l l o w i n g c o n t i n u a t i o n of a1 for h E L [ t , / t 2 1 :

214 M . Hille arid H. Wefelscheid

h, ri(hl) for f = - r hl h ,h EL[tl,t21 h2 1 2

and define T , (-) := I h2 -r;(hZJ

h , A l s o let be gradi€ = gradi$-:= gradihl - gradih2 the degree function

of the polynomials hlrh2 EL[tlrt21 with respect. to ti.

Now we define for € , g E F :

grad f grad2f fog:= fmfw(g) with fQ:= T 2

Then Fr+,oru) is a planar KT-field (to be exact: (F,+,oru) is a planar

KT-nearfield; o(a) = a the inverse with respect to ( - 1 1 . We note thatA = Kern cp = {fEF*l gradifZO mod 2 for i=Ir2}cS

and [F* : A] = 4 = 211[. The 3 other cosets are: t,*A, tZ-Ar t,-t2-A

The Dickson-group r:= If EAut F / f EF*] consists of the 4 automor-

phisms: r = {id, ~~r -c2, T ~ T ~ } . F O ~ any T €rand any z E Itl,t2,tl.t21 we have T(Z) = z and therefore Z O U ( Z ) = 1. * z E S and

z - A = zoAcSoS for z E {t,,t2rtl-t2). * F* = S o S .

This shows that this example satisfies the conditions of Theorem 7, whence the induced group G fulfills G o . c J 2 and therefore G = J .

-1

cp

3

References

[l] H. KARZEL: Ztmsammenhlnge zwischen Fastbereichen,scharf 2-fach transitiven Permctationsgruppen und 2-Strukturen mit RechtecksaTiom, Abh. Math. Sem. Hamburg, 31 (1967) 191-208

[2] W. KERBY: Infiilite sharply mul'iply transitive groups. Hamburger Mathe- matische Einzelschriften. Neue Folge Heft 6, GSttingen 1974, Vandenhoek und Ruprecht

[3] W. KERBY und H. WEPELSCHEID: Uber eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur. Abh. Math. Seminar Hamburg 37(1972) 225-235

[ 4 ] H. WEFELSCHEID: ZT-subgroups of sharply 3-transitive groups, proceedings of the Edinburgh Mathematical Society 1980, Vol. 23, 9-14

[ 5 ] H. WEFELSCHEID: Zur Planaritat von KT-Fastksrpern, Archiv der Mathematik, Vol. 36(1981) 302-304

Fachbereich Mathematik der Universitlt Duisburg D - 4100 Duisburg 1

Annals of Discrete Mathematics 30 (1986) 275-284 0 Elsevier Science Publishers B.V. (North-Holland)

OY T I E GENERALIZED CIIROMATIC NUMBER

275

F l o r i c a Kramer and I l o r s t K r a m e r Computer Technique Resea rch I n s t i t u t e

s t r . R e p u b l i c i i 109 3400 Cluj-Mapoca

ROFIANIA

The c h r o m a t i c number re la t ive t o d i s t a n c e p,de- n o t e d by r(n,C) i s t h e minimum number o f colors s u f f i c i n g f o r c o l o r i n g t h e vertices o f G i n s u c h a way t h a t any t w o vertices of d i s t a n c e n o t q r e a - ter t h a n p have d i s t i n c t colors. We q i v e uppe r bounds f o r t h e c h r o m a t i c nur.ber 'f(3,G) f o r h i - p a r t i t e ( p l a n a r ) g r a p h s and g e n e r a l i z e a r e s u l t o f S .Antonucc i q i v i n q a lower bound f o r r ( 2 , C ) .

I. INTFODITCTION

I n t h e f o l l o w i n g w e s h a l l u s e some c o n c e p t s and n o t i o n s i n t r o d u c e d i n t h e book L 5 ] of C.Oerqe. We res t r ic t o u r s e l v e s t o s i m p l e g r a p h s , i .e . connec ted u n d i r e c t e d f i n i t e g r a p h s which have no l o o p s o r mul- t i p l e edqes. L e t G = ( V , E ) b e a s i m p l e g r a p h , where 17 i s t h e v e r t e x s e t and R i s t!ie ec'ge se t o f G. I f x , y e V , w e s h a l l d e n o t e by q ( x ) the deqree o f t h e v e r t e x x , by b (G)=max Lg(x) ; x ~ V 3 t h e maximum d e g r e e o f t h e vertices of G , by d G ( x , y ) t h e d i s t a n c e be tween t h e v e r t i c e s x and y i n t h e g r a p h G , i.e. t h e l e n g t h of t h e s h o r t e s t p a t h c o n n e c t i n g v e r t i c e s x and y i n t h e q r a p h G ( t h e l e n g t h of a p a t h is cons i r i e r ed t o be the number o f e d g e s which form t h e p a t h ) . it d e n o t e s t h e comple t e g r a p h w i t h n vertices and K t h e c o m p l e t e b i p a r t i t e q raph .

I n 1969 w e have c o n s i d e r e d i n t h e p a p e r s [111 and 1123 t h e fo l low- i n ? c o l o r i n g problem of a g r a p h :

DEFINITION 1. L e t p he a g i v e n i n t e q e r , p z 1 . An a d m i s s i b l e k-colo- r i n q r e l a t i v e t o d i s t a n c e p o f t h e g r a p h G=(T ' ,E) i s a f u n c t i o n f : V 9 {1,2,...,k) s u c h t h a t

The s m a l l e s t i n t e g e r k f o r which t h e r e e x i s t s an a r ' rn i s s ih l e k-colo-

n m*n

v x , y € V w i t h 1 ~ d C ( x , v ) 2 p => f ( x ) # f ( y ) .

216 F. Kramer and H. Kramer

r i n g r e l a t i v e t o d i s t a n c e p is denoted by y(p ,C) and i s c a l l e d t h e chromatic number r e l a t i v e t o d i s t a n c e p of t h e qraph G .

DEFINITION 2 . L e t G=(V,E) b e a g i v e n graph and p an i n t e g e r , p 2 1. W e s h a l l deno te by GP=(V,E ) t h e g raph , which h a s t h e same v e r t e x s e t 1 7 a s t h e graph G whereas t h e edge set E is d e f i n e d by:

P P

( x , y ) E Lp i f and on ly i f 1 s d G ( x l y ) 5 p.

There e x i s t s t hen t h e fo l lowinq r e l a t i o n between t h e ch romat i c num- b e r r ( p , C ) and t h e o r d i n a r y ch romat i c number g(C) :

I n t h e pape r s [ill, [121 and [133 w e o b t a i n e d some r e s u l t s r e l a t i v e t o t h e chromatic number graphs G f o r which we have r ( p , G ) = p + l . The problem of c o l o r i n q a graph r e l a t i v e t o a d i s t a n c e s w a s recon- s i d e r e d i n 1975 by P.Speranza [ l G ] under t h e name of Ls-color ing of a graph; t h e same oroblem was a l s o s t u d i e d by M.Gionfriddo [6], [7]

and [f311 S.Antonucci [11 and by G.Wegner L171.

2 . UPPER BOUNDS FOR THE CHROMaTIC NUMBER r ( 3 , G )

G.Weqner 1111 proved t h e fo l lowing theorem:

T!IEOREY 1. L e t G=(V,F) be a s imple p l a n a r graph such t h a t A (C) f 3. W e have then :

A s t i l l open problem is t h a t of f i n d i n g a p l a n a r graph C, w i t h maximal deq ree 3 such t h a t w e have $ - ( 2 , G ) = 8 or else t o prove F i r s t w e s h a l l g i v e some bounds f o r t h e g e n e r a l i z e d chromatic number

) $ ( P I C ) = $-(l,CP) = K(C,P).

r ( p , G ) , e s p e c i a l l y one c h a r a c t e r i z i n g t h o s e

s ( 2 , G ) f - 8.

&-(2,G)f7.

) ( " ( 3 , G ) of b i p a r t i t e g raphs . L e t C=(A,B;E) be a q iven b i p a r t i t e graph. W e s h a l l deno te by Gp=(A,EA) t h e g raph , which h a s t h e v e r t e x se t A

and t h e cdqe set EA is d e f i n e d by:

The graph GA can be o b t a i n e d from t h e graph C, by t h e e l i m i n a t i o n of t h e v e r t i c e s of 13 and by t h e a p p l i c a t i o n of t h e f o l l o w i n g t h r e e ope- r a t i o n s : a ) I f w e have a " 3 - s t a r " w i t h t h e c e n t e r i n a v e r t e x y of B (it f o l - l o w s t h a t t h e v e r t i c e s x1,x2,x3 a d j a c e n t t o y are i n A) t h e n w e con- s t r u c t i n GA t h e c i r cu i t x1x2x3x1. h) I f a verte:: y o f B h a s deq ree 2 i n G and x1,x2 a r e t h e two v e r t i - ces of A a d j a c e n t t o y i n G I t h e n w e r e p l a c e t h e p a t h xlyx2 of l e n g t h 2 i n C by an edge (x1,x2) i n GA.

c) I f by t h e a p p l i c a t i o n of t h e p reced ing two o p e r a t i o n s appea r mul- t i p l e edges between t w o v e r t i c e s x and y, w e s h a l l r e t a i n o n l y one

( x , y ) E E A i f and on ly if d G ( x l y ) = 2 .

011 the Generalized Chromatic Number 211

of t h e s e edges and delete t h e o t h e r s . Analogously w e i n t r o d u c e t h e graph G B = ( B , E R ) w i t h t h e v e r t e x set 8 and t h e edge set EB d e f i n e d by:

An upper hound f o r t h e ch romat i c number r ( 3 , G ) i n f u n c t i o n of t h e o r d i n a r y ch romat i c numbers $-(C,) and r ( C R ) i s q iven i n t h e f o l l o - w i n q :

TlICOREM 2 . L e t G=(A,B;E) be a b i p a r t i t e qraph. Then

( x , y ) E E B i f and on ly i f d c ( x , y ) = 2 .

(1)

Proof . $ ( G A ) be ing t h e o r d i n a r y ch romat i c number of t h e graph CA

t h e r e is a c o l o r i n g f l : A 3 Ll,? ,..., r ( G A ) ) o f i t s v e r t i c e s such t h a t (x ,y ) E EA i m p l i e s f l ( x ) # f l ( y ) . We c o n s i d e r a l s o a c o l o r i n g w i t h )-'(CB) c o l o r s of t h e v e r t i c e s of t h e graph GB f 2 : B + [bp(GA)+l, bb(GR) + 2 , . . . , y(GA) + ?,G(GB))

The f u n c t i o n f : A U l 3 -3 ( 1 , 2 , ..., r ( C A ) + g ( C B ) j d e f i n e d by

( 2 ) f ( x ) =

is then a c o l o r i n q of t h e v e r t i c e s of t h e b i p a r t i t e graph G=(A,B;E) w i t h '$"(GA) + r ( G B ) c o l o r s . We s h a l l now v e r i f y t h a t t h e s o o h t a i - ned c o l o r i n g is an a d m i s s i h l e ( f-(GA) t

d i s t a n c e 3 of t h e graph C,, i .e . t h a t x ,y 6 A V B and 15 d G ( x , y ) 6 3

i m p l i e s f ( x ) # f ( y ) . R e a l l y , i f d G ( x I y ) = l or d C ( x , y ) = 3 it f o l l o w s t h a t one v e r t e x , le t it be x , be longs t o A and t h e o t h e r , y , be longs t o B. Then o b v i o u s l y f ( x ) # f ( y ) . On t h e o t h e r hand i f dG(x ,y )=2 it fo l lows from t h e b i p a r t i t e c h a r a c t e r of C t h a t bo th vertices x and y a r e i n A , o r bo th a r e i n R . Cons ide r now t h e c a s e s : a )

T ( 3 , G ) 5 pA) + K ( C B ) .

such t h a t (x ,y ) 6 EB i m p l i e s f2 ( x ) # f 2 (y) . f l ( x ) , i f x E A { f 2 ( x ) , i f x E B

(C,) ) - c o l o r i n g r e l a t i v e t o

i f x , y E A and AG(x ,y )=2 , i t r e s u l t s t h a t ( x , y ) E E A and t h e r e f o r e

if x , y E B and d G ( x , y ) = 2 , i t r e s u l t s t h a t ( x , y ) c E B and t h e r e f o r e f ( x ) = f 1 ( x ) # f p = f ( y ) .

f ( X I = f 2 ( x ) f f 2 ( y ) = f ( y ) . b )

Thus w e have proved i n e q u a l i t y (1).

THEOREM 3 . L e t G = ( A , B ; E ) be a s imple p l a n a r b i p a r t i t e graph w i t h A(G) $ 3 . Then w e have ( 3 ) r ( 3 , G ) 5 8 .

T h i s bound is s h a r p i n t h e s e n s e t h a t t h e r e are c u b i c p l a n a r b i p a r - t i t e g raphs f o r which X ( 3 , C ) = S .

Proof . From t h e p l a n a r i t y of t h e i n i t i a l g raph G and t h e c o n s t r u c t i o n way of t h e g raphs GA and GB r e s u l t s t h e p l a n a r i t y of t h e g r a p h s GA

and CB. By t h e f o u r - c o l o r theorem proved by K.Appe1 and W.Haken([Z],

218 I;: Krumer and ti. Krumer

L31 and [41) w e have t ( G A ) 5 4 and $(GB) 5 4. B y Theorem 2 f o l l o w s then immediately i n e q u a l i t y ( 3 ) .

'rl: Fig. 1

i n Theorem 3 we o b t a i n :

The f a c t t h a t t h e so o b t a i n e d bound f o r $ ( 3 , C ) can n o t be lowered , can be seen from t h e fo l lowing example: L e t G=(A,B;E) be t h e graph of t h e hexahedron, which is a b i p a r t i t e p l a - n a r c u b i c graph w i t h 8 v e r t i c e s and d i ame te r 3 . I t r e s u l t s t h a t w e have x ( 3 r G ) = I A l J B l = 8 . I f w e renounce t o t h e p l a n a r i t y of G

THEOREM 4 . L e t C = ( A , B ; E ) be a b i p a r t i t e graph such t h a t (GI f 3 .

Then w e have: ( 4 ) p ( 3 , C ) 5 1 4 . The o b t a i n e d bound is s h a r p i n t h e s e n s e t h a t t h e r e e x i s t b i p a r t i t e cub ic qraphs f o r which r ( 3 , C ) = 1 4 .

Proof . Consider a b i n a r t i t e qraph G = ( A , B ; E ) w i t h n ( G ) f 3 and t h e correspondinq q raphs an(' c,., d e f i n e d above. From A ( C ) L, 3 r e s u l t s i pmed ia t e ly A ( C , ) f 6 and A(C,) L, 6 . A well-known t.!ieorem (see f o r inn tance [ l5 ] vo l .1 , S a t z I V . 2 . 1 ) asserts t h a t f o r any graph G, t h e o r d i n a r y ch romat i c number i s a t most one g r e a t e r t h a n t h e maximum degree A (GI. We have t h e n Theorem 2 w e have Y ( ? , G ) 5 14.

(GA) 5 7 and F(GB) f 7. Applying aga in

The f a c t t h a t t h i s bound i s s h a r p , can be seen from t h e example of t h e well-known Ileawood graph (see F i q . 2 ) , which is a c u b i c b i p a r t i t e graph w i t h 1 4 v e r t i c e s and of d i ame te r 3 . We have then : A d i r e c t g e n e r a l i z a t i o n of Theorem 4 i s t h e fo l lowinq theorem.

r ( 3 , C ) = I A U B I = 1 4 .

F i g . 2

THEORE'fi 5 . L e t C: be a s imple b i p a r t i t e qraDh w i t h maxinum clegree

(5) ~ ( 3 ~ ~ 1 f + A ( A - 1)).

Proof . From A = max I g ( x ) ; x E A U B) w e o b t a i n immediately t h a t t h e number of v e r t i c e s y which a r e a t d i s t a n c e 2 from a g iven v e r t e x

A = a (c ) . Then w e have:

On the Generalized Chromatic Number 279

x i n t h e q raph G i s a t most A ( A - 1 ) .\ve have t h e n A(c;,) = L A ( A - 1 ) and b ( G B ) 2 n ( A - 1 ) and t h e r e b y g ( G A ) 5 A( A - 1 ) +1 and r e s p e c t i v e l y r ( C B ) f - a(a-1)+1. Applying now Theorem 2 w e o b t a i n i n e q u a l i t y ( 5 ) .

COROLLARY 1. L e t G = ( A , B ; E ) be a s i m p l e A - r e g u l a r b i p a r t i t e g raph o f d i a m e t e r 3. Then w e have :

16) ( A \ + ! B 1 5 2 ( 1 + h ( a - 1).

Examples o f g r a p h s f o r which w e have t h e e q u a l i t y s i q n i n (5) and a l s o i n ( 6 ) a r e : 1) f o r b = 2 t h e c i r c u i t C6 o f l e n q t h 6 ; 2 ) f o r

a = 3 t h e above d i s c u s s e d ifeawood g raph and 3 ) f o r A = 4 t h e 4 - r e g u l a r b i p a r t i t e q raph of d i a m e t e r 3 w i t h 2 6 v e r t i c e s from F i q . 3. D(C) = 3

impl ies a g a i n r ( 3 , C ) = I A U R \ = 2 6 . An improvement of Theorem 5 and o f Theorem 4 can b e o b t a i n e d i f w e app- l y Brooks theorem (see f o r i n s t a n c e [ 9 1 Theorem 12 .3 ) : I f A(C)= A t h e n G i s A - c o l o r a b l e u n l e s s : ( i ) = 2 and G h a s a comnonent which i s an odd c y c l e , or ( i i ) A > 2 and

i s a component o f G . K A + l F i s . 3 The c a s e A.2 w a s d i s c u s s e d i n [ _11 ] ,1121 and L131.Let u s assume t h a t A 7 2 . The g r a p h s CA and GB a r e connec ted s i n c e C, i s connec ted . I f

now a t l e a s t one of GA, GB - s a y , GA - i s n o t a K t h e n by

Brooks theo rem, g(GA)f A ( A - 1 ) and by Theorem 2 w e have t h e inequa- l i t y have ( A 1 = \ B \ = n(n - l )+l and C, i s a b i p a r t i t e A - r e g u l a r q r a p h o f d i a m e t e r 3. I f none o f GA,GB i s a K A ( A -1) +1 t h e n )$'(GA) f a(n -1), b b ( G B ) 5 n(n-1) and by Theorem 2 x ( 3 , G ) f 2 a ( A - 1 ) . W e c an now

enounce :

COROLLARY 2 . L e t C = ( A , B ; E ) b e a s i m p l e b i p a r t i t e q r a p h such t h a t

K ( 3 , G ) 5 2 A ( A - 1 ) + 1 . Thus i f e q u a l i t y h o l d s i n Theorem 5 w e

A ( G ) = A > 2 and min ( I A I , \ B \ ) > A ( a - l ) + l . Then w e have : )9(3,G) 5 2 4 ( A - 1) .

1 ; (3 ,G) 5 - 1 2 .

I n p a r t i c u l a r , i f A = 3 and I A \ > 7 , \ R 1 > 7 t h e n w e have :


3 . LOWER BOIJNDS FOR THE CHROMATIC NUMBER $'(2 , G )

F i r s t we s h a l l p rove a r e s u l t needed i n t h e s e q u e l .

THEOREM 6 . L e t G=(V,E) b e a s i m p l e g raph w i t h t h e p r o p e r t i e s : ( i) t h e derjree o f e a c h ver tex is a t l eas t 2 , (ii) t h e d i a m e t e r o f t h e g r a p h D(G)=2, (iii) G does n o t c o n t a i n c i r c u i t s of l e n g t h 3 and 4 . Then G is a Moore g raph .

P roof . P r o p e r t y (i) i m p l i e s t h e e x i s t e n c e of a t l eas t one c i r c u i t i n t h e g r a p h G. By (ii) and (iii) r e s u l t s t h a t t h e g i r t h o f G is 5 . (7

is t h e n a Moore q raph by a r e s u l t o f R .S ing le ton l 1 4 1 which a s s e r t s t h a t a s i m g l e g raph w i t h d i a m e t e r k 2 1 and g i r t h 2 k + l I s a l s o regu- l a r and hence a Moore a r a p h .

I n 1978 S . A n t o n u c c i L 1 1 o b t a i n e d t h e f o l l o w i n g lower bound for t h e c h r o m a t i c number y ( 2 , G ) a s a f u n c t i o n of t h e number of v e r t i c e s and t h e number of e d q e s o f t h e g r a p h G:

THEOREY 7. L e t C=(V,E) be a s i m p l e g r a p h w i t h n v e r t i c e s and m e d g e s and w i t h o u t c i r c u i t s of l e n g t h 3 and 4 . Then w e have :

(7)

But S. Antonucci d i d n ' t g i v e hound is a t t a i n e d . We s h a l l

3 n

n 3- 4m2

any example o f g r a p h s f o r which t h i s p r o v e t h e f o l l o w i n g theorem:

TIIE0RE:I 8. The o n l y g r a p h s of d i a m e t e r D ( C , ) f 2 , w i t h o u t c i r c u i t s o f l e n g t h 3 and 4 , w i t h n v e r t i c e s and m e d g e s f o r which w e have

a r e t h e g r a p h s K1, K2 and t h e Moore g r a p h s of d i a m e t e r 2 .

Proof . The o n l y g r a p h of d i a m e t e r D(C)=O i s t h e g r a p h K1, f o r which we have n=1, m=O,

I f D ( G ) = l , G is a comple t e g r a p h K w i t h n - 2 . The o n l y comple t e

g raph K n l n which w e have n=2, m=1, ) f ' (2 ,K2)=2 and t h e r e f o r e ( 8 ) is v e r i f i e d . I f D(G)=2, w e have t o d i s t i n g u i s h t w o cases: a ) g = min c q ( x ) : x G V } = 1. Then t h e r e is a v e r t e x al o f d e g r e e 1 and t h e v e r t e x b a d j a c e n t t o a l h a s t o b e a d j a c e n t t o a l l t h e o t h e r v e r t i c e s o f V b e c a u s e D ( G ) = 2 . But a s G d o e s n ' t c o n t a i n c i r c u i t s o f

'$'(2,G)=1.K1 v e r i f i e s t h e n e v i d e n t l y (8).

n 2 , w i t h o u t c i r c u i t s o f l e n g t h 3 is t h e g raph K2, f o r

On rhe Generalized Chromaric Number 28 I

l e n g t h 3 , G C a 1 * a 2 * i s a ( n - 1 ) - s t a r w i t h n 2 3 , i .e. G=(V,E) w i t h V=

=n. R e l a t i o n (El becomes n(n3-4(n-1) ) = n . The o n l y so-

1 3 4

and E= [ ( a i , h ) , i = 1 , 2 ,..., n-1J . We have t h e n m=n-1 2 3

l u t i o n s of t h i s e q u a t i o n a r e n =0, n 2 = 1 and n =n =2, none o f which c o r r e s p o n d s b e c a u s e as w e have s e e n above w e have n 'I, 3. The conc lu - s i o n is t h a t w e can n o t have $= 1. b) x= min {g(x) ; x E V 3 2 2 . G is t h e n a Moore g r a p h o f d i a m e t e r 2 by Theorem 6 . A r - r e g u l a r Moore g r a p h o f d i a m e t e r 2 h a s n = 1 + J2 v e r t i c e s and m = n. x/2 = $(l+ x 2 ) / 2 edqes . Because D ( G ) = 2 w e have

x ( 2 , G ) = n = 1+ J2. I t f o l l o w s t h a t

With t h a t Theorem 7 i s p roved .

REMARK. By a well-known r e s u l t o f A.J.Hoffman and R.R.Sinqle ton (101

a Moore g r a p h of d i a m e t e r 2 h a s one o f t h e d e g r e e s 2 , 3 , 7 o r 5 7 ; f o r e a c h o f t h e d e q r e e s 2 , 3 , 7 t h e r e i s e x a c t l y one Moore g r a p h of d i a m e t e r 2 ( i t is. n o t known whe the r o r n o t t h e r e i s a Moore q r a p h of d i a m e t e r 2 and d e g r e e 5 7 ) .

A lower bound f o r can a l s o b e deduced f o r g r a p h s which have c i r c u i t s of l e n g t h 3 or 4 .

b " ( 2 , G ) s i m i l a r t o t h a t o b t a i n e d by S.P.ntonucci

TIIFOREM 9. L e t C=(V,E) be a s i m p l e connec ted g r a p h w i t h n v e r t i c e s and m e d g e s i n which w e d e n o t e by: (i)

(i i)

c3 t h e number of c i r c u i t s o f l e n q t h 3 i n G ;

c: t h e number of c i r c u i t s o f l e n g t h 4 , f o r which no p a i r of o p p o s i t e vertices i n t h e c i r c u i t are a d j a c e n t i n G ;

(iii) c4 t h e number o f c i r c u i t s o f l e n g t h 4 , f o r which one p a i r o f o p p o s i t e vertices i n t h e c i r c u i t are a d j a c e n t i n C and t h e o t h e r p a i r of o p p o s i t e v e r t i c e s a r e n o t a d j a c e n t i n G .

1

I f G d o e s n ' t c o n t a i n any suba raph of t h e t y p e K t h e n t h e chroma- t i c number X ( 2 , G ) v e r i f i e s t h e i n e q u a l i t y

( 9 )

2 , 3

I * *

3 n 3 0 1 2 n +n (6c3+4c4+2c4) - 4 m

T h i s bound i s s h a r p i n t h e s @ n s e t h a t t h e r e e x i s t s s r a p h s v e r i f y i n q t h e h y p o t h e s e s o f t h e theorem and f o r which w e have t h e e q u a l i t y s i q n i n ( 9 ) .

c P r o o f . The proof o f t h i s theorem can b e o b t a i n e d hy a m o d i f i c a t i o n of t h e p roof g i v e n by S .Antonucc i f o r Theorem 7 . A s w e have obse rved


2 2 above w e have Y ( 2 , C ) = $ ( l l C 2 ) = r(C; ) , where c; = ( V I E ) i s t h e s q u a r e of t h e q r a p h G. I f w e d e n o t e by m2 t h e c a r d i n a l i t y of t h e edqe set E 2 , t h e n w e have by a Theorem of C . B e r q e ( c 5 J l p.321)

(10) r(2,C;) = g(G2) 7 = . The number o f a l l p o s s i b l e p a t h s of l e n g t h 2 i n t h e q raph G i s q iven

by t h e sum 2 (q(ii)) . I f w e i n t r o d u c e c o r r e s n o n d i n q t o e a c h p a t h xyz of lenq&' 2 i n C an edqe ( x , z ) w e s h a l l o h t a i n a q raph C " = ( V , E " ) .

Obvious ly E2 C E", h u t t h e r e may h e e d q e s i n E 2 which are m u l t i p l e edges i n El'. L e t a , b EV h e a p a i r of v e r t i c e s , which i s connec ted i n C; by a t least one p a t h of l e n q t h 2 . We have t o d i s t i n q u i s h t h e cases: 1) a and h are a e j a c e n t v e r t i c e s i n G . Then t h e ec'qe ( a ,b ) i s con- t a i n e d i n a t l e a s t one c i r c u i t of l e n q t h 3 i n G and t h e o r d e r of mul- t i p l i c i t y of t h e edqe ( a , b ) i n E" w i l l be e q u a l w i t h t h e numher of c i r c u i t s o f l e n n t h 3 which c o n t a i n t h e edqe ( a , b ) . As e a c h c i r c u i t of l e n g t h 3 c o n t r i b u t e s t o t h e i n c r e a s e of t h e m u l t i n l i c i t y o f e a c h edge of t h i s c i r c u i t by one u n i t y , w e have t o d e l e t e 3c3 edqes from El' i n o r d e r t o make a l l e d g e s a p a r t a i n i n q t o c i r c u i t s of l e n q t h 3 simple edges . 2 ) a and 13 a r e n o t a d j a c e n t i n C. Because G does n o t c o n t a i n any subgrauh of t h e t y p e K 2 , 3 , t h e v e r t i c e s a and h can h e connec ted i n G by a t most t w o p a f h s of l e n g t h 2. W e d i s t i n q u i s h t h e n t h e s u b c a s e s : 2 a ) a and b a r e connec ted i n C: by e x a c t l v one p a t h of l e n g t h 2 . Then ( a , b ) i s o b v i o u s l y a s imple edge o f t h e g raph G " .

2b) a and b a r e connec ted i n G by two p a t h s of l e n q t h 2 . The e d g e ( a , h ) w i l l b e a d o u b l e edge i n C". But i n t h i s c a s e a and b form a p a i r of o p p o s i t e ver t ices i n a c i r c u i t of l e n q t h 4 i n G . Because a c i r c u i t of l e n q t h 4 of t h e tyrJe (ii) leads t o t h e d u p l i c a t i n g of b o t h d i a g o n a l s of t h e c i r c u i t , and a c i r c u i t of l e n g t h 4 o f t h e t y p e ( i i i) l e a d s t o t h e d u p l i c a t i n q o f o n l y one d i a q o n a l , i n o r d e r t o ob- t a i n the q raph C2 w e have t o d e l e t e 2c:+c: edges from El' b e s i d e t h e

2

2

n2-2m2

3c3 edctes a l r e a d y d e l e t e d . W e have t h u s

i=1

n g(xi) 1 ) - (3c3+2cy+c4) . m2 = m + t ( I t r e s u l t s t hen :

On the Generalized Chromatic Number

1 2 - - - - 2m ( 3c3+2c>c4) . n

283

T h i s i n e q u a l i t y and. (10) v ie l c l s

As r ( 2 , C ) i s an i n t e g e r w e o b t a i n immedia t e ly t h e i n e q u a l i t y ( 9 ) i n which [r]* d e n o t e s t h e smal les t i n t e q e r 2 r.

An example o f a n raph f o r which w e have t h e e q u a l i t y s i g n i n ( 9 ) i s t h e q r a p h from F i g . 4 f o r which w e have n=7, m = l l , c =3, c4= l , c0=2, 4 3 D ( G ) = 2 and )f(2,C)=n=7. It is e a s y t o v e r i f y t h a t f o r t h i s q r a p h w e have t h e r e l a t i o n s :

1

F i q . 4

n = [343/551* = 7 = $ ( 2 , C , ) . 'i' 3

n +n (6c3+4c4+2c4) - 4 m 0 1 i 3 ACKNOWLEDCYENT. The a u t h o r s wi sh t o t h a n k t h e r e f e r e e f o r t h e h e l p - f u l comments. The second a u t h o r would l i k e t o t h a n k a l s o t o t h e Alexande r von Humbold t -S t i f tunq f o r t h e f i n a n c i a l s u p p o r t d u r i n g t h e y e a r s 1981-1982.

REFERENCES

[l) Antonucc i ,S . , G e n e r a l i z z a z i o n i d e l c o n c e t t o d i cromatismo d ' u n

[23 Appe1,K. ,Haken,W., Every p l a n a r map is f o u r c o l o r a b l e , B u l l .

Eq Appe1,K. ,Ifaken,IJ., Every p l a n a r map is f o u r c o l o r a h l e , P a r t I.

[q Appe1,K. ,Iiaken,W. ,Koch,J . , Every p l a n a r map is f o u r colorable, ? a r t 11. R e d u c i b i l i t y , I l l i n o i s J.Math. 2 1 (1977) 491-567.

ti51 Rerqe,C. , Craphes e t hype rg raphes (Dunod, P a r i s , 1 9 7 0 ) . [6! Gionfr iddo ,M. , S u l l e c o l o r a z i o n i Ls d ' u n g r a f o f i n i t o , Boll .Un.

[7] Gionfr iddo ,M. , A l c u n i r i s u l t a t i r e l a t i v i a l l e c o l o r a z i o n i Ls

[8] Gionfr iddo ,M. , Su un problema r e l a t i v o a l l e c o l o r a z i o n i L2 d ' u n q r a f o p l a n a r e e c o l o r a z i o n i t s I Riv.Mat.Univ.Parma (4) 6 (1980)

q r a f o , B o l l . U n . V a t . I t a 1 . ( 5 ) 15-B (1978) 20-31.

Amer.Vath.Soc. 82 (1976) 711-712.

D i s c h a r g i n g , I l l i n o i s J.Math. 2 1 (1977) 429-490.

Ma t . I t a1 . (5) 15-A (1978) 444-454.

d ' un q r a f o , Riv.Mat.Univ.Parma ( 4 ) 6 (1980) 125-133.

284 I;. Krarner and H . Kramer

151-160.

1 IIarary,F., Graph Theory (Addison-Wesley Publ.Comp. ,Mass. 1969). [lo1 Hoffman,A.J. ,Singleton,R.R., On Moore graphs with diameters 2 [l<l Kramer,F. ,Kramer,H., Un probleme de coloration des sommets d'un

graphe, C.R.Acad.Sci.Paris 268 A (1969) 46-48.

[lq Kramer,F. ,Kramer,H., Ein Firbungsproblem der Knotenpunkte eines

Graphen bezGqlich der Distanz p, Revue Roum.Math.Pures et Appl.

14 (1969) 7, 1031-1038.

[lgKramer,F., Sur le nombre chromatique K(p,G) des graphes, R.A.I.

rlq Singleton,R.R., There is no irregular Moore qraph, American

rlq Sachs,N., Einfchrung in die Theorie der endlichen Graphen, vol.1. [lg Speranza,F. , Colorazioni di specie superiore d'un grafo, Boll.

[l-g Weqner,G., Graphs with qiven diameter and a coloring problem,

- - and 3, IBM J.Res.Deve1op. 4 (1960) 497-504.

R.O., R-1 (1972) 67-70.

Math.Monthly 75 (1968) 42-43.

- (Teubner Verlagsg., Leipzig, 1970).

Un.Mat.Ita1. (4) 12 Supp1,fasc. 3 (1975) 53-62.

(Preprint , Dortmund , 1976) .

A CONSTRUCTION OF SETS OF PAIRWISE ORTHOGONAL F-SQUARES OF COMPOSITE ORDER

Paola L a n c e l l o t t i and Consolato P e l l e g r i n o

D i p a r t i mento d i Ma temat i ca V ia Campi 213/B

41100 MODENA (ITALY)

I n t h i s no te complete systems o f o r thogona l F-squares a r e cons t ruc ted i n which t h e number o f symbols i s v a r i a b l e and t h e o r d e r n i s a pr ime power. Fu r the r , we g i v e an ex tens ion o f t h e MacNeish theorem by c o n s t r u c t i n g systems o f o r thogona l

e hav ing

el e2 m F-squares o f composi te o r d e r n = p1 p2 ... p, a v a r i a b l e number o f symbols: such c o n s t r u c t i o n improves t h e r e s u l t s t h a t have been reached so f a r .

1 - DEFINITIONS AND PRELIMINARY RESULTS

A square m a t r i x F = [aij] o f o r d e r n , d e f i n e d on a s e t A , i s s a i d t o be an

F-square o f t y p e (n,1) (and we s h a l l w r i t e s h o r t l y F(n,A) i f each element o f A appears A t imes i n each row and i n each column o f F . Given t h e F-squares F1(n,’xl)

and F2(n,A2) d e f i n e d r e s p e c t i v e l y on t h e se ts A1 and A2 , we s h a l l say t h a t they

a re or thogona l i f a f t e r super imposing them each p a i r (x,y)€A,xA2 appears ; \1x2 t imes. I n such case we s h a l l w r i t e F 1 ( n y ‘ x 1 ) ~ F p ( n y ’ x 2 ) .

Given t h e F-squares

fo rm an or thogona l system i f F i (n , ’ x i )LF. (n , ’ x j ) f o r a l l i , j = 1 y 2 y . . . y t y i# j . I f I tAl, ‘x 2 , . . . , ~ t l l z 1 we s h a l l say t h a t t h e or thogona l system has a v a r i a b l e

number o f symbols, I n

THEOREM 1 .l. The maximal number t o f o r thogona l F-squares F1 (n,hl), F2(n,’x2),

..., Ft(n,’xt) s a t i s f i e s t h e f o l l o w i n g i n e q u a l i t y (we s e t n = l . m i i f o r

i=1,2, ..., t) :

Fl(n,’xl) , F2(n,’x2) , ... , Ft(n,’xt) , we s h a l l say t h a t t hey

J

1 3 t h e f o l l o w i n g theorem has been proved:

2 i mi - t < (n-1) . i =1

An or thogona l system o f F-squares F1(n,.“,), F2(n,X2), ... , Ft(n,At) i s c a l l e d

complete i f

5 mi - t = (n-1) 2 . i =1

Complete systems o f F-squares a r e cons t ruc ted i n [ 2 ] , [ 3 ] .

286 P. Laricellotti and C. Pellegritlo

2 - CONSTRUCTION OF COMPLETE SYSTEMS OF F-SQUARES

We s h a l l say t h a t a p a r t i t i o n .d= I XI,X2 ,..., Xr 1 o f an m-set A=Ial,a2 ,..., am)

i s h - regu la r i f 1 Xi \ = h ( i=1 ,2 ,..,, r) . I f I,= I 1 , 2 ,..., r 1 we denote by

@ = O ( & ) t h e mapping f rom A t o 1, de f i ned by

o(ai) = j i f and o n l y i f aie x j '

The mapping

mapping o assoc ia ted w i t h an h - regu la r p a r t i t i o n .d o f A , we rep lace each element x o f F by ~ ( x ) and o b t a i n an F-square o f type a square the descendant o f F by o and denote i t by o(F) .

OBSERVAT 10'1

I f F i s an F-square over t h e s e t A and @ i s t h e canon ica l mapping assoc ia ted w i t h a n 1 - regu la r p a r t i t i o n o f A we can i d e n t i f y @(F) w i t h F . PROPOSITION 2.1. L e t F1 , F2 be F-squares o f type (n,x,) and (n,h2) r e s p e c t i v e l y ,

de f i ned on the s e t s Al and A2 ; l e t o1 be t h e canon ica l mapping assoc ia ted

w i t h an h l - regu la r p a r t i t i o n dl = I X1 ,X2, , , . ,Xr l o f A1 and l e t o2 be the

canon ica l mapping assoc ia ted w i t h an

o f A2 ; i f F1 1 F2 then o,(F1) 1 a2(F2) . PROOF. L e t us observe f i r s t o f a l l t h a t t h e s e t s

..., s ) c o n s t i t u t e a p a r t i t i o n

element (a ,b )cX i x Y . appears ~~h~ t imes i n t h e super impos i t i on o f F1 w i t h

F2 t h e p a i r ( i y j ) 6 1, x Is appears i 1 x 2 hlh2 t imes i n t h e super impos i t i on o f

o w i l l be c a l l e d t h e canon ica l mapping assoc ia ted w i t h t h e p a r t i t i o n Sg . Given an F-square F o f t y p e (n,h) d e f i n e d on a s e t A and t h e c a n o n i c a l

such (n,ih).We s h a l l c a l l

y S } h2 - regu la r p a r t i t i o n d2 = { Y1 ,Y2,...,

Xi x V ( i=1,2, ..., r ; j=1,2, j

hlh2 - r e g u l a r o f A1 x A2 ; f u r t h e r , s ince each

J

ol(F1) w i t h e2(F2) and t h a t proves t h e a s s e r t i o n .

PROPOSITION 2.2. Given an or thogona l system .F o f F-squares, i f one rep laces one p a r t i c u l a r square o f 9 by a s e t o f descendants which a r e p a i r w i s e or thogona l , one obtains an or thogona l system again.

PROOF. I t f o l l o w s immedia te ly f rom P r o p o s i t i o n 2.1. and f rom t h e obse rva t i on preced ing i t .

PROPOSITION 2.3. L e t F be an F-square o f type (n,x) de f i ned on a s e t A ; l e t Q be t h e canon ica l mapping assoc ia ted w i t h an h - regu la r p a r t i t i o n d = { X1,X2,

..., X r } o f A and l e t YJ be the canon ica l mapping assoc ia ted w i t h a k - regu la r

p a r t i t i o n B = { Y l , Y 2 , ..., Y s 1 o f A ; then i f t h e c o n d i t i o n

( I ) lxin y j I = ?-- - ( i=1 ,2 ,..., r ; j=1,2 ,..., s ) r

holds, t hen o (F ) - I Y(F) and converse ly .

PROOF. Superimposing @(F) and Y(F) each p a i r ( i 7 j ) e 1, x Is appears A I X i n V . 1

t imes i n each row and hence nAlXi n Y . 1 J

t imes a l t o g e t h e r ; i t f o l l o w s t h a t J

Sets of Painvise Orthogonal F-Squares 287

Q(F) Y ( F ) i f and o n l y i f the c a r d i n a l i t y o f t h e se ts Xi n Y i s t h e same j

f o r each i=l ,Z, . . .,r ; j=1,2,. ..,s and t h a t proves t h e a s s e r t i o n .

Two mappings Q and 'Y s a t i s f y i n g t h e hypotheses o f P r o p o s i t i o n 2.3. t oge the r w i t h t h e c o n d i t i o n ( I ) w i l l be c a l l e d or thogona l and denoted b r i e f l y by Q _L Y I

PROPOSITION 2.4. L e t (G, t ) be an a b e l i a n group; l e t H, K be sobgroups o f G and Q , Y t h e canon ica l mappings o f G on to t h e q u o t i e n t groups G/H ang G/K r e s p e c t i v e l y . I f

(11) H t K I X t y : X E H , y 6 K = G 1

then Q 1 Y . PROOF. S ince H n K i s a subgroup o f G , i t s u f f i c i e s t o p rove t h a t f o r each X E G / H and f o r each Y a G / K t h e c o n d i t i o n X n Y E G / H . n K ho lds . I n f a c t i f X = a t H and Y = b t K , then i t f o l l o w s f rom (11 ) t h a t X = y t H f o r some element y c K and Y = x t K f o r some element x e H . I f ZE X n Y then we have

2 y t x ' = x t y '

w i t h

X ' - x = y ' - Y E H f l K . Hence

z = ( X t y ' - y t y ) E (X t y ) t H f I K

and t h a t means

X A Y c _ ( x t y ) t H l l K . S i m i l a r l y one shows t h a t ( x t y ) t H A K S X n Y and t h a t proves t h e a s s e r t i o n .

PROPOSITION 2.5. For each pr ime power q and f o r each F-square F o f t ype (n,A)

de f i ned on a s e t A o f c a r d i n a l i t y m = q k , t h e r e e x i s t s r = (qk - l ) / ( q - 1 )

p a i r w i s e or thogona l descendants o f F of t y p e (n,Aqk-l) d e f i n e d on t h e s e t I

PROOF. L e t us cons ide r a k-dimensional v e c t o r space V over GF(q); f o r any two d i s t i n c t (k -1 ) -d imens iona l v e c t o r subspaces W1 , W 2 o f V we have W1 @ W2 = V .

Since t h e r e a r e r = (qk - l ) / ( q - 1) d i s t i n c t (k -1 ) -d imens iona l v e c t o r subspaces o f V , i t f o l l o w s f rom t h e p rev ious p r o p o s i t i o n t h a t F admits r p a i r w i s e

or thogona l descendants o f t ype (n,Aq ) .

PROPOSITION 2.6. L e t 9 be a complete system o f t F-squares o f o r d e r n ; i f

one o f t h e squares of t h e system i s de f ined ove r a c a r d i n a l i t y m = qk f o r a

pr ime power q , then t h e r e e x i s t s a complete system o f s = t - 1 t ( q - l ) / ( q - 1 ) F-squares o f o rde r n ,

PROOF, An immediate consequence o f P r o p o s i t i o n 2.2. and 2 .5 .

q '

k -1

k

The P r o p o s i t i o n 2.6. permi ts , i n a s imp le way, t o c o n s t r u c t complete systems e i t h e r o f known t ype o r w i t h a v a r i a b l e number o f symbols. For i s t a n c e we can o b t a i n complete systems o f bo th types by r e p l a c i n g one o r more l a t i n squares, w r i t t e n i n t h e f i r s t column o f t h e f o l l o w i n g t a b l e , w i t h t h e r e s p e c t i v e descendant w r i t t e n beside each o f them.

288 P. Lancellotti und C. Pellegrino

0 1 2 3 0 0 1 1 0 1 0 1 0 1 1 0 1 0 3 2 0 0 1 1 1 0 1 0 1 0 0 1 2 3 0 1 1 1 0 0 0 1 0 1 1 0 0 1 3 2 1 0 1 1 0 0 1 0 1 0 0 1 1 0

0 2 3 1 0 1 1 0 0 0 1 1 0 1 0 1 1 3 2 0 0 1 1 0 1 1 0 0 1 0 1 0 2 0 1 3 1 0 0 1 0 0 1 1 1 0 1 0 3 1 0 2 1 0 0 1 1 1 0 0 0 1 0 1

0 3 1 2 0 1 0 1 0 1 1 0 0 0 1 1 1 2 0 3 0 1 0 1 1 0 0 1 1 1 0 0 2 1 3 0 1 0 1 0 0 1 1 0 1 1 0 0 3 0 2 1 1 0 1 0 1 0 0 1 0 0 1 1

3 - EXTENSION OF THE MACNEISH THEOREM

MacNeish i n 1922 proved the fo l l ow ing theorem

THEOREM 3.1. I f we l e t the prime decomposition o f a number n be

( c f . [ 4 ] ) :

e el e2 el e2

n = p p2 ... pmem then there e x i s t s a se t o f min(pl ,p2 , ... ,p,

pa i rwise orthogonal l a t i n squares o f order n .

3.P.Mandeli and W.T.Federer i n 1983 orthogonal systems o f F-squares o f composite order n w i t h a va r iab le number o f symbols, using a technique which i s an extension o f the MacNeish theorem f o r l a t i n

squares. If n = p1 p2 ... p, defined as fo l lows:

( c f . [ 5 ] ) gave a method f o r const ruct ing

e then the system consis ts o f the F-squares el e2

L i l 8 LA c ... c LJ 2 'm

j = 1, 2, ... nl-1

j = n 1' 1 2 n tl, ..., n -1 0 0 LJ c ... B LJ "1 "m

i t 1 - l j = n n.+l,...,n On1n2.. .nmm-l B LAitl 0 ... c LJ "m i' i

On1n2., .nmm-l c LJ j = nmm-lynmm-,tly ..., n m -1 ; "m

n.-1 we have denoted by Yi = Lni Lni , ... L '

"i a complete system o f l a t i n 1 2

Sets of Pairwise Orthogonal F-Squares 289

ei squares o f o r d e r ni := pi , by On t h e square m a t r i x o f o r d e r n w i t h a l l

e n t r i e s equal t o 0 and by @ t h e d i r e c t sum of ma t r i ces . By P r o p o s i t i o n 2.6. i t e .

1 i s p o s s i b l e t o rep lace each square o f o r d e r pi w i t h a system o f t

e . el F-squares. I f s denotes t h e g r e a t e s t o f t h e number si=(pi '-p, ) (

i=2,3, ..., m we can then s t a t e t h e f o l l o w i n g Theorem

THEOREM 3.2. I f t h e pr ime decomposi t ion o f a number n i s q i v e n by e el e2 ( p1 < p2 < ... < p, rn ) then t h e r e e x i s t s a system el e2 em

n = p 1 P2 * * . P,

o f t = s + p 1 F-squares of o r d e r n w i t h a v a r i a b l e number o f symbols;

such system con ta ins t h e system o f p a i r w i s e or thogona l l a t i n squares cons t ruc ted by MacNeish.

EXAMPLE. If n = 23-33-31 we c o n s t r u c t a system o f 76 F-squares which con ta ins :

a ) t h e system o f 7 p a i r w i s e or thogona l l a t i n squares cons t ruc ted by MacNeish;

b ) 23 F-squares o f t y p e (n,2 . 3 ) ;

c ) 46 F-squares o f t y p e (n,2 - 3 - 3 1 ) .

1

3 2

3 2

ACKNOWLEDGEMENTS. Work done w i t h i n t h e sphere o f GNSAGA o f CNR, p a r t i a l l y supported by M P I .

REFERENCES

J.P.Mandeli, F.C.H.Lee, W.T.Federer, On t h e c o n s t r u c t i o n o f o r thogona l F-squares of orde r n f rom an or thogona l a r r a y (n,k,s,2) and an OL(s,t) s e t ; J . S t a t i s t . P lan . In fe rence 5 (1981) 267-272.

A.Hedayat, D.Raghavarao, E.Seiden, F u r t h e r c o n t r i b u t i o n s t o t h e theo ry o f F-squares des ign ; Ann. S t a t i s t . 3 (1975) 712-716 . W.T.Federer, On t h e ex i s tence and c o n s t r u c t i o n o f a complete s e t o f o r thogona l F(4 t ;2 t ,Z t ) -squares des ign ; Ann .S ta t i s t . 3 (1977) 561-564 . H.F.MacNeish, E u l e r ' s squares ; Ann. Math. 23 (1922) 221-227 . J.P.Mandeli, W.T.Federer, An ex tens ion o f MacNeish's Theorem t o t h e c o n s t r u c t i o n o f s e t s o f p a i r w i s e or thogona l F-squares o f composi te o rder ; U t i l . Math, 24 (1983) 87-96 .

Annals of Discrete Mathematics 30 (1986) 291 -296 0 Elsevier Science Publishers B.V. (North-Holland) 29 1

RIGHT S-n-PARTITIONS OF A GROUP AND REPRESENTATION OF GEOMETRICAL SPACES OF TYPE "n-STEINER"

Domenico Lenzi Dipartimrnto di Matematica

Universith degli Studi LECCE (Italy)

SUMMARY. In [9] we gave a generalization of thr notions of

subgroup and of S-partition of a group, in order to

obtain a description of all the linear spaces having

a transitive group of collineations, by means of a method

that generalizes some methods used in particular cases l i k e transitivr projective planes, transitive linear spaces with

a parallelism (in the sense of AndrB) and others (see

bibl.iography) . Hrre, after some considerations on gromrtrical spaces,

I V O rxtrnd our method, by means of the concept of right

S-n-partition of a group ( s e e def.6), in order t o obtain a

description of all geometrical spaces (E,Q), where &jcz,y(E),

of type "n-Steiner" (i.e.: if P1 , . . . , Pn+l arr n arbitraty

and pairwisr different elrmrnts of E, then there is a unique Be%? such that {P1,...,Pn+l}c B) having a transitive group of automorphisms.

N.1.PRELIMINARIES AND RECALLS. Let G be a group and S a subgroup

of G . Wr shall say that a subset Q of q ( G ) is a partial right

S-covering of G if the following properties hold:

REMARK 1. Onr can immediately verify that:

(a) Sa c A , for every AeB. lndeed S c Aa-l,by (2) and(1). (b) Let I:=AzBA. Then, by ( 2 ) , A1-l 2 I for every

AeB and i e I ; thus A 2 I 1 and hence I is a

submonoid of G . This ensurrs that t h e set

of t h e subgroups of G included in I has a maximum

292 D. Lenzi

element H = {ieG: Ii=I } = {ieG: iI=I} = { ieG;VAeQ ,iA=A}. 1

(c) If we set &G:={Ag:AeQ , geG1, then the group of the right translations determined by the elements of

G is a group of automorphisms of the geometrical

space ( G , 0- G ) . Moreover Q coincides with the

set of the elements BE& G such that leB;

as a consequence, for xeG and BeQG, xeB if

and only if Bx-le8.

For every A,A'eQ , let us set AiA' when A'=Aa-', where a is a suitable element of A. In such a

(d)

manner we define an equivalence relation on & .

Now. for an arbitrary right S-covering &of G , let: E:=ISx:xeC}

(for Be(tG) and W : = { : B e Q G I .

If we set (Sx)*y=S(xy) (for every x,yeC). then G becomes a

transitive generalized automorphism group of the geometrical

space (E,g)(i.e.: the function that associates to every geG

the permutation 8 acting on E like g is an homomorphism from G

(the set of the right cosets of S in G ) ; : ={Sx:Sx B}

into the automorphism group of ( E , O ) ) .

Conversely. every geometrical space (E' a') having a transitive

generalized automorphism group G can be obtained (but for

an isomorphism) in the previous manner. Let indeed P be a

fixed element of E' and S:= {geG:Pg=P1 (the stabilizer of P in C); if we associate to every Q=PgeE' the right coset Sg.

then we obtain a bijective function f from E' onto the set E * of the right cosets of S in G ; as a consequence we can construct

a geomrtrical space (E*, a * ) (by setting g*:= W'f): furthermore

G becomes in a natural way (by setting (Sx)*y:=(Sx)f-'yf=S(xy))

a transitive generalized automorphism group of (E*, a*), Now let us B; : = s x y B ~ S ~ (for every B*e W*) and : = { B* :leBL} . It is easy to verify that Q is a partial right S-covering of G

and (E*,B*) can be obtained from Q. in the previous manner.

If ( E . 9 ) is a geometrical space let u s set, for PeE. Wp=IBeg:PeBl B(the closure of P in W ) , Then (E,g) is said to be

and = a To space when, for every X,YeE, x=y implies that X = Y ; moreover,

( E . B ) is said to be a T1 space when. f o r every XeE,IX(=l.

REMARK 2. Obviously, if ( E , B ) has a transitive generalized

automorphism group and P is a fixed element of E , then (E,g)

U

Right S-n-Partitions of a Group 293

is a To space if and only if, for every X e E , x=P implies that

X=P; moreover, (E,.B) is a TI space if and only if P = { PI.

Now let us fix a subgroup S and a right S-covering (Z of G.

The following propositions hold.

PROPOSITION 3 . For every geG, A70.(Ag)=Apo-A if and only if geHl

( s e e (b) in remark 1).

PROOF. T h e assert is obvious, since ,4za(Ag)=(A2fiA)g= Ig.

PROPOSITION 4. Let (E,.%?) be the geometrical space associated to

Q with respect to S ; then:

Q.E.D.

i) ( E L B ) is a TI space if and only if S= n A .

ii) ( E 3 ) is a To space if and only if S=H1 (see (b) in

A€&

remark 1).

PROOF. i) It is obvious. ii) For every e 3 , Sge if

and only if geB; then the closure of Sg in B is the stit of

the right cosets o f S included in ge'&B B= A2m,(Ag). As a

consequence, is equal to the closure of S in if and only

if (Ag) = Apa A ; whence the thesis by proposition 3 and

remark 2.

Q.E.D.

In [9] ( s e e thror. 9 ) we proved that a group G is a transitive

generalized group of automorphism (collineation) of a linear

space if and only if it admits a right generalized S-partition.

We can say that a subset of 9 ( G ) is a right generalized

S-partition of G if it is a partial right S-covering of G 'and the following properties hold:

(3) AYaA = G ;

(4) VA1,A2eQ: Al#A2 a A l n A 2 = S ; ( 5 ) w a .

By (1),(2) and (4) it is easy to verify that, for every element A

of a right generalized S-partition 0 of G , the following property

holds (cfr. 191. def.2):

(6) S c A and (Vx,yeA:A~-~#Ay-l = Ax-l n Ay-l=S).

Wr observe that the foregoing conditions can be reformuled, by

virtue of the following

PROPOSITION 5 . Let Q c 9 ( G ) and I:= Aga A; furthermore let

294 D. Lenzi

p r o p e r t y ( 2 ) h o l d , a n d :

( 4 ' ) V A 1 , A 2 e C 2 : A1 # A 2 = A 1 n A 2 = I .

Then I i s a s u b g r o u p o f G.

PROOF. We c a n s u p p o s e 101 # 1. Now l e t A1 .A2 d i f f e r e n t e l e m e n t s

o f Q . T h e n , f o r e v e r y i , j e I , i j eAl j nA2j . B u t A , j - ' # A 2 J ,

h t ~ n c t ~ ( b y ( 2 ) and ( 4 ' ) ) A 1 j Q . E . D .

N . 2 . THE CASE OF THE GEOMETRICAL SPACES OF TYPE "n-STEINER". Le t u s g i v e t h e f o l l o w i n g

DEFINITION 6 . We s h a l l s a y t h a t a s u b s e t fl+ o f t h r p o w e r se t ? ( G ) o f < I g r o u p 6 i s a r i g h t S - n - p a r t i t i o n o l G i f i t i s ;1 p c i r t i a l r i g h t S - c o v e r i n g s u c h t h a t t h e f o l l o w i n g p r o p e r t i e s h o l d :

-1 -1 -1 -1

-1 n A Z j - ' = I , t h e r e f o r e i j - l e I .

( 7 ) Vgl , . . . , g n e G ] A € & : { g l , . . . , 9 , ) s A . ( 8 ) VA1,A2e& : A1 # A 2 - A1 n A 2 i s tht . u n i o n

o f m 5 n r i g h t c o s e t s o f S i n G ( w h r r e m i s d r p e n - d i n g on A1 a n d A 2 ) .

I t i s e a s y t o v e r i f y t h a t i f ( E ' , B ' ) i s a g e o m e t r i c a l s p a c e o f t y p e " n - S t r i n e r " ( s e e summary) h a v i n g a t r a n s i t i v e g e n e r a l i z r d a u t o m o r p h i s m g r o u p G , t h e n t h e s e t I'lc P(C) f r o m w h i c h ( E ' , 2 8 ' ) c a n b r o b t a i n e d (*I i s a r i g h t S - n - p a r t i t i o n o f G .

C o n v r r s r l y , l e t ('I b e a r i g h t S - n - p a r t i t i o n o f G; t h e n t h e a s s o c i a t e d g r o m r t r i c a l s p a c e ( E , % ) ( s r e N . l ) i s o f t y p e " n - S t r i n e r " .

T h i s I S a n i m m e d i a t e c o n s e q u e n c e o f t h e f o l l o w i n g

PROPOSITION 7 . I f Sh l , . . . , S h n t l a r e n a r b i t r a r y a n d p a i r w i s e d i f f e r e n t r i g h t c o s e t s o f S i n G , t h e n t h e r e i s a u n i q u e s u b s r t

B o f G o f t y p e Ag , w i t h AeQ a n d geG, s u c h t h a t lil S h l c B . n

-1 -1 PKOOF. By ( 7 ) a n d ( l ) , t h e r e i s AeQ s u c h

- c A , h e n c e I h l , . . . , 1 ~ ~ ~ ~ 1 ~ , t h u s U S h l 4 Ahntl ( s i n c e

SA=A).

On t h e o t h e r h a n d s , n + l

g l , g 2 e G ) t h e n S l!lShlh-:tl 4. Alglh, i ln A2g2h,:l. As a c o n s e q u e n c e

( s e e t h e s e c o n d p a r t o f ( c ) i n remark 1 ) A lg lhn+ l=AZg2hn t l

by v i r t u e o f (8), a n d h r n c e A1g1=A2g2.

( ) C u t f o r a n i s o m o r p h i s m ; s r r N . l .

t h a t { h l h n + l . . . . , h n h n t l , l ) c n + l

i f Algl n A2g22Y# S h l ( w i t h A l , A 2 e @ a n d

-1 -1

* Q . E . D .

Right S-ri-Partitions of a Group 295

REMARK 8 . W r o b s r r v r t h a t i f i s a r i g h t S - n - p a r t i t i o n o f G ,

t h e n ( b y (8 ) and (2)) f o r e v e r y AeQ t h e f o l l o w i n g p r o p e r t y h o l d s :

( 9 ) V x , y e A : A ~ - ~ # A y - l - A x - ' n Ay-I i s t h e i in ion of m - < n r i g h t c o s e t s of S i n G ( w h r r r m i s d e p e n d i n g on A) .

Wr s h a l l s ay t h a t a s u b s r t A of C i s a r i g h t S - n - b l o c k o f G i f S c A and p r o p e r t y ( 9 ) h o l d s ( * * ) ; m o r e o v e r i f A i s a r i g h t S - n - b l o c k o f G b u t n o t a r i g h t S - ( n - 1 ) - b l o c k , we s h a l l s a y t h a t A i s a p r o p e r r i g h t S - n - b l o c k o f G .

O b v i o u s l y , e v e r y r i g h t S - o - b l o c k i s a s u b g r o u p of G .

PROPOSITION 9 . A n e c e s s a r y ( a n d t r i v i a l l y s u f f i c i e n t ) c o n d i t i o n f o r a s u b s e t A of G t o b r a r i g h t S - n - b l o c k o f G i s t h a t :

( 1 0 ) S c A ,

( 1 1 ) Vgl.g2 e G : Agl # Ag2 - Agl n Ag2 i s t h e and

u n i o n o f m 5 n r i g h t c o s e t s o f S i n G .

PROOF. I t i s enough t o p r o v e t h a t t h e c o n d i t i o n ( 1 1 ) i s n e c e s s a r y . H r n c r l r t A b r a r i g h t S - n - b l o c k o f G , l e t Agl#Ag2 ( w h e r e

gl .g2eG) and l e t t e A g l g i l n A . Then A g , g , ' t - l # A t - l and t g 2 g i 1

e A n Ag2g;t C o n s e q u e n t l y , by ( 9 1 , Aglg2 -' t -' n A t - ' ( a n d h r n c r a l s o Agl n AgZ) i s t h e u n i o n of m 5 n r i g h t c o s e t s o f S i n G .

We c o n c l u d e by o b s e r v i n g t h a t e v e r y r i g h t S - n - b l o c k A of G d r t r r m i n r s i n a n a t u r a l manner a r i g h t S - n - p a r t i t i o n o f G . I n f a c t o n e c a n c o n s i d e r t h e s e t 0. : = Q1 1 . ~ Q 2 , w h e r e f f l =

= { A1 e b ( G ) i s t h e c l a s s of t h r s u b s e t A 2 o f G s u c h t h a t :

e

Q . E . D .

: A1 = Aa-', w i t h a e A } and C 2

1) s A z ; j j ) A 2 i s t h r u n i o n of n + l p a i r w i s e

I(il : A 2 L A1.

d i f f e r e n t r i g h t c o s r t s o f S i n G ;

j j j ) V A 1 e

C o n s r q u r n t l y , f o r r v r r y g r o u p C and f o r r v r r y s u b g r o u p S o f G ,

(**) I n 191 and 1101 w r c a l l e d r i g h t S - b l o c k s t h e r i g h t S - 1 - b l o c k s ( c f r . t h r p r r v i o u s p r o p e r t y ( 6 ) ) and p r o v e d s e v e r a l a l g r b r a i c a l and g r o m r t r i c a l p r o p r r t i r s .

296 D. Lerizi

wr can have several (trivial) geometrical spaces of typr

n-Strinrr, since if m 5 n+l then the union A of m pairwisr

different right cosrts of S in G such that S 4 A is a right

S-n-block of G .

B I B L I OG RA PHY

1. 11 Andre' J., Ubrr Parallrlstrukturrn , Tril I , Tril 11, Tril 111, Tril IV, Math. Z.E(lY61). 85-102,155-163,240-256,

[2] Biliotti.M., S-spazi ed 0-partizioni. Boll. U.M.I.(5)

[3] mliotti,M., Sullr strutturr di traslazionr, Boll.U.M.1.

1_4] Biliotti, M., Strutturr di Andre rd S-spazi con traslazioni, Grom. Drdicata, lo (1981), 113,128.

151 Biliotti, M., Strutturr di Andre rd S-spazi con traslazioni 11, Ann. di Mat. pura ed appl., (IV), Vol. CXXXV.(1983),

1-61 Biliotti, M., Hrrzrr A., Zur Gromrtrir drr Translationsstrukturrn mit rigrntlichrn Dilatationen, Abh. Math.Sem.Hamburg 53

1:7] Karzrl, H., Bericht uber projrktive Inzidrnzgruppen, Jbrr. Drutsch.Math. Vrrein. E(1964) 58-92.

[ S J Lingenberg,R.. Uebrr Gruppen projectivrr Kollinrationrn, wrlchr rine prrsprktive Dualitat invariant lassen, Archiv. der Math. E(lY62) 385-400.

191 Lrnzi, D., Rrprrsrntation of a linear space with a transitive group of coll inrat ions by a general izrd S-part it ion of a group, Atti Convrgno "Grom.combin." La Mrndola-Italy

1,lOJLrnzi , D . , On a charactrrization of finitr projrctive planes

llllScarsrllj, A. , Sullr S-partizioni rrgolari di un gruppo finito, Rrnd.

11121 Zappa,G., Sui piani grafici finiti transitivi e quasi transiti-

1-13.1 Zappa,G., Sugli spazi grnrrali quasi di traslazionr, Le Matematichr,

11141 Zappa, G.. Sulle S-partizioni di un gruppo finito, Ann.di Mat.2, (1966).

1151Zappa, C . , Partizioni ed S-partizioni dri gruppi finiti,

..

311-333.

14-A(1977), 333-342.

(5) G-A(1978), 667,677.

151-172.

(1983). 1-27.

(1982), 471-482.

having a transitive group of collinrations, to apprar.

Acc.Naz.Lincri, LXII (1977). 300-304.

vi, Ricrrchr di Matematica,II (lY53), 274-287.

19 (1Y64), 127-143.

1-24.

Symposia Math. I(196Y), 8 5 - 9 4 .

Annals of Discrete Mathematics 30 (1986) 297-302 0 Elsevier Science Publisliers B.V. (North-Holland) 297

ON BLOCK S H A R I N G S T E I N E R QUADRUPLE SYSTEMS

Giovanni LO FAR0 ("1

Dipartimento di Matematica dell'Universitl, Via C. Battisti 90, 98100 MESSINA, Italy

A b s t r a c t ,

We detarmine, f o r a l l u :4 or 8 (mod. 121 , the se t of a l l those f o r which there exist two S t e i n e r quadruple systems of or- numbers

der u on the same s e t that share exac t ly k bZocks. k

Introduction.

A Steiner quadruple system (SQS) is a pair (Q,qJ where Q is a finite

set and (I is a collection of four element subsets of Q (called blocks) s u c h

that every three element subset of Q belongs to exactly one block of q . The number IQI = v is called the order of the SQS((Q,q! and u E 2 or 4

(mod. 61 is an obvious necessary existence condition for an SQS of order u

( S Q S l v ) ) . On the other hand, in 1960 Hanani proved 181 that this condition is also

sufficient. Therefore in saying that a certain property concerning SQS(v) is

true for all u it is understood that u -2 or 4 (mod. 6) . If iQ,ql is an SQSiu) then 141 = 7, =uiv-l)(U-2)/24 . A quadruple system (Q,qj has a proper subsystem if there exist sets R C Q

and rcu such that ( R , r ) is an SQS with Irl < 141 . A . Hartman 191 proved that for every v L 1 6 there exists an SQSlV) with a

proper subsystem of order 8 . Earlier results on subsystems of SQSs

In their excellent survey of Steiner quadruple systems, C . C . Lindner and A.

can be found in 131.

Rosa 1111 rise a series of questions. One of them is: "For a given

k 2 q v is it possible to construct a pair of SQS(ujs having exactly k blocks

in common?".

u , for which

Denote by J ( v ) = { k : j SQSSIV) (Q,qll, (Q,q2) such that 1q1nq21 = k ) ;

I ~ v ) = IO, I, 2,. . . ,qv-14,qu-12,qv-8,qu} , v '8 .

298 G. Lo Far0

So the author's best knowledge, the only results concerning this problem are:

(i) J ( 4 ) = I ; J ( 8 ) = { 0 , 2 , 6 , 1 4 } ; J f l O ) ={0 ,2 ,4 ,6 ,8 ,12 ,14 ,301;

J ( 1 4 ) - 3 1 ( 1 4 ) - (48,50,52,57,58,60,62,63,. . ., 77,?9,83} ;

{ q - 8 = 8 3 ; q 1 4 - 1 2 = 7 9 ; q 1 4 - 1 4 = 7 7 -15 = 7 6 I n J ( 1 4 1 =0 14 " 1 4

(ii) J ( u ) E I ( u ) , for all 028

(iii) J ( V ~ = ~ ( u i , for all u = ~ n+l n

; 5 - 2 ; 7 . 2 n , n 2 2 . The aim of this paper is the investigation of the block intersection problem

for SQSs of order 2u obtained by doubling suitable systems of order 2, . In particular it is obtained that J l Z v ) = 1 ( 2 u l .

Although he was not able to prove it yet, the author feels that the following

conjecture makes sense.

Conjecture: J ( U ) = I ( v l , for all v 2 1 6 .

2. Prel i minaries .

In this section we describe two constructions for quadruple systems o r order

2u which are the main tool used in what follows.

C o n s t r u c t i o n A (well known e.g. see 1111).

Let (X,al and tY,b) be any two SQSS(vl with X n Y = B . Let F = { F F

..., F 3 and G = I G 7, G ,..., G v - l } be any two I-factorizations of K (the u-1 U

complete graph on vertices) on X and Y respectively, and let be any

permutation on the set

I' 2' ' ' *

{7,2 ,..., u-21 . Define a collection s of blocks of S = X u Y , as follows:

(1) Any block belonging to a or b belongs to s ;

( 2 ) If x ,x E X (x #x ) and y l , y 2 E Y ( y l # y 2 ) then fx 3: ,y ,y )ES if 1 2 1 2 1 ' 2 7 2

and only if {x ,x ) € F i , { y l , y Z ~ E G j and a ( < ) =j . 1 2

It is a routine matter to check that (S,sl is an SQSf2uI , We will denotc

iS,sl by l X u Y l Ia,b,F,G,al ,

C o n s t r u c t i o n B (compare 13 1 ) . Let iQ,ql be an SQSlu) , Q' a finite set such that IQI = I & ' \ , Q n Q ' = @

and let cp be a bijection from Q onto Q r with x'.=cp(x.) , for every ziEQ . z

Block Sliaririg Steiner Quadruple Systems 299

Obviously, ( Q ' , q ' ) is an SQS(v) (the S@S(v) obtained from (Q,q.1) where

q ' = V ( q ) ={vie/ : e c q l . If q l c q we define a collection p(ql) of blocks of P=QuQ' as follows:

(1) Any block belonging to q1 or 4; (=cp(q I) belongs to p(qlI ; 1

( 2 ) I (z ,xq, xi, x i ) ; (x , t , xL,x'); ( x , x , x ',a- ' ); ( 2 ,z ,x 1, x 1); (x

( 3 ) { i s .x ,x7,x'); ix ,x9,,x ,x'I; ix ,x .x ,xLI; (x ,x .x .XI); (x',x',x',J: I ;

z ,x ',x 'I; 1 A 1 3 d 4 1 4 2 3 2 3 1 4 2 , 4 1 3

(s,,x u 4 1 2 , . r ' , . r ' l ~ c p ( q 1 ) if and only if (x*,x2,x3,x41 '? I' '

1 2 2 4 1 - 4 3 1 3 4 6 2 3 4 1 1 2 3 4 (x',x',x',x J.(x',x',x' 3: );(x',x',x',x I } c p ( q ) 1 2 4 3 " 1 3 4 ' 2 2 3 4 1 1

1 2 - 4

1 1 2 1

if and only if

(3: ,x , x 7 , x I E q-q1 ;

( 4 ) i x , . ~ ~ , x ' , x ' i t p ( q ) , for every x l , x 2 t Q , x l # x 2 . It is a routine matter to check that ( P , p ( q )I is an SQS(2vl . We will de-

1 note (P,p(ailI by ( ( Q u Q ' l , ( q , q l ) ) . If q i = O then we denote (P,p(@II by

iP,pI .

3 . SQS(2vls w i t h blocks i n common.

In this section we will determine J ( 2 v ) . Take X = { Z , 2 ,..., v} , v an even positive integer, and

two 2-factorizations of X where F = {FlJF2,. . .aFv-Il and

We will say that F and G have k edges in common if and

In 1121 C.C. Lindner and W.D. Wallis gave a complete solution to the interse5

tion problem for I-factorizations. In particular, they showed that for any v - > 8

there exist two I-factorizations F and G with k edges in common for every

v iv-1 I k E i 0 , 1 , 2 , .... -- - n} - in-l,n-2,n-3,n-51 . Take Q = { I , Z ,..., v} and Q' = { 7 ' , 2 ' ,..., 0 ' 1 with Q n Q ' = 0 and let

F = { F F ,..., F 1 be 1-factorization of 4 , then F ' = { F ' F' 2J...J~i-ll is a

I-factorization of Q' (called the 1-factorization derived from F) where

{ x i , x i } E F ' . if and only if {xi,x ) E F .

v- 1

3 h i

v (v-1) 2

THEOREM 3 . 1 . Assme ~ 2 8 . If X = { O , 1 , 2 ,..., -=nl--in--Ian-2,n-3,n-5I

then H C J(2v) .

Proof. Let iQ,q) be an SQSiv) . If k € H then there exist two l-factoriza-

300 G. Lo Faro

I f e i s the i d e n t i t y permuta t ion , i t i s a r o u t i n e ma t t e r t o s ee t h a t

[ Q U Q ' I [q,q',F,G',e] and ((Qu&!'l,(q,@)) a r e two SQS(2v)s wi th e x a c t l y k

blocks i n common. The statement fo l lows .

I t i s wel l known 111 t h a t i f 1 and y a r e even p o s i t i v e i n t e g e r s and

32'2~ , then the re e x i s t s a 1 - f a c t o r i z a t i o n of o rde r z con ta in ing a sub-

1 - f a c t o r i z a t i o n of o r d e r y ( a 2 - f ac to r i za t ion of o rde r n i s a I - f ac to r i za -

t i o n of K ) .

v iv-1) 2 THEOREM 3 . 2 . Let n =- , v - > 16 . I f k E {n-I,n-Z,n-3,n-5,n+l,n+Z,. . .

. . . ,n+98,n+lOO,n+104,n+112) 1 , t hen k E J ( 2 v ) . Proof. L e t (Q,q) be an SQS(v) con ta in ing a n SQS(8) (R,rJ as a subsystem.

It i s s t r a igh t fo rward t o see t h a t t h e SQS(2vl ((QuQ'), (q,0)) = ( P , p ) con-

t a i n s t h e SQSS(16) ( ( , ? U R ' ) , (?,'$)) = ( T , z ) . Let F={FIJFg, ... ,F 1 and F " = { F " F" 2,.. ,,F"} be two I - f a c t o r i z a t i o n s

v-2 7 of Q and R r e s p e c t i v e l y wi th F$GPi , f o r eve ry i = l , 2 , , . . , 7 . It is equal-

l y easy t o see t h a t t h e SQS(2vl I Q u Q ' I Iq,q ' ,F,F' , i] = ( P , s ) con ta ins t h e

SQS(l6) I R u R ' I I r , r ' ,F" ,F" ' , i ] = ( T , t ) . Obviously, we have:

Let k E J ( 1 6 ) = 1 ( 1 6 ) . It i s p o s s i b l e t o c o n s t r u c t two SQS(

such t h a t l anb l = k ,

I f f p - z l u a = p ' and I s - t l U b = s ' , then ( P , p ' l and

v (v-1 I 2

SQS(2vls w i th e x a c t l y - -28+k b locks in common.

This completes t h e proof of t h e theorem.,

= 28

6 ) (i",al and (T,b)

( P , s ' ) are two

v Iv-1 ) (v-2) =qv . I f 1 =q -81k-hl then ZEJ12vl .

24 2v LEMMA 3 . 3 . O < h < k

Proof. Let (Q,qi be an SQS(V) and take q l Jq2 wi th q 1 4 q 2 C . q , l q l / = h ,

_ _ _

/ q I = k . Clea r ly , t he number o f d i s t i n c t b locks of (P,p(ql)) and (P,p(qz))

i s : Z(k-hl +6(:c-h) =8(k-h) and then Ip (q11np(q ) I =qgv-8(k-h) . 2

2

REMARK. L e t q = 8 and q2Cq , )q21 = t L q v ; from Lemma 3 . 3 , i t fo l lows 1

Block Sharing Steiner Quadruple Systems 30 1

that q - 8 t ~ J ( d v l , f o r e v e r y i : E { O , 1 , 2 ,..., . 2V

v ( v - 1 ) 2

THEOREM 3.4. Assume

Proof. We s t a r t n o t i c i n g t h a t i f

v , l G . If - + 98 2 <q2v-140 , then l € J ( 2 v ) . qZv-7. =8t+h , h=O,1,2 ,..., 7, t h e n

I7 + 4-71

e - h+2

q -12-- -0 8

Let (Q,qJ be an SQS(v.i c o n t a i n i n g a n SQS(8) (R,ri and l e t r C q l C q w i t h

( q 1 =t+12(<q I . By Lemma 3.3 ( P , p ( r l l and ( P , p ( q I ! a r e two SQS(2u) s such

t h a t I p i r ) n p ( q I 1 =y -8(t-21 . I V I

1 2V

I t i s s t r a i g h t f o r w a r d t h a t b o t h ( P , p ( r ) ) and ( P , p ( q ) ) c o n t a i n t h e 1

SQS(16) ((RuR'), (F,P)) ; t h e r e f o r e -8( t - i?) - (140-k)€ J ( 2 v ) , f o r a11

k E J 1 1 6 ) = I i l 6 1 . qZV

The s t a t e m e n t f o l l o w s by c h o o s i n g k E ~ l 1 7 , 1 1 8 , . . . , 1 2 4 ) C . 1 ~ 1 6 / .. THEOREM 3 .5 . Assume v, lC . I f l E I ( 2 u ) and L F q Z V - l 4 C then l E J ( 2 v ) ,

Proof. I t i s w e l l known 111 t h a t i f t h e r e e x i s t s a n SQS(u! w i t h a sub-SQS(u),

t h e n t h e r e e x i s t s a n SQS(2ul w i t h a sub-SQSS(2u) . Thus, t h e r e e x i s t s a n SQS(2Vj

w i t h a sub-SQS(16) . I t i s e a s y t o check t h a t q -(140-h.)€ J ( 2 v ) , f o r a l l h c J ( 1 6 ) =I(161 . T h i s comple tes t h e proof..

21)

Combining t o g e t h e r ( i i i ) and Theorems 3 .1 , 3 .2 , 3 .4 , 3 . 5 we g e t o u r main

r e s u l t :

THEOREM 3.6. J l v ) = I ( v ) for a l l v E 4 o r 6 (mod 1 2 ) .

REFERENCES

111 A. CRUSE, On embedding incomplete symetric l a t i n squares, J. Comb. Theory

121 J. DOYEN and A . ROSA, An u l ' h t e d bibliography and surV?y of Steiner systems,

S e r . A ( 1 9 7 4 ) , 19-22.

Annals of D i s c r e t e Math. 7 (1980) , 317-349.

131 J. DOYEN and M. VANDENSAVEL, Non isomorphic Steiner quadruple systems, B u l l . SOC. Math. B e l g i q u e 33 (1971), 393-410.

141 M. GIONFRIDDO, On the block intersection problem for Steiner quadruple systems, A r s Combina tor ia 1 5 ( 1 9 8 3 ) , 301-314.

302 G. Lo Far0

151 M. GIONFRIDDO and C.C. L I N D N E R , Construction of S te iner quadruple SySi%ms having a prescribed number o f blocks i n common, Disc re t e Math. 34 ( 1 9 8 1 ) , 31-42.

161 M. GIONFRIDDO and G. LO FARO, Cn S te incr systems Si.?,4,14! , t o ap-

pear .

171 M. GIONFRIDDO and M.C. MARINO, On S te iner Systems S(3,4,20) and S(3 ,4 ,32) , U t i l i t a s Math., 25 ( 1 9 8 4 ) , 331-338.

181 H. H A N A N I , On quadrupie systems, Canad. J. Math. 1 2 ( 1 9 6 0 ) , 145-157.

19 I A. HAR'I'EIAN, Quadruple systems contuining AG/3,2) , D i s c r e t e Math. , 39 (1982) ( 3 ) , 293-299.

1101 C . C . L I N D N E R , On the construct ion of non-isomorphic S t e iner quadrupZe 651-

stems, C o l l o q . Math. 29 ( 1 9 7 4 ) , 303-306.

1111 C . C . LINDNER and A. ROSA, S te iner quadruple systems - n survey, D i s c r e t e Ma- th . 2 1 ( 1 9 7 8 ) , 147-181.

112 1 C . C . LINDNER and W.D. WALLIS, A note on one-factorizat ions having a prescr i -

1131 G. LO FARO, On t h e s e t for S te iner qumdmpZc systems of order v=7-2"

1141 G. LO FARO, S te iner yuadrupie systems having a prescribed nwnber o f quadru-

1151 G. LO FARO and L. PUCCIO, Su l l ' i n s i eme

bed number o f edges i n common, Annals of D i s c r e t e Math. 12 ( 1 9 8 2 ) , 203-209.

J ( v ) w i t h n - > 2 , A r s Combinatoria, 1 7 (1984) , 39-47.

p l e s i n cononon, t o appear.

J ( 1 4 ) d e i sistsmi d i quaterne di S te iner , t o appear .

("1 Lavoro e segu i to ne l l ' a rnbi to d e l GNSAGA e con finanziamento MPI ( 1 9 8 4 , 4 0 % ) .

Annals of Discrete Mathematics 30 (1986) 303-310 0 Elsevier Science Puhlishers B.V (North-Holland) 303

ROOTS OF AFFINE POLYNOMIALS

Giampaolo Men iche t t i (-)

Dipar t imento d i Matematica, Un ive r s i tP d i Bologna, I t a l y

INTRODUCTION. Let F = GF(q) be a Galo is f i e l d of o r d e r q = ph, where p is a pr ime,

and l e t K = GF(qn) be an a l g e b r a i c ex tens ion of a g iven degree n > l . An a f f i n e

pziynon;iaZ (of K[x]over F) i s a polynomial of type

(1 ) P (x ) = L(x) - b , b € K ,

with

n- 1 5 x . u . . i =O

I f a b a s i s {uo,ul, ..., u } of K ove r F i s f i x e d then we can pu t x = n-1 1 1

Hence, t h e de t e rmina t ion of ( even tua l ) r o o t s of t h e polynomial (1 ) i n K can be

reduced t o thc de t e rmina t ion of s o l u t i o n s o f a l i n e a r sys tem of equa t ions i n

inde te rmina te s xi and wi th c o e f f i c i e n t s i n F ( c f . [ l l , Chap.11). This procedure i s ,

however, t ed ious a l s o in t h e most simply cases and does not d e c i s e "a p r i o r i " how

many r o o t s i n K e x i s t .

I n t h i s pape r , we prove t h a t the equa t ion (1) has r o o t s i n K i f and on ly i f

t he fo l lowing sys tem o f l i n e a r equa t ions

= b n-1Yn-1 +.. .+ 1

+. . .+ 1~-2yn- l = bq

loyo + l l Y l

+ l:yl l:-lYO ( 3 )

..................... I . , .

n- 1 n-1 n- 1 1qn-ly0 + 1; y1 +. . . + 1; yn-l = bq

has s o l u t i o n s . Moreover, i f ( 3 ) i s s o l v a b l e and i f r is t h e rank of t h e ma t r i ces

be longing t o ( 3 ) , then qn-' g ives us t h e number of s o l u t i o n s of (1) i n K. Besides

t h i s , we show t h a t t h e r o o t s of (1) a r e e x p r e s s i b l e as f u n c t i o n s of c e r t a i n

s o l u t i o n s of t he l i n e a r sys tem ( 3 ) . In p a r t i c u l a r , t h e ob ta ined r e s u l t s a r e u s e f u l

a l s o i n t h e case t h a t t h e c o e f f i c i e n t s of t h e polynomial (1) are n o t cons t an t

( c f . f . e . [2 ] ).

(-) This r e sea rch was suppor ted i n p a r t by a g r a n t from t h e M.P.I.(40% f u n d s ) .

304 G. Menicketti

1. An n x n matrix of the type

n-1 ... a

- A( a. , al , . . . , an- ) =

is called autocirculant . Each row of A(ao,al,...,an-l) is obtained by the previous

one if we permute the elements under the cyclic permutation ( 0 1. ..n-1) and then

apply the field automorphism a : K -+ K , a I+ aq.

The sum and the product of autocirculant matrices are autocirculant matrices.

In addition, if A is autocirculant, the transpose it is also autocirculant. Let

- T = &(O,l,O, ..., 0 )

and Aq(aO,al ,... ,a ) = A(aq,aq, ..., a:-1). It is easy to verify n-1 - 0 1 Aq= T A T-l , - --- ( 4 )

LEMMA 1. An autocirculant matrix A has rank r, 15rGn-1, i f and only if i t s

f i r s t r FOWS (colwnns) are l i near l y K-independent and i t s (r+l)-th row (column) is

a K-linear cornbination o f t h e preceding POWS (coLwmzs).

Proof. The transpose of an autocirculant matrix is itself autocirculant. Thus

it is sufficient to prove the statement for the columns of A. - The assertion is an obvious consequence of the following observation,

Let A = (%,,A1,...,$I-l). ~f

s-1 (5) --s A = E kiAi , kfK,

i=O

0 < s < n-1, then A' = C kqAq and therefore s-1

-s i=o 1-1 s-1 s-2 s-1 s-1

1=0 i =O = 1 kqii+l + kq 1 k . A . = c k!A. ,k!E K.

&+I=. 1 kqAi+l s-liz0 l-li=o 1-1 1

n-s-1 and use previous 2 3 If we raise both sides of ( 5 ) t o the powers q ,q ,...,q

arguments, we see that the columns &+2, & + 3 , , , , , &-l can be expressed as a

K-linear combination of %, A1 ,..., A .n

Linear system

( 6 )

has so lu t ions of the type 5 = (zo zl,,. z

given A' = A(zo, zl,. . . , z

s PROPOSITION 2 . I$& i s an autocirculant ma t r i x o f rank r then the homogeneous

-- A Y = 0 , y = (Yo Yl Y,-l)t

t

0,. . . , O), wi h m e rank(&')= n-r and LL' t = 2. 0 .., 0) wi th zr # 0 . Furthermore,

r' Proof. If one has A = (A+, 4,. . . , 411) then Lemma 1 guaranties the

n-1 existence of the elements a ! E K such that A = T a!A. holds. And, this proves the

-K i=o 1-1

Roots of Affine Polynomials

f i r s t p a r t of t h e a s s e r t i o n .

305

Let - A t t = (s, A; ,..., A&). By d e f i n i t i o n , one has % = - z and t h e r e f o r e n- 1

-(n-1Izq -1 A'= - T-lzq, - ..., A'-l= r - If we cons ide r t h e q-th power of A?= 0 and t ake i n account a l s o ( 4 ) , w e

deduce 0 =

and have f i n a l l y

= T A A i . Thus h i = 0. Analogously, we prove A A ' = 0,. , . ,A$-l=g, - -2

- A($, hi,. . . , g-l) = Ah' = 0. We deduce, i n p a r t i c u l a r , t h a t eve ry element Ai,, A;, . . . , A ' €Kn i s a

-n- 1 s o l u t i o n of t h e l i n e a r sys tem ( 6 ) and hence , i t fo l lows r ank(A ' t ) ,<n - r .

To see t h e i n e q u a l i t y rank(A' )> ,n- r , w e remember t h a t t h e m a t r i x which ag rees

i n t h e f i r s t n-r rows and las t n-r columns w i t h A' i s non-s ingular .0

LEMMA 3 . T4e elements w .€ K , i = 0,1,. . . , s , are l inear ly F-independent if und

t n n-1 o n l y if the vectoras w . = (wi wq . . . w! ) E K , i = 0 , 1 , . . . ,s, are Zinearly

-1

K-independent . Proof. Let us examine the cond i t ion

S

1 k.w. = 0, kiEK, 1-1

( 7 ) i =O

under t h e hypo thes i s t h a t t h e e lements wi a r e F-independent.

I f w e suppose t h a t a t least one c o e f f i c i e n t k i , f o r example ko, i s not z e r o ,

then w e can de te rmine k E K such t h a t h = kk

ob ta in

t r ( h o ) # 0 ( ') ho lds . From ( 7 ) , w e 0 0'

S

T: h.w. = 0, h . = kk 1-1 - I i '

i = O s j Rais ing t h e l e f t s i d e o f t h i s equa t ion t o t h e powers qJ we o b t a i n ? h: wi = 0,

i =O S

j = 0,1, ..., n-1. I f we add these n expres s ions , w e f i n d C ( t r ( h i ) ) w i = 2; i n

p a r t i c u l a r i =O

S

Y(t r (h i ) )wi= 0, t r ( h o ) # 0, i = O

i n c o n t r a s t w i th the hypo thes i s .

It i s ev iden t how t h e second p a r t of t he t h e s i s may be proved.!

COROLLARY 4 . Y7ie n x n matrix 2 = (%,El ,... ,%-,), li= (ui u: ... uq i

For any polynomial ( 2 ) , t h e s e t

n-1 ) t , i z

nm-singular if and onlg if { u o , ul,. . . , u ~ - ~ } is a bas i s o f the vector F-space K.!

Z(L) = { x € K : L(x) = 01 is obvious ly a v e c t o r subspace of K . Moreover, i f Z(P) = { x E K : P(x) = 01 # d then

(8) Z(P) = xo+ Z ( L ) , x0E Z ( P ) .

n-1 ( l ) t r ( x ) = t r (x) = x + xq + ... + xq , v X E K .

F

306 G. Menichetti

Given an a f f i n c polynomial ( l ) , l e t

- A(P) = G ( L ) : = A(lo , l l , . . . , l n - l ) .

PROPOSITION 5. If rank(A(L)) = r then din$Z(L) = n-r.

Proof. Let {wo,wl, ..., w ] be a b a s i s of Z(L) and l e t V C K n be t h e s o l u t i o n

space of t he homogeneous l i n e a r sys tem

(9) - A(L)JI- = 0, 11. = (yo y1 . . . yn-l)t . n-1

From Lemma 3 , i t fo l lows t h a t the v e c t o r s w.= (w. wq .,. w: ) t , i = O , l ,..., s , -1 1 1

a r e l i n e a r l y K-independent and i t i s e a s i l y v e r i f i e d t h a t each of them i s a

s o l u t i o n of ( 9 ) .

Thus ,

(10) di%Z(L) < di%V = n-r.

Let A' be an a u t o c i r c u l a n t ma t r ix which s a t i s f i e s the cond i t ions

(11)

( c f . Prop .2) and l e t 2 = (%,,ul, ..., u Coro l l . 4 ) .

i \ (L)Af t = - 0, rank(A' ) = n-r

) be an n x n non-singular m a t r i x ( c f . -n-1

From ( l l ) , we deduce

t t - A ( L ) ( A ' g) = 0, rank(&' 2) = n-r.

n-1 With t h e obse rva t ion t h a t Attg = (I&,?; ,..., $I&), u! = (u! u! '... u!'

can conclude t h a t u ! E Z ( L ) , i = O , l , ..., n-1 and d i % = n-r

t COROLLARY 6 . Suppot~e rank(A(L)) = r . Tf - z = ( z o zl...z 0 O...O) is a

sd lu i ion of the Zineura system (9 ) for any choose of t he basis Iuo ,u l , . , , , u

o f t l ie vector F-space K, the eZements

( 1 2 ) x. = z u + 29, us + zq uq + ... + 2; u4 , i = O , l , ..., n-1,

f o m n se t of generators of Z(L). Hence, oiie has

( 1 2 ) ' Z(L) = { x = z k + zq kq + ... + zq kq : kEK}.

1 n-l

n - r n-r n - r+l n - r+ l n-1 n-1 1 O i r-1 1

n-r n-r n-1 n-1

O r 1

Proof. The Coro l l a ry fo l lows from the proof o f t h e prev ious P ropos i t i on i f n-r n-r+l n- 1

one observes t h a t A ' t= A(zo,O ,..., 0,z: .z:-~ ,.... zq ). 0 1 t PROPOSITION 7 . If z = ( z o z l . . . z ~ - ~ ) E K" is a sokction of the linear system

n-1 ( 1 3 ) A(L)y = b, y = ( y o y1 ... = (b bq ... bq I t , then, for every v E K w i t h t r ( v ) # 0 ,

2 2 n-1 n-1 (14) x = (z v + z:-lvq + zq vq +...+ zq vq ) / t r ( v ) 0 0 n-2 1

Roots of Affine Polynomials 307

is a root of the poZynomiaZ ( 1 ) .

Proof . Let A(?) = A ( z o , z l , , s . , z n - l ) . Then

2 n- 1 t T- (n - l ) zq ). - A (5) = (2, -'zq, T 2 z q - ,..., - -

2 n-1 Rais ing L ( L ) z = b t o t he power q , we o b t a i n A q ( L ) z q =bq=(bq bq ... bq b ) t . - -

Using ( 4 ) , we have , t h e r e f o r e , -- T A ( L ) L 1 z q = Lq o r A(L)(X-'L~) = 2. I t e r a t i n g t h i s ,

we f i n d

Thus, i t fo l lows ,

t - A(L)A ( 2 ) = (b b ... b). n-1

Now, the r i g h t m u l t i p l i c a t i o n o f t h i s equa t ion by

l t

= ( v vq . . . vq ) t g ives

n-1 - A(L) 1' = ( t r ( v ) ) b ,I1= ( v ' v ' q . , , v f q

COROLLARY 8. The poZyizorniaZ (1) hus roots i ? i K ,if and onZy if rank(A(1,)) =

rank(A(L) Ib-) = r . If t h i s condi t ion holds , one izus IZ(P)I = qn-r.

Proof. I f ( 1 ) h a s a r o o t x E K then r a i s i n g bo th s i d e s of t h e e q u a l i t y 0

2 n-1 t o the powers q , q ,..., q , we f i n d

n- 1 l:-l~o + 1' xq + . . . + 1:-2x: = b q ,

0 0 .....................

n-1 n- 1 n-1 n-1 n-1 1: xo+ 1; x i + . . . + 1: xq = bq

0 n- 1

Thus, &= (xo X: ... xq 0 fo l low t h a t (1) has r o o t s i n K i f and on ly i f (13) has s o l u t i o n s .

) t is a s o l u t i o n of ( 1 3 ) . From he re and from Prop.7, i t

Taking i n account (8), t he las t p a r t o f the a s s e r t i o n fo l lows from Prop.5.u

I n p a r t i c u l a r , w e f i n d the fo l lowing

RESULT (Dickson 1 3 ) ) . T k ma[) L: K -t K , x + L ( x ) is n p r m u t a t i o n on K f,f c r n d onz$ If det (A(L)) # 0. -

Moreover, we observe t h a t i f de t (A(L)) # 0, t h e on ly r o o t x E K of the 0

polynomial ( 1 ) can be determined u s i n g Cramer's r u l e , t h a t i s

x 0 = d e t ( b , - 4 ,..., $-l)/det(%, A 1 9 . - * s S - 1 ) 3

($, Al.. .. ,&-l) = A(L) . I n g e n e r a l , the a f f i n e s u b v a r i e t y of R c o n s i s t i n g of t he s o l u t i o n s of

polynomial ( 1 ) is given by (8) wi th xo and Z ( L ) expressed by ( 1 4 ) and (12 ) '

r e s p e c t i v e l y .

308 G. Meniclietti

From Corol la ry 8 , we deduce t h e fo l lowing u s e f u l

OBSERVATION. A polynomiaz (1) W i t h deg(L(x) ) = qd , 0 ,< d < n-1, is

corripZetcZy redueible in K if and on2y if rank(L(1) j b) = rank(A(L)) = n-d.

Another consequence i s the fo l lowing

PROPOSITION 9 . Tuo a f f i n e poZynomiaZs, (1) and P ' ( x ) = L'(x) - b ' , have

common u>oots in K if and onZy ?'f t h e equalions of t he Zinear s is tems (13) and

A(L') - - y = b ' m e compatible. - Proof. If xEK i s acommon r o o t of both P(x) and P'(x) then x = O 4

) t i s a s o l u t i o n f o r both l i n e a r systems i n the a s s e r t i o n . n-1

(xo x: ... x: Conversely, i f t he equa t ions of bo th systems a r e compat ib le , we f i n d , by ( 1 4 ) , a

common r o o t f o r t he given polynomials.[

I t i s easy t o prove t h a t , when the cond i t ion of the prev ious p ropos i t i on i s

s a t i s f i e d , t he s e t of common r o o t s f o r P(x) and P ' ( x ) i s an a f f i n e subva r i e ty of

K whose dimension is n - r ' , where

A(L) A(L) b

- - A(L')I b' r ' = rank (- Gi1: ,-;I = rank (-: - - -I - :- ) .

Now we want t o use the prev ious r e s u l t s t o d i scuss the equa t ion

m (15) xq - x = b , b E K , 1 ,< m , < n-1 .

F i r s t we observe t h a t , given d = (n,m) and k = n / d , the i n t e g e r s i m + j ,

i = O , l , ..., k-1, j = O , l , . . . ,d -1 , a r e pa i rwi se incongruent modulo n. In t h i s ca se , t h e l i n e a r sys tem (13) becomes

'+m = bq'

Y2m+ j - 'm+j (16) ...................

'+(k-2)m = bqJ

'( k-1 ) m+ j - (k-2 )m+ j

'i - (k-1 )m+ j

j + (k-1 ) m , j = 0.1, ..., d-1, = bq

k-1 j+ im k-1 i m j and thus i t s equa t ions a r e compatible i f and on ly i f C bq = ( C bq )' = 0.

i = O i = O

From t h i s , we deduce t h a t (15) has some roo t s i n K i f and on ly i f

(17) C bq = t r F , ( b ) = 0,

d where F' = GF(q ) C_ GF(qn) ( 2 ) ,

k-1 i m

i =O

(') The i n t e g e r s hd , h = 0 , 1 , ... ,k-1, and i m , i = 0,1, ..., k-1, a r e congruent

modulo m and t h e r e f o r e I: bq = C bq . k-1 i m k-1 hd

i =O h=O

Roots of Affine Polynomials 309

m L(x) = xq - x imp l i e s obvious ly

d (18) Z ( L ) = GF(q ) , d = ( n , m ) .

The re fo re , w e can de te rmine a r o o t x E K of (15) us ing Prop .7 and supposing t h a t ( 1 7 ) i s s a t i s f i e d .

C

From (16), by success ive s u b s t i t u t i o n s , we f i n d

i-1 hm j

h=O yim+l = y j + ( C bq )q , i = 1 , 2 ,..., k-1, j = 0,1, ..., d-1,

and by (17)

k-1 hm j yim+j = A. - ( C bq )' , X.EK, i = 0,1, ..., k-1, j = O,l , . . . ,d - l .

1 h=i J

Let us cons ide r t he p a r t i c u l a r s o l u t i o n

k-1 hm j

h = i = - ( C bq ) q , i = O,l, ..., k-1, j = 0,1, ..., d-1, ' im+j

ob ta ined f o r X = 0, j = 0 ,1 , ..., d-1. j

From ( 1 4 ) w e o b t a i n

n-1 h h k- ld-1 qn-(im+j) n-(im+j) x t r ( v ) = E 24 vq =

h=O "4 ' 'im+j i = O j=o n-h 0

d- 1 Hence, s e t t i n g v = wq , we have

k-1 d-1 n-(im+j) n-(im+j)+d-1

i = O j = O x t r (w) = C C z;m+j wq 0

where t r ( w ) = t r ( v ) # 0.

I f we observe t h a t

n-(im+j) k-1 n+(h-i)m k-i-1 rm = - C b q = - C bq ,

h = i r=O zSm+ j

then , s u b s t i t u t i n g i n t o t h e prev ious e q u a l i t y , one h a s

k-1 k-i-1 rm d-1 n-im+(d-1-j)

i = O r=O j =O k-1 k-i-1 r m d-1 n-im+s

= - C C bq C w q i = O r=O s =o k-1 k-i-1 rm d-1 s n-im

i = O r = O s =o

x t r (w) = - Z C bq C wq 0

= - C Z bq ( C w ' ) ~ . From h e r e , p u t t i n g

a = C W ~

d-1 s

s=o and observ ing

k-1 d-1 i m + j k-1 i m

i=o j = O i = O t r (w) = c I: wq = c a' = t r F , ( a ) ,

we deduce

310 G. Menichetti

k-1 k-i-1 rm n-im

i = O r = O x O t r F I ( a ) = - C C bq aq

k h-1 r m hm = - C C b q aq

h=1 r=O k-1 h-1 rm hm

= - c ~ b q aq . h = l r=O

Therefore : The equation (15) has roots i n K = GF(qn) i f and only i f b s a t i s f i e s

the condition ( 1 7 ) . If such condition i s s a t i s f i e d , the s e t of roots i s the

a f f i n e subvariety ( 8 ) in which Z(L) i s given by (18) and

k-1 h-1 rm hm x = - - C C bq a' , t r F l ( a ) # 0.

t r F , ( a ) h = l r = O

I f (k ,p ) = 1 (p = char K ) then t r F l ( l ) = k # 0 and t h e r e f o r e , we can se t a = l .

The previous r e s u l t a l lows u s t o determine the r o o t s of a second degree

equa t ion i n a f i e l d K of char 2 . I n f a c t , f o r q = 2 , m = 1, w e f i n d the we l l -

known cond i t ion t r ( b ) = 0 i n o r d e r t h a t t he equat ion

X' + x t b = 0

has a r o o t in K = GF(2"). Moreover, from (18) and ( 1 9 ) , we deduce t h a t t he r o o t s

of the above equat ion a r e

n-1 h-1 2r 2h x = - - C C b a and xo+ 1,

t r ( a ) h = l r = O

where a E K i s a f ixed element w i th t r ( a ) # 0.

REFERENCES

[ l ] Berlekamp, E . R . , AZgebraic coding theory (Mc Graw Book Company,New York,1968).

121 B i l i o t t i M. and Meniche t t i G . , On a general izat ion of Kantor's l ikeable planes,

Geom. Dedica ta , 1 7 (1985) 253-277.

[ 3 ] Dickson, L . E . , Linear Groups wi th an exposi t ion o f the Galois fieZd theory

(Teubner, Le ipz ig . Repr in t Dover, New York, 1958).

Annals of Discrete Mathematics 30 (1986) 31 1-330 0 Elsevier Science Publishers B.V. (North-Holland) 31 I

On the parameter n ( v , t -13) for Steiner triple systems ( " )

Salvatore Milici ("")

Abstract. L e t D ( v , k l ([l], [ 8 ] ) be t h e maximum number of S t e i n e r T r i p l e S y s t e m s of o r d e r v t h a t con be c o n s t r u c t e d i n s u c h a way t h a t an3 two of them have e x a c t l y k b l o c k s i n common, t h e s e k bZocks b e i n g moreover i n each o f t h e S T S ( v ) , . Let t v = v l v - 1 1 / 6 . I n t h i s paper we prove t h a t D ( v , t v - 1 3 ) = 3 f o r everg ( a d m i s s i b l e ) v ,1: .

1 . Introduction a n d definitions.

A Pnr t iaZ T r i p l e S y s t e m (PTS) is a pair (P,P) where P is

a finite non-empty set and P is a collection of 3-subset of P , called blocks, such that any 2-subset of P is contained in at

most one block of P . Using graph theoretic terminology, we will say that an element

x of P has d e g r e e d ( x ) = h if x belongs to exactly h blocks

of P . Clearly d ( x ) =31PI . We will call the d e g r e e - s e t ( D S l X E P

of a PTS (P,P) the n - u p l e DS = I d ( x ) , d ( y i , . . . ] , where x , y , . . . are the elements of p . If there are r elements of P having i degree h , for i = I , . . . ,s , we will write DS =

i

, where r + . . . + r = IPI . If r . = I , for some i , 1 S

then we will write I h I = h i~ i '

Two P T S s (P,PII and IP,P21 are said d i s j o i n t and mutuaZlg

ba lanced (DMB ) if P n P = 0 and a 2-subset o f P is contained

in a block of Pl

A set of s PTSs (P,P1), iP,P2i,. . ., ( P , P s l is said to be a set of

1 2

p2 * if and only if it is contained in a block of

( A ) Lavoro eseguito nell'ambito del GNSAGA (CNR) e con contributo finanziario MPI (1983).

( *$<) Dipartimento di Matematica dell'Universit5, Viale A . Doria, 6 9 5 1 2 5 - Catania.

312 S. Milici

s D M B P T S s i f ( P , P i ) and ( P , P . ) a r e DMB f o r e v e r y

i , j E { 1 , 2 , 3 ,..., s l and i # j . We d e n o t e by ( P ; P I , P 2 J , . . , P ) a

s e t of s D M B P T S s . J

Two ( s + l ) - t u p l e s ( P ; P l , P 2 , . . .,Psi and ( P ; P ; , P ' , . . . , P L ) a r e

i s o m o r p h i c i f I P , P i ) i s i somorph ic t o ( P , P ; I by t h e same isomor-

phism, f o r e v e r y i = 1 , 2 , . . .,s .

2

A d e g r e e se t DS i s a s s o c i a t e d w i t h e v e r y ( P ; P I , P z , ..., PSl . However i f ( P ; P , P 2 , , . . , P s ) and ( P ; P ' , P i , . . . , P ' I a r e n o t isomor-

p h i c , i t i s p o s s i b l e t h a t t h e y have t h e same DS . 1 1

4 S t e i n e r t r i p Z e s y s t e m of o r d e r v ( o r more b r i e f l y a S T S ( v ) )

i s a P T S ( S , B ) such t h a t IS1 = v and e v e r y 2 - s u b s e t o f S i s

c o n t a i n e d i n e x a c t l y one b l o c k o f B . I t i s well-known t h a t a r . eces

s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e of an S T S I v ) i s

u : I o r 3 (mod 6 ) ( v a d m i s s i b l e ) and t h a t t h e number o f t h e

b l o c k s i s I B I = t = v ( v - l ) / 6 , V

We w i l l p u t

S

A ( a , I a , b , c l l = A ( a ) - { b , c l w i t h { a , b , c } E u Pi .

The P T S (P,?) i s s a i d t o be embedded i n t h e t r i p l e sys tem

i = l

( S , B l prov ided t h a t P S S and P C B . Given an i n t e g e r k such t h a t 0 - - k < t o , l e t u s deno te by

D ( v , k ) t h e maximum number o f STS(vls t h a t can b e c o n s t r u c t e d on

a se t o f c a r d i n a l i t y v i n s u c h a way t h a t any two o f them have

e x a c t l y k b l o c k s i n common, t h e s e k b l o c k s b e i n g moreover i n

each o f t h e D l v , k ) sys t ems . I n I l l , Doyen asked t o de t e rmine

D ( u , k ) , ( s e e a l s o IS]).

Much r e s u l t s a r e a l r e a d y known i n t h e c a s e k = O 181 . For

k f o , same r e s u l t s a r e c o n t a i n e d i n 131, 1 4 1 , 151 , 161 , 1-71. I n

p a r t i c u l a r , i n t h e s e p a p e r s D ( v , k ) h a s been de te rmined f o r e v e r y

u a d m i s s i b l e and k = t -m w i t h m(12 . V

Parameter D(v, tv-13) for Steiner Triple Systems 313

. . . I . * . I

I n t h i s p a p e r w e p r o v e t h a t D l v , t - 1 3 ) = 3 f o r e v e r y v z 1 5 . 2,

. . .

2 . P r e l i m i n a r r e s u l t s .

We now g i v e some p r o p e r t i e s w h i c h w i l l b e u s e d i n t h e r e m a i n i n g

s e c t i o n s .

Let 1x1 b e t h e g r e a t e s t i n t e g e r lesser t h a n o r e q u a l t o s . I n a n y fp;P , P 2 , . . . , P ) w e h a v e (31:

( 2 . 1 ) I P I = I P 2 1 = . . . = I P s l 24 , IPI ~6 ; we w i l l p u t n = I P I a n d

1

1 r n = I P i l ( i = 2 , 2 , ..., s) ;

( 2 . 2 ) I f h =muz { d l s ) : X E P } , t h e n m 2 2 h a n d n 2 2 h + 1 ;

( 2 . 3 ) d(x) > 2 , p = m i n {dlx) : S E P } 5 and s 5 2 ~ - 1 ;

( 2 . 4 ) s c 2 d l u l - r l - 2 , w h e r e n = I A ( u , { u , u , w } l - A ( v , { u . u , w ~ l l ;

-

( 2 . 5 ) I f R i s a b l o c k s u c h t h a t l R n M p I '2 w i t h P = 2 , 3 , then s < 2 r - 2 . -

3

j = l X = u A ( j , { l , 2 , 3 } ) a n d 3 ' Lemma 2 . 1 . ~ e t R = I I , ~ , ~ } = M

3

j = l B = n A ( j , I l , 2 , 3 } 1 . I n a ( P ; P , P 2 , P , P ) if R E P i f o r some

i = I, 2 , 3 , 4 , t h e n :

1 3 4

i ) 1x1 = 161 = 4 0 1 7 i i ) 1x1 = 5 a n d I B l = 2 .

Proof. L e t R E P 1 , w i t h o u t l o s s o f g e n e r a l i t y . I f A l l ) = { 1 , 2 , 3 ,

4 ,5 ,b ' ,aI a n d A l 2 ) = { 1 , 3 , 4 , 5 , 6 , b } , we h a v e n e c e s s a r i l y

1 2 3 1 4 a 1 5 6 2 4 6 2 5 b . . .

1 2 4 1 5 u 1 6 3 2 3 5 2 6 b . . .

1 2 5 1 4 3 1 6 a 2 4 h 2 3 6 . . .

p 4

1 2 6 1 4 5 1 3 a 2 4 5 2 3 b . . . * . .

314

1 2 a 1 4 5 1 6 7 2 4 6 2 5 7

S. Milici

1 6 a 1 6 4 1 4 7 1 7 5 1 7 a 1 5 a 1 2 4 1 2 5 1 2 6 2 5 a 2 4 7 2 5 4 2 6 7 2 6 a 2 7 a

o r

i i )

1 2 a 1 4 5 1 6 7 2 4 6 2 5 7

--

P I

1 2 3 1 4 a 1 5 6 2 4 6 2 5 b ... ...

1 2 4 1 6 4 1 2 7 1 6 a i 7 a 1 4 a 1 7 5 1 2 5 1 5 6 2 5 6 2 4 a 2 4 5 2 7 a 2 7 6 2 6 a

p 4

1 2 6 1 4 3 1 5 a 2 3 5 2 4 b ... . . .

...

I n c a s e i ) we o b t a i n 3 e M 4 . T h i s i s i m p o s s i b l e . I n c a s e i i )

wc o b t a i n A ( 3 ) = { l J t ' , 4 , 5 , a , b } and t h i s c o m p l e t e t h e p r o o f o f t h e

lemma.

... ... ...

Lemma 2 . 2 . L e t R = { 1 , 2 , a } , w i t h 1 , 2 E M and a E M In a 3 4 .

( P ; P 1 , P , P , P .J , if 2 3 1 R E Pi , f o r some i = 1 , 2 , 3 , 4 , t h e n

i ) A ( l , { 1 , 2 , a l l = . 4 ( 2 , { 1 , 2 , u } ~ a n d ~ A l a , { 1 , 2 , a } i n A ~ j , { 1 , 2 , a } l 1 = 3 , 4 , f o r e v e r y j = 1 , 2 ,

...

i i ) if x l , x 2 E P a r e such that x . E A l j , { Z , 2 , a ~ l and 3

... ... ...

x . e A ( 3 - j J { 1 , 2 , a } ) , f o r e v e r y j = 1 , 2 , t h e n n e c e s s a r i Z y

x x 2 E A l a ) . 3

I'

Proof. Let ~ = i l , 2 , a l E P , w i t h o u t l o s s o f g e n e r a l i t y . I f 1

A ( l , { l , Z , a } l = A / Z , { I , Z , a } i = { 4 , 5 , 6 , 7 } , we h a v e n e c e s s a r i l y

I I

o r

I ,

o r

Parameter D(v, tv-13) for Steiner Triple Systems

pl

3 1 5

p2 p 3 p4

1 2 a 1 4 5 1 6 7 2 4 6 2 5 7

1 2 4 1 2 6 1 2 7 1 6 5 1 4 a 1 4 6 1 7 a 1 5 7 1 5 a 2 5 a 2 5 4 2 4 a 2 6 7 2 7 a 2 6 5

... ... I ... I ... I

1 2 0 1 4 5 1 6 7 2 4 6 2 5 7 ...

o r

1 2 5 1 6 a 1 4 7 2 4 0 2 6 7 ...

1 5 6

2 5 x ,

2 4 6 L1

I t i s a r o u t i n e m a t t e r t o see t h a t

1 1 6 x l a x

2 5 6 2 4 a

2 a x 2 f i x

1

2 2

1 - I

2 5 a

2 6 x 2

...

1 2 a 1 2 4 1 2 5 1 4 x 1 1 1 5 a 1 1 4 6

2 4 2 2 4 5

2 6 a 2 a x 2

2 ... ...

p4

1 2 6 1 4 a

1 5 x

2 5 a

2 4 x

1

2 ...

o r

p 1

1 2 0 1 4 x

1 5 6

2 5 s

2 4 6

1

2

...

t lence X ~ , X , , E A C a ) . A t t h i s p o i n t t h e ii

c o m p l e t e .,

3. STS w i t h b l o c k s i n c o m m o n

1 6 a 1 1 6 x 1 1 l a x l

p r o o f o f t h e lemma i s

I n t h i s s e c t i o n we wil l p r o v e t h a t D ( u , t - 1 3 1 2 3 f o r e v e r y U

u 2 1 . i . Let P = { 1 , 2 , . . . , 8 , a , b , c l a n d l e t PI , P 2 a n d P 3 b e t h e

f o l l o w i n g t h r e e s e t s o f 1 3 t r i p l e s e a c h :

316 S. Milici

P = 1

1 2 3 l a b 1 4 c 2 a c 2 4 b 3 b c 3 4 a a 5 6 a 7 8 b 5 7 b 6 8 c 5 8 c 6 7

P = 2

, 1 2 4 l a c 1 3 b 2 3 a 2 c b 3 4 c 4 a b a 5 7 a 6 8 b 5 8 b 6 7 c 5 6 c 7 8

Lemma 3 . 1 . D l v , t v - 1 3 1 2 3 , f o r v = l E

Let S = I 1 , 2 ,..., 8 ,9 ,0 ,a ,b , c ,d , e } and

c 9 d c e O

T =

C l e a r l y , ( S , P ~ U T ) ,

1 5 9 1 6 0 1 7 d 1 8 e 2 5 e 2 6 9

P = 3

1 2 c 1 3 a 1 4 b 2 3 4 2 b a 3 b c 4 c a a 5 8 a 6 7 b 5 6 b 7 8 c 5 7 0 6 8

and f o r e v e r y v 2 3 1 .

i = 1 , 2 , 3 , a r e t h r e e S T S l l S l s s u c h

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t T w i t h

IT1 = t - 1 3 . 1 5

I t f o l l o w t h a t D ( 1 5 , t - 1 3 ) ' 3 . I n [ Z ] , J . Doyen and R . M . 15

Wilson h a v e shown t h a t a n y S T S f v ) c a n b e embedded i n t o an S T S ( u I

f o r e v e r y u , B v t l . Then D ( v , t v - 1 3 ) 2 3 f o r e v e r y v 2 3 1 .. Lemma 3 . 2 . D f 1 9 , t -131 2 3 .

1 9

L e t S = { 1 , 2 ,..., 8,9,O,a,b,c ,d ,e ,x ,y ,Z, t l a n d

I d x l e y 2 5 0 2 6 9 2 7 t 2 8 2

2 e x 2 d Y

3 5 x 3 6 Y 3 7 d 3 8 e 3 9 2 3 0 t

4 6 2 4 5 Y

6 e z

Parameter D(v. tv-131 for Steiner Triple Systems 317

x y z x d a x e b

x f c x l g x 2 h 1 5 9 x 7 0 x 3 8 x 4 6

C l e a r l y , ( S , P , u F ) , i = l , 2 , 3 , a r e t h r e e S T S ( 1 9 ) s s u c h z

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t F w i t h

I F 1 = t -13 . Then D ( 1 9 , t l 3 - 1 3 ) - > 3 .. 1 9

y d c y a e y b f y 3 g y 4 h

y 6 9 y 5 0 y 2 8 y 1 7 d e f

Lemma 3 . 3 . ~ ( 2 1 , t - 1 3 ) 1 3 . 2 1

d 1 8 d 5 h d 2 9 d 3 0 d 7 4

e 7 g e 8 h e 4 3 e l 0 e Z 6

L e t S = I l , 2 ,..., 9 , 0 , a , b , c , x , y , z , d , e , $ , g , h ~ a n d

e 3 5 b h O

f 8 g c g o f 7 h c 9 h f 1 9 g 2 5 f 2 0 h 1 6 f 3 6 9 3 7 $ 4 5 4 8 0 a 9 0

u g h b 9 g

l =

C l e a r l y , ( S , P . u L / , i = 1 , 2 , 3 , a r e t h r e e STS(2l)s s u c h z

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t L w i t h

[ L I = t Z 1 - 1 3 . Then D ( 2 1 , t Z 1 - 1 3 1 ' 3 .,

Lemma 3 . 4 . D ( 2 7 , t - 1 3 ) ' 3 . 2,

Let S = i 1 , 2 , , . . , 8 ,9 ,O ,x , y , z , a, b , c, d , e , f. g , i , p , q , F, s , t , M I and

hl

a 0 9 a d e a x y a z t a u s a z r a f P a g q b o d b 9 e

1 5 0 1 6 3 1 7 d 1 8 e 1 x u l y i l z r I t s '

I f s I P q

2 5 9 2 6 0 2 7 e 2 8 d 2 x s

2 Y U 2 z i 2 t r

2 f - s 2 P g

3 5 x

3 7 2 3 8 t

3 O P 3 9 q 3 d r 3 e s

3 u g 3 i f

8 x q

4 5 Y 4 6 x 4 7 t 4 8 2

4 o q 4 9 P 4 d s 4 e r 4 i g

4 u f B Y P

C l e a r l y , ( S , P . u H ) , i = 1 , 2 , 3 , a re t h r e e S T ' S ( 2 7 1 s s u c h

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t M w i t h

/ M I = t g 7 - 1 3 . Then D ( 2 7 , t - 1 3 ) ' 3 ., 2 7

318 S . Milici

2 4 b Z a l

2 c p Z t d Z e f 2 g r 2 h s Z i n

Lemma 3 . 5 . D 1 2 5 , t - 1 3 ) - > 3 . 2 5

- - - - Let P = { I , ~ ,..., 9 , 0 } a n d l e t P I , P 2 , P g b e t h e f o l l o w i n g

t h r e e s e t s o f 1 3 t r i p l e s e a c h :

3 5 c 3 a n 3 b r

3 d f 3 e s

3 g t 3 h l

3 p i 1

7 0 a ? b e ? c h 7 d s

7 f i 7 g n 7 1 t

7 p r

- P =

2

8 O b 8 a i 8 c s 8 n d 8 e g 8 f l

8 r t 8 h p

0 1 2 0 3 4 0 5 6 7 8 3 7 9 6 8 9 2 7 1 4 7 2 5 8 1 5 8 4 6 9 1 3 9 4 5 2 3 6

c i Z

c t f c n r c d e 4 a c 4 d l 4 t e

4 f s

- P =

3

g i g 4 h r

4 n p 5 a b

5 p d 5 e n

5 f h 5 g s

8 1 4 8 2 5 8 3 6 9 1 2 9 3 4 9 5 6

S i t 6 b c 6 d r 6 e i

6 f p 6 g Z 6 h n

Let S = { 1 , 2 , . . . , S , O , a , b , c , d , e , f , g , h , i , l , n , p , r , s , t I and

O d i

O e p O f g O h t O l n O s r

9 O c 9 a t

9 b S 9 d h 9 e Z 9 g P 9 s n 9 r i

C l e a r l y , i S , P i u Q ) , i = 1 , 2 , 3 , a r e t h r e e STS(25/s s u c h

t h a t a n y two o f them i n t e r s e c t i n t h e same b l o c k - s e t Q w i t h

IQi = t Z 5 - 1 3 . Then D ( 2 5 , t - 1 3 ) 2 3 .. 2 5

I n c o n c l u s i o n , by Lemmas 3 . 1 , 3 . 2 , 3 . 3 , 3 . 4 a n d 3 . 5 w e o b t a i n

t h e f o l l o w i n g t h e o r e m .

Theorem 3 . 1 . D ( v , t - 1 3 ) 2 3 f o r e v e r y v 215 , -

4 . Dlv,t - 1 3 1 f o r e v e r y 1 1 2 1 5 . V

I n t h i s s e c t i o n we w i l l p r o v e t h a t t h e r e d o e s n o t e x i s t a

Parameter D(v, tv-13) for Steiner Triple Systems 3 19

P . = z

( p ; P , P , P , P I w i t h m = 1 3 . F u r t h e r w e w i l l d e t e r m i n e D ( v , t - 1 3 1 . From P r o p e r t y 2 . 3 , t h e e x i s t e n c e o f a IP ;P , P , P , P i i m p l i e s

M 2 = @ . I t i s e a s y t o see t h a t a IP;P , P , P , P , I , w i t h M = @ and

m = I 3 , c a n h a v e t h e f o l l o w i n g p a r a m e t e r s :

1 2 3 4 V

1 2 3 4

1 2 3 2

1 ) n = 1 0 a n d D S = [ ( 4 / $ , 3 ] ;

2 ) n = I 1 and D S = L ( 4 1 6 , 1 3 / 5 ] , o r DS = [ 5 , ( 4 / 4 J ( 3 ) 6 ] , o r

DS = [ ( 5 ) , , ( 4 ) 2 , ( 3 ~ ~ 1 , o r DS = [ (5 )3 , ( ~ 1 ~ 1 ;

3 ) n = 1 2 a n d DS = [ ( 4 1 3 , / 3 ) 9 ] o r D S = [ 5 , 4 , f 3 I l 0 - 1 ;

4 ) ~1 = 2 3 a n d D S = [ ( 3 I l 3 ] .

7 8 1 7 3 4 8 4 5 x l a 1 2 3 7 9 2 7 5 6 9 1 5 2 2 1 3 8 9 3 8 2 6 9 4 6 x 3 y

Lemma 4 .1 . T h e r e is no ( P ; P , P 2 , P , P I w i t h DS = [ 1 4 / 9 J 3 ] . 1 3 4

P r o o f . Suppose t h a t t h e r e e x i s t s a ( P ; P l J P 2 , P , P I w i t h

o s = [ f 4 1 31 . Let M = { X I , M = f l , z ,..., 9 1 , P = M u M and

A = A ( x l = { 1 , 2 , 3 , 4 , 5 , 6 } .

3 4

9’ 3 4 3 4

I f { 7 , 8 , 9 } E Pi , f o r some i = 1 , 2 , 3 , 4 , t h e n n e c e s s a r i l y

w h e r e F i ( A ) , i = l , 2 , 3 , 4 , t h e r e a r e f o u r d i s t i n c t 1 - f a c t o r s on

A .

I f { 7 , 8 , 9 } @ P , t h e n n e c e s s a r i l y t h e r e e x i s t s a b l o c k , ? € P i i

s u c h t h a t R E A . Let R = { l , 2 , 3 } , w e h a v e

w i t h (a = 4 , B = 5 , y = 6 1 o r ( a = 6 , B = 4 , y = 5 ) . P a r t i c u l a r l y , i f { 7 , 8 , 9 } $ P I w e o b t a i n

320

X I 6 x 2 4 x 3 5

S. MiIici

€ P I or

Let F = I F i I i = 1 , 2 , . ..,5} b e the I-factorization on A gi-

ven by:

5 . I t f o l l o w s t h a t Afxl = F o r A(xl = F

I

i) Let Alx) = F , . Then, up t o isomorphism, we o b t a i n 1

p l

x 1 4 x 2 5 x 3 6

7 8 1 7 9 2 8 9 3

7 3 4 7 5 6 8 2 6

8 4 5 9 1 5 9 4 6

1 2 3 --

p2

x 1 2 x 3 4 x 5 6

7 8 5 7 9 4 8 9 1

7 1 3 7 2 6 8 2 3

8 4 6 9 2 5 9 3 6

1 4 5

7 8 3 7 9 6 8 9 2

7 1 4 7 2 5 8 1 5

8 4 6 9 2 3 9 4 5

2 3 6

Parameter D(v, t,,-13) for Steiner Triple Systems 32 I

p 3

x 1 3 x 2 6 1

x 4 5

7 8 5 7 9 1 8 9 2

7 2 3 7 4 6 8 1 4

8 3 6 9 3 4 9 5 6

1 2 5

7 8 6 7 9 3 8 9 4

7 1 4 7 2 5 8 ' 1 5

8 2 3 9 1 2 9 5 6

3 4 6

7 8 3 7 9 1 8 9 4

7 2 6 7 4 5 8 1 2

8 5 6 9 3 6 9 2 5

1 3 4

7 8 2 7 9 6 8 9 5

7 1 3 7 4 5 8 1 4

8 3 6 9 1 2 9 3 4

2 5 6

where j i E { 1 , 3 , 4 } , r € { 1 , 2 , 4 } , 8 . € { 1 , 2 , 3 } . i z

i i ) Let A ( x ) = F . Then, up to isomorphism, we obtain 5

x 1 6 x 2 4 2 3 5

7 8 1 7 9 2 8 9 3

7 3 4 7 5 6 8 2 6

8 4 5 9 1 5 9 4 6

1 2 3

7 8 5 7 9 3 8 9 1

7 1 6 7 2 4 8 2 3

8 4 6 9 2 6 9 4 5

1 3 5

7 8 4 7 9 6 8 9 2

7 1 5 7 2 . 3 8 1 6

8 3 5 9 1 3 9 4 5

2 4 6

322 S. Milici

7 8 5 7 9 1 8 9 6

7 2 3 7 4 6 8 1 2

8 3 4 9 2 4 9 3 5

1 5 6

7 8 2 7 9 3 8 9 4

7 1 5 7 4 6 8 1 6

8 3 5 9 1 2 9 5 6

2 3 4

p 4

2 3 5 /:::i 7 8 2 7 9 6

8 9 1

7 1 3 7 4 5 8 3 4

8 5 6 9 2 4 9 3 5

1 2 6

7 8 3 7 9 5

8 9 4

7 1 6 7 2 4 8 1 2

8 5 6 9 1 3 9 2 6

3 4 5

where j l E I 1 , 3 , 4 1 , r E { 1 , 2 , 4 1 , s E { 1 , 2 , 3 } . i i

I t i s a r o u t i n e m a t t e r t o s e e t h a t , i n i ) and i i ) , t h e r e i s no

a ( P ; P , P , P , P w i t h D S = [ 3 , f 4 1 1 . 1 2 3 4 9

Lemma 4 . 2 . T h e r e i s no ( P ; P 1 , P J P , P I w i t h DS = I . ( 4 J 6 , ( 3 1 5 ] . 2 3 4

P r o o f . Suppose t h a t t h e r e e x i s t s a ( P ; P , P , P , P ) w i t h

M = { 1 , 2 , . . . , 5 ) , 1 2 3 4

DS = 1 , ( 4 1 6 , ( 3 1 5 ] . Let

P = M U M

M = { a , b , e , d , e , t } and 3 4

3 4 . A t f i r s t , suppose t h a t t h e r e e x i s t s a b l o c k R G M . Let

R = { 1 , 2 , 3 1 E P w i t h o u t l o s s o f g e n e r a l i t y . Apply ing Lemma 2 . 1 , we

o b t a i n U A l i , { l , 2 , 3 } 1 = 4 , 5 . Let Y = P - U A f i l , we o b t a i n

( Y ( = 3 , 4 w i t h I Y n M 4 ( ) l . Then i t must b e ( P I ( 2 1 4 . T h i s i s i m p o s s i b l e .

3

1 3

i 3 i= 1 i=l

Now, suppose t h a t t h e r e e x i s t s a b l o c k R such t h a t

3 l R n M I = 2 . Let { 1 , 2 , a ) E P I , X = A ( l ) u A ( 2 ) u A f a ) - { 1 , 2 , a } and

Y = P - { A ( l ) u A ( 2 1 u A ( a l } , I t f o l l o w s t h a t 1x1 = 6 , 7 , 8 ,

A t f i r s t , suppose 1x1 = 8 . Let X - A l a l = { z 1 , z 2 1 , from Lemma

Parameter D(v , tv-13) for Steiwer Triple Systems 323

2 . 2 w e o b t a i n x x E A ( i , { l , Z , u } l , f o r some i = 1 , 2 a n d h e n c e

~ A l i , { 1 , 2 , a ? I n A l a , { 1 , 2 , a } l I - < 2 . T h i s i s i m p o s s i b l e . 7' 2

Now, s u p p o s e 1x1 = 6 . S i n c e Y n M 4 # @ i m p l i e s I P l I ~ 1 4 , t h e n

we h a v e Y = { 3 , 4 } . flence { 3 , 4 , t l E P 1 w i t h t E A ( l l n A l 2 l n A ( a i , s i n c e o t h e r w i s e m > 1 3 . F u r t h e r , f r o m P r o p e r t y 2 . 4 i t f o l l o w s t h a t

1 A ( 3 , { 3 , 4 , t } l n A ( 4 , 1 3 , 4 , t 1 ) n I M - { a , t i ) l ~3 a n d I A ( l , { l , Z , u l ) n

n A ( 2 , { 2 , 2 , u l ) n ( M - { u , t l ) l 1 .2 . Then 4

1M41 2 7 . T h i s i s i m p o s s i b l e .

Now, s u p p o s e 1x1 = 7 . S i n c e n = I l , i t f o l l o w s t h a t Y n M = @ .

Let Y = I 3 1 , t h e n , s i n c e n = I 1 , ( 3 , 4 1 C _ P f o r e v e r y i = 1 , 2 , 3 , 4 . Let { 3 , 4 , t } E P . S i n c e 4 e Y , w e h a v e 4 E A ( l l u A ( 2 ) u A ( a l . I t

f o l l o w s t h a t 5 e A l 4 1 , o t h e r w i s e , f o r some j = 2 , 3 , 4 , R = { 3 , 4 , 5 } E P w i t h R C M . S i n c e 1 , 2 , a $ A ( 3 ) , i t f o l l o w s t h a t

a ~ A ( 4 1 a n d 1 , 2 # A ( 4 / . I n f a c t , l e t x E { l , 2 ] , i f x , a E A ( 4 ) we

must h a v e I A i 3 , 1 3 , 4 , t l ) n A ( 4 , { 3 , 4 , t } l l (2 . I f x c E A f 4 1 a n d

a $ A ( 4 1 , we must have w i t h

y E M 4 - { a , t } . Let { 4 , a , b ? , { 4 , c , d } E P . S i n c e 3 4 A l a ) , w e h a v e

{ b , c , d } G A ( 3 ) a n d { c , d , t } E A ( a ) . F u r t h e r , s i n c e e , d # A ( l l n A ( 2 1

( o t h e r w i s e e , d E M ) , i t f o l l o w s t h a t I b , t , e } c A ( l ) n A ( z ) . Let

R = { 2 4 , u , r } E P w i t h { u , u , r } C { 5 , b , c , d , e l . A t t h i s p o i n t we h a v e

n e i t h e r c , d $ A ( l ) u A ( 2 ) n o r c , d E A ( l ) c , A ( 2 / , o t h e r w i s e

( c , d ) I = R 2 o r u = v = 5 o r u = u = e .

4

4

i

1

1 j 1 - 3

1,414, { 4 , x , y 1 ) n A ( x , { 4 , x , y I ) I - < 2

1

5

2 1

I t f o l l o w s t h a t S , a E A ( l ) t ~ A ( 2 ) a n d $ A ( l I u A ( 2 ) w i t h

(cr,B) = ( c , d ) . I f a = c a n d B = d C r i s p . u = d a n d B = c ] , t h e n

R = { 5 , e , d } . From P r o p e r t y 2 . 4 i t i s n o t p o s s i b l e t h a t

I A ( x , { x , 5 , y } l n A ( S , { x , 5 , y } l l < 3 f o r x = l , 2 and y E M 4 . Then

{ 5 , a , t } E P o r { 5 , a , b I E P

2

I ' 1 T h i s i s i m p o s s i b l e a n d t h e p r o o f o f t h e lemma i s comple te . .

Lemma 4 . 3 . T h e r e i s no f P ; P , P 2 , P , P I with DS = [ 5 , ( 4 / q , ( 3 / 6 ] . 1 3 4

Proof. S u p p o s e t h a t t h e r e e x i s t s a f P ; P , P , P , P I w i t h 1 2 3 4

DS = is , / 4 ) 4 , ( 3 ) 6 ] . Let

and P = u M

M = { I , z , . . . ,6 '1 , M = { a , b , c , d l , M = I s } 3 4 5

5

i = 3 i *

A t f i r s t , s u p p o s e t h a t t h e r e e x i s t s a b l o c k REE4 . L e t , 3

3 24 S. Milici

2

R = { 1 , 2 , 3 } E P w i t h o u t l o s s o f g e n e r a l i t y . Apply ing Lemma 2 . 1 we

o b t a i n I u A ( i , { 1 , 2 , 3 ] 1 ) = 4 , 5 . Let Y = P - ( A f l ) u A f 2 l u A ( 3 ) ) ,

t h e n ( n e c e s s a r i l y ) IY( = 3 w i t h Y c M 3 , s i n c e o t h e r w i s e l P l l 2 1 4 .

I t f o l l o w s t h a t

2 3

i=l

1 2 3 2 x b x 4 5 4 . . 3 . . P = l x a 2 . . x 6 d 5 6 b

1 . . 3 x c 4 6 a 5 . .

w i t h d ~ A ( 4 J u A I 5 l , o t h e r w i s e 1 A ( 6 , { 6 , d J + l I n A I d J C 6 , d , x } ) I ( 2 . Suppose , f i r s t , d ~ A ( 4 ) . N e c e s s a r i l y 1 5 , c , a l E P . S i n c e

1 ( 4 , b , d ) $ P , ( o t h e r w i s e I A ( 5 , { 5 , c , a } l n A l c , { 5 , c , a } ) I ZZ), i t fo l -

3 lows t h a t 1 4 , d , c ) E P and b E n A l i ) .

i=l

S i n c e I A ( 5 , { 5 , 6 , b } l n A ( b , { 5 , 6 , b } l l 1 3 , i t f o l l o w s t h a t

1

{ b , I J c } , { b , 3 , a } E P 2 and s o d e M 3 . T h i s i s i m p o s s i b l e .

Now, suppose d ~ A ( 4 1 n A f S J . I t f o l l o w s t h a t a 4 A 1 5 1 , o t h e r -

3

i=l 1 wise { 4 , d , c l E P l , c E n A ( i l and s o { l J c , b } , ~ 2 , c , a } , { 3 , d J b } E P

and ~ A f l , { I , a , x } ) n A f a , ~ l , a , x ~ ) ( ( 2 . I f a $ A ( 5 / , f o r P r o p e r t y 2 . 4

i t i s n o t p o s s i b l e t h a t I A f 4 , [ 4 , 6 , a } ) n A ( a , { 4 , 6 , a l ) l < 3 . Then,

( 4 , b J d } , { a , 2 , d 3 , { a , 3 , b } E P and hence { l , b , c } E P w i t h c E M

T h i s i s i m p o s s i b l e . 3 ' 1 1

Now, suppose t h a t t h e r e e x i s t s a b l o c k R such t h a t I R n M 3 1 = 2 .

X = A ( l ) u A ( 2 l u A l a l - { l J Z J a } and 1 '

L e t { 1 , 2 , a } E P

Y = P - ( A ( l ) u A ( 2 ) u A ( a ) l . I t f o l l o w s t h a t 1x1 =6,7,8 . From Lemma

4 . 2 we o b t a i n 1x1 f 8 . Now, suppose 1x1 = 6 . S i n c e Y n M = @ i m p l i e s l P l l 2 1 4 ,

4 t h e n we have Y = { 3 , 4 1 . Hence { 3 , 4 , b } E Pl w i t h b E A ( l ) n A f 2 1 n

n A ( a ) s i n c e o t h e r w i s e m > 2 3 . F u r t h e r , s i n c e I A ( Z ) n A ( 2 ) A

n ( M 4 - { a l l I 2 2

1 M 4 1 > 4 . T h i s i s i m p o s s i b l e .

and I A ( 3 ) n . 4 ( 4 1 n (M4 - { a , b ) ) I ' 2 , i t f o l l o w s

Now, suppose 1x1 = 7 . S i n c e n = I 1 , i t f o l l o w s t h a t Y n M 4 = @ .

Parameter D(v, t,-l31 for Steiner Triple Systems 325

Let Y = { 3 } , t h e n , s i n c e n = I 1 , ( 3 , 4 ) E P i f o r e v e r y i = 1 , 2 , 3 , 4 . I t f o l l o w s t h a t I A ( 3 ) n M 1 = 2 , o t h e r w i s e , f o r some j = 2 , 3 , 4 , R E P w i t h R G M . Let A ( 3 ! = { 4 , 5 , x , b , c , d } . N e c e s s a r i l y

{ 3 , 4 , x } 4 PI , o t h e r w i s e A ( 4 1 = { 3 , x , a , b , c , d } a n d so l A ( 1 , { 1 , 2 , u } ) n

n A f 2 , { 1 , 2 , a } ) n ( M 4 u M5) I '2 . Let { 3 , 4 , b } ~ P ~ . S i n c e 1 , 2 , a $ A ( 3 ) , i t fo l lows t h a t a ~ A ( y )

3

J- 3

a n d 1 , 2 $ A ( y ) f o r y = 4 , 5 . I n f a c t , l e t z E { 1 , 2 } , i f a , r ~ A ( y ) w e must h a v e

] A ( 3 , { 3 , y , 6 j I n A ( y , 1 3 , y , 6 } 1 I 5 2 w i t h 6 = b f o r y = 4 a n d 6 E I x , ~ }

f o r y = 5 . I f a # A ( y ) a n d x ~ A l y ) we must h a v e l A ( y , { y , x , c ~ } ) ( 3

n A ( x , { y , x , a } ) 1 ( 2 w i t h ~ E J M - { a } ) u M 5 . Let a ~ A ( 4 ) . { 3 , 5 , x l , { 3 , c , d } E P I . I t f o l l o w s t h a t

4

S u p p o s e , f i r s t ,

x , c , d ~ A ( 4 ) a n d c , b , d € A f 5 ) . Then { 4 , x , c } , I 4 , a , d } , I S , c , ~ } ,

{ 5 , d , b } E P , ce

{ u , x , 6 } , { I , X, b ) , { I , 6 , c ) , { 2 , x , d ) , { 2 , 6 , b } E P l a n d hen-

I A f 1 , { 1 , 2 , a } ) n ~ ( 2 , { 1 , 2 , a l ) n ( M 4 u M 5 ) I ' 2 . T h i s i s i m p o s s i b l e .

Now, s u p p o s e { 3 , x , c } , { 3 , 5 , d I E P I t f o l l o w s t h a t I ' { x , c , d , a } c A ( 4 1 a n d [ z , c , b , a } C A ( 5 1 w i t h { 4 , 5 , c , z } E A l a l . Fur-

t h e r i t f o l l o w s t h a t 6 E A ( I ) r \ A ( 2 ) A A ( a ) a n d h e n c e I A ( i , { l , 2 , a } ) n

n A l a , { 1 , 2 , u } ) n ( M 4 - {a]) 1 ' 2 , f o r some

b l e and t h e p r o o f i s complete . .

i = 1 , 2 . T h i s is i m p o s s i -

Lemma 4 . 4 . There i s no ( P ; P P , P , P w i t h D S = [ ( 5 ) 2 , ( 4 ) 2 , ( 3 ) 7 ] . 1 ' 2 3 4

Proof. S u p p o s e t h a t t h e r e e x i s t s a ( P ; P , P 2 , P , P ) w i t h 1 3 4

DS = [ f 5 1 ( 4 ) 2 , ( 3 / 7 ] . Let M = { I , Z ,..., 7 1 , M = { a , b } , M = { z , y } 2' 3 4 5

a n d n = 1 1 , i t f o l l o w s t h a t 5

i = 3 I ' P = u M i , S i n c e

A t f i r s t , s u p p o s e t h a t e x i s t s a b l o c k R G M . Let

( x , y l C _ P

3 ~ = { 1 , 2 , 3 } € P I , w i t h o u t l o s s o f g e n e r a l i t y . A p p l y i n g Lemma 2 . 1 , we

o b t a i n u A ( i , { l , 2 , 3 } ) = 1x1 = 4 , 5 . Let Y = P - U A l i ) . Then 3

i 3 i= I i = l

. F u r t h e r , i t f o l l o w s Y G M o t h e r w i s e 3

1 3 i f = 5 l y l =

1 4 i f 1x1 = 4

P I , I - > 1 4 ,

326

=i:lI I

S. Mil id

l a b 2 x b 3 y b 4 6 y 5 6 b 6 7 x

2 a y 3 x a 4 5 x 4 7 a 5 7 y

I f 1x1 = 4 , w e o b t a i n X = { a , b , x , y } , Y = { 4 , 5 , 6 , 7 } ,

a n d h e n c e l A ( 7 , f 4 , 7 , a } l n A ( a , 1 4 , 7 , a } ) l = 2 . T h i s i s i m p o s s i b l e .

I f I X I = 5 , we o b t a i n Y = { 4 , 5 , 6 1 and X = I 7 , a , b , z , y l . I t

3

i = l f o l l o w s 7 E A l i ) , o t h e r w i s e (i, j , k l C A ( 7 1 w i t h i , j E { 1 , 2 , 3 1

and k ~ 1 4 , 5 , 6 l o r i,j E { 4 , 5 , 6 1 and k E { 1 , 2 , 3 1 and h e n c e

I A ( 7 , { 7 , i : , y l l n A ( k , { 7 , k , y } i l ' 2 w i t h y E { a , b , z , y l . Then n e c e s s a -

r i l y w e h a v e z E A ( ~ J and z # A ( j ) w i t h i E { l , 2 , 3 1 , j E { { 1 , 2 , 3 1 - { i l l and z E { a , b l ,

I f z = b , we o b t a i n

and h e n c e I A i i , { i , b , y } I n A I b , { i , h , y } i I = 2 w i t h y E { 7 , a , x , y } . 3

Now, s u p p o s e t h a t t h e r e ex i s t s a b l o c k R s u c h t h a t l R n M I = 2 .

Let R = { 1 , 2 , a } E P I , X = A ( l l u A ( 2 ) u A f a l - { 1 , 2 , a } , Y = P - ( A ( I ) U

u A ( 2 ) u A ( a ) l . I f f o l l o w s 1x1 = 6 , 7 , 8 . From Lemma 4 . 2 w e o b t a i n

1x1 # 8 . I t f o l l o w s 1x1 # 6 , s i n c e o t h e r w i s e Y E M , w i t h Y = { 3 , 4 ) ,

{ 3 , 4 , b I E P w i t h b E A ( l ) n A I 2 J n A ( a ) a n d h e n c e I A ( 3 , { 3 , 4 , b I ) n

n ~ l b , t 3 , 4 , b I ) n ~ ~ I '2 f o r i 2 4 .

w i t h 6 E M u M a n d h e n c e I A ( 3 , { 3 , 4 , 6 1 ) n A ( 4 , { 3 , 4 , 6 } 1 n M i ) ( 2 f o r

i 2 4 .

3

I

F u r t h e r , i t f o l l o w s 1x1 f 7 , s i n c e o t h e r w i s e { 3 , 4 , 6 3 E P I ,

4 s

T h i s c o m p l e t e s t h e p r o o f o f t h e lemma.,

Lemma 4 . 5 . T h e r e i s n o ( P ; P , P , P , P w i t h DS = [ ( S 1 3 , ( 3 J g ] . 1 2 3 4

P r o o f . Suppose t h a t t h e r e e x i s t s a ( P ; P , P , P , P I w i t h 1 2 3 4

D S = [ ( 5 j 3 , ( 3 j 8 ] , Let M ={1,2,.,.,8} , M = { a , b , c } and P = M u M 3 5 3 5 .

Parameter D(v. t,-13) for Steiner Triple Systems

S i n c e n = I 1 , i t f o l l o w s t h a t ( x , y l C _ P , f o r x , y E i a , b , c l . 1

L e t { a , b , c } $ P , , w i t h o u t l o s s o f g e n e r a l i t y . Then

1 2 3 l a b 1 4 c 2 a c 2 4 b 3 4 a 3 b c

w i t h ct,B,y ~ { 4 , 5 , 6 , 7 , 8 } . S i n c e n ( A ( i , { 1 , 2 , 3 } j ‘ 3 , f r o m Lemma 2 . 1 we o b t a i n i 3 i=l

u = B = y = 4 ;

A t f i r s t , s u p p o s e { 1 , 2 , 4 ) E P2 . N e c e s s a r i l y

1 2 4 l a 3 I b c 2 a b 2 3 c 3 4 b 4 a c

or

i i )

1 2 3 l a b 1 4 c 2 a c 2 4 b 3 4 a 3 b c

1 2 4 l u c 1 3 b 2 3 a 2 c b 3 4 c . . .

. . . ...

1 2 a 1 4 b 1 3 c 2 4 c 2 3 b 3 4 a

1 2 c 1 4 a 1 3 b 2 4 3 2 b a 3 a c

1 2 b 1 4 3 l e a 2 3 a 2 4 c 3 b c

... ... ...

... I . . . I ... 1

- p 3

1 2 a 1 4 3 I b c 2 4 c 2 3 b 3 a c ... . . .

1 2 b 1 4 a 1 3 c 2 3 a 2 4 c 3 4 b

1 2 c 1 3 a 1 4 b 2 4 3 2 b a 3 b c

. . . I . * * I * * *

* . .

I n case i ) we h a v e PI n F 5 # @ and PI n F 3 # 0 .

327

I n case i i ) we

h a v e P a n p 4 # @ a n d P I A f 0 . Then a ( P ; P , P , P , P ) c a n n o t

e x i s t . 1 2 3 4

Now s u p p o s e { I , 2 , 4 } 4 P f o r e v e r y j = 2 , 3 , 4 . N e c e s s a r i l y j

S. Milici

p Z

1 2 3 1 2 a l a b 1 4 3 1 4 c l b c 2 a c 2 4 c 2 4 b 2 3 b 3 4 a 3 a c 3 b c ...

a , . ... _I

328

i ) p 3 p 4

1 2 b 7 2 c 1 4 a 1 3 0 1 3 c 1 4 b 2 a 3 2 3 4 2 4 c 2 b a 3 4 b 3 b c ... . . . ... . I .

o r

i i )

1 2 3 l a b 1 4 e 2 a c 2 4 b 3 4 a 3 b c . . .

p 2 p 3 p 4

1 Z a 1 2 b 1 2 e 1 4 b 1 3 4 1 4 a 1 3 c l e a 1 3 b 2 4 c 2 4 a 2 3 4 2 3 b 2 3 c 2 b a 3 4 a 3 a b 3 a c ... ... ... ... ... ...

I I I I

I t f o l l o w s t h a t P n P # @ i n c a s e i ) and P l n P 2 # @ i n c a s e 1 4

( P ; P , P z , P , P ) i i ) . Then a c a n n o t e x i s t s a n d t h e p r o o f i s comple- 1 3 4

L e m m a

o r D S =

t e .

Pro0 f

4 . 6 . T h e r e i a no ( P ; P , P , P , P I v i t h DS = [ 5 , 4 , ( 3 1 1 1 2 3 4 10

1 4 1 3 , ( 3 1 9 ] .

Suppose t h a t t h e r e e x i s t s a ( p ; P , P P , P q l w i t h 1 2’ 3

DS = 1 5 , 4 , ( 3 1

R c M . Let { 2 , 2 , 3 } € P I w i t h o u t l o s s o f g e n e r a l i t y . A p p l y i n g Lemma

2 . 1 we o b t a i n U A ( i , { 1 , 2 , 3 } ) = 1x1 = 4 , 5 . Let Y = P - U A ( i l ,

c l e a r l y (XI = 4 + k a n d ( Y ( = 5 - k f o r k = 0 , 1 . I t f o l l o w s t h a t

1x1 # 4 , o t h e r w i s e ( Y I = 5 a n d l P , l 224 . Then we h a v e 1x1 = 5

a n d ( Y I = 4 w i t h Y A M i = @ f o r i = 4 , 5 .

I 1 0 - o r DS = [ ( 4 ) 3 , ( 3 ) 9 ] . I n e v e r y case e x i s t s a b l o c k

3 3

I 3 i=l i=l

1 Let Y = { 4 , 5 , 6 , 7 } ~ M . S i n c e m = 1 3 , we o b t a i n t h a t ( x , y ) C P 3

Parameter D f v , tv-13) for Steiner Triple Systems 329

w i t h x J y E Y . Let 2 = { z E P - { 1 , 2 , 3 , . .., 7 1 : 7 { z J y , x } E P I w i t h s , y E l'} .

I t f o l l o w s t h a t I Z I = 3 , o t h e r w i s e { 4 , 5 , z I l , { 6 , 7 , z 1 E P , w i t h 2 1

a n d h e n c e I A ( 4 , { 4 , 5 , z 1 ) n A ( z l , { 4 , 5 , z 1 ) ) I 5 2 . O b s e r v e t h a t z l f z 2 1

Z nM = a . O t h e r w i s e we o b t a i n l A f l , { l , z l , y } l n a f z l , { l , z l J y I I I = O 3

3 E Z ~ M ~ . A t t h i s p o i n t we h a v e M = @ , n A ( i , { l J 2 , 3 1 1 =

i=l z l 5

w i t h

= { 8 , 9 ] E M 3 a n d h e n c e DS = ( 1 4 ) 1 3 J 9 ] . 3'

4

i = 3 i '

Let M = { 1 , 2 ,..., 9 ) , M = { a , b , c ] a n d P = UM

Then

3 4

o r

I n c a s e i ) we h a v e l ~ ( l , { ~ , O , ~ } ) n ~ f u J { ~ , u , ~ } l l 5 2 , In case

i i ) , l ~ f ~ , { l , 8 , a } ) n A f u J { l J 8 , 0 } ) l 5 2 . Then a ( P ; P ,P , P , P ) c a n -

n o t e x i s t s and t h i s c o m p l e t e s t h e p r o o f . , 1 2 3 4

Lemma 4 . 7 . T h e r e is no (P ;P l ,P2 ,P . P w i t h D S = [ ( 3 ) 1 3 ] . 3 4

P r o o f . The s t a t e m e n t f o l l o w s i m m e d i a t e l y from Theorem 2 . 1 o f [ S ] .

Theorem 4 . 1 . D ( v , t - 1 3 ) = 3 f o r e v e r y v z 1 5 .

Proof . A p p l y i n g Lemmas 4 . 1 , 4 . 2 , 4 . 3 , 4 . 4 , 4 . 5 , 4 . 6 a n d 4 . 7 , w e

V

o b t a i n t h a t a I F ; PI , P2 , . . . , P ) w i t h m = 1 3 a n d s > 3 c a n n o t

e x i s t , Then , s i n c e t h e e x i s t e n c e o f s STS(vls , s u c h t h a t a n y two

o f them i n t e r s e c t i n t u - 1 3 b l o c k s ( t h e s e t - 1 3 b l o c k s o c c u r r i n g , V

330 S. Milici

moreover, in each of the S T S f v l s ) implies the existence of a

(P;PIJP2,...,PSl , from Theorem 3.1 we obtain every v L 1 5 ..

D(v , t - 1 3 1 = 3 for V

R E F E R E N C E S

111 J . Doyen, C o n s t r u c t i o n of d i s j o i n t S t e i n e r t r i p l e s y s t e m s ,

L2-1 J. Doyen and R . M . Wilson, Embeddings of S t e i n e r t r i p l e s y s t e m s ,

[3] S . Milici and G. Quattrocchi, Some r e s u l t s on t h e maximum num-

Proc. Amer. Math. SOC., 32 (1972), 409-416.

Discrete Math., 5 (1972), 229-239.

b e r of S T S s s u c h t h a t any two of t h e m i n t e r s e c t in t h e same b l o c k - s e t , preprint.

Z ' e s i s t e n z a di t r e D M B P T S e o n e l e m e n t i d i grado 2 , Le Mate matiche (to appear).

b l o c c h i e i m m e r g i b i l i i n un STS , Riv. Mat. Univ. Parma (to appear).

di S t e i n e r , Le Matematiche (to appear).

14.1 S. Milici and G. Quattrocchi, A l c u n e c o n d i z i o n i n e e e s s a r i e p e r

[ 5 ] G . Quattrocchi, S u l massimo numero d i D M B P T S a v e n t i 1 2

161 G. Quattrocchi, SuZ p a r a m e t r o D ( 1 3 , 1 4 1 p e r S i s t e m i d i T e r n e

1.71 G . Quattrocchi, SuZ p a r a m e t r o D ( v , t v - l O l , 1 9 ' v - < 33 p e r

S i s t e m i di T e r n e di S t e i n e r , Quaderni del Dipartimento di Mate- matica di Catania, Rapport0 interno.

Discrete Math., 7 (1980), 115-128. 181 A . Rosa, I n t e r s e c t i o n p r o p e r t i e s of S t e i n e r s y s t e m s , Annals

Annals of Discrete Mathematics 30 (1986) 331-334 0 Elsevier Science Publishers B.V. (North-Holland) 33 1

A NEW CONSTRUCTION OF DOUBLY DIAGONAL ORTHOGONAL LATIN SQUARES

Consolato P e l l e g r i n o and Paola L a n c e l l o t t i

D ipa r t imen to d i Matematica V ia Campi, 213/B

41 100 MODENA ( ITALY) .

We g i v e a new s imp le c o n s t r u c t i o n o f p a i r s o f doub ly d iagona l o r thogona l L a t i n squares o f o r d e r n, DDOLS(n), f o r some n=3k i n c l u d i n g t h e case n=12.

A p a i r o f doub ly d iagona l o r thogona l L a t i n squares o f o rde r n, DDOLS(n), i s a p a i r o f o r thogona l L a t i n squares o f o rde r n w i t h t h e p r o p e r t y t h a t each square has a t ransve rsa l bo th on t h e f r o n t d iagona l D1 and on t h e back d iagona l D2 . The reader i s r e f e r r e d t o t h e monograph [I] by J.Denes and A.D.Keedwel1 f o r t h e d e f i n i t i o n s which a r e n o t g i ven here. W.D.Wallis and L.Zhu proved t h e ex i s tence o f 4 DDOLS(12) i n [ Z ] . The problem was posed by K .He in r i ch and A.J.W.Hilton i n [ 3 ] .

L e t Q be a L a t i n square o f o rde r n based on t h e s e t In={O,l ,..., n-1) and l e t

S, T be t r a n s v e r s a l s o f r) . We fo rm a pe rmuta t i on on In as f o l l o w s :

t o t h e element o f In occupying t h e c e l l ( h , i ) o f S we assoc ia te t h e element

o f In occupying t h e c e l l ( k , i ) o f T ( i . e . t h e c e l l o f T t h a t l i e s i n t h e same

column). s o f Q w i t h t h e element a S a T ( s ) . Obv ious ly we have:

(a ) i f U i s a t r a n s v e r s a l o f Q then U i s a l s o a t r a n s v e r s a l o f Q(S,T); (b ) i f R i s a L a t i n square which i s o r thogona l t o Q t hen R i s a l s o or thogona l

L e t Q be a L a t i n square and l e t h be a symbol; we denote by Qh t h e copy o f Q

ob ta ined by r e p l a c i n g each e n t r y s o f Q w i t h t h e ordered p a i r (h , s ) .

THEOREM. For an even p o s i t i v e i n t e g e r k l e t A, B be a p a i r o f DDOLS(k) and l e t T1, T2 be two common t r a n s v e r s a l s o f A and B . I f T1 and T2 have no

common c e l l w i t h each o t h e r and w i t h each d iagona l D1 and D2 , then t h e r e

e x i s t s a p a i r o f DDOLS(3k).

P roo f . Consider t h e two or thogona l L a t i n squares o f o r d e r 3k

We denote by Q(S,T) t h e L a t i n square ob ta ined by r e p l a c i n g each e n t r y

t o Q(S,T).

332 C. Pellegrino and P. Luncellotti

O f course 8 possesses a t ranSversa1 on t h e f r o n t d iagona l , w h i l e t h e back d iagona l i s a t ransve rsa l o f B . S t a r t i n g f rom A and B we fo rm t h e f o l l o w i n g L a t i n squares o f o rde r 3k

From (a ) and (b ) i t f o l l o w s immedia te ly t h a t t h e square s t i l l has a t r a n s v e r s a l

on t h e f r o n t d iagona l w h i l e i has a t ransve rsa l on t h e back d iagona l . I n a d d i t i o n we have :

( c ) each subsquare Aij o f and Bij o f 6 i s a doubly d iagona l L a t i n square

(d ) A” and a re o r thogona l .

Since t h e square i s ob ta ined f rom 7i by s u i t a b l y renaming symbols i n t h e subsquares, we have f o r j=1,2, ..., k :

hav ing T,,T, ,D, ,D, as p a i r w i s e d i s j o i n t t ransve rsa l s ;

t h e s e t H j o f t h e e n t r i e s o f t h e j - t h column o f which l i e on t h e

t ransve rsa l s D, o f A,, , T, o f A,, and D, o f A,, co inc ides w h i t t h e s e t

H’. o f t h e e n t r i e s o f t h e ( k t j ) - t h column o f a l y i n g on t h e t r a n s v e r s a l s

D, o f A,, T, o f A,, and D, o f A,, ;

t h e s e t K o f t h e e n t r i e s o f t h e ( k t j ) - t h column o f which l i e on t h e

t r a n s v e r s a l s D, o f A,,, T, o f A,, and D, o f A,, co inc ides w h i t t h e s e t

K j o f t h e e n t r i e s o f t h e ( 2 k t j ) - t h column o f ’li l y i n g on t h e t r a n s v e r s a l s

D, o f A,, , T, o f A,, and D, o f A,, .

each j = 1 , 2 , ..., k exchange i n ?i t h e elements o f H and Hj appear ing on j

same row; s i m i l a r l y we exchange t h e elements o f Ki and K: appear ing on t h e

J

j

same row o f A” ( p r o p e r t y ( c ) i m p l i e s t h a t t he e lem in ts o f i j d i s t i n c t c e l l s ) : f rom (e ) and ( f ) i t f o l l o w s immediately t h a t t h e r e s u l t i n g

and K! occupy J

m a t r i x A ̂ i s a L a t i n square. F u r t h e r 8 i s doub ly d iagona l as t h e c o n s t r u c t i o n e a s i l y shows.

Doubly Diagonal Orthogonal Latin Squares 333

Observing t h a t

exchange elements i n B as we d i d i n d e r i v i n g A” f rom and thus o b t a i n a

doub ly d iagona l L a t i n square 8 which i s o r thogona l t o R because o f p r o p e r t y

( d ) . Hence 8 and a r e p a i r o f DDOLS(3k).

EXAMPLE. S ince f o r each r s 2 t h e r e e x i s t s a p a i r o f DDOLS(2r) s a t i s f y i n g t h e hypothes is o f t h e p rev ious Theorem, we have t h a t f o r each r 3 2 we can c o n s t r u c t

a p a i r o f DDOLS(3-2r).

has p r o p e r t i e s which a r e analogous t o ( e ) and ( f ) we can c

A

ACKNOWLEDGEMENTS. Work done w i t h i n t h e sphere og GNSAGA o f CNR, p a r t i a l l y suppor ted by MPI .

REFERENCES

J.Denes and A.D.Keedwel1, L a t i n squares and t h e i r A p p l i c a t i o n s (Academic Press New York, 1974).

W.D.Wallis and L.Zhu, Four p a i r w i s e or thogona l d iagona l L a t i n squares o f s i d e 12, U t i l . Math. 21 (1982) 205-207.

K .He in r i ch and A.J .H i l ton , Doub1.y d iagona l o r thogona l L a t i n squares, D i s c r . Math. 46 (1983) 173-182.

Annals of Discrete Mathematics 30 (1986) 335-338 0 Elsevier Science Publishers B.V. (North-Holland) 3 3 5

ON THE MAXIMAL NUMBER OF MUTUALLY ORTHOGONAL F-SQUARES

Consolato PELLEGRINO and N i c o l i n a A . MALARA

D i p a r t i mento d i Ma tema ti ca V ia Campi, 213/B

41 100 MOOENA ( ITALY)

I n t h i s paper we pi-ov and Lee and Federer 131 f o r t h e number F1(n;xl), F2(n;;x2), ... , Ft(n;ht) t h e number t o f o r thogona l F1(n;x1,1,x1,2 ,..., x 1 9

F*("12,1 , A * , 2 ' * * . J ~ , ~ ~ L ... , Ft(n;ht,l,xt,2 ,... ,A t,mt 1 squares.

t h a t t h e upper bound, g i ven by Mandel i t o f o r thogona l

squares, a l s o ho lds f o r

1 ,ml

1 - DEFINITIONS AND PRELIMINARY RESULTS

DEFINITION 1 . L e t F = k . 1 b

..., a 1 . We say t h a t F i s an

w r i t e b r i e f l y F(n;xl,x2 ,..., A,,,),

A appears p r e c i s e l y Ak t imes

I n p a r t i c u l a r i f

L ':J m

A ~ = X ~ = . . . = A = A m

Hedayat and Seiden, i n connect ion w i t h some r e s u l t s by o t h e r au thors , g i v e i n 111 a g e n e r a l i z a t i o n o f t h e concept o f l a t i n square: t h e c o n d i t i o n t h a t each element appear e x a c t l y once i n each row and i n each column i s s u b s t i t u t e d by t h e c o n d i t i o n t h a t each element appear one and t h e same f i x e d r lumber o f t i m e s i n each row and i n each column. They c a l l such squares f requency-square o r s h o r t l y F-squares. More p r e c i s e l y they g i v e the f o l l o w i n g d e f i n i t i o n :

a nxn m a t r i x d e f i n e d on a m-set A = Ial,a2,

i f f o r each k=1,2,. ..,m t h e element ak o f

i n each row and i n each column o f F .

F-square o f t ype (n;xl,x2 ,..., A,), and we

ik 2 1)

then m i s determined u n i q u e l y by n and h , hence we s imp ly w r i t e F(n;A) . Note t h a t an F ( n ; l ) square i s s imp ly a l a t i n square of o rde r n . I t i s easy t o prove t h a t an F(n;Al,x 2,...,hm) square e x i s t s

m

i f and o n l y i f & X i = n

I n [l; Hedayat and Seiden a l s o ex tend t o F-squares t h e concept o f o r t h o g o n a l i t y o f l a t i n squares th rough t h e f o l l o w i n g d e f i n i t i o n s :

DEFINITION 2. Given an F(n;xl,A 2 2 ' . . . , A ) square on a r - s e t U = {u,,u2,.,.,ur}

and F2(n;p1,p2 ,..., p s ) square on a s - s e t V = {vl,v2 ,..., v I , we say t h a t F,

i s o r thogona l t o F1 on 5 t h e p a i r (ui,v.) o f UxV appears xipj t imes, f o r each i=1,2, ..., r and f o r

J each J=1,2, ..., s .

OEFINITION 3. L e t Ai be a m.-set, i=1,2 ,..., t. For each i, l e t Fi be on

F-square o f t ype (n;xi ,1 , A ~ ,2,. . "i ,mi

, ,.,Ft i s a s e t o f t m u t u a l l y ( p a i r w i s e ) o r thogona l F-squares i f F i L F j , i p j , i, j=1,2 ,..., t.

r

S F 2 , and w r i t e F1 1. F2 , i f upon super impos i t i on o f

1 ) on t h e s e t Ai . We say t h a t F,,F2,

336 C. Pellegrino and N. A . Malara

I n [2] orthogonal F(n;x) squares i s (n-1) /(m-1) , where m=n/A. I n [3] Mandeli, Lee and Federer proved t h a t the maximal number t o f mutual ly orthogonal F1(n;;X1), F2(n;;x2), ... , Ft(n;xt) squares (where f o r each i = l y 2 a...,t F i s

defined on a mi-set and n=himi) s a t i s f i e s the i n e q u a l i t y

Hedayat, Raghavarao and Seiden proved t h a t the maximal number o f mutual ly 2

i

2. ON THE MAXIMAL NUMBER OF MUTUALLY ORTHOGONAL F-SQUARES

I n analogy t o [3] we prove the f o l l o w i n g

THEOREM. Let Fl(~;~l,lyhl,2a...a;xl,ml) , F2(n;;x2,1,h2,2y ..., hZam2) , ... , Ft(n;;xtal,xt,2,...,A ) be t mutual ly orthogonal F-squares, where f o r each

t a m t m i

i = l a 2 , ..., t Fi i s def ined on the mi-set Ai and n = & xi,j . Then the number t s a t i s f i e s t h e i n e q u a l i t y

t ‘&mi - t 5 (n-1) 2 . 1=

Proof. From Fh(n;;xh,lyhh,2y...yx ) we de f i ne a n 2 xmh ma t r i x Mh = [a:ja,], hamh

where ah -1 i f the k - th symbol o f Ah occurs i n the c e l l ( i , j ) ( i a j = l , 2 ,

..., n) o f Fh and 0 otherwise. Le t M = [MlIM21 .. . I M t ] . By the property o f the

F-squares, the number o f l i n e a r l y independent rows i n M i s a t most (n-1) t 1 and so we ob ta in

i j , k -

2

Now, we can w r i t e the product o f t he transpose o f M w i t h M i n t h i s manner:

2 Jm2xmtLt . . . L2Jm xm L~ L2N2

:~ 2 1

M ’ M =

I

where Li = [ U ~ , ~ J ’ ( i = l y 2 y , . . a t ) i s a diagonal ma t r i x o f order mi w i t h urlr= i

w i t h nr,r = n f o r each r=1,2, ...ami

( i , j=1,2,. . . ,t) w i t h the element 1 everywhere.

f o r each r=1,2 ,..., mi, Ni = [nk,s] i s a diagonal ma t r i x o f order mi

j i s a m a t r i x o f s i ze mixm

i and Jmixmj

Maximal Number of Mutually Orthogonal F-Squares 337

L e t

* . . Om2xml L2

- 1 Lt J

. . .

i s t h e m a t r i x o f s i z e m.xm. ( i y j = l y Z ,..., t ) w i t h t h e element 0 1 . l where Omixm

everywhere. As A . . # O , A i s i n v e r t i b l e and t h e m a t r i x M ' M has t h e same rank

as t h e m a t r i x

j

1 ,J

! JmtxmlL1 - Jmtxm2L2 * * * Nt j .

The e igenva lues o f M a r e t n , n and 0 w i t h r e s p e c t i v e m u l t i p l i c i t i e s 1 , t

t- (mi-1) and t - 1 . Then GT

t

1 + (mi-1) = rank(M) = rank(M'M) =

t

rank (# ) 5 min { (n-1)

Hence t

2 mi - t < (n-1) . 1 =

When XiYl=Xiy2= ...= 1. = A ( i=1,2, ..., t) we have t h e r e s u l t by Mandel i , Lee

and Federer [3] . Furthermore, t h e p rev ious theorem suggests t h a t we c a l l a s e t

of m u t u a l l y o r thogona l F-squares F1(n;;xl ,l,;xl y 2 ,..., A,,,,~), F2(n;;x2,1yx2,2y...y

A ~ , , , ~ ) , . . .

i ,m. i 1

Ft(n;;h t , l ,~t ,2, , . ,y~t ,mt) a complete s e t i f

338 C. Pellegrino and N . A . Malara

where n=hiYl+xiy2t ...+ x. 1 ,mi (i=l,2,. . . , t ) .

ACKNOWLEDGEMENTS. Work done w i t h i n t h e sphere o f GNSAGA o f CNR, p a r t i a l l y supported by M P I . REFERENCES

(1 1 A.Hedayat, E.Seiden, F-squares and or thogona l z a t i o n o f l a t i n square and or thogona l l a t i n squares design; Ann. Math. S t a t i s t . 41 (1970) 2035-2044.

121 A.Hedayat, D.Raghavarao, E.Seiden, F u r t h e r c o n t r i b u t i o n s t o t h e t h e o r y of F-squares design, Ann. S t a t i s t . 3 (1975) 712-716.

131 J.P.Mandoli ,F.C.H.Leey W.T.Federer, On t h e c o n s t r u c t i o n o f o r thogona l F- squares o f o rde r n f rom an or thogona l a r r a y (n,k,s,2) and an OL(s, t ) se t , J. S t a t i s t . Plann. I n fe rence 5 (1981) 267-272.

F-squares design: a genera lL


CARTESIAN PRODUCTS OF GRAPHS AND T H E I R CROSSING NUMBERS

G i u s t i n a P ica +

D ipa r t imen to d i Matematica e A p p l i c a z i o n i U n i v e r s i t a d i Napo l i , Naples, I t a l y

Tomat P isansk i ++

Oddelek za Matemat ik0,Univerza v L j u b l j a n i L j u b l j a n a , Yugoslavia

A ldo G.S.Ventre +

I s t i t u t o d i Matemat ica,Facol ta d i A r c h i t e t t u r a U n i v e r s i t a d i Napo l i , Naples, I t a l y

Kainen and White have determined exac t c r o s s i n g numbers o f some i n f i n i t e f a m i l i e s o f graphs. T h e i r process uses repeated Car tes ian produc ts o f r e g u l a r graphs. I t i s shown how t h i s process can be s u b s t a n t i a l l y genera l i zed y i e l d i n g exac t c r o s s i n g numbers and bounds f o r va r ious f a m i l i e s o f graphs.

INTRODUCTION

I n t h i s paper graph embeddings and imners ions a r e s tud ied . I n o r d e r t o keep i t s h o r t we adopt s tandard d e f i n i t i o n s o f t o p o l o g i c a l graph theory t h a t can be found, say i n [ 2,3,4,5,13] .Usua l l y o n l y normal imnersions o f graphs i n t o su r faces a r e considered, i . e . imnersions i n which no two edges c ross more than once and no edge crosses i t s e l f . I n p a r t i c u l a r , t h i s means t h a t two edges t h a t a r e ad iacen t do n o t c ross . We r e q u i r e i n a d d i t i o n the imners ion t o be a 2 - c e l l immersion which means t h a t t h e complement o f t he immersed graph i s a d i s j o i n t un ion o f open d i s k s (2 - c e l l s ) and t h a t t h e r e e x i s t s a s e t o f edges t h a t can be removed f rom the immersed graph i n o rde r t o o b t a i n a 2 - c e l l embedding o f i t s spanning subgraph i n t o t h e same sur face . The connected components o f t h e complement o f t h e immersion a r e c a l l e d faces. I n a 2 - c e l l imners ion o r embedding a l l faces a r e open d i s k s . A f a c e i s s a i d t o be p a r t i a l i f i t has a t l e a s t one c r o s s i n g p o i n t on i t s boundary o the rw ise i t i s s a i d t o be t o t a l . We w i l l make use o f t he d e f i n i t i o n o f an ( s k)-embedding o f [ 1 1 1 t h a t we repea t here f o r convenience (see a l s o [ l o ] and 1125).

A 2 - c e l l embedding o f a graph G i n t o a su r face S i s s a i d t o be an (s,k)-embedding i f we can p a r t i t i o n t h e s e t o f faces o f t he embedding i n t o s+l se ts F1yF2y,..,FS,R i n such a way t h a t t he boundary o f each s e t Fi, l( i(s, i . e . t h e un ion o f bounda- r i e s of faces be long ing t o F o f G c o n s i s t i n g o f cyc les o f even leng ths ; fu r thermore , k o u t o f t h e s 2 - f a c t o r s c o n s i s t o f q u a d r i l a t e r a l s o n l y and a l l f aces o f R ( i f t h e r e a r e any) a r e q u a d r i l a - t e r a l s . R i s c a l l e d t h e s e t of r e s i d u a l f aces and may be empty. I f k=s we a r e dea-

i s an even 2 - f a c t o r o f G, i . e . a spanning subgraph i’

340 G. Pica. T. Pisanski and A. C.S. Ventre

l i n g w i t h q u a d r i l a t e r a l embedding. I f G has no t r i a n g l e s the embedding i s a l s o m i - n imal, y i e l d i n g t h e genus o r nonor ien tab le genus o f G (depending on the o r i e n t a b i - l i t y t ype o f S), see [Ill . L e t GI have an (s,k)-embedding i n t o S ' and l e t G2 have an (s,k)-embedding i n t o S " . We say t h a t t he two (s,k)-embeddings agree i f t h e r e e x i s t s a b i j e c t i o n between the ve r tex s e t s o f G o f a l l s sets o f nonres idua l faces .

Example 1 . P a r t ( a ) o f F i g u r e 1 shows an (1,O)-embedding o f K sphere. The o u t e r f ace i s hexagonal and t h e r e a r e two r e ~ i d u a 1 ~ ' ~ f a c e s . P a r t (b ) of F igu re 1 shows an (l,O)-embedding of K i n t o the p r o j e c t i v e plane. There i s one hexagonal f ace and t h r e e r e s i d u a l 3 9 3 faces . The two ( 1 ,D)-embeddings agree, which i s shown by an approp r ia te numbering o f v e r t i c e s i n b o t h graphs.

and G wh ich induces a b i j e c t i o n 1 2

-2K2 i n t o t h e

An imners ion o f a g raph G i n t o a

( b ) F i g u r e 1

su r face S i s s a i d t o be an (s,k,c,e)-immersion i f i t i s a 2 - c e l l immersion w i t h c c r o s s i n g p o i n t s and i t . i s p o s s i b l e t o o b t a i n an (s,k)-embedding o f a spanning subgraph H o f G i n t o S by removal o f e edges, and by removal o f any e-1 edges t h e r e remain some c r o s s i n g p o i n t s ( e i s m in ima l ) . H i s s a i d t o be a reduced graph o f t he (s,k,c,e)-immersion o f G. L e t G have an (s,k,c,e) -immersion i n t o S and an (s,k,c ' ,e ' ) - immersion i n t o S ' . We say t h a t t he two immer- s ions agree, i f t h e cor respond ing (s,k)-embeddings o f reduced graphs agree. The f o l l o w i n g examples h e l p e x p l a i n t h e above d e f i n i t i o n s .

Example 2. , The reduced graph and i t s (l,O)-embedding i s dep ic ted o n , F i g u r e l ( a ) . Note 3 y 3 t h a t t he immer- s ions o f K

The f o l l o w i n g two examples were f i r s t used by Kainen [6,7] and Kainen and White[9].

Example 3. P a r t (a ) o f F igu re 3 shows an (1,1,4,4)-immersion o f K sphere. I f the edges 1-6, 2-7, 3-8, and 4-5 a r e removed a (3 ,3 ) -emf jdd ing wh ich i s o f course a l s o an (1,l)-embedding of t h e 3-cube graph Q i n t o t h e sphere re - 3 s u l t s ; see F i g u r e 3(b). P a r t ( c ) o f F igu re 3 rep resen ts t h e well-known genus embedding o f K i n t o t h e t o r u s which i s a (4,4)-embedding.

494

F igu re 2 represents p l a n a r (1,0,3,2)-immersion o f K

on F igu res 2 and l ( b 1 agree. 3,3

i n t o t h e

Cartesiati Products of Graphs 34 I

1 2

6 3

5 4

F i g u r e 2

No te t h a t immers ions on F i g u r e s 3 ( a ) and 3 ( c ) a g r e e as (1,1,4,4) - and ( l , l ,O ,O) - immers ions , as t h e y have f a c e s 1-2-3-4 and 5-6-7-8 i n comnon.

( C )

F i g u r e 3

342 G. Pica, T. Pisanski and A.G.S. Ventre

Example 4. p roduc t C x C 4 i n t o the sphere f o r t he case m = 3. Pa r t s ( b ) and ( c ) o f F igu re 4 a r e a p p r o p r i a t e f o u r edges A-B, C-0, E-F, and G-H we o b t a i n a p lana r , res dual ( 3 , 3 ) - embedding o f PPm x C4 (which i s o f course a l s o a (2 ,2 ) - and even ( 1 , 1 -embedding) as dep ic ted by t o r o i d a l (4,4)-embedding o f C 2m x C4. Note t h a t ( a ) and ( c ) agree as - and ( 1 , I ,O,O)- immersions P and 4).

Recent ly Beineke and Ringeisen have shown [ I ] t h a t c r (C x C ) = 2m. Th i s means t h a t t he immersion o f F igu re 4 ( a ) i s f a r f rom op t ima l .

F igu re 4 (a ) represents a (2,2,8m - 8,4)-immersion o f the Car tes ian

2m analogous t o p a r t s ( b ) and ( c ) o f F igu re 3. Namely, by removing

F igu re 4 ( b ) . F i n a l l y , F igu re 4 ( c ) represents the f a m i l i a r

f o r any 1,1,8m - 8,4)

(and n o t as (2,2,p,q)- immersions

m 4

Cartesian Products of Graphs 343

F i g u r e 4

CONSTRUCTION OF IMMERSIONS

When d e a l i n g w i t h c r o s s i n g numbers on su r faces the f o l l o w i n g two comb ina to r ia l i n v a r i a n t s a r e handy.

d (G) = q - g (p - 2 ( 1 - k ) ) / ( g - 2 )

Jk(G) = q - g (p - 2 + k ) / ( g - 2 ) k

Here p denotes the number o f v e r t i c e s , q denotes the number o f edges, and g deno- t e s t h e g i r t h o f G.In b o t h cases k i s a nonnegat ive i n t e g e r r e p r e s e n t i n g i n t h e f i r s t case the ( o r i e n t a b l e ) genus and i n t h e second case t h e nonor ien tab le genus o f some sur face . They a r e sometimes c a l l e d E u l e r d e f i c i e n c i e s as they r e f e r t o t h e graph and t h e sur face , and o n l y t h e E u l e r c h a r a c t e r i s t i c o f t h e su r face i s i n v o l - ved. They were i n t roduced by Kainen. The o r i e n t a b l e v e r s i o n was i n t roduced i n [ 6 ] w h i l e the nonor ien tab le one was de f i ned i n [ 8 ] and l a t e r used by Kainen and White [ 9 1 . Eu le r d e f i c i e n c y t e l l s us t h e number o f super f l uous edges which o b s t r u c t t he embedding o f a graph i n t o the sur face . The Po l l cw ing lemma shows how E u l e r de- f i c i e n c i e s

Lemma 5.

serve as lower bounds f o r c r o s s i n g numbers c r ( G ) and Cr (G). k k

crk(G)>dk(G) and Zk(G)>dk(G) . F o r p r o o f o f t h e o r i e n t a b l e case see [ 6 ] . The nonor ien tab le case i s e s s e n t i a l l y t h e same; see a l s o [8,9] . To o b t a i n an upper bound f o r t h e c r o s s i n g number we need t h e f o l l o w i n g lemna.

Lemma 6, o r i e n t a b l e (s,s,c,e)-immersion, then c r (G),<c, where k = 1 - p /2 + (q-e)/4.

I f G admits a nonor ien tab le (s,s,c,e)- imnersion, then h e ) /2 .

L e t G be a connected graph w i t h p v e r t i c e s and q edges. I f G admi ts an

k (G),<c, where h=2-p+(q-

3 44 G. Pica, T. Pisanski and A.G.S. Ventre

Proof . o f t h e reduced subgraph H. There fore t h e s u b s c r i p t k (h ) corresponds t o t h e genus (nonor ien tab le genus) o f t h e su r face i n which the immersion o f G and t h e embedding of H takes p lace . Since the re a r e c c ross ing p o i n t s the statement o f Lemma fo l l ows .

The f o l l o w i n g lemma w i l l he lp us t o f i n d graphs w i t h (s,s,c,e)- immersions, namely the Car tes ian produc ts o f o t h e r graphs. These w i l l be the graphs f o r wh ich Lemmas 5 and 6 w i l l app ly y i e l d i n g lower and upper bounds f o r t he c ross ing numbers.

Lemma 7. L e t G and H be two connected graphs and 0 1 , s>,k)O two i n t e g e r s . L e t t h e r e e x i s t a p roper edge c o l o r i n g o f H us ing a t most s c o l o r s such t h a t d o f the c o l o r s used determine d 1 - f a c t o r s o f H. L e t t h e r e be f o r each ve r tex v o f H an (s,k,c e )-immersion o f G i n t o some su r face S(v). Furthermore, l e t a l l t h e i rmer - s ions ” ’agree. De f ine C =xc and E = x e . Then t h e Car tes ian p roduc t G x H admits an (s+d, min(k+Zd ,s+d)y C,E)-immerxion i n t o some su r face T. (Here T i s o r i e n t a b l e i f and o n l y i f a l l sur faces S (v )a re o r i e n t a b l e and H i s b i p a r t i t e . )

The p r o o f i s e s s e n t i a l l y t h a t o f Lemma 2.3 o f [II] and i s om i t ted . I n t u i t i v e l y , t he c ross ing p o i n t s do n o t i n t e r f e r e w i t h t h e argument used i n the p r o o f , as they occur o n l y i n the r e s i d u a l faces and a l l embeddings o f G agree on a l l nonres idua l faces.

S t a r t i n g now w i t h graphs such as those o f Examples 1,2,3 and 4 and u s i n g repeated- l y Lemma 7 i t i s p o s s i b l e t o o b t a i n composi te graphs t h a t s a t i s f y Lemma 6. I n some cases t h e upper bound o f Lemma 6 and t h e l ower bound o f Lemna 5 c o i n c i d e t o g i v e e x a c t c r o s s i n g number; see [9] f o r examples!

CONSEQUENCES

I n t h i s sec t i on we app ly our lemmas t o o b t a i n bound f o r c ross ing numbers and exac t c ross ing numbers i n some spec ia l cases.

Theorem 8. I f a connected graph G admits an (s,s,c,c)-immersion i n t o a su r face o f genus k then c r (GI = c if the sur face i s o r i e n t a b l e o r 5 (G) = c i f i t i s non- o r i e n t a b l e , where k= 1 -p/Z+(q-e)/4 and h=Z-p+( q-e) /2.

Proo f . Combining Lemnas 5 and 6 we o b t a i n t h e i n e q u a l i t i e s e(cr (G)<c i n t h e o r i - en tab le case, and t h e i n e q u a l i t i e s e ( G ( G I @ i n the h case. As we have c=e the r e s u l t f o l l ows .

Us ing our lemmas and the Theorem above i t i s then poss ib le t o prove a l l theorems o f [ g ] and bo th theorems of [7]. I n a d d i t i o n we o b t a i n the f o l l o w i n g r e s u l t s .

Theorem 9.

By removal of e edges we o b t a i n i n b o t h cases a q u a d r i l a t e r a l embedding

k h

knonor ien tab le

L e t G be an a r b i t r a r y connected b i p a r t i t e graph. For each n ) A ( G ) - l and f o r each m, such t h a t O<m<Zn t h e r e i s c r t h e genus o f K x Q, x G.

( K x Q x G) = 4m, where g i s g-m 4,4 n 4,4

Proof . We f i r s t p rove t h a t K x Q admi ts an o r i e n t a b l e (n+ l ,n+l,4m,4m)-immer- s ion , f o r each m, O<m<Zn. c e r t a i n l y t r u e f o r n = 0, as shown by F i g u r e 3(a,c). The i n d u c t i o n s tep f o l l o w s by Lemna 7 if we take K edge-colorable w i t h duc t o f K4,4 x Q and G t o o b t a i n an o r i e n t a b l e (n+ l ,n+l,4m,4m)-immersion o f K4,4 x Q, x G.

p r s o f i s by i n d u c t i o n on n. The s ta tement i s

x Q f o r G and K2 f o r H. As each b i p a r t i t e graph i s 4 s 4 % ( G ) c o l o r s we app ly Lemma 7 t o the Car tes ian pro-

n Using Theorem 8 the r e s u l t now f o l l o w s r e a d i l y .

Cartesian Products of Graphs 345

Note t h a t t he r e s u l t o f Theorem 6 i n case o f G = K was ob ta ined a l ready by Ka i - 1 nen and White [9 ] .

Theorem 10. t ege r . For each i n t e g e r n > d ( G ) - 1 and f o r each m such t h a t 0,<M2n t h e r e i s

L e t G be an a r b i t r a r y connected b i p a r t i t e graph and l e t k),2 be an i n -

4mGrg-m(C21. x Qnt2 x G) < 8 ( k - l ) m , x G. where g i s t h e genus o f CZk x

Proo f . d u c t i o n on n t h a t C 2k x Qn+2 admi ts an o r i e n t a b l e ( n t l ,n+l,8(k-1 )m,4m)-immersion and then as i n t h e admi ts an immersion o f t h e same type. By the o r i e n t a b l e p a r t of Lemmas 5 and 6 t h e r e s u l t f o l lows.

Note t h a t by t a k i n g k = 2 i n Theorem 10 we o b t a i n t h e i n e q u a l i t y : 4m\<crg-m(Qnt4 x G)(8 wh ich f u r t h e r reduces f o r G = K t o t h e i n e q u a l i t y o f Kai- nen Note one can take a n o n b i p a r t i t e graph w i t h g i r t h a t l e a s t f o u r i n Theorems 9 and 10 i n o r d e r t o o b t a i n r e s u l t s o f t he same t ype f o r

g-m' Theorem 11. n> A ( G ) and f o r each m such t h a t 0(m<2n t h e r e i s 2m(E- where g denotes the nonor ien tab le genus o f

P roo f . The p r o o f i s e s s e n t i a l l y t he same as t h a t o f Theorem 9 t h a t i s by induc- t i o n on n. The b a s i s o f i n d u c t i o n i s p rov ided by F i g u r e l ( b ) and F i g u r e 2. A s m<Zn t he re has t o be a t l e a s t one nonor ien tab le imners ion of K i n v o l v e d and the f i n a l su r face i s nonor ien tab le . Note t h a t i f G i s n o t b i p a r t i t Z y 3 then t h e Theo- rem can be extended a l s o t o the case m = 2n. I f G i s 1 - f a c t o r a b l e t h e bound f o r n can be lowered by 1 .

S t a r t i n g w i t h immersions as i n F i g u r e 4(a,c) we cou ld e a s i l y p rove by i n -

2k 'n+2 p r o o f o f Theorem 9 we observe t h a t a l s o C

m 1 [7 ] .

L e t G be an a r b i t r a r y connected graph w i t h o u t t r i a n g l e s . F o r each x Qn x G)( 3m, (K g-m 3,3

K3,3 x Qn x G.

REFERENCES

I ] L.W.Beineke and R.D.Ringeisen, On the c r o s s i n g numbers o f p roduc ts o f cyc les

[ 21 R.K.Guy, Cross ing numbers o f graphs, i n "Graph Theory and A p p l i c a t i o n s " , Lec t .

[ 31 R.K.Guy and T.A.Jenkyns, The t o r o i d a l c r o s s i n g number o f K J.Combinator ia1

[ 41 M.Jungerman, The non-o r ien tab le genus o f t h e n-cube, P a c i f i c J.Math. 76 (1978)

[ 53 P.C.Kainen, Embeddings and o r i e n t a t i o n s o f graphs, i n "Combinator ia l S t r u c t u -

[6 ] P.C.Kainen, A l ower bound f o r c r o s s i n g numbers o f graphs w i t h a p p l i c a t i o n s t o

[ 71 P.C.Kainen, On t h e s t a b l e c r o s s i n g numbers o f cubes, Proc. Amer.Math.Soc. 36

[8] P.C.Kainen, Some r e c e n t r e s u l t s i n t o p o l o g i c a l graph theory , i n "Graphs and

and graphs o f o r d e r f o u r , J.Graph Theory 4 (1980) 145-155.

Notes i n Mathematics 303 (ed. Y.Alavi , e t a l . ) , Spr inger -Ver lag , B e r l i n , Hei- de lberg , New York, 1972, 553-569.

Theory 6 (1969) 235-250. n ,my

443-451.

r e s and t h e i r A p p l i c a t i o n s " , Gordon and Breach, New York, 1970,193-196.

Kn*Kp,q and Q(d), J.Combinator ia1 Theory B 12 (1972) 287-298.

(1972) 55-62.

Combinator ics", Lect .Notes i n Mathematics 406 (ed. R.A.Bari and F.Harary) ,

346 G. Pica, T. Pisanski and A.G.S. Ventre

Spr inger-Ver lag, B e r l i n , Heidelberg, New York, 1973, 76-108. C 91 P.C. Kainen and A.T. White, On s t a b l e c ross ing numbers, 3. Graph Theory 2

[ l o ] T. P isansk i , Genus of Cartes ian products o f r e g u l a r b i p a r t i t e graphs, J. Graph Theory 4 (1980) 31-42.

C111 T. P isansk i , Nonor ientable genus o f Car tes ian products o f r e g u l a r graphs, J . Graph Theory 6 (1 982) 391 -402.

C121 G. Pica, T. Pisansk i and A.G.S. Ventre, The genera o f amalgamations o f cube graphs, Glasnik Mat. 19 (39) (1984) 21-26.

C131 A.T . White, Graphs, groups and surfaces, North-Holland, Amsterdam, London, 1984.

(1978) 181-187.

t Work performed under the auspices of CNR-GNSAGA.

t+ The author was p a r t i a l l y supported by 6. K i d r i E Fund, Slovenia, Yugoslavia.

Annals o f Discrete Mathematics 30 (1986) 347-354 @ Elsevier Science Publishers B.V. (North-Holland) 347

OVOIDS AND CAPS I N PLANAR SPACES

Giuseppe T a l l i n i (Roma)

1. - INTRODUCTION

Let ( S , Y J be a l i n e a r space , t h a t is a s e t S whose elements we c a l l p o i n t s ,

Y a fami ly o f p a r t s i n S , whose elements we c a l l l i n e s , such t h a t any

l i n e has a t least two p o i n t s and two d i s t i n c t p o i n t s are con ta ined i n j u s t one

l i n e . A subspace i n (S,p) is a s u b s e t S ‘ i n S such t h a t f o r any x , y E S ‘ ,

x f y , t h e l i n e j o i n i n g them belongs t o S ’ . Obviously t h e s e t t h e o r e t i c a l

i n t e r s e c t i o n of subspaces i n ( S , y j is a subspace , so t h a t t h e fami ly o f sub-

spaces is a c l o s u r e system.

Suppose a fami ly 9 o f subspaces i n ( S , Y ) e x i s t s such t h a t 19 1 2 2 ,

every JZ E c o n t a i n s t h r e e independent p o i n t s and through t h r e e independent

p o i n t s t h e r e is only one element o f .p . The t r i p l e (S ,&?,p) is c a l l e d

p l a n a r space , t h e e lements o f 9 a r e c a l l e d p l anes . St ra igh t fo rward examples

of p l ana r spaces a r e : t h e a f f i n e o r p r o j e c t i v e spaces o f dimension n 2 3 ,

with r e s p e c t t o t h e i r l i n e s and t h e i r p l a n e s ; any s u b s e t H i n PG(n ,q) ,

n 2 3 , [HI = PG(n,q) , wi th r e s p e c t t o t h e i n t e r s e c t i o n s of H w i th its s e c a n t

l i n e s and p l anes having t h r e e independent p o i n t s of H . A f u r t h e r example is

t h e fo l lowing : cons ide r a whatever fami ly 9‘ of d-dimensional subspaces i n

PG(n,q) , n 2 4 , two by two meeting i n a t most one l i n e , l e t 9’’ be t h e

fami ly of p l anes i n PG(n ,q) each o f them n o t be longing t o any S d E 9‘. S e t

S = PG(n,q) and 9 t h e fami ly of l i n e s i n PG(n ,q) . The t r i p l e ( S , y $ % p ? q ’ ?

is a p l a n a r space .

Let (S,x9) be a p l a n a r space . We denote cap H i n ( S , z % a s e t o f p o i n t s

t h r e e by t h r e e n o t c o l l i n e a r . We c a l l ovoid i n (S,L?,!?) a cap 9 such t h a t :

(1.1) Given any p o i n t

l i n e s t o 51 through P is a subspace Z p &I ( S , a such t h a t

PE a t h e se t t h e o r e t i c a l union o f t h e t angen t

348 G. Tallini

e v e r y p l a n e t h r o u g h P n o t i n 5 meets Z i n a l i n e . The

space Z is c a l l e d t a n g e n t s p a c e t o a t h r o u g h P . P - P

P

I n t h e f o l l o w i n g we suppose t h a t t h e l i n e s o f t h e p l a n a r s p a c e ( S , Y , y )

have t h e same s i z e k 2 3 , and t h e p l a n e s have t h e same s i z e v , t h a t is:

(1.2)

Then ( S , a is a S t e i n e r sys tem S(2,k,V) , IS1 = V and for any J-C €9, i f

p’denotes t h e s e t o f l i n e s i n JT , (JT, is a S t e i n e r sys tem S ( Z , k , v ) ,

If R d e n o t e s t h e number o f l i n e s t h r o u g h a p o i n t o f ( S , Y ) and r is

t h e number of l i n e s through a p o i n t o f

(1.3) R = ( V - l ) / ( k - l ) , r = ( v - l ) / ( k - l ) ,

(JT,%) , i t is :

moreover :

( 1 . 4 ) = V(V-l ) /k(k-1) , lyd = v ( V - l ) / k ( k - l ) .

If s is t h e number o f p l a n e s through a l i n e i n ( S ? g q ) , we e a s i l y o b t a i n

(1 .5) s = (V-k) / (v-k) = (R-l)/(r-l)

Count ing i n two d i f f e r e n t ways t h e number of p a i r s c o n s i s t i n g of a p l a n e

J-C and a l i n e t h r o u g h i t , w e have :

( 1 . 5 ) ) :

/.!??I la = 14 s , t h a t is (see ( 1 . 4 ) , ( 1 . 3 ) ,

(1.6) 191 = V ( V - l ) s / v ( v - l ) = VHs/vr = V R ( R - l ) / v r ( r - l ) .

If l.ypl i s t h e number o f p l a n e s t h r o u g h a p o i n t P , c o u n t i n g i n two d i f f e r e n t

ways t h e number of p a i r s e a c h of them c o n s i s t i n g o f a p l a n e t h r o u g h P and a

l i n e on i t n o t t h r o u g h P , w e o b t a i n : l~pl([yJ-Cl-r) = 1YI-R , t h a t is (see

( 1 . 4 ) , ( 1 . 3 ) , ( 1 . 5 ) ) :

( 1 . 7 ) Iqp1 = ( V - l ) ( V - k ) / ( v - l ) ( v - k ) = Rs/r = R(R-l)/r(r-l) .

If (s,Y’,~) is a p l a n a r s p a c e s a t i s f y i n g ( 1 . 2 ) , we p r o v e :

Theorem 1. - If i n a (S ,y , . !??) a n o v o i d a e x i s t s , it is (S,y,a=

= P G ( 3 , q ) i s a n e l l i p t i c q u a d r i c i f q = k-1 &

o d d , an o v o i d i n P G ( 3 , q ) , if q is even .

Ovoids and Caps in Planar Spaces 349

Theorem 2 . - Let H be a h-cap o f a p l a n a r s p a c e (S,SP) , Then:

(1.8) h<.R+l , r odd.

I f r i s e v e n , o r if r is odd and 1n76H l # r+l f o r any -

E.9, we have:

( 1 . 9 ) h l R - s + l ,

t h e e q u a l i t y h o l d s i f f H is a n o v o i d , t h a t is ( s e e Theorem 1)

- iff (S,xfl = PG(3,q) .

Theorem 3. - If i n a p l a n a r s p a c e (S,Zy) H o f t y p e ( 0 , n ) , n >1 , w i t h r e s p e c t t o l i n e s e x i s t s , t h e n m = ( r - l ) / n must be

a n i n t e g e r and r - m d i v i d e s s-1 , Moreover n < r - m g r - 1 , the e q u a l i t i e s h o l d i n g i f f ( S z , . q ) = PG(d,q) , H is t h e cowplement

o f a pr ime i n P G ( d , q ) @ n = q .

We remark t h a t , i f r is odd, a (R+l)-cap i n ( S , y , p ) is a se t of t y p e

(0,2) , so by Theorem 3, we have:

Theorem 4. - If r is odd, i n (1.8) t h e e q u a l i t y n e v e r h o l d s i f

(r+1)/2 d o e s n o t d i v i d e s-1 . f MCD(s-l,(r+l)/Z) = 2 @

i n (1.8) e q u a l i t y h o l d s , t h e n ( S , y + V ) = P G ( d , 2 ) and t h e cap

H is t h e complement o f a pr ime i n PG(d ,2) .

-

2 . - O V O I D S I N (S ,6q ty )

L e t be a n o v o i d i n a p l a n a r s p a c e (S,y,% s a t i s f y i n g (1 .2) . If P , Q E a , P # Q , any p l a n e n th rough PQ does n o t b e l o n g t o t h e t a n g e n t s p a c e

rp t o Sz a t P ( b e c a u s e Q E 3t and P # Q ) , so i t meets 'C i n a l i n e t ,

I t f o l l o w s t h a t t h e r-1 l i n e s t h r o u g h P i n JT , d i f f e r e n t from t

a r e s e c a n t o f 3t (7 and s o InnfiI = r . I t f o l l o w s t h a t e v e r y p l a n e o f

( S , Y , * q ) is e i t h e r e x t e r i o r , or t a n g e n t , or r - s e c a n t t o fi and ( b e i n g

s t h e number o f p l a n e s t h r o u g h P Q ) : I = s(r-2)+2 , i . e . by (1 .5 ) :

P P

P '

(2.1) [ I = R - s + l . L e t to , tl , tr b e t h e numbers of e x t e r i o r , t a n g e n t and r - s e c a n t p l a n e s t o

350 G. Tallini

s1 respectively. We have (see (1.6),(2.1)5(1.7)):

(2.2ll is obvious. (2.2) and (2.2) follow by computing in two different ways

the pairs consisting of a point of a and a plane through it and two points of fi and a plane through them. By (2.2) we have (see (1.5)):

2 3

t = (R-s+l)(R-s)s/r(r-l) , t = (R-s+l)s(s-l)/r(r-l) , t = (s/r)[(V/v)R + s - (R+l)s] . 1

0

(2.3) 2

Being t 20 , by (2.3) we have: 0 3

(2.4) vcs - s(R+1)] + V R 2 0 . 2

By (1.3) it is V = (k-l)R+l ; v = (k-l)r+l , whence we obtain:

By (1.5) we have s-1 = (R-r)/(r-l) , R-s = (rR-2Rtl)/(r-l) , whence we have (see also (2.5) and dividing it by R-s):

(2.6) k - 1 2 ER(r-2)+l]/(R-r) = r-2 + (r-1) /(R-r) ,

Being (r-1) /(R-r)> 0 , by (2.6) it is k > , r , so k = r and every plane JG

in 9 is a projective plane of order q = k-1 . It follows that (S,zrzq.% is

a Galois space PG(n,q) , n 2 3 and a is a ( q t1)-cap in PG(n,q) (being

2

2

n-1

n-1 n-2

qi - i=O i=O

n-1 I = R-s+l = qi + 1 = q + 1). It is known (see [5]) that in

n-1 PG(n,q) , n > 4 , (q +l)-caps don't exist, whence n = 3 and is a

(q tl)-cap in PG(3,q) . So Theorem 1 is proved. 2

3 . - CAPS IN ( S , z . q )

Let H be a h-cap in a planar space (S,y,p) satisfying (1.21. Given

nE9, n = IJGAH(L 2 , $7 n H is a n-arc in . We easily prove that:

(3.1) n s r + 1 ,

(3.2) n = r+l a H has no tangent lines 3 r odd .

Let P,Q E H , P # Q . By (3.1) each of the s planes through PQ meet

H in at most r-1 points different from P and (3 , whence h = I t 1 1 1 (r-l)s+2,

that is (see (1.5)):

(3.3) h < R + 1 ,

the equality holding iff:

J7 E ~ 9 , I J d n H I 2 2 = I n n H I = r+ l ,

so that, by (3.2):

h = R + l 0 H has no tangent lines F )

@ H is of type ( O , r + l ) with respect to planes (3.4)

Let us now suppose:

we remark rha-c, by ( 3 . 2 ) , (3.5) is fulfilled, if r is even. F o r any P , Q E H ,

P f Q , by (3.5) and (3.11, each of the s planes through PQ meets H in

at most r-2 points different from P and (3 , whence h = I H ( S ( r - 2 ) s + 2 ,

that is (see (1.5)):

(3.6) h l R-s+l ,

the equality holds iff

u n ~ @ , [nnH122 a lnnql = r ,

that is:

(3.7) h = R-s+l e H is of class [O,l,r] with respect to planes.

If (3.5) holds and h = R - s + l , for any plane Jd through PQ , J7 n H is

a r-arc having only one tangent at P , As the number of planes through PO

is s , there are just s tangents to H at P . The set theoretical union of such s tangents by (3.7) is a subspace z of ( S a meeting every plane

through P , not in z p , in a line. So H is an ovoid. Conversely, if

H = fi is an ovoid, by ( Z . l ) , / H I = R-s+l and (3.5) holds (see sect. Z ) , i.e.:

P

h = R-stl F ) H is an ovoid . So Theorem 2 is proved.

352 G. Tallini

4. - SETS OF TYPE ( 0 , n ) WITH RESPECT TO LINES I N (s,2',.9)

L e t H be a s e t o f t y p e ( 0 , n ) w i t h r e s p e c t t o l i n e s i n a p l a n a r s p a c e ( S , y , , f l

s a t i s f y i n g (1.2). Let S ' be a s u b s p a c e o f ( S S s u c h t h a t S ' n H f 0 and

r ' = (lS'l-l)/(k-l) be t h e number o f l i n e s through a p o i n t i n S ' . If

P E S ' n H , every l i n e i n S ' t h r o u g h P meets S i n H i n n-1 p o i n t s d i f f e r -

e n t from P , whence:

( 4 . 1 ) JslnHI = r l ( n - 1 ) + 1 . If w e se t S ' = S , o r S ' = J'C r e s p e c t i v e l y i n (4.11, w e have :

( 4 . 2 ) ( H I = R ( n - 1 ) + 1 ,

l o r ( n - l ) t l , (4 .3) VJGEY, 1nnHI =

whence H is of t y p e ( O , r ( n - l ) + l ) w i t h r e s p e c t t o p l a n e s .

If n = 1 , H r e d u c e s t o a p o i n t , so it is n o t n = k . T h e r e f o r e :

( 4 . 4 ) 2 S n l k - 1 . We remark t h a t i f n = k-1 , H is t h e complement o f a subspace S ' i n

( S , Y ) , meet ing any l i n e n o t b e l o n g i n g t o S ' i n a p o i n t . We d e n o t e s u c h a

subspace prime o f ( S , Y ) . A p l a n a r s p a c e ( S , y , . ! ? ) is c a l l e d p r o p e r , i f

e v e r y n E p i s t h e l i n e a r c l o s u r e o f whatever t r i p l e t of i t s i n d e p e n d e n t p o i n t s .

We p r o v e :

Theorem 5. - ( S , y , . ! ? ) H of type (O,k-l)l e x i s t s , t h e n ( S , 2 ? , 9 ) = PG(d,q) & H

is t h e complement o f a pr ime i n P G ( d , q ) . I t f o l l o w s t h a t a

p r o p e r p l a n a r s p a c e is a Galois s p a c e i f f i t c o n t a i n s a pr ime.

Proof - S' = S-H is a s u b s p a c e of ( S , a meet ing any l i n e n o t i n S ' i n a

p o i n t . Let P , Q € S ' and T E H . The p l a n e J'C j o i n i n g P ,Q,T meets S ' i n

t h e l i n e PQ , b e i n g ( S , 2 ' , 9 ) p r o p e r . Each o f t h e r l i n e s through T

i n 3t meets PQ = S'nn i n a p o i n t , s o r = k and e v e r y p l a n e i n ( S , z f l

is p r o j e c t i v e , i . e . ( S , y , Y , = P G ( d , q ) , w i t h q = k-1 . Thus t h e theorem

is proved .

-

Let n be a n i n t e g e r s a t i s f y i n g (4 .4) and 3t E 9 s u c h t h a t 3t n H # 0 , i . e .

( s e e ( 4 . 3 ) ) /nnHI = r ( n - l ) + l . If P € n -H , t h e number o f l i n e s throi igh P

in 3c , n-secant of 3cnH , is u = (3cnHI/n , so that:

hence m = (r-l)/n m u s t be an integer. If e is an exterior line of H , the

number of planes through 1 meeting H is v = /Hl/(r(n-l)+l) , that is (by

(4.2),(1.5) and being r-1 = mn) we have:

therefore r-m divides s-1 .

It is r - m > n . In fact, if r-m<n (being m = (r-l)/n and n12) we should

have n+l > r ; by (4.4), k 2 n + l , whence the contradiction k > r . So it is n<r-mI r-1 , the equalities hold iff n = r-1 , but by (4.41, r>k2n+l , whence k = r , so that every plane is projective. Then (S,Y,YJ is a Galois

space PG(d,q) and H is a set of type (O,k-lIl , that is the complement of

a prime. Thus Theorem 3 is completely proved.

REFERENCES

r_l] F. Buekenhout, Une caract6risation des espaces affins bas6e s u r la notion de droite, Math. Z. 111 (19691, 367-371.

[Z] F. Buekenhout et R . Deherder, Espaces linCaires finis Q plans isomorphes, B u l l . SOC. Math. Belg. 23 (1971), 348-359.

[3] M.Hall, Jr., Automorphisms of Steiner triple systems, IBM J. Res. Develop. 4 (1960), 460-472 ( = Amer. Math. SOC. Proc. Symp. Pure Math. 6 (1962) ,47-66).

[4] H. Hanani, On the number of lines and planes determined by d points,

Technion. Israel Inst. Tech, Sci. Publ. 6 (1954/5), 58-63.

Is] G. Tallini, Sulle k-calotte di uno spazio lineare finito, Ann. Mat. (4) 42 (1956), 119-164.

[6j G. Tallini, La categoria degli spazi di i-ette, 1st. Mat. Univ. L'Aquila,

(1979/80), 1-25.

[7] L. Teirlinck, On linear spaces in which every plane is either projective

or affine, Geometriae Dedicata, 4 (1975), 39-44.

(k,n;f)-ARCS AND CAPS I N FINITE PROJECTIVE SPACES

B. J . Wilson

Chelsea C o l l e g e U n i v e r s i t y of London

552 K i n g ' s Road London SWlO OUA

I n 5 1 some theorems which g e n e r a l i s e r e s u l t s o f E. d ' A g o s t i n i are g i v e n . Some p a r t i c u l a r examples of ( k , n ; f ) - a r c s are d i s c u s s e d i n 52 and t w o i n f i n i t e classes are d e s c r i b e d i n 13. I n 14 t h e e x t e n s i o n of t h e r e s u l t s of 5 1 t o h i g h e r dimensions is n o t e d .

1 . I n t h e pape r s of Barnabei [ 2 ] and d ' A g o s t i n i [ 4 ] [ 5 ] a n account h a s been g iven of some r e s u l t s conce rn ing weighted ( k , n ) - a r c s i n f i n i t e and p a r t i c u l a r l y G a l o i s p l a n e s . These o b j e c t s a r e a l s o c a l l e d ( k , n ; f ) - a r c s i n [4] and [ 5 ] . I n t h i s n o t e w e are concerned wi th ex tend ing t h e work i n [ 5 ] . However w e f i r s t mention t h a t t h e i d e a of a weighted ( k , n ) - a r c was o r i g i n a l l y proposed by M. T a l l i n i - S c a f a t i [16] and t h a t t h e theme of t h a t pape r , namely t h e embedding of t h e a r c i n a n a l g e b r a i c c u r v e was c o n t i n u e d by Keedwell i n 191 and [ l o ] .

We s h a l l work w i t h t h e d e f i n i t i o n of a weighted arc g i v e n i n [ 5 ] acknowledging t h a t t h e d e f i n i t i o n g i v e n i n [ I 6 1 i s e q u i v a l e n t and h a s p r i o r i t y . Thus w e are concerned w i t h a s e t K of k > 0 p o i n t s i n PG(2,q) t o each p o i n t P of which i s a s s i g n e d a n a t u r a l number f ( P ) c a l l e d i t s weight and such t h a t t h e t o t a l weight of t h e p o i n t s on any l i n e does n o t exceed a g iven n a t u r a l number n , i . e . f o r each l i n e R of PG(2,q) w e have

C f (P ) 5 n P E R

A l i n e hav ing t o t a l we igh t i is c a l l e d a n i - s e c a n t of K . P o i n t s n o t i n c l u d e d i n K are a s s i g n e d t h e we igh t ze ro . Fol lowing t h e n o t a t i o n of [5] w e l e t

w = max f (P) PEK

and u s e L. t o d e n o t e number of p o i n t s of we igh t j f o r j = O,l,...,w.

Following t h e n o t a t i o n and t e rmino logy o r i g i n a t i n g i n [ I31 l e t ti deno te t h e number of i - s e c a n t s of K f o r which i = O,l,...,n. W e c a l l t h e ti t h e c h a r a c t e r s of K and , i f e x a c t l y u of them a r e non-zero w e s ay t h a t t h e a r c K h a s u c h a r a c t e r s . I f t h e v a l u e s of i f o r which t i is non-zero a r e m l < ... < n then K i s s a i d t o b e of t y p e ( m ~ , r n 2 , . . . , n ) . I n [ 1 3 ] , [ 1 4 ] , [71 and subsequen t ly i n l a t e r p a p e r s , one of which i s [ I S ] c o n s i d e r a b l e a t t e n t i o n h a s been g iven t o ( k , n ) - a r c s having e x a c t l y two c h a r a c t e r s . I n p a r t i c u l a r t h e connec t ion between such a r c s and Hermit ian c u r v e s h a s been exp lo red and g e n e r a l i s a t i o n s i n t o h i g h e r dimensions d i s c u s s e d i n [ 1 5 ] . A good b i b l i o g r a p h y i s inc luded i n [ 1 6 ] .

3

350 B. J. Wilson

I t i s proved i n [ 5 ] t h a t f o r a ( k , n ; f ) - a r c K of t y p e ( m , n ) , when 0 < m < n , it is n e c e s s a r y t h a t

w S n - m (i)

and q 5 0 mod (n - m ) . (ii)

L e t W = C f ( P ) = C f ( P ) , and then it may e a s i l y be seen [51 PEK P E E (2,q)

t h a t

m ( q t 1 ) 5 W 5 ( n - w ) q + n . (iii)

A r c s f o r which e q u a l i t y h o l d s on t h e l e f t a r e c a l l e d minimal and a r c s for which e q u a l i t y h o l d s on t h e r i g h t a r e c a l l e d maximal. The case of m = n - 2 w a s d i s c u s s e d a t l e n g t h i n [ 5 ] . By (ii) w e must have g = 2h and t h e n (i) r e q u i r e s t h a t w 5 2. I n order to have an arc which i s n o t s imply a ( k , n ) - a r c w e t h u s must have w = 2 so t h a t (iii) g i v e s

( n - 2 ) ( q + 1 ) 5 W 2 ( n - 2 ) ( q + 1 ) + 2 . ( i v ) I t may e a s i l y be shown t h a t W * (n - 2 ) ( q + 1 ) + 1 and t h e o t h e r two p o s s i b l e v a l u e s of w a r e d i s c u s s e d i n [ 5 ] . Such arcs have p o i n t s having p o s s i b l e we igh t s 0 , l and 2; wi th t h i s i n mind w e can s t a t e t h e f o l l o w i n g r e s u l t s which are g e n e r a l i s a t i o n s of theorems i n [ 5 ] and which can be proved by s i m i l a r methods.

THEOREM 1.

Let K be a (k,n;fl-arc of type (m,n) wi th n > m > 0 of minimal weight FI=m(q+ 1) having some points of weight w = n - m , some poin ts of weight a for exactly one value of a sa t i s fy ing both 1 2 u S w - 1 and ( w , d = I and a t least one point of weight 0.

la ) points of weight w form a l w q + w - q - 1,wl-arc of which the ( w - 1)-secants are concurrent i n the s ingle points of weight 0.

( b ) weight w and the q2 points not co2lincar with them, each of weight a. n = a q + w .

THEOREM 2.

Let K be a (k,n;f)-arc of type h , n l wi th n > m > 0 of maximal weight W = ( n - w ) l q + 1 ) + W having some points of weight 0 , some points of weight a. f o r exactly one value of a sa t i s fy ing both 1 s a ~ w - 1 and ( w , d = 1 and a t least one point of weight w .

la) Then a = 1 and the points of weight 0 form a Iwq+w-q- l,w)-arc of which the ( w - l ) -secants are concurrent i n the single point of weight w.

Ibl Then K cons is t s of a q / w + 1 col l inear poin ts each of weight w and the q2 poznts not col t inear with them, each of weight a. Further n = a q + w .

I t h a s been shown [ l ] t h a t t h e e x i s t e n c e of a ( n o q + n o - q - 1 , n o ) - a r c with n , > 2 r e q u i r e s t h a t q - 0 (mod n o ) and t h a t i n t h a t c a s e t h e arc p o s s e s s e s e x a c t l y q + 1 ( n o - 1 ) - s e c a n t s which are c o n c u r r e n t i n a p o i n t N c a l l e d t h e nuc leus of t h e arc. The a d d i t i o n of t h e p o i n t N t o t h e a r c then g i v e s a ( n , , q + n , - q , n , ) - a r c , known as a maximal

Suppose there i s exactly one point of weight 0 . Then a = w - 1 and the

Suppose that 1, > 1 . Then K consis ts of a q / w col l inear points each of Further,

Suppose there i s exactly one point of weight w.

Suppose k w > l .

( k * ri;f)-Arcs und Cups in Finite Projective Spaces 357

( k , n , ) - a r c . For even v a l u e s of q such a r c s have been c o n s t r u c t e d i n G a l o i s p l a n e s by Denniston 161. I t h a s a l s o been shown t h a t no such arcs can be c o n s t r u c t e d i n G a l o i s p l a n e s f o r n o = 3 by Cossu f o r q = 9 [3] and Thas [ 1 9 ] . Hence t h e theorems 1 and 2 are of i n t e r e s t o n l y i f n - m > 3 .

2 . I t f o l l o w s need t o c o n s i d e r ( k , n ; f ) - a r c s t h e

from t h e r e s u l t s of § 1 t h a t f o r t h e c a s e n - m = 3 w e f u r t h e r i n a d i s c u s s i o n of maximal and minimal c a s e s

and

I n p a r t i c u l a r w e R, > 0; II, > 0; 1, > 0; (1, > 0.

d i s c u s s ( v ) . I n t h a t case ( v i )

( n - 3 ) ( q + 1 ) 5 W 5 ( n - 2 ) q t n . ( v i i ) Fol lowing t h e n o t a t i o n of [ 5 ] w e u s e t h e symbol v! t o d e n o t e t h e number of i - s e c a n t s which p a s s through a p o i n t of weight j . Using t h e arguments i n [5 ] t h e v a l u e s of v: and v A - ~ , j = 0,1,2 a r e f i x e d independen t ly of t h e p o i n t under c o n s i d e r a t i o n .

Fo r W minimal, i .e . W = ( n - 3 ) ( q + 1 ) w e have i n p a r t i c u l a r

v o = q + l n-3 vo = 0 n

( v i i i )

It f o l l o w s immediately t h a t no p o i n t of we igh t 0 l ies on a n n-secant . W e now a t t e m p t t o c o n s t r u c t examples of ( k , n ; f ) - a r c s which s a t i s f y t h e s e c r i t e r i a , i . e . ( v ) , n - m = 3 , and W = ( n - 3 ) ( q + 1). Easy c o u n t i n g arguments g i v e

So lv ing ( i x ) g i v e s tn = ( n - 3 ) q / 3

= ( 3 q 2 + 6 q - n q + 3 ) / 3 tn-3 Now l e t a be a n - secan t on which t h e r e are no p o i n t s of weight 0 so w e suppose t h a t on a a r e a p o i n t s of weight 1 and 6 p o i n t s of weight 2 . Counting p o i n t s of u and we igh t s of p o i n t s on a g i v e s

a + B = q + l c c + B = n

s o l v i n g ( x i ) g i v e s a = 2 ( q + 1 ) - n

= n - ( q + l )

( x i )

( x i i )

358 B. J . Wilson

Counting i n c i d e n c e s between p o i n t s of we igh t 2 and n - secan t s g i v e s

Hence, u s i n g ( v i i i ) , ( x ) and ( x i i ) w e have II, = ( n - 3 ) ( n - q - 1 ) / 2 ( x i i i )

S i m i l a r l y coun t ing i n c i d e n c e s between p o i n t s of we igh t 1 and n - secan t s g i v e s

R,vk = t n C l

whence, u s ing ( v i i i ) , ( x ) and ( x i i ) w e have L 1 = ( n - 3) ( 2 q + 2 - n ) ( x i v )

From ( x i i i ) and ( x i v ) , c o u n t i n g t h e p o i n t s i n t h e p l a n e w e o b t a i n 2 q 2 + ( 1 1 - 3 n ) q + n 2 - 6 n + 1 1 - 2 R o = 0 (xv )

It i s t h u s n e c e s s a r y t h a t ( n - q ) , - ( 4 8 - 1 6 R 0 )

should be a s q u a r e . Whi l s t it i s n o t t h e on ly c a s e f o r i n v e s t i g a t i o n an obvious v a l u e t o t r y i s L o = 3 . The r e s u l t i n g s o l u t i o n s f o r (xv) a r e t h e n n = 2 q + 1 and n = q + 5.

By coun t ing t h e p o i n t s of an ( n - 3 ) - s e c a n t c o n t a i n i n g t h r e e c o l l i n e a r p o i n t s of weight 0 i t may e a s i l y be seen t h a t f o r n = 2 q + 1 it i s imposs ib l e f o r t h e t h r e e p o i n t s of weight 0 t o be c o l l i n e a r . A simple example may be found i n PG(2 ,3 ) . Assign t h e weight 0 t o t h e p o i n t s ~1,0,0) , (0 ,1,0) and ( O , O , I ) , t h e weight 1 t o t h e p o i n t s ~ 2 , 1 , 1 ~ , ~ 1 , 2 , 1 ) , ( 1 , 1 , 2 ) and ( l , l , l ) and t h e we igh t 2 t o a l l o t h e r p o i n t s . Th i s y i e l d s a ( 1 0 , 7 ; f ) - a r c of t y p e (4,7).

For t h i s c a s e of R, = 3 wi th n = 2 q + 1 w e may o b t a i n from ( x )

Thus by ( x i i i ) t h e n - secan t s form t h e d u a l of a

The e x i s t e n c e of such a n arc would, i f q > 3, v i o l a t e t h e Lune l l i -Sce c o n j e c t u r e [ I l l t h a t f o r a ( k , n , ) - a r c w i t h q i 0 (mod n ) it i s n e c e s s a r y t h a t k 5 ( n o - l ) q + 1 . Mason [ 8 ] t h a t counterexamples t o t h i s c o n j e c t u r e c a n be found f o r an i n f i n i t e number of v a l u e s of q .

We now c o n s i d e r t h e c a s e i n which (v ) h o l d s , w i t h t o = 3 , W minimal and n = q + 5 . I f t h e t h r e e p o i n t s of weight 0 a r e c o l l i n e a r t hen i t may be shown t h a t t h e p o i n t s of weight 2 form a complete ( 2 q + 3 , 4 ) - a r c of t h e type ( 1 , 2 , 4 ) . Examples of such a c o n f i g u r a t i o n have been found i n PG(2,3 ' ) . I f w e d e f i n e GF(3') by t h e r e l a t i o n a'= 2 a + 1 ove r GF(3) t h e n one such example i s c o n s t r u c t e d as f o l l o w s :

However i t was shown by H i l l and

Ik,n;fl-Arcs and Cops in Finite Projective Spaces 359

The t h r e e p o i n t s ( 0 , l , a 3 ) , ( 0 , l , a 6 ) I ( 0 , l , a 7 ) have weight 0 and t h e twenty two p o i n t s

Q, ( 0 , 0 , 1 ) (1 ,a2 ,a3) ( 1 , a S I a : ) Q l ( O t 1iO) ( 1 ,a; ,a;) ( ~ , a ~ ~ a

Q 3 ( 0 i l r a 1 ( 1 ( 1 , a6 ,a) ( 1 , a l l 1 ( 1 , a 3 ,a4 1 ( 1 ,a67,a7)

( l , a , a 1 (1,a5,a 1 ( 1 ,a7 ,a) ( I , a 7 , a 7 )

Q, ( 0 , 1 ,a:) ( 1 , a l a 1 ( 1 ,a6 , 1 )

( 1 ,a,a)7 (1,a3 ( 1 , a r l )

have weight 2 . The remaining p o i n t s of PG(2,3 ) a r e a s s i g n e d we igh t 1 g i v i n g an (88 ,14 ; f ) -arc K O of t y p e ( 1 1 , 1 4 ) w i th t h e p o i n t s of weight 2 forming a complete ( 2 2 , 4 ) - a r c of t y p e ( 1 , 2 , 4 ) . L e t v be t h e l i n e , x , = O , of c o l l i n e a r i t y of t h e t h r e e p o i n t s of we igh t 0 . On v are f o u r p o i n t s R,,R,,R,,R, of weight 1 and f o u r p o i n t s Q a , Q l r Q 2 , Q 3 , w i t h c o o r d i n a t e s as i n d i c a t e d above, of we igh t 2. F u r t h e r , t h e r e a r e p r e c i s e l y t h r e e p o i n t s P 1 , P 2 , P J of weight 1 which do n o t l i e on v and which a r e j o i n e d t o t h e p o i n t s R i o n l y by 14-secants . T h e i r l i n e of c o l l i n e a r i t y p a s s e s through Q , and t h e n i n e l i n e s PiQ, are 11- secan t s meet ing i n t h r e e s a t t h e s i x p o i n t s P l , P 2 , P 3 , Q l , Q 2 , Q 3 and o t h e r w i s e o n l y i n p a i r s . T h i s c o n f i g u r a t i o n is, i n P G ( 2 , 3 2 ) , determined comple t e ly by f i v e of t h e p o i n t s

* . - r Q , .

3. The ( 1 O l 7 ; f ) - a r c of t y p e ( 4 , 7 ) i n PG(2,3’) which w a s c o n s t r u c t e d i n 5 2 i s a p a r t i c u l a r c a s e of two o t h e r w i s e d i s t i n c t i n f i n i t e c l a s s e s . F i r s t l y w e n o t e t h a t t h e p o i n t s of weight 1 form a $-arc, t h i s be ing t h e i r r e d u c i b l e c o n i c

x i t x: + x ; = 0.

I n P G ( 2 , q ) , w i t h q odd, l e t C be an i r r e d u c i b l e c o n i c . There a r e e x a c t l y q t 1 p o i n t s on C a t each of which t h e r e i s a unique t a n g e n t l i n e t o C. The p o i n t s of t h e p l a n e which are n o t on C may then be p a r t i t i o n e d i n t o t h e d i s j o i n t classes of q ( q + 1 ) / 2 e x t e r i o r p o i n t s , through each of which p a s s e x a c t l y t w o t a n g e n t s t o C and q (q - 2 ) /2 i n t e r i o r p o i n t s , through each of which t h e r e a r e n o t t a n g e n t s t o C . Assigning weight 0 t o each i n t e r i o r p o i n t , weight 1 t o each p o i n t of C and weight 2 t o each e x t e r i o r p o i n t g i v e s a minimal ( ( q 2 + 39 t 2 ) / 2 , 2 q ; f ) - a r c of type ( q + 1 , 2 q + 1 ) . By a s s i g n i n g d i f f e r e n t we igh t s t o t h e p o i n t s of t h i s c o n f i g u r a t i o n o t h e r ( k , n ; f ) - a r c s may be o b t a i n e d . For example a s s i g n i n g t h e weight 0 t o each i n t e r i o r p o i n t of C , t h e weight 1 t o each e x t e r i o r p o i n t of C and t h e weight ( q + 1 ) / 2 t o each p o i n t of c a maximal ((q2 + 3 q t 2 ) / 2 , ( 3 q + 1 ) / 2 ; f ) - a r c of t y p e ( q t 1 ,2q + 1 ) is o b t a i n e d .

A second i n f i n i t e c l a s s of ( k , n ; f ) - a r c s i s sugges t ed by r e g a r d i n g t h e t r i a n g l e of p o i n t s of weight 0 i n t h e ( 1 0 , 7 ; f ) - a r c of 12 as a subp lane . G e n e r a l l y l e t n o be a subp lane of o r d e r q o of a ( n o t n e c e s s a r i l y G a l o i s ) f i n i t e p r o j e c t i v e p l a n e n of o r d e r q w i t h q > q ; + q , . I n t h i s c a s e t h e r e a r e some l i n e s of n which do n o t c o n t a i n any p o i n t of n o . Assign weight 0 t o p o i n t s of n o , weight u t o p o i n t s of TT which a r e n o t on l i n e s of n o and weight v t o t h e remaining p o i n t s where u/v = (q - q:) / (q - q , - q i ) i n i t s lowes t terms Then t h e r e i s formed a minimal

of t ype ( ( q + q o ) u , ( q : + q o + l ) u + ( q - q i - q , ) v ) .

( ( q - 9 0 1 ( g + 9, + 1 ) I (4 + 9, + l ) u + ( q - q i - q , ) v ; f ) - a r c

3 60 B.J. Wilson

As in the case of the previous example reassignment of other weights to the sets of points involved leads to further (k,n;f)- arcs.

4 . The definition of a (k,n;f)-arc given in § I may be extended to that of a (k,n;f)-cap [5] by substituting PG(r,q) for PG(2,q) with r > 2 . In [ 5 ] it was shown that (k,n;f)-caps of type (n- 2,n) , with r 2 3 do not exist. Segre [12] p 166 concerning the non-existence of certain k-caps in PG(r,q) with r 2 3.

If we use the notation Qr to denote the number of points in PG(r,q) then the results in [I21 showed that the number of points on a k-cap cannot be Qr-l. For a (k,n;f)-cap of type (m- n) with O < m < n the minimal weight is mQy-l. analogous arguments to those indicated above that a (k,n;f)-cap of minimal weight mQr-l and otherwise satisfying the conditions of theorem 1 cannot exist.

This proof required results listed by

However it may be shown using

A similar result can be obtained for maximal arcs.

REFERENCES

Barlotti, A., Su {k;n}-archi di un piano lineare finito, Boll. Un. Mat. Ital. 1 1 (1956) 553-556.

Barnabei, M., On arcs with weighted points, Journal of Statistical Planning and Inference, 3 (19791, 279-286.

Cossu, A., Su alcune proprieta dei {k;n}-archi di un piano proiettivo sopra un corpo finito, Rend. Mat. e Appl. 20 ( 1 9 6 1 ) , 271-277.

d'Agostini, E., Alcune osservazioni sui (k,n;f)-archi di un piano finito, Atti dell' Accademia della Scienze di Bologna, Rendiconti, Serie XIII, 6 (19791, 211-218.

d'Agostini, E., Sulla caratterizzazione delle (k,n;f)-calotte di tipo (n-2,n), Atti Sem. Mat. Fis. Univ. Modena, XXIX, (1980), 263-275.

Denniston, R.H.F., Some maximal arcs in finite projective planes, J. Combinatorial Theory 6 (1969), 317-319.

Halder, H.R., h e r Kurven vom Typ (m;n) und Beispiele total m-regularer (k,n)-Kurven, J. Geometry 8, (19761, 163-170.

Hill, R. and Mason, J., On (k,n)-arcs and the falsity of the Lunelli-Sce Conjecture, London Math. Soc. Lecture Note Series 49 (1981) , 153-169.

Keedwell, A.D., When is a (k,n)-arc of PG(2,q) embeddable in a unique algebraic plane curve of order n?, Rend. Mat. (Roma) Serie VI, 12 (19791,397-410.

1 : ) [lo] Keedwell, A.D., Comment on "When is a (k,n)-arc of PG(2

embeddable in a unique algebraic plane curve of order n? , Rend. Mat. (Roma) Serie VII, 2 (19821, 371-376.

( k , n;fl-Arcs and Caps in Finite Projective Spaces 36 1

L u n e l l i , L. and Sce, M . , Cons ide raz ione a r i t h m e t i c h e e v i s u l t a t i s p e r i m e n t a l i s u i {K;nlq-archi , 1st. Lombard0 Accad. S c i . Rend. A 98 (1964), 3-52.

S e g r e , B. , I n t r o d u c t i o n t o Galois Geometries, A t t i . Accad. Naz. L i n c e i Mem. 8 (1967), 133-236.

T a l l i n i S c a f a t i , M . , { k , n } - a r c h i d i un p i a n o g r a f i c o f i n i t o con p a r t i c o l a r e r i g u a r d o a q u e l l i con due cara t te r i (Nota I ) , A t t i . Accad. Naz. L i n c e i Rend. 40 (1966), 812-818.

T a l l i n i S c a f a t i , M . , { k , n ) - a r c h i d i un p i ano g r a f i c o f i n i t o con p a r t i c o l a r e r i g u a r d o a q u e l l i con due c a r a t t e r i (Nota 111, A t t i . Accad. Naz. L i n c e i Rend. 4 0 (19661, 1020-1025.

T a l l i n i S c a f a t i , M . , C a t t e r i z z a z i o n e g r a f i c a d e l l e forme h e r m i t i a n e d i un S r , q . Rend. Mat. e Appl. 26 (19671, 273-303.

T a l l i n i S c a f a t i , M . , Graphic Curves on a Galois p l a n e , A t t i d e l convegno d i Geometria Combinator ia e s u e A p p l i c a z i o n i P e r u g i a (1971) , 413-419.

T a l l i n i S c a f a t i , M., k - in s i emi d i t i p 0 (m,n) d i uno s p a z i o a f f i n e A r l q , Rend. M a t . ( R o m a ) S e r i e V I I , 1 (1981), 63-80.

T a l l i n i S c a f a t i , M., d-Dimensional two-cha rac t e r k - s e t s i n an a f f i n e space A G ( r , q ) , J. Geometry 22 (19841, 75-82.

Thas, J.A. , Some r e s u l t s c o n c e r n i n g ( q + 1 ) (n-1) - 1 , n ) - a r c s and { ( q + 1 ) ( n - 1 ) + l , n } - a r c s i n f i n i t e p r o j e c t i v e p l a n e s of o r d e r q, J. Combina to r i a l Theory A 19 (19751, 228-232.


N. Zagaglia Salvi Diparthnto di Matematica Politecnico di Milano, Milano, Italy

Let C be a circulant (0,l)-matrix and let us arrange the elements of the first row of C regularly on a circle. If there exists a diameter of the circle with respect to which 1's are synanetric, we call C reflective. In this papr we prove some properties of the reflective circulant ( 0 , l ) -matrices and of certain corresponding cam binatorial structures.

INIXOWrnION

A matrix C of order n is called circulant if C P = P C, where P represents the permutation ( 1 2 . . . n 1 . Let C be a circulant (Ofl)-mtrix and let us arrange the e l m t s of the first row regularly on a circle, so that they are on the vertices of a regular polygon. If there exists a diameter of the circle with respect to which 1's are symnetric, we call C reflective. In this paper we prove some properties of the reflective circulant (O,l)-mtrices and of certain corresponding carbinatorial structures. In particular, % is proved that a circulant (O,l)-mtrix C of order n satisfies the equation C P = CT, 0s h 2 n-1, if and only if it is reflective. Moreover we determine the number of such C for every h. It is proved in certain cases the conjecture of the non-existence of circulant Hadamard matrices and, therefore, of the non-existence of certain Barker sequences. We also give a sufficient condition that the autcmrphism group of a directed graph is C Finally we determine a characterization for the tournaments with reflective circulant adjacency matrix. For the notations, I and J denote, as usual, the unit and all-one matrices: the matrix C

the cyclic group of order n. n'

denote the transpose of C. T

I . L e t c be a circulant matrix. If it follows [2] that the eigenvalues of c are

[co, cl, ... , c 3 n-1 is the first nm of C,

n- 1 jr x = c c,bJ

and w = exp( - 1 .

(1) j=o 3 r

where 0 5 r 5 n-1 2 a i n

Consider the circulant matrix A = C P. The first row of A is obtained frcm the

364 N . Zagaglia Salvi

first row of C by shifting it cyclically one position to the right.

since the eigenmws of A are pr = 1 c

scripts are reduced mil n, it follm po=Xo, p =A w , * . . r lJn-I=An-lW h h r

So the matrix B = C P , 1 2 h 5 n-I , is circulant with eigenvalues u =A (w . PFOKSITtCN 1.1 - !he eigenvalues Ar and vr , 0 5 r 5 n-I , of the cir trices C and CP , 0 2 h 5 n-I , as in (1) , satisfy the relations v =A (w ) .

r r

PROKSITION 1.2 - The eigenvalues A and p trices c and D, as in (11, satisfy h e relations p =A if and d y if D = cT.

Roof. If A= diag( 1, w, w , . . . , w unitary mtrh U such that P = U A U*. If

we have C = .C c.P3 = U r Ux where r= ;i!,cjA7 and D = U( nf'd,AJ)U*.

If p equals 'i for all r, it follm jgodjA7 = 7 and hence D = T'

TIE L e r s e i5 easily proved. Q

DEFINITICN 1 . 3 - Let w , up two nth roots of unity , 0 5 r,p 2 n-I, and w =

exp ( -I . If d = I r-p I , we say that such roots are at distance d and we write

dist( wr, wP)=d. Clearly, then, dist( Gr, Gp)=d.

THMlREM 1.4 - A circulant (0,l)-mtrix C satisfies the equation CP =C 0 5 h 5 n-I, if and only if it is reflective.

n- 1 j=O j-1

Jrr where 06 r <n-1 and the sub- n-I

*

r r

ant ma- h "3ir

0 5 r 5 n-I , of the circulans ma- r'

r r

n 2 n-I 2ri 1, where w=exp(-), then there exists an

1 are the first rows of C and D, [c0, c , ..., c 1 and [do, dl, ..., d .n-1 n: 1

n-I '

' n-1 3=0 3 7=0 7

r

2 ~ i n

h T'

proof. k t C be a reflective circulant (O,l)-matrix of order n; let A 0 I r 5 n-I, be the eigenvalues of C as in (1).

l = w + w +...+a , 0 5 p < p 2 < ... <ps5n-l. 1

Suppose that A "hen the elements 1 on the first rcw of C are m the positions p1 +I, p2+1 , .. . , P,+l. Pi Since C is reflective, it follms that also for the s nth roots of unity w , 1 5 i I s, there exists a dimter 2 of the mit circle with respect to which such m t s are symnetric. If 2 contains at most o m of the above roots, consider an ordering H = I w

. . . , w of such roots obtained by traversing the unit circle counterclockwise,

r '

p2 PS

a a 1 2 ,u , a

so that the diameter 2 is encountered only once. In case a does not contain any

,w ) , I ~ i s 5 - 1 . roots, s is even and dist( w , w )=dist( w

In case 2 containrone root, s is odd and that rwt is w ai a

?hen dist( w , w )=&st( w

o! ci- "i ' "i+l s-(i-l) s-i 2

5%

a i+ 1 's-(i-I) s-i s-1

, w ),ISis--. 2

If g contains two roots, we ignore one of these, then we proceed as before. a

So the two roots on g are wax and w .

Cornbinatorial Structures 365

ci a -011 - 2 ,s

It is easy to see that the roots w , w , ..., w are syrmmetric for the diameter b, where b is the reflection of g in the real axis.

There are two cases:

1) the diameter a contains at mst one of the roots w , lSiSs.

k t h be the minirmnn positive integer such that w

By the symnetry of the roots with respect to g,we obtain: dist( w

a i

a +h a s - 1

= w . cis "s-1 , w ) =

a a a a a +h -a2 a +h s- 1 s-2 - dist( w w 'I= dist( ', '1. It follows that w = w , 0 -

S h - h r - a al+h a - 3

w , ..., w = ij andhence hlw = h Thus A ( w ) =A ISrk-1. 1- r r t h

Consider the matrix D = 8 . Let p- be the eigenvalues of D, as in (1 ) . By Prop. - 1.1, we have p = ( whir so thai, the a v e statements, pr = Ar.

r r

a. i T' By Prop. 1.2 D=C

2) The diameter 5 contains two of the roots w , l<i<s. ci +h a1

Let h be the minirmnn positive integer such that w = G . ~y the w e a.+h ci 1 - s-i

statmts we have that w = w , 15i5s-1, and, in particular, w

h a +h ci s - s Hence w = w and 8 =C,.

I h

h

Suppose, m, that the circulant (O,l)-matrix C satisfies the equation 8 =CT, 06hSn-1; we prove that C is reflective. If [co, cl, ..., c is the first rcw of C, then the first rows of 8 and

c are respectively [c-h, cl-h, ..., c J and [cot cn-l, ...,cJ . Frm the relation C P = C we obtain the equalities c = c where O5i5n-1

and the indices are mod n, so that C is ref1ective.a

Recall that, if A = [a. ,I and B are two matrices of order m and n respectively, then the direct product of A with B is a matrix of order rm defined by

3 n-1

h n-l- T

TI i-h n-it

1 3

,-

a B a B ... 21 22

A X B = ...

Let C be a reflective circulant (0,l)-matrix. If there are two diameters of the circle with respect to wich the 1's are symnetric, then there are t m integers

h k and k kl k such that C P = C , i = 1,2. I& C satisfies such equations and 1% <;f <n, then C P = C, where h=k2-klt [l,n-1).

In this regard we have the following theoren.

THKlREM 1.5 - A circulant (O,l)-matrix C of order n satisfies C P = C, IhSn-I,

2' 2' 1 2

h


i f and only i f it is C = 0 or C = J for (h,n)=l and C = J n where q = - and D is circulant (0,l)-matrix of order t. t

Proof. The f i r s t rrm of 8 cyclically h p x i t i o n s to the right. If [cl, C2' . . . I CJ

x D for ( h , n ) = t > 1 , q

h is obtained f ran the f i r s t row of C by sh i f t ing it

h is the f i r s t row of C, f ran 8 =C we obtain

where the indices are mod n. W e have t w cases to consider:

a) (h,n)=l

b) (h ,n )> l .

Case a ) . I f (h ,n )= l , then the n en t r i e s of H = { I , l+h, ..., l t (n - l )h} are a l l d i s t i nc t and H is equal to the set {1,2,. . . ,n} . 1 Hence in the sequence (2) there are a l l the entries of the f i r s t row of C and it is or C = 0 or C = J.

Case b). Let (h,n)=t>l w i t h n=tq. I n HI only q elements of the set K ={ l , l+h , 1 . . . , I + (q-I) hl are d i s t inc t .

It is easy t o prove that a lso the elements of the set K =I i , i + h , . . . , i+ (q-1) h l , i 6 [l,t] are d i s t inc t and, mreover, K . is d i s jo in t fran K

c2 ... c repeated q times determines the f i r s t I t follows that the sequence c row of c. hen, i f D is the c i rcu lan t matrix whose f i r s t r m is [cl c2 . . . ct] , we have C = J xD.

q Conversely, i f it is c = J x D, then it is easy t o prove that ah = C. II

q

Remark t h a t , i f a circulant (0,l)-matrix C of order n=hq s a t i s f i e s the re la t ion

c P = C, then there are also sa t i s f i ed the relations c ph" = C, s=1,2,. . . ,q. let C be a circulant (0,l)-matrix such that C P = CT, k€[O,n-l].

r r r I f k is even, say k=2r, then we obtain CP = ( CP IT; denoted CP =E, it follaws E = E

If k i s odd, k=2q+l, 920, then we obtain (CPq)P=(mq)Tr where, denoted CPq=E, it is EP= .

'IIIEORFM 1.6- A n-square c i rcu lan t (0,l)-matrix C = J x D, where D has order t and n=tqr is ref lec t ive i f and only i f D is reflectiye.

1

i i f j . 1 1'

t

h

k

T'

ET

Proof. Let C = J is ( 0 , 1 ) - c i r c u l d t , of order t, and by Theorem 1.5 it is sa t i s f i ed CP =C.

X D, of order n=tq, t , q > l , be a circulant (O,l)-IMTx; then D

If C is reflective, there exists an integer r such that, denoted d = E , we have E = %or E P = ET.

L

Since it is El?= = E, by Theorem 1.5, there exists a circulant matrix F of order t such that E = J x F.

9

Combinatorial Structures 367

.- It is easy to prove that E = E iff F = F and E P = E iff F P = F where? represents the permutation ( 1 2 . . . t) .

r fir Since C P = F and C is reflective iff D is ref1ective.a

T T T TI

= E, then it is D P

DEFINITICN 1.7 - A set D = {d if the circulant d,, d2, ...,%, is reflective. THMlRFM 1.8 - If the set of integers mod n D = {dll d , ..., $2 then it is reflective the set t~ = {tdl , td2, ...,tc$ , for every t prime to n.

Proof. Let D = {d

of k integers mod n is reflective to w h i c h the 1's are in the positions

is reflective,

} be a reflective set and let C be the circulant (0,l)-matrix with ch the 1's are in the positions If we arrange the e l m t s of the first rcw of C rqlarly on a exists a diameter g of the circle with respect to which the 1's are symnetric.

} an ordering of the elements of D obtained by tra- L e t H = { a versing the unit circle counterclockwise, so that the diameter 2 is encountered only once. Let ~1 be the last element of H before traversing a Let i Consider the set tH ={ kll ta2, ..., ta ments of t€l are distinct and tcci f toi' (mod n) I but perhaps for i=r.

~remer, as we have a ki' - kr' , where td dif#e&ces &e mod n and i C [ltr-11.

Then tcci', l$isr, is synanetric of tcr. as regards to the diameter that reduces to half the distance between tclr and to:'; hence tD is reflective.0

, , a2, . . . , a k

( a r r c1 ' the syrrpnetric of a. as regards tog , iC[l,r].

can belong to a).

1 . Because t is prime to n, the ele- k

- a,= a ' - a ' , then tar - h.= t(a -cr.)=t(ai' -a ' I = r i r

2. Let C be a reflective circulant (0,l)-mtrix. If [col c1 , ...,c hc [O,n-d , by the above remarks we obtain

1 is the first rcw of C, fran the relation ah = cTI n- 1

c = c i-h n-i

(3)

where Oliln-I and the indices are rrcd n. The relation (3) is an equality between two different elements except for i-h : n-i ( mod n ) ; in that case there exists an integer k 5 0 such that

2i = h + kn. ( 4 )

Let n and h odd, n=2t+l, h=2r+l and r't; then we have 2i=2r+l+k(2t+l). For k=l we obtain i=r+t+l and clearly this is the d y possible value.

If n is d d and h even, n=2t+l and h=2r, we have 2i=2r+k(2t+l) and i=r is the only possible value.

If n is even and h odd, n=2t and h=2r+l, we have 2i=2r+l+k(2t); this equation is clearly impossible.

368 N. Zagaglia Salvi

For n and h even, n=2t and h=2r, the equation 2i=2r+k(2t) has two solutions: i=r and i=r+t, corresponding to k=O and k=l.

THEXlFEM 2.1 - Let h,n be two integers , OSh5n-1. When n is odd, there are 2

circulant (0,l)-matrices C that satisfy the equation C P = C,, while , when n

n%

h

n + A is even, there are ? such matrices for h cdd and 2 for h wen.

n- 1 Proof. Let n be odd. By the above remarks, from (3) we obtain - equalities

n-1 + 1 = - n+? elements are 2 and one identity. So in the first row of C exactly - 2

arbitray.

circulant (0 , l ) -matrices that "+A As every e l m t can be 0 or 1, there are 2

h satisfy the equation C P = CT, for n odd.

n n-2 If n is even, the relations (3) correspond to 5 equalities for h odd and to 2 equalities and two identities for h even.

Hence, there are 2 when h is even.0

% n+32 distinct mtrices when h is odd and 2 such matrices

3. Recall that a difference set D = {d 1 l d2, . . . , $ I

rnod v such that every a f 0 (mod v) can be exspressed in exactly X ways in the form d - d. E a (mOa v) , where d. and d . are in D. We suppose O<X<k<v-l to e x c d c e A n degenerate confic$rationJ. It is well h a m that a difference set D is equivalent to a (v,k,X)-canfiqura- tion with incidence matrix a circulant.

is a k-set of integers

PROPOSITICN 3.1 - If C is a circulant incidence matrix of a (v,k,A)-configuration, it never satisfies the equation 8 = CT.

Praf.If [corcl, ... ,c row of c is

T indices are mod n. k t pl, p2, ..., p be the positions of the elements equal to 1 on the first row

of C. If C is an incidence matrix of a (v,k,X)-canfiguration, then fp ,p ,...,psi is a difference set. Since -1 is not a multiplier for a difference set [37 , then there exists no integer hL [O,n-I] such that { p,+h, p2+h, ...,pS+h) =

i.e. c d ~ e s not satisfy c ph = cT.a

h

] n-1 is the first r m of a circulant matrix C, the first

[cofcn-l, .. ., cl] , i . e . c. has been replaced by c-~, where the

S

1 2

-pl r -p2, .. ., -ps) r

Recall that a (v,k,X)-mtrix is a (O,l)-mtriY A of order v which satisfies

A % = ( k - X ) I + X J 2

v- 1 where 0 < X < k < v-1. It follows that detA = k(k-X ) .

Combinatorial Structures 369

A Hadamird matrix H of order v is a matrix of + I ' s and -1's such that H HpI . It has been conjectured [6] that a Hadarnard mtrix cannot be circulant for v > 4 . Remrk that we can define the notion of reflective circulant Hadmard matrices, analagous to the corresponding notion for the (0,l)-matrices.

PR(IPOS1TION 3.2- A Hadarmrd matrix of order v cannot be reflective circulant or coincident block circulant , for v > 4.

Proof. If H is a --aiiective circulmt or coiiiiident block circulant Hadamrd matrix, then K = -( H + J \ is a reflective o coincident block circulant (v,k,A )-matrix, w i t h k = $ v ~ r v ) and A = z( v ~f i 1 . By m-0~. 3.1 a reflective circulant matrix cannot be the incidence m a t r i x of a (v,k,X )-configuration, then cannot be a (v,k,X )-matrix. Moreaver, since it is clear that the determinant of a coincident block circulant matrix is zero, such a matrix cannot be a (v,k,X )-matrix. [I

f 2 f

The hrop. 3.2 is a partial answer to the problem, in digital c d c a t i o n s , of the existence of f in i te sequences of 1 and -1 {b.Iv w i t h the property that

1 1 their aperiodic auto-correlatim- coefficients

c . = C b.b

1 2 j Lv-1, should be 1 ,O,-I. It is w e l l knm that there is a one-to-one correspondence between such s e q u e n ~ s and circulant Hadamrd matrices [I ,pg. 981 . W e note that the results of the Prop. 3.1 and the Prop. 3.2 were implicitely proved in [ 3 ] .

V

3 i.1 1 i + j '

4. Le t G be a directed graph with n vertices;we car. suppose that its adjacency matrix is not symnetric. We call cyclic a graph w i t h circulant adjacency matrix. For a cyclic graph the mapping i + i + l , and its powers, are clearly autanor- phim by definition. Hence, these graphs always have the cyclic group of order n, C , as a subgroup of their autcmorphisn group. In f5lthe problem is posed h w t o give a procedure for determining the auk- morphism group of a cyclic graph. The follmdng Theorem 4.2 gives a subset of the graphs for which the autcmorphisn group i s exactly C

LDW?I 4.1- If C f 0,J is a circulant matrix of order prime n,then det C * 0. n'

Proof. I f [co , c , . . . ,c m t r i x are as i n 11). If det c = 0, there exists an e i envalue A Hence F( w ) = O , where F(x)= 1 c , x ' ~ with iq 40d n, I s a rational plyncxnial of degree <n. Then, P(x) is divisible by the minimum polynmial of w over the rational num

for n prime is knm t o be ~y ( X I = x or k=l; then the f i r s t T(XII of% is k[1 1 ... 1)and C is or 0 or J.0

7 is the f i r s t row of C, the eigenvalues of such

= c c .w '' = 0.

n- 1

7

bers. This minimum plyncxnial is the Ciy"lot0mic polyncmial of order n, which n- +...+ x+l. W e obtain F(x)=kVn(x), with k=O

370 N. Zagaglia Salvi

l%3m 4.2 - Let G be a directed cyclic graph having a prim number n of vertices and reflective circulant adjacency matrix. Then the automrphim group of G is precisely C .

n

Proof. Let G be a directed graph with a prime number n of vertices. If A is the adjacency matrix of G and Q is the permutation matrix Corresponding to an automrphim of G, we have

If A is reflective circulant, there exists a kt [I ,n-11 for h ich A Pk = \. men we have

4 , A Q = A . (5)

k k 4 y A P Q P T = A .

By LemM 4.1, (5) and (6) we obtain k k P Q P T = Q .

It follows prkuprk = Q, r c [I ,.I. Since n is prime, The integers rk, rt[l,n], mod n are distinct; then there exists jt[l,n]such that jk- 1 (md n). Hence we have P Q P circulant. 0

= Q, i. e. Q is T

In [5] it is proved that if a directed graph with a prim n m h r of vertices has C as autawrphism group, then the adjacency matrix A of G has distinct n eigenvalues.

COROLIARY 4.3- If A is the reflective circulant adjacency matrix of a directed graph G with a p r h n&r of vertices, then the eigenvalues of A are distinct.

If R is the mtrix corresponding to the pewtation i+ n-i, we have the fol- lowinq

COFOLTARY 4.4 - Let G be a directed graph with a prime number of vertices and reflective circulant adjacency matrix. Then G is self-converse an has exactly n ismrphistns with the converse, corresponding to the matrices RP , hC[l ,nJ . Proof. Let G be a directed graph with a prime nunher of vertices and reflective circulant adjacency matrix.

4,

holds, R Corresponds to an isanorphisn of G w'th t& conver- f?.y;ekm 4 .2 , every autmrphim of G corresponds to P ti , h Cll ,n] ;

then every matrix RP corresponds to an isamorphim of G with G'. It is well knm that the number of imrphisns of a self-converse graph G w i t h the converse GI is equal tc the nunber of autanorptusms of G.U

5. Recall that a tournwent T of order n is a directed graph in which every pair of vertices is joined by exactly one arc. A tournament matrix A is the adjacency matrix of a tournament T and satisfies A + % = J - I .

Combinatorial Structures 37 1

LetR the reflective circulant tournament matrix, of odd order n, that sati-

T' sfiesRP = A It is easy to prove that the I Is of the first row ofR are in even positions, the 0's in odd positions.

THEy)REM 5.1 - The autmrphisn group of the tournament 2 corresponding to P is C

n'

Proof. Let Q be the permutation matrix corresponding to an autmrphisn cf then we have

Q p Q = 4 ( 8 )

(9)

;

and, being A P =AT , it follows QTA P Q PT =a.

Since obtain P Q P = Q, i. e. Q is circulant. a

is regular, it follaws ~ n t R is nonsingular [ 9 ] ; by (3) and ( 9 ) w

T

In 183 the follawing theorems are proved. THFxlRFM 5 . 2 - If a tournament matrix A satisfies A Q = i+ where Q is a permutation matrix, then Q corresponds to a n-cycle.

PROPOSITIW 5 . 3 - Every tournament matrix A for wfiich AQ, where Q corresponds to a n-cycle, is also a tournament matrix, is penrutationally similar toR . Now, we have the following

THMRFM 5.4 - A reflective circulant toumamntmatrix is reflective circulant if and only it is pennutationally similar t o A . Proof. Let A be a reflective circulant tournament matrix of order n. Since the row sums of A are constant, it follaws that the tournmt corresponding to A is regular; hence n is odd. Bec use A is reflective circulant, there exists an integer kC[l ,n-1] for which A P = % . By Theorem 5.2 P corresponds to a n-cycle; then, by Proposition 4 . 3 A is per- nutationally similar toR. The converse is easily pr0~e~1.O

r: k

COROLWIY 5 . 5 - The automorphisn groq of a tournament with reflective circulant adjacency matrix is C .

Proof. The proof follaws imnediately fran Theorems 5.1 and 5 . 4 .

n

CD-Y 5.6 - The determinant of a reflective circulant tournament matrix of order n is -. Proof. Since the matrix of cdd order n R satisfies the relations + It =

n-1 2


J - I andRP =AT, it follm A (I + P)= J - I; then we have detR.det(I+P)= det(J - I). It is immediate to prove that det(1 t P)=2 and det(J - I)=n-I for n odd. BY Proposition 5.3, the prmf is capletai.CJ

[I] Bamrt, L.D., Cyclic Diffemce Sets (Lscture Notes in Mathematics, n. 182, Springer Verlag, 1971).

[2] Biggs, N., Algebraic Graph Theory (Cambridge University Press,1974).

131 Brualdi, R.A., A note on multipliers of difference sets, J. of bs. Nat. B. of Standards, ~01. 69 B (1965) 87-89.

[4] Davis, P.J., Circulant matrices ( A Wiley-Interscience Publication, 1979) . [5] Elspas, B. and Turner, J., Graphs with circulant adjacency matrices,

J. Cunbinatorial Theory 9(1970) 297-307.

[6] Ryser, H.J., Canbinatorial Mathematics ( Cams Math. Monograph, N. 74, New York, 1963).

Turner, J., Point-Symwtric Graphs with a Prim N d r of Points, J. Cmbi- natorial Theory 3 (1967) 136-145.

Zagaglia Salvi, N., Sulle matrici tome0 associate a matrici di pennuta- zione, Note dihtematica, vol. I1 (1982) 177-188.

Zagaglia Salvi, N., Alcune proprieta delle matrici torneo regolari, in: Atti del Convegno "Geanetria Cconbinatoria e di incidenza: fondamenti e applicazioni" La Mendola, 4-11 Luglio 1982 ( Editr. Vita e Pensiero) 635- 643.

[7]

[8]


OVALS I N STEINER TRIPLE SYSTEMS

Herbert Zeit ler Mathemat i sches I n s t i t u t , U n i v e r s i t g t Bayreu th

D-8580 Bayreu th , West Germany

T h i s a r t i c l e i s a r e p o r t o n some j o i n t works w i t h H. Lenz, B e r l i n [ 2 , 7 , 8 ] . Using t h e n o t i o n o f a r e g u l a r o v a l , S t e i n e r t r i p l e s y s t e m s o f o r d e r v = 9 o r 13 + 12n w i t h n E INo are c o n s t r u c t e d .

INTRODUCTION

L e t V w i t h IVI = v > 3 be a f i n i t e s e t and B w i t h I B I = b a set o f 3 - s u b s e t s of V. The e l e m e n t s of V are called p o i n t s , t h e e l e m e n t s o f B l i n e s o r b l o c k s . I f a n y 2 - subse t o f V i s c o n t a i n e d i n e x a c t l y o n e l i n e , t h e n t h e i n c i d e n c e s t r u c t u r e ( V , B , E ) i s ca l led a S t e i n e r t r i p l e sys t em of o r d e r v , b r i e f l y S T S ( v ) . Each p o i n t o f a n STS(v) l i e s on e x a c t l y r = ~ ( v - 1 ) l i n e s and t h e r e are e x a c t l y b = - v ( v - I ) l i n e s . The c o n d i t i o n v = 7 o r 9 + 6 n , n E INo

c i e n t f o r t h e e x i s t e n c e of a n S T S ( v ) . The se t of t h e s e " S t e i n e r numbers" v w i l l b e d e n o t e d by STS.

Any non empty s u b s e t H c V i n a STS(v) i s c a l l e d a h y p e r o v a 2 i f f e a c h l i n e of STS(v) h a s e x a c t l y e i t h e r t w o p o i n t s ( s e c a n t ) o r no p o i n t ( e r t e r n a 2 l i n e ) i n common w i t h H. Thus w e o b t a i n I H I = l + r . The complement = V x H t o g e t h e r w i t h a l l e x t e r n a l l i n e s of H forms a subsys t em S T S ( r ) and , v i c e versa, t h e complement of a subsys t em STS ( r ) i s a h y p e r o v a l i n STS ( v ) . P r e c i s e l y f o r any v = 7 o r 15 + 1211, n E INo t h e r e e x i s t s a n STS ( v ) w i t h a t least o n e h y p e r o v a l [ 2 1 , 1 4 I , [7]. The set of a l l t h e s e s p e c i a l S t e i n e r numbers w i l l be d e n o t e d by HSTS. The r e m a i n i n g S t e i n e r numbers i n RSTS = STS\HSTS a re v = 9

or 1 3 + 12n, n E INo.

1 1 6

is n e c e s s a r y and s u f f i -

374 H. Zeitler

1 . OVALS

1 . 1 DEFINITIONS

Any non empty s u b s e t 0 c V i n a STS(v) w i l l b e ca l l ed a n o v a l i f f t h e r e e x i s t s e x a c t l y o n e t a n g e n t a t e a c h p o i n t o f 0 ( t h i s t a n g e n t meets 0 i n e x a c t l y o n e p o i n t , t h e s o - c a l l e d p o i n t o f c o n t a c t ) and each l i n e of STS(v) h a s a t most t w o p o i n t s i n common w i t h 0. Some- t imes t h e complement 5 = V \ O t o g e t h e r w i t h a l l t h e e x t e r n a l l i n e s t o 0 w i l l b e r e f e r r e d t o as t h e “ c o u n t e r o v a l ” . The p o i n t s o f 0

w i l l b e c a l l e d t h e o n - p o i n t s , t h e p o i n t s o n t h e t a n g e n t s t o 0 w i t h t h e p o i n t s o f c o n t a c t d e l e t e d w i l l b e c a l l e d t h e e x - p o i n t s and a l l t h e r e m a i n i n g p o i n t s t h e i n - p o i n t s .

1 . 2 ENUMERATION THEOREMS

1 .2 .1 CLASSES OF POINTS

Each o v a l has e x a c t l y r = - ( v - l ) p o i n t s and t h e

c o u n t e r o v a l e x a c t l y r + 1 = ~ ( v + l ) p o i n t s .

1 2 1

P roof : Each p o i n t on t h e oval l i e s o n r l i n e s , of them i s a t a n g e n t .

c o r r e s p o n d i n g

and p r e c i s e l y o n e

1 .2 .2 CLASSES O F LINES

1 = ~ ( v - 1 ) t a n g e n t s , W i t h r e s p e c t t o a n o v a l 0 t h e r e e x i s t e x a c t l y r

1 1 e x a c t l y (;) = s ( v - l ) (v-3) s e c a n t s and e x a c t l y -r(r-I) =’(v-I) (v-3) e x t e r n a l l i n e s .

6 24

Proof : An e a s y c o u n t i n g a rgumen t .

1 . 3 THEOREM

The number of t a n g e n t s t h r o u g h a n e x - p o i n t o f a n oval - 0 i s e v e n o r

odd, a c c o r d i n g t o r b e i n g even o r odd .

Proof : L e t s b e t h e number of s e c a n t s , t t h e number o f t a n g e n t s t h r o u g h a n e x - p o i n t o f 0 . By 1 . 2 . 1 , t + 2 s = r , and t h e s t a t e m e n t f o l l o w s .

1 . 4 THEOREM

For any v E HSTS t h e r e e x i s t s a n STS(v) w i t h a t l e a s t one o v a l .

Ovals in Steincr Triple Systems 375

Proof : When v E HSTS t h e r e e x i s t s a n STS(v) w i t h a t l ea s t o n e h y p e r o v a l . Any r p o i n t s of t h i s h y p e r o v a l form a n ova l . These ovals are s p e c i a l o n e s , b e c a u s e a l l t h e t a n g e n t s m e e t i n e x a c t l y one p o i n t , t h e k n o t of t h e o v a l .

1 .5 DEFINITION

O v a l s a l l whose t a n g e n t s a r e c o n c u r r e n t will be c a l l e d k n o t o v a l s .

I f t h r o u g h each e x - p o i n t of a n o v a l t h e r e p a s s e x a c t l y two t a n g e n t s ,

t h e n t h e o v a l w i l l b e c a l l e d a r e g u l a r o v a l .

1 . 6 THEOREM

In an STS(v) w i t h v E HSTS t h e r e a r e no r e g u l a r o v a l s , i n an STS(v) w i t h v t RSTS t h e r e e x i s t no k n o t o v a l s .

Proof : I f v E HSTS, t h e n r = 3 o r 7 + 6n, n E I N o , and r i s odd. By 1.3, r e g u l a r o v a l s c a n n o t e x i s t i n s u c h a n S T S ( v ) . I f i n a n STS(v) w i t h v E RSTS a k n o t o v a l e x i s t e d , t h e n a d d i n g t h e k n o t t o t h e p o i n t s of t h e oval w e would o b t a i n a h y p e r o v a l ; t h e r e f o r e , v E HSTS, a c o n t r a d i c t i o n .

1 .7 THEOREM

For any r e g u l a r o v a L 0 i n at? STS(v) t h e r e e x i s t s e x a c t l y one i n -

p o i n t .

Proof : L e t e be t h e number o f e x - p o i n t s and i t h e number o f i n - p o i n t s of 0 . The e x - p o i n t s and i n - p o i n t s t o g e t h e r a re t h e p o i n t s o n t h e c o u n t e r o v a l 6. T h e r e f o r e , e + i = = r + '1. By 1.5 e a c h ex- p o i n t l ies on e x a c t l y two t a n g e n t s and o n e a c h of t h e r t a n g e n t s to 0 t h e r e a re e x a c t l y t w o e x - p o i n t s . Consequen t ly , e = r and i = 1 .

1 .8 THEOREM

For a r e g u l a r o v a l t h e numbers of l i n e s i n t h e d i f f e r e n t c l a s s e s

th rough p o i n t s of d i f f e r e n t c Z a s s e s a r e a s f o l l o w s :

316 H. Zeitler

s e c a n t s t a n g e n t s e x t e r n a l l i n e s

1 1 2 ~ ( r - 2 ) = T(v-5) 1 1 ? ( r -2 ) = T(v-5) 1 1 0 1 1

1

ex- po i n t

i n - p o i n t = x(v-1) zr = p - 1 ) Tr on- po i n t r - 1 =?(v-3) 1 0

i

The proof fo l lows immediately from t h e p reced ing theorems.

2 . THE M A I N THEOREM

For a n y v E RSTS t h e r e e x i s t s a n STS(v) w i t h at l e a s t one regular

o v a l .

Now t h e S t e i n e r numbers v E HSTS and v E RSTS w i l l b e c l a s s i f i e d i n a g e o m e t r i c a l way u s i n g t h e e x i s t e n c e o f S T S ( v ) ' s w i t h knot o v a l s and w i t h r e g u l a r o v a l s , r e s p e c t i v e l y . D i s rega rd ing t h i s d i s t i n c t i o n between o v a l s w e c a n summarize: For any S t e i n e r number v E STS t h e r e e x i s t s a n STS(v) w i t h a t least one o v a l . I n [ 2 ] t h i s theorem is proved by c o n t r a d i c t i o n . We now g i v e a d i r e c t proof by c o n s t r u c t i o n [8 I . Throughout t h e proof by a n o v a l w e always mean a r e g u l a r o v a l . Taking i n t o account theorem 1 . 6 , w e s t a r t w i t h a S t e i n e r number v E RSTS and c o n s t r u c t a n STS(v) w i t h a t l eas t one r e g u l a r o v a l .

2 . 1 THE POINTS

1 2 With v E RSTS, 101 = r = - ( v - l ) = 4 o r 6 + 6n, n E INo , and

1 161 = r + 1 = z ( v + l ) = 5 or 7 + 6n, n E INo. The re fo re , t h e number o f p o i n t s on t h e o v a l 0 is even and t h e number of p o i n t s on t h e c o u n t e r o v a l i s odd.

W e c o n s i d e r two r e g u l a r r-gons w i t h t h e same c e n t r e M whose edges have d i f f e r e n t l e n g t h s . Furthermore, t h e r-gons are r o t a t e d w i t h r e s p e c t t o each o t h e r by a n a n g l e f . The vertices Po,P, , .. . ,Pr-, of t h e i n n e r polygon a r e t aken a s on -po in t s , t h e v e r t i c e s Qo,Q, , . . . ,Qr- l of t h e o u t e r polygon as ex -po in t s and f i n a l l y t h e p o i n t M i s t a k e n a s t h e i n - p o i n t of t h e o v a l 0 . W e coun t t h e v e r t i c e s of each polygon c o u n t e r c lockwise . F igu re 1 shows t h i s i n t e r p r e t a t i o n f o r r = 1 6 .

Ovals in Steiner Triple Systems 377

0

M

F i g u r e 1

I t t u r n s o u t u s e f u l t o r e p r e s e n t t h e numbers r modulo 1 2 , Conse- q u e n t l y , r = 1 2 or 4 or 10 o r 6 + 12n. n E INo. I n t h i s pape r w e restrict o u r s e l f t o t h e case r = 4 + 12n. The p r o o f s i n t h e o t h e r cases may be done i n a s i m i l a r way.

2 . 2 CONSTRUCTION OF SPECIAL 3-SUBSETS

F i r s t of a l l w e c o n s t r u c t s p e c i a l p o i n t sets hav ing no c o n n e c t i o n w i t h a n ova l .

L e t a r e g u l a r r-gon be g i v e n wi th v e r t i c e s O,l, ..., r-1 and c e n t r e M .

( a ) F i r s t c lass of s u b s e t s I 1 There a r e = T(v-l) d i a m e t e r s i n t h e g i v e n polygon. The two end-

378 H, Zeitler

p o i n t s o f each d i a m e t e r t o g e t h e r w i t h t h e c e n t r e M f o r m a 3 - subse t o f t h e f i r s t class.

( b ) Second c lass o f s u b s e t s

L e t i , j w i t h i < j be t w o ver t ices o f t h e g i v e n polygon. Then t h e minimum o f j - i, r + i - j i s c a l l e d t h e " d i f f e r e n c e " o f t h e t w o v e r - t i c e s . By t h i s d e f i n i t i o n , t h e f o l l o w i n g d i f f e r e n c e s are p o s s i b l e : 1 , 2 , . . . , p - 1 = a ( " - 5 ) . w i t h t h e f i r s t c lass!) Now w e form o r d e r e d " d i f f e r e n c e t r i p l e s " (d l , d2 .d3 ) i n s u c h a way t h a t each o f t h e ment ioned d i f f e r e n c e s oc- c u r s a t m o s t once and d l + d 2 = d3.

Which d i f f e r e n c e s c a n o c c u r i n t h e t r i p l e s ? How many d i f f e r e n c e t r i p l e s o f t h i s k i n d are p o s s i b l e ? I n [ I ] , [ 2 1 , [ 3 1 , [ 6 1 t h e s e q u e s t i o n s are answered . Here w e g i v e t h e r e s u l t o n l y - w i t h o u t any p r o o f .

1 1 1 (The d i f f e r e n c e ?r is a l r e a d y e l i m i n a t e d

r = 4 t 12n, n E IN

( 1 , 5 n + l , 5n+21 , ( 2 , 3n, 3n+2) , (31 5n, 5n+3) , ( 4 , 3n-1, 3n+3) ,

(2n-1, 4n+2, 6n+ l ) , (2n , 2n+ l , 4 n + l ) .

1 4 The d i f f e r e n c e 3 n + 1 = -r i s m i s s i n g , any o t h e r d i f f e r e n c e o c c u r s

e x a c t l y once as r e q u i r e d . A l t o g e t h e r w e have 2n d i f f e r e n c e t r i p l e s .

The l e a s t v a l u e f o r r , i . e . r = 4 1 d o e s n o t p r o v i d e d i f f e r e n c e t r i p l e s . We o b t a i n t h e a f f i n e p l a n e AG(2 ,3 ) . It i s e a s y t o show t h a t i n t h i s sys t em t h e r e e x i s t r e g u l a r ovals ( f o r i n s t a n c e t h e se t o f p o i n t s ( x , y ) w i t h x , y E GF(3) and x2 - y 2 = 1 ) . I n t he f o l l o w i n g w e o m i t t h i s s p e c i a l case.

2.3 THE EXTERNAL L I N E S

Now t h e p o i n t s 0 , 1 , . . . , r - 1 i n 2 . 2 a r e t h e e x - p o i n t s Q o I Q 1 f . . . f Q , - l and M i s t h e i n - p o i n t o f a n oval 0.

Next, t h e e x t e r n a l l i n e s t o 0 are c o n s t r u c t e d u s i n g t h e d i f f e r e n c e t r i p l e s g i v e n i n 2.2. F o r better t y p i n g w e f r e q u e n t l y w r i t e ( P , x ) and (Q,x) i n s t e a d o f Px and Qx.

(3) The e x t e r n a l l i n e s c o n t a i n i n g M

The s u b s e t s o f t h e f i r s t c lass d e t e r m i n e d by t h e polygon d i a m e t e r s

Ovals in Steiner Triple Systems 319

are t h e e x t e r n a l l i n e s o f 0 c o n t a i n i n g M . We c a n w r i t e { M , ( Q , x ) , ( Q , x + 1 w i t h x t { O , 1 , . . . , r - I 1. W e a l w a y s u n d e r s t a n d t h e number x modulo r .

2

(b ) The r e m a i n i n g e x t e r n a l l i n e s

x t { 0 , 1 , ..., r - I }

( Q , x ) , ( Q , x + 2 ) , ( Q , x+3n+2

( Q , x ) , ( Q , x + 4 ) , (Q1x+3n+3

( Q , x ) , ( Q f x + 2 n ) , ( Q l x + 4 n + l

1 2 4 I n t h i s way w e o b t a i n a l t o g e t h e r r . 2 n = - - ( v - I ) (v-9) a d d i t i o n a l 3-

s u b s e t s o f p o i n t s . T h e s e l i n e s are e x t e r n a l l i n e s o f 0 , t o o . T o g e t h e r 1 1 w i t h t h e -r = - ( v - l ) e x t e r n a l l i n e s c o n t a i n i n g M w e have g o t 2 4

1 1 1 ~ ( v - 1 ) +? ; r (v- I ) (v-9) = ~ ( v - 1 ) (v-3) e x t e r n a l l i n e s . By 1 . 2 . 2 , t h i s i s t h e se t o f a l l e x t e r n a l l i n e s .

I n t h e d i f f e r e n c e t r i p l e s 2 . 2 t h e d i f f e r e n c e 3 n + 1 i s m i s s i n g . The a s s o c i a t e d p a i r of e x - p o i n t s ( ( Q , x ) , ( Q , x + 3 n + l ) ) w i t h x E { O , I , . . . , r - l } w i l l b e i n v e s t i g a t e d l a t e r on i n c o n n e c t i o n w i t h t h e t a n g e n t s .

2 .4 THE SECANTS

I n f i g u r e 2 w e have t h e s e c a n t s o f a n o v a l c o n t a i r . i n q t h e e x - p o i n t Q

f o r r = 16. B 0 -

* 0

0 0

M . , , , , , , , , ,. .... 0.. ..... .. . ..... ... . ..... -0 A

-0 D -- -z ----_

-0 --- -- -- 01,- -- -Q

- -+

-Q

--- 0- -

o--

o Q

F i g u r e 2

3 80 H. Zeitler

S e c a n t s c o n t a i n i n g a n e x - p o i n t

( a ) First c lass o f po lygon c h o r d s

W e s t a r t w i t h t h e polygon v e r t i c e s A and B ( f rom B t o A c lockwise ! ) hav ing t h e d i f f e r e n c e Tr . I n t h e se t o f po lygon c h o r d s o r t h o g o n a l t o t h e d i a m e t e r t h rough B w e choose t h o s e T(r-4) c h o r d s which a re n e a r e s t t o B ( f u l l l i n e s ) . Then t h e n e x t po lygon chord o f t h i s k i n d i s a polyqon d i a m e t e r CA. The e n d p o i n t s o f t h e polygon c h o r d s

1 1

choosen i n t h i s way have t h e d i f f e r e n c e s 2 , 4 , . . . ,-!-r-2, r e s p e c t i v e l y .

( b ) Second c lass o f po lygon c h o r d s

The polygon v e r t e x which f o l l o w s A g o i n g c l o c k w i s e from B t o A i s D . I n t h e se t o f po lygon c h o r d s p a r a l l e l t o CD w e choose t h o s e -r c h o r d s which are f a r t h e s t from B ( d o t t e d l i n e s ) . The e n d p o i n t s o f t h e s e polygon c h o r d s have d i f f e r e n c e s l 1 3 ~ . . . r ~ r - l ~ r e s p e c t i v e l y . One o f t h e s e c h o r d s i s a n edge o f t h e polygon. The e x - p o i n t l y i n g close t o t h i s edge i s Q .

The e n d p o i n t s o f e a c h polygon c h o r d , b o t h i n ( a ) and i n ( b ) , t o g e t h -

2

1 4

1

e r w i t h Q form a n o v a l s e c a n t . I n t h i s way w e o b t a i n a l t o g e t h e r 1 1 - ( r - 4 ) + ar = ; ( r - 2 ) s e c a n t s c o n t a i n i n g Q. By 1 .8 w e have found a l l 4 s e c a n t s o f t h i s k i n d .

The re i s no s e c a n t c o n t a i n i n g Q and t h e t w o o n - p o i n t s A and B as w e l l . T h e r e f o r e , t h e s e t w o p o i n t s must b e t h e p o i n t s o f c o n t a c t o n t h e t w o t a n g e n t s t h r o u g h Q . By o u r c o n s t r u c t i o n t h e d i f f e r e n c e o f A

and B i s e x a c t l y -r.

I n t h e same way it i s p o s s i b l e t o c o n s t r u c t t h e s e c a n t s t h rough any ex -po in t ( Q , x ) . Now w e c a n e x p l i c i t l y w r i t e down a l l t h e s e c a n t s c o n t a i n i n g a n ex- p o i n t ( Q r x ) :

1 4

1 1 ( Q l x ) , ( P , x + ~ r - l ) , ( P r x + - r + l ) } . 2

1 1 3 4 Fu r the rmore , (A,x) = ( P r x + , r ) , (B ,x ) = ( P , x + ~ r ) , ( C , X ) = ( P , x + - r ) .

We f u r t h e r see t h a t t h e t a n g e n t c o n t a i n i n g ( Q l x ) and ( Q , x + + ) h a s t h e p o i n t o f c o n t a c t ( B , x ) . The d i f f e r e n c e o f t h e s e t w o e x - p o i n t s i s 1 Tr . Again , x E { O r l t . . . , r - l } .

Ovals in Steiner Triple Systems 38 1

S e c a n t s c o n t a i n i n g t h e i n - p o i n t

The e n d p o i n t s o f a n y d i a m e t e r o f t h e i n n e r po lygon form a n oval s e c a n t t o g e t h e r w i t h M . I n t h i s way w e o b t a i n e x a c t l y zr s e c a n t s c o n t a i n i n g M. By 1 . 8 w e have found a l l t h e s e c a n t s o f t h i s k ind : namely, t h e y a re {M, ( P , x ) , ( P , X + ~ ~ ) 1 , x E { O , l , ..., r-1).

Adding t h e numbers o f oval s e c a n t s i n t h e t w o d i f f e r e n t classes w e o b t a i n r--(r-2) +-r = -r(r-1) = - ( v - l ) ( v - 3 ) .

1

1

1 1 1 1 2 2 2 8

2 .5 THE TANGENTS

When c o n s t r u c t i n g t h e e x t e r n a l l i n e s i n 2 . 3 some p a i r s o f e x - p o i n t s w e r e l e f t w i t h o u t a c o n n e c t i n g l i n e . The d i f f e r e n c e be tween t w o p o i n t s o f such a p a i r w a s d = -r = 1 + 3n. The c o n s t r u c t i o n o f t h e o v a l s e c a n t s i n 2 . 4 was done i n s u c h a way t h a t t h e d i f f e r e n c e o f t h e t w o e x - p o i n t s on each t a n g e n t g i v e s t h e number d . T h e r e f o r e , t h e oval t a n g e n t s are c o m p l e t e l y de t e rmined : { ( Q , x ) , ( Q , x + T r ) , ( P , x + z r ) } , x E { O t l r - . . f r - l } . Our c o n s t r u c t i o n i n case r = 4 + 12n w i t h n E IN i s comple t ed .

1 4

1 1

CONCLUDING REMARKS

Long ago P . Erdos a s k e d t h e f o l l o w i n g q u e s t i o n s : Which c a r d i n a l i t y i s t h e maximal one f o r a p o i n t se t M i n a n STS(v) c o n t a i n i n g no l i n e ? For which S t e i n e r numbers v may t h i s maximal case o c c u r ? A l l t h e s e q u e s t i o n s w e r e a l r e a d y answered i n [ 5 ] . H e r e w e relate t h e s e problems t o h y p e r o v a l s and o v a l s . The r e q u i r e d maximal c a r d i - n a l i t y o c c u r s i f each p o i n t o f M l i e s on s e c a n t s o n l y . T h i s y i e l d s I M I i; l + r . T h e r e f o r e , i n case o f maximal c a r d i n a l i t y M i s a hyper - o v a l . P r e c i s e l y , f o r any v € HSTS t h e r e e x i s t s a n STS(v) w i t h s u c h a s e t o f maximal s i z e l + r . I f s u c h h y p e r o v a l s do n o t e x i s t , t h e n 1M[ 5 r . The upper bound i s a c h i e v e d i f M i s a r e g u l a r oval . F o r a l l S t e i n e r numbers v E RSTS t h e r e e x i s t S T S ( v ) ' s w i t h se t s o f t h e r e q u i r e d maximal c a r d i n a l i t y r .

Another matter i s t h e i n v e s t i g a t i o n o f f i n i t e a f f i n e a n d p r o j e c t i v e s p a c e s w i t h r e g a r d t o ovals. W e men t ion some r e s u l t s : I n PG(d ,2 ) w i t h d > 2 t h e r e e x i s t e x a c t l y 2 ( 2 d+l - 1 ) k n o t o v a l s and no r e g u l a r o v a l s . I n AG(2,3) t h e r e e x i s t e x a c t l y 54 r e g u l a r o v a l s and no k n o t o v a l s . I n AG(d, 3 ) w i t h d 2 3 t h e r e e x i s t n e i t h e r k n o t

382 H. Zeitler

o v a l s n o r r e g u l a r ovals.

We n o t i c e t h a t t h e r e a re s t i l l many unso lved problems a b o u t o v a l s i n a n STS(v) . (The to ta l number of ovals i n a g i v e n STS (v) ; t h e number of non-isomorphic S T S ( v ) ' s w i t h o v a l s i f t h e S t e i n e r number i s g i v e n ; i n v e s t i g a t i o n o f automorphism g r o u p s ; ... ) F o r S t e i n e r sys t ems S ( k , v ) w i t h k Z 3 t h e r e are o n l y a few r e s u l t s c o n c e r n i n g o v a l s , h y p e r o v a l s and similar n o t i o n s [ 4 ] . E x t e n s i o n s o f t h e ErdBs problems t o g e n e r a l sys t ems of t h i s k i n d are unknown, too.

REFERENCES

H i l t o n , A.J.W., On S t e i n e r and s imi la r t r i p l e s y s t e m s , Math. Scand. 2 4 ( 1 9 6 9 1 2 0 8 - 2 1 6

Lenz , H . and Ze i t l e r , H . , A r c s and Ova l s i n S t e i n e r T r i p l e Sys tems, Combinator iaZ T h e o r y , L e c t u r e No t e a i n M a t h e m a t i c s 9 6 9 ( 1 9 8 2 ) 2 2 9 - 2 5 0 ( S p r i n g e r V e r l a g , B e r l i n , H e i d e l b e r g , N e w York)

P e l t e s o n , R . , E i n e Losung d e r be iden H e f f t e r s c h e n D i f f e r e n z e n - probleme, C o m p o a i t i o Math. 6 1 1 9 3 9 ) 2 5 1 - 2 5 7

Resmini , M. d e , On k - s e t s of t y p e (m,n) i n a S t e i n e r sys t em S ( Z , l , v ) , F i n i t e G e o m e t r i e s and D e s i g n s , L . M . S . L e c t u r e N o t e S e r i e s 4 9 ( 1 9 8 1 ) 1 0 4 - 1 1 3

S a u e r , N . and Schonheim, J . , Maximal s u b s e t s o f a g i v e n set hav ing no t r i p l e i n common w i t h a S t e i n e r t r i p l e sys t em o n t h e set, Canad. Math. BUZZ. 1 2 ( 1 9 6 9 ) 7 7 7 - 7 7 8

Skolem, T . , Some remarks o n t h e t r i p l e sys t ems o f S t e i n e r , Math. S c a n d . 6 ( 1 9 5 8 ) 2 7 3 - 2 8 0

Zeit ler , H . and Lenz, H . , Hyperovale i n S t e i n e r - T r i p e l - Systemen, Math. Sem. B e r . 3 2 ( 1 9 8 5 ) 1 9 - 4 9

Zeit ler , H . and Lenz, H . , Regu la r Ovals i n S t e i n e r T r i p l e Sys tems, JournaZ of C o m b i n a t o r i c s , I n f o r m a t i o n and S y s t e m S c i e n c e s , D e l h i ( to a p p e a r )

383

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Modena

B a r i

Lecce

C a t a n i a

Messina

Napol i

B a r i

Napol i

K a i s e r s l a u t e r n

Modena

B r e s c i a

P e s c a r a

Case r t a

Napoli

Roma

Bologna

Lecce

Roma

C a t a n i a

C a t a n i a

Lecce

Napol i

B a r i

Participan Is

A.M. P a s t o r e

S . P e l l e g r i n i

C. P e l l e g r i n o

G. P e l l e g r i n o

C. P e r e l l i Cippo

M . P e r t i c h i n o

G. P i c a

F . P i r a s

A . I . Pornil io

L. Porcu

L . Pucc io

G . Q u a t t r o c c h i

P . Q u a t t r o c c h i

G. Raguso

L . R e l l a

T . Roman

L . A . R o s a t i

H . G . Samaga

M . S c a f a t i

R . S c a p e l l a t o

E . Schroder

R . Schu lz

D . Sena to

H . Siernon

C . S o m a

R. S p a n i c c i a t i

A.G. S p e r a

R . S tangarone

K . Strarnbach

G . T a l l i n i

P . . T e r r u s i

J . Thas

M . Ughi

V . V a c i r c a

B a r i

Bresc i a

Modena

Pe r u g i a

B r e s c i a

B a r i

Napoli

Cagl i a r i

Rorna

Milano

Mess i n a

C a t an i a

Modena

Bari

B a r i

Buc a r e s t i

F i r enze

Hamburg

Roma

Parma

Hamburg

B e r l i n

Napoli

Ludwigsburg

Roma

Rorna

P a l e rmo

B a r i

Er langen

Roma

Bari

Gent

Pe rug ia

C a t an i a.

Participants 387

K . Vedder

A. Venezia

F . Verroca

R . V incen t i

A . Ventre

H . Wefelscheid

B. Wilson

N . Lagag l i a Salvi

H . Z e i t l e r

E . Z i z i o l i

R. Z u c c h e t t i

Gissen

Roma

B a r i

Pe rugi a

Napoli

Essen

London

M i 1 ano

Bay r e u t h

B r e s c i a

Pav ia

389

ANNALS OF DISCRETE MATHEMATICS

VoI. 1 : Studies in Integer Programming edited by P.L HAMMER, E.L. JOHNSON, B.H. KORTE and G.L NEMHAUSER 1977 v i i i + 562 pages

VoI. 2: Algorithmic Aspects of Combinatorics edited by B. ALSPACH, P. HELL and D.J. MILLER 1978 out of print

Vol. 3: Advances in Graph Theory edited by B. BOLLOBAS 1978 viii t 296 pages

Vol. 4: Discrete Optimization, Part I edited by P.L HAMMER, E.L JOHNSON and B. KORTE 1979 xii + 300 pages

Vol. 5 : Discrete Optimization, Part I1 edited by P.L HAMMER, E.L JOHNSON and B. KORTE 1979 vi + 454 pages

Vol. 6: Combinatorial Mathematics, Optimal Designs and their Applications edited by J. SWASTAVA 1980 viii + 392 pages

Vol. 7: Topics on Steiner Systems edited by C.C. LINDNER and A. ROSA 1980 x + 350 pages

VoI. 8 : Combinatorics 79, Part I edited by M. DEZA and I.G. ROSENBERG 1980 xxii + 310 pages

Vol. 9: Combinatorics 79, Part I1 edited by M. DEZA and I.G. ROSENBERG 1980 viii t 310 pages

Vol. 10: linear and Combinatorial Optimization in Ordered Algebraic Structures edited by U. ZWERMANN 1981 x + 380 pages

390

VoI. 1 1 : Studies on Graphs and Discrete Programming edited by P. HANSEN 198 1 viii t 396 pages

Vol. 12: Theory and Practice of Combinatorics edited by A. ROSA, G. SABIDUSI and J. TURGEON 1982 x t 266 pages

Vol. 13: Graph Theory edited by B. BOLLOBAS 1982 vi i i t 204 pages

Vol. 14: Combinatorial and Geometric Structures and their Applications edited by A. BARLOTTI 1982 viii t 292 pages

VoI. 15: Ngebraic and Geometric Combinatorics edited by E. MENDELSOHN 1982 xiv -+ 378 pages

Vol. 16: Bonn Workshop on Combinatorial Optimization edited by A. BACHEM, M. GROTSCHELL and B. KORTE 1982 x t 312 pages

Vol. 17: Combinatorial Mathematics edited by C. BERGE, D. BRESSON, P. CAMION and F. STERBOUL 1983 x t 660 pages

Vol. 18: Cornbinatorics '81 : In honour of Beniamino Segre edited by A. BARLOTTI, P.V. CECCHERINI and G. TALLINI 1983 xii t 824 pages

Vol. 19: Algebraic and Combinatorial Methods in Operations Research edited by RE. BURKARD, R.A. CUNINGHAME-GREEN and U. ZIMMERMA" 1984 v i i i t 382 pages

Vol. 20: Convexity and Graph Theory edited by M. ROSENFELD and J. ZAKS 1984 xii t 340 pages

VoI. 2 1 : Topics on Perfect Graphs edited by C. BERGE and V. CHVATAL 1984 xiv t 370 pages

39 1

Vol. 22: Trees and Hills: Methodology for Maximizing Functions of Systems of Linear Relations R. GREER 1984 xiv t 352 pages

Vol. 23: Orders: Description and Roles edited by M. POUZET and D. RICHARD 1984 xxviii t 548 pages

Vol. 24: Topics in the Theory of Computation edited by M. KARPINSKI and J. VAN LEEUWEN 1984 x t 188 pages

Vol. 25: Analysis and Design of Algorithms for CombinatoriaI ProbIems edited by G. AUSIELLO and M. LUCERTINI 1985 x t 320 pages

Vol. 26: Algorithms in Combinatorial Design Theory edited by C.J. COLBOURN and M.J. COLBOURN 1985 viii t 334 pages

Vol. 27: Cycles in Graphs edited by B.R. ALSPACH and C.D. GODSIL 1985 x t 468 pages

edited by M. KARONSKI and A. RUCIrjSKI 1986 approx. 5 10 pages

Vol. 28: Random Graphs '83

Vol. 29: Matching Theory L. LOVASZ and M.D. PLUMMEK 1986 in preparation

combinatorics 1984: finite geometries and combinatorial structures: colloquium proceedings: finite...

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