Combinatorics 1984: Finite Geometries and Combinatorial Structures: Colloquium Proceedings: Finite Geometries and Combinatorial Structures

COMBINATORICS '84

NORTH-HOLLAND MATHEMATICS STUDIES 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 CaliforniaInstitute 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

123

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

edited by

A. BARLOlTI Universita di Firenze, Firenze, Italy

M. BILIOITI Universitadi Lecce, Lecce, Italy

A. COSSU Universitadi Bart Bari, Italy

G. KORCHMAROS Universitadelta 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

All rights reserved. No part of this publication may be reproduced, storedin a retrieval system, or transmitted, in any form orbyanymeans, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87962 5

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Library of Congress Catalogingin-PublicationData

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. 1 1 1 . Title: Combinatorics eighty-four. IV. Series. V. Series: North-Holland mathematics studies ; 1 2 3 . QA167.158 1984 511l.6 85-3 1 121 ISBN 0-444-87962-5

PRINTED I N 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 Combinatorics 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

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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, importante 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 pubblicazione 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.

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ix

CONTENTS Preface G. TALLINI, Intervento di apertura a1 Convegno

V Vii

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

1

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

9

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

15

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

31

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

39

A. BICHARA, Veronese quadruples

57

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

69

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

85

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

99

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

105

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

107

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

115

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

125

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

137

Contents

X

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

143

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

159

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

171

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

185

G. FAINA, Pascalian configurations in projective planes

203

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

217

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

225

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

243

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

25 1

A. HERZER, On finite translation structures with proper dilatations

263

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

269

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

275

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

285

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

29 I

G. LO FARO, On block sharing Steiner quadruple systems

297

G. MENICHETTI, Roots of affine polynomials

303

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

311

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

331

Contents

xi

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

335

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

339

G. TALLINI, Ovoids and caps in planar spaces

347

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

355

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

363

H. ZEITLER, Ovals in Steiner triple systems

373

PARTICIPANTS

383

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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 Universith di Bari Italia

Larato

q , In this paper translation planes of odd order , are constructed. Their main interest consists in the fact that their translation complement contains At first these planes a group isomorphic to SL(2,5) 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 2 is not a near-field plane of order 11

2 5 1 q -1

.

.

1. ORDER

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

DESARGUESIAN PLANE OF

2 Set K = GF(q J , q odd. We may assume that the elements of K can be 2 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 x = c with m,b,c elements of K The affine the form y = mx+b or no coordinatiz@d by F is an affine Baer subplane; the image of subplane 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 K u { oo} of the form:

.

(1.1)

[

ap+b

I

a,b€K , p runs over F u { m }

j

or (1.2)

.

Let v and r be any two elements of K F o r any d E K such that gq+l = 1 , the set of all points (x,y) for which (1.3)

and

g

EK ,

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

i s t h e p o i n t - s e 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 h a s e q u a t i o n ( 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 o n l y p o i n t ( 0 , d ) a c c o r d i n g as ( 1 . 2 ) h o l d s o r does n o t hold. The a f f i n e Baer s u b p l a n e s with e q u a t i o n s 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 points of 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 a n e M(q) t a k e a s c i r c l e s t h e sets (1.1) and ( 1 . 2 ) o f elements o f K ~ { c o } . isomorphic t o I n o r d e r t o c o n s i d e r a n automorphism group o f $C d i s t i n g u i s h two c a s e s , a c c o r d i n g a s 51q+l o r 51q-1

.

CASE

51qtl

a

t h e r e e x i s t s an element

of

( c f . [4]). W e p u t b = (a-aq)-' and c such t h a t l e t u s c o n s i d e r t h e f o l l o w i n g a f f i n e mappings o f a

fl: and t h e i d e n t i t y Note t h a t < a ,

fl>

a x ' = bx

t

x , SL(2,5) , (cf.

& : x' = 2

,

: x ' = ax

,

cy

we

y' = y

141,

-

1. and

: mqtl

1 and

Ci+2

: (m-2a

t

. >

on t h e l i n e a t i n f i n i t y , r.l(q).

...

}

i(1-q) q 2 -1 q t l 2 -2 b c ( 2 b +1) ) = (2b +1)

-

: (C1)(C2C3C4C5C6)

, fl

,

a

( i = 0,

and

: (C1C2)(C3C6)(C4)(C5)

PROPOSITION 2 . - The group I' a c t s on t w o - t r a n s i t i v e 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 .

-

bqy

p. 1 9 9 ) .

PROOF Some l o n g and e a s y c a l c u l a t i o n s prove t h a t i t s e l f and a c t on t h e s e t $f as f o l l o w s

a

b q + l = 1. After t h i s

q+1 I n M(q) t h e group I' maps t h e c i r c l e C 0 :. = -l l e a v e s t h e s e t V = { C1 , C 2 , ,C6 i n v a r i a n t , where

PROPOSITION onto i t s e l f =

t

a5 = 1

y t = aqy

,p

PROOF (1.4) and (1.5)

SL(2,5)

such t h a t

K

cq+l

y ' = -cqx

Now o u r purpose is t o s t u d y t h e a c t i o n o f < a i . e . t h e a c t i o n o f I' = leaves invariant each of the PROPOSITION components corresponding with the derivation set

co

PROOF

B

-

The

q+l

*

components are the Baer subplanes

.

9

B

with equations g A straigh forward cal-

: y = rx g where rqtl = -1 and g runs over A g culation shows that U as well as p leaves each B invariant. g PROPOSITION 9. - The group < U , p >splits the set of the 6(q+l corresponding with the multiple derivation set C I U C 2 U (q+1)/2 orbits each of length 12.

components into

...

...

q+l

u c6

PROOF - Let H , (i = 1,2, ,6) be the set consisting of the q+l components which corrispond with the derivation set Then < a , p > acts on the Ci set { H 1 , ,H6 } in the same way as on the set { C1, . ,C6 } By Prcp. 2,

.

...

...

a,p>acts transitively on { H1, 3 2 is< a , l >with A = p a p a p , i.e. 2 A : x ' = c(2b -acq+l-aqb2)y




Y' = - ~ ~ ( 2 b ~ - a ~ c ~ + ~ - a b ~ ) x

V,Abatangelo and B. Larato

6

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 invariant and 1 maps E onto E shows that a leaves each E g g -g PROPOSITION 10.- The line-orbit of < a.8 > containing the line joining the origin to Ym has length 12.

.

-

.

-

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

.

3,

We point out that for q = 9 tions 8,9 and 10 yield:

,

which is the first non trivial case, Proposi-

PROPOSITION 11.- (i) The group r =/ splits the line at infinity of fC into one orbit of length ten and six orbits of length twelve; into six (ii) tie group r = / < - E > splits the line at infinity of f~ 3 orbits of length twelve and ten orbits of length one.

8

Now we state the following PROPOSITION 12.2,3). If SL(2,5) R or f C 3 . 2

Let (I be a plane obtained by derivation from a . (j = is an automorphism group of (I , then (I coincided with

-

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 C1 ,C2, , ,C6 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 i C E{Co ,C1, ,C6 } , then necessarily C = C ; on the other hand C cannot

...

...

0

stay on Hg because, by Prop. 8 , no orbit of than 10 in H 3 '

SL(2,5)

is long 10 or less

2 q , 5(q-1 , CONTAINING 4. - A TRANSLATION PLANE OF ORDER TRANSLATION COMPLEMENT AN AUTOMORPHISM GROUP ISOMORPHIC TO

IN ITS 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 of M(11)

,

,

y' =

c ~ m :- m l 1 = 0 , C; : (m

-

2

(8+7t)x + (2tt)y

9 is equivalent to the following chain:

we can check that

-2i 11 )

--

C' : m

P

1+1

(-2)

t

, i

ml1=0 =

1,2,

,

... ,5

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

, 9

determines a

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.

Translation Planes

7

Finally we want to remark that Pellegrino and Korchmlros plane surely is not 2 a near-field plane of order 11 (cf. [q, p. 8 8 ) , 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 Pellegrino and Korchmlros plane has an orbit of length 42 = ( H 1 on its line at 2 infinity

.

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

A.A.

P. Dembowski, Finite geometries (Springer-Verlag, Berlin-Heidelberg-New York. 1968). D.R. Hughes-F. Piper, Projective planes (Springer-Verlag, Berlin-Heidelberg-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-Heidelberg-New York, 1970).

T.G. Ostrom, Lectures on finite translation planes, Conf. Sem. Mat. Univ. Bari, n. 191, 1983. G. Pellegrino-G. KorchmSros, Translation planes of order of Discrete Math., 14 (19821, 249-264.

112, Annals

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

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Annals of Discrete Mathematics 30 (1986) 9-14

0 Elsevier Science Publishers B.V. (North-Holland)

SYMPLECTIC GEOMETRY,

QUASIGROUPS,

9

A N D STEINER SYSTEMS

L u c i e n Beneteau UER-M.I.G. Universite Paul Sabatier 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 g e n e r a l i z e d . I t t u r n s o u t t h a t t h e r e i s a c a n o n i c a l correspondence between e q u i v a l e n c e c l a s s e s o f non z e r o 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 l a s s e s o f r a n k ( 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 t h e s e designs and t h e r e l a t e d S t e i n e r q u a s i groups may be p r e s e n t e d as a s p e c i a l case o f a more g e n e r a 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 l g e b r a . As an i l l u s t r a t i o n o f these i d e a s we s h a l l d e a l c o m p l e t e l y w i t h t h e case ns6. F o r n=6 one o b t a i n s e x a c t l y 5 isomorphism c l a s s e s o f HTSs

.

1-INTRODUCTION

-

S e c t i o n 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 i d e , and t h e c u b i c h y p e r s u r f a c e quasigroups and t h e F i s c h e r groups on t h e o t h e r s i d e . We r e f e r t h e r e a d e r t o t h e l i t e r a t u r e f o r t h e c o n n e c t i o n s w i t h o t h e r p a r t s o f a l g e b r a and d e s i g n 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 ) . T h i s process i s n o t c a n o n i c a l . B u t i t a l l o w s t o g e t a l l t h e non a f f i n e HTSs whose 3 - o r d e r s e q u a l s t h e r a n k p . As u s u a l t h e r a n k 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 g e n e r a t o r subset. The e q u a l i t y s = p corresponds t o an e x t r e m a l s i t u a t i o n , t h e non a f f i n e HTSs o b e y i n g sdp, w h i l e t h e a f f i n e ones 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 v e n r a n k obey s=p-1 (see [ll). whose o r d e r i s minimal t h a t l e d us t o a problem o f s y m p l e c 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 symp 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 c o u n t i n g o r b i t s i n some s p e c i a l cases. F o r 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 t h e n 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 l a s s e s o f some HTSs. T h i s w i l l be used h e r e t o o b t a i n an e x h a u s t i v e l i s t o f t h e HTSs o f o r d e r g2187 whose r a n k s 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 c o r r e s o n d i n g S t e i n e r q u a s i g r o u p 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 l a w : 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 Steiner quasigroups can be a l g e b r a i c a 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 said 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 permut a t i o n of x,y,z ; t h i s i s equivalent t o the conjunction o f the commutativity 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 a m i l y o f l i n e s L such t h a t (E,L) be a Steiner T r i p l e System i s equivalent t o provide E w i t h a s t r u c t u r e o f Steiner quasigroup. So i n what f o l l o w s we s h a l l i d e n t i f y (E,L) w i t h (E,o). A H a l l T r i p l e System (HTS) i s a Steiner 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 p o i n t s i s an a f f i n e plane =AG(2,3). This a d d i t i o n a l assumption i s equivalent t o the f a c t t h a t the corresponding S t e i n e 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 Steiner quasi groups.

Let K be a commutative f i e l d . Consider an absolutely i r r e d u c i b l e cubic hypersurface V o f the p r o j e c t i v e space Pn(K). Let E be t h e set o f i t s non-singular K-points. Three p o i n t s x,y,z o f V w i l l be s a i d t o be c o l l i n e a r ( n o t a t i o n : L(x,y,z)) i f there e x i s t s a l i n e L containing 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 c y c l e ) . The best known case i s when dim V = l , and n=2 : V i s then a plane curve, i t does not contain 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 e x a c 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 symmet r i c . The set of the idempotent p o i n t s o f (E,o) i s the set o f f l e x e s ; i t i s isomorphic t o AG(t,3) w i t h t1. Assume t h a t K i s i n f i n i t e . We have the f o l l o w i n g f a c t t h a t we mention here without a l l t h e required d e f i n i t i o n s ( f o r a more complete account see Manin [9] pp. 46-57, e s p e c i a 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 s u i t a b l e 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 :

2

C o r o l l a r y : 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 Steiner quasigroup. A l l the f i b r e s A=p-l(e) o f p are isomorphic elementary abelian 2-groupsY and

(E,o) = . I x A ( d i r e c t product). Let us say t h a t a Fischer qroup i s a group o f the form G=<S> where S i s a conjugacy class o f i n v o l u t i o n s o f G such t h a t O(xy)h3 f o r any two elements x and y from S ( i n other 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 special Fischer qroup. I n any special Fischer group G t h e r e i s j u s t one class o f i n v o l u t i o n s S (namely, the s e t o f a l l the i n v o l u t i o n s from G ) , and S may be provided w i t h a g t r u c t u r e o f HTS by s e t t i n g xoy=xY=yxy(=xyx). We c a l l ,o) the HTS corresponding t o 6. This group-theoretic construction o f HTSs canonical. More precisely :

(5

i4

Theorem : Given any HTS E, the (non-empty) family 7 o f special Fischer groups whose corresponding HTS i s E admits : (i) a universal o b j e c t U ; any G i n i s o f the form G=U/C where CcZ(U). ( i i ) a smallest object I = U / Z ( U ) , which i s a l s o t h e unique centerless element o f F .

11

Symplectic Geometry, Quasigroups, and Steiner Systems

3-A PROCESS OF EXPLICIT CONSTRUCTION : L e t E be a v e c t o 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 eyements from G$(3f, say

(i)

.

) i j k lGi<j7 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, indecomposable. and f o r s=2nt2=p t h e r e i s j u s t one such system Jn

BIBLIOGRAPHY

C11 L. Beneteau ; Topics about Moufang loops and H a l l t r i p l e systems, Simon Stevin 54 (1980) 107-124. 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 79", Annals o f Discrete Math. 8 (1980) 229-232. C31 L. Beneteau ; Les Systemes T r i p l e s de H a l l de dimension 4, European J. Combinatorics (1981) 2, 205-212. [41 L. Beneteau ; H a l l T r i p l e systems and r e l a t e d t o p i c s . Proc. Interna. Conf. on Combinatorial Geometries and t h e i r applications, 1981 ; Annals o f Discrete Math. 18 (1983) 55-60. C51 G. Bol ; Gewebe und Gruppen, Math. Ann. 114 (1937) 414-431 ; Zbl. 16, 226. C61 Marshall HALL Jr. ; Automorphisms o f S t e i n e r T r i p l e systems, IBM J. Res. Develop. (1960), 406-472. MR 23AX1282 ; Zbl. 100, p. 18.

Symplectic Geometry, Quasigroups, and Steiner Systems

C7 Marshall H a l l J r . ; Group theory and block 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. C8 T. Kepka ; D i s t r i b u t i v e E i n e r quasigroups o f order 35, Comment. Math. Univ. Carolinae 19,2 (1978) ; MR 58 X 6032. [ 9 Yu. I. Manin ; Cubics forms, North-Holland Publishing Company, AmsterdamLondon ; Amercian E l s e v i e r Publishing Company, Inc. New-York (1974).

[I0 H.P. Young ; A f f i n e t r i p l e systems and matroid designs, Math. Z. 132 (1973) 343-366. MU. 50 # 142.

13

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

IS

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 g r o u p s g s G1, Gr of o b j e c t s ( p e o p l e , g o o d s ) c o n c e r n i n g a p r o p e r t y E ( o r p r o p e r t i e s E l , ..., E n ) and t o o r d e r them c o n c e r n i n g E s t a r t i n g w i t h t h e dominant g r o u p i n g . T h i s c a n 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 r o u p i n g i n one e l e m e n t ) and by j u d g i n g 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 " i s E - b e t t e r t h a n " o r " i s 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 r o u p i n g and one h a s 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 - o r d e r i n g f o r a l l t h e groupings.

...,

.

U s u a l l y t h e r e are t o o many i n t e r a c t i o n s , and t h e problem i s t o f i n d a b a l a n c e d 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 s h o u l d b e based. When I became c o n f r o n t e d w i t h t h i s problem i n a s p e c i a l c a s e , my p r o p o s a l w a s t o u s e g e n e r a l i z e d L a g u e r r e g e o m e t r i e s i n c a s e o f one s i n p l e n r o p e r t y E ( a n d g e n e r a l i z e d Minkowski g e o m e t r i e s for s e v e r a l p r q erties E l , . E n ) : T h e p a r a l l e l c l a s s e s o f p o i n t s o f a p l a n e La-guerre geometry may f o r i n s t a n c e r e p r e s e n t t h e g r o u p i n g s , and t h e s e l e c 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 t h e geometry. The f a c t t h a t t h r o u g h t h r e e F a i r w i s e 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 c o n c e r n i n g t h e chosen subclass of i n t e r a c t i o n s . A s a matter of f a c t those generalized L a g u e r r e 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 u n d e r 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 ] ) , o p t i m a l g e o m e t r i e s ( % l c k r , & h [4])and t h e y p l a y an i m p o r t a n t r81e i n c o n n e c t i o n w i t h optir,ial c o d e s i n c o d i n g t h e o r y ( H a l d e r , 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 u n d e r 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 decompos i t i o n of t h e s e t o f b l o c k s o f an o p t i m a l geometry s u c h t h a t a l l t h e components o f t h e d e c o m p o s i t i o n 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 s u c h a d e c o m p o s i t i o n i n a p r a c t i c a l c a s e one c a n 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 s u c h t h a t a l l o b j e c t s are i n v o l v e d i n a subtest. 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 o f a s e t o f okj e c t s c o n c e r n i n g 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

,...,

W.Benz

16

c e r t a i n c h a i n g e o m e t r i e s ( [ l ] ) . The c l a s s o f o p t i m a l g e o m e t r i e s can be i d e n t i f i e d w i t h t h e c l a s s o f T ( t , q , r , l ) . The Minkowski-m-strucm 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 t h a t a n e c e s s a r y 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 d e t e r m i n e t h e number of b l o c k s o f 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 ) X > 1 ) by a p p l y i n g p e r m u t a t i o n s e t s and s p e c i a l c l a s s e s (cases X = l , of f u n c t i o n s . 2 . Let q , r , t be i n t e g e r s s u c h that q > 1 and 2 I t 5 r. C o n s i d e r t h e matrix

where t h e o r d e r e d 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 p u t ( 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 g r o u p i n g s . 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 e v e r y 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 o f B ( t ) b l o c k s . We a r e t h e n i n t e r e s t e d i n t h e f o l l o w i n g p r o p e r t y of b a l a n c e S h a v i n g 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 g r o u p i n g s o f M t h e r e is e x a c t l y one b l o c k c o n t a i n i n g S.

(*) To every t - s e t

By T ( t , q , r ) ( o r T ( t , q , r , l ) w i t h 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 c e r t a i n t , q , r a r e known ( s . f o r i n s t a n c e H a l d e r , Heise [ 4 1 , Heise 151, H e i s e , 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 i s a b i j e c t i o n ( c a l l e d isomorphism) o f t h e s e t o f p o i n t s of T ( t , q , r ) o n t o t h e s e t o f p o i n t s o f T ( t l , q B , r t )s u c h t h a t t h e b l o c k s o f t h e first s t r u c t u r e are mapped o n t o b l o c k s o f t h e second s t r u c t u r e . S i n c e 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 samegmupi n g i f f t h e r e i s no b l o c k j o i n i n g them ( n o t e t L 2 ) isomorphisms map columns o n t o columns. Obviously, isomorphic T ( t , q , r ) , T ( t l , q t , r l ) c o i n c i d e i n t h e p a r a m e t e r s , 1 . e . t = t l ,q = q ' , r = r ' . I n [ l ] we have s t u d i e d c h a i n g e o m e t r i e s , The f i n i t e c h a i n g e o m e t r i e s of Laguerre t y p e ( [ l ] , p . 144) C ( K , L ) are s t r u c t u r e s T ( t , q , r ) . Here K i s a G a l o i s 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 w i t h L / N 2 K , where N d e n o t e s t h e maximal i d e a l o f L. The p a r a m e t e r s a r e Ni, r = ~ + 1 . The c l a s s o f c h a i n g e o m e t r i e s of g i v e n by t = 3 4 = != 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 o f m i q u e l i a n Laguerre p l a n e s . Two Laguerre g e o m e t r i e s C ( K , L ) , C ' ( K ' , L ' ) w i t h c h a r K $: 2 4 char K ' a r e isomorphic i f f t h e r e i s an isomorphism 0 : L -t L ' such t h a t o l K is an isomorphism o f K o n t o K 1 ( [ l ] , p . 176,

17

On a Test of Dominance

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

(kl

vl) + (k2

, v2)

:=

(kl

V1)

.

, v2)

:=

+

(k2

+

Hence L L , because of N n - l 0 , ( N 1 ) 2 = 0 , where r e s p e c t i v e l y N , N ' a r e t h e maximal i d e a l s of L , L , . We t h u s g e t two non isomorphic s t r u c y +, l ) , T I ( 3 , y n - 1 , y + l ) . tures ~ ( 3 , ~ - 1 The s t r a t e g i c decomposition we have announced i n s e c t i o n 1 c o n c e r n s t h e f o l l o w i n g c l a s s of s t r u c t u r e s T ( t , q , r ) which i s a s u b c l a s s o f t h o s e s t r u c t u r e s d e f i n e d i n H a l d e r , Heise [ 4 ] on pages 268, 269 by u s i n g l i n e a r forms. L e t K be a G a l o i s f i e l d GF(Y) and l e t V be a vector space o v e r K w i t h 1 < dim V < For an i n t e g e r t such t h a t 3 2 t < y + 1 now d e f i n e 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 t h e b l o c k s a r e given by

-.

By u s i n g Vandermonde's d e t e r m i n a n t i t i s e a s y t o check t h a t t h e p r o p e r t y of balance ( * ) is s a t i s f i e d f o r t . t-1)

Theorem 1: L e t A be t h e G a l o i s f i e l d G F ( y and l e t f E K[x] be t h e minimal polynomial of a p r i m i t i v e element 6 of A o v e r K . Assume V ~ , . . . , V ~ - ~ V . Then t h e s e t B ( v l , v t - 1 ) of b l o c k s

...,

r(a,vf(a)

t-1 t-1-v E v = l vva

+

) l a E

K ) U I(m,v)l

, v

E

V,

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

is a d i s j o i n t decomposition o f B ( t ) .

( N o t i c e t h a t t h e degree o f f is

t-1).

P r o o f . Let v,w be two d i s t i n c t elements of V. We l i k e t o show t h a t t h e two b l o c k s t(a,vf(a) + t(r:,wf(6)

+

o f B(v l s . . . , v t - l ) have no p o i n t i n common. Assume t o t h e c o n t r a r y t h a t (C,x),C E K ' , x E V , is a p o i n t i n b o t h b l o c k s . T h i s i m p l i e s 5$ because of v w. Hence

-

W.Benz

18

i . e . f ( 6 ) = 0 which i s n o t t r u e s i n c e f i s i r r e d u c i b l e o v e r K.‘Ihe s e t E(vl,. ,vtdl) c o n t a i n s a s many b l o c k s as t h e r e a r e e l e m e n t s i n V. The number of p o i n t s on a b l o c k i s 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 i s t h u s a p a r t i t i o r . o f t h e s e t of points.

..

...,

Now w e l i k e t o show B(vlI..

.

,V

t-1

)

fl

B(wl,.

. ., W t-1

=

8

i n c a s e t h a t t h e two o r d e r e d ( t - 1 ) - p l e t s ( V ~ , . . . , V ~ 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 h a t t h e b l o c k s

- ~ ) (,

~ ~ , . . . , w ~ - ~ )

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

v=l

(wu

-vu)a

t-1-u

= o

for a l l a E K.Because of t < Y + 1 t h e r e e x i s t p a i r w i s e d i s t i n c t e l e ments 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

...,

The Vandermonde m a t r i x P h e r e i s r e g u l a r b e c a u s e of # ( a l , a t:l} = = 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 i s n o t t r u e . - There are 6 c f V ) t - 1 many s e t s B ( v l , v t - 1 ) . Every B ( v l , . . . , v t - l ) c o n t a i n s #V many b l o c k s . S i n c e 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 ) t h e s e t

...,

U

B(vl,

...,v t-1 1

contains a l l the blocks. Example. C o n s i d e r K = GF(3). V = K and t = 3. A r e q u i r e d decomposit i o n h e r e i s g i v e n by

On a Test of Dominance

19

aApP bBqQ cCrR

aBrP bCpQ cAqR

aCqP bArQ cBpR

aArR bBpP cCqQ

aBqR bCrP CAPQ

aCpR bAqP cBrQ

aAqQ bBrR cCpP

aBpQ bC qR cArP

aCrQ bApR cBqP

3. Let q,r,t,nbe 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

...,(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, P

=

t(l,l,i),

...,

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):

W.Benz

20

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 t h e columns o f 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 w i t h wine of t h e 2 . , 3 . , 1 . , 4 . wine p r o d u c e r r e s p e c t i v e l y . Now c h e c k b o t h q u a l i t i e s a l o n g t h e f o l l o w i n g s e t of b l o c k s : bBqQ Ap aQ rcCQ brAQ qCaQ BcpQ

bqDS cpSD cBPs aAP s pBdR rCdR

rAPd BqPd brRD paRD bAsS ccss

bBRs CaRs CqSd ApSd aqDP rcDP.

( T h i s example is t h e m i q u e l i a n Minkowski p l a n e o f o r d e r 3 . ) Theorem 2 : Let K b e t h e G a l o i s f i e l d G F ( r ) and l e t n > 1 be an i n t e w i t h n f a c t o r s . The c h a i n geomeg e r . Denote by Ln t h e r i n g K x . . . x K t r y Z(K, Ln) is then a s t r u c t u r e T ( 3 , ( u + l ) n - l I ~ + l , n ) . P r o o f . a ) C o n s i d e r t h e f o l l o w i n g maximal i d e a l s

J.

:=

{(kl,

...,kn)

of Ln f o r i = l , . . . , n . A p o i n t of R ( p l , P,)

:=

E

Ln

1 ki

= 01

E(K,Ln) i s g i v e n by

{(rpl, rp2)

I r

E

R},

where R d e n o t e s t h e group of u n i t s of Ln and where p l r p 2 a r e e l e m e n t s o f Ln s u c h t h a t t h e i d e a l g e n e r a t e d 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 ) w i t h a = ( a l , . . . , a n ) , b = ( b l l a . . , b n ) K(an,bn)) can be i d e n t i f i e d w i t h t h e o r d e r e d n - p l e t s ( K ( a l l b l ) , of p o i n t s K ( a i , b i ) o f t h e p r o j e c t i v e l i n e n o v e r K . Moreover: R ( a , b ) I l i R ( c , d ) i f f K ( a i , b i ) = K ( c i , d i ) . Hence11 i i s an e q u i v a l e n c e r e l a t i o n on t h e s e t o f p o i n t s and t h e r e are ( y + l ) " - l many o r d e r e d ,Pn), Pl,..,,Pn e n , s u c h t h a t P i = c o n s t . Thus t h e n-plets (PI, is (y +l)n-I. number of p o i n t s i n an e q u i v a l e n c e c l a s s c o n c e r n i n g 1 1 The number o f e q u i v a l e n c e c l a s s e s c o n c e r n i n g 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(ai,bi) i n n , 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 ) with (y+l) 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 b e t h e 1 1 - e q u i v a l e n c e c l a s s e s .

...,

...

i

21

On a Test of Dominance d ) Two p o i n t s R ( a 1 , a 2 ) , R ( b l r b 2 ) a r e o b v i o u s l y c o m p e t i t o r s

iff

- I n c h a i n geometry [ l ] i t i s proved t h a t t h r o u g h p o i n t s A , B , C such t h a t

lal a 2 1 c R .

three b1 b2

t h e r e i s e x a c t l y one c h a i n 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 t h e element o f R . Chains a r e hence g l o b a l

...&.

( n o t e t h a t any c h a i n c o n t a i n s u + l points)ofM,, i n t e r a c tions Define t h e s e t o f c h a i n s t o be t h e s e t B ( 3 ) . Then ( i ) h o l d s t r u e f o r t = 3 . I n o r d e r 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 , . , n i be j d i s t i n c t i n t e g e r s . Consider e q u i v a l e n c e E ( i j ) of the relations ( ( i l , IIij r e s p e c t i v e l y . Then c l a s s e s E(il),

..

..

...,

...,

...,

i j in (Pi, f o r j = l , ...,n . For i f we f i x t h e components i l , 6 n , t h e n t h e number of t h e remaining n - p l e t s i s (Y + l ) n - j .

...,P n ) ,

Pi

Remark: A b i j e c t i o n o f t h e s e t o f p o i n t s o f 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 i n v e r s e images o f b l o c k s a r e b l o c k s . A s a s p e c i a l c a s e of a theorem of S c h a e f f e r [ l o ] t h e automorphism group o f Z ( K , L n ) i s known for Ln s e m i l o c a l , #K > 3 and char K#2(inmse t h a t K i s f i n i t e , o b v i o u s l y , Ln must be s e m i l o c a l ) : T h i s i s t h e group p r L ~ ( 2 . L ~ ) -. TheZ(K,L2) are the m i q u e l i a n Minkowski p l a n e s , 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 o f q . Moreover: If D i s t h e p o i n t i n t e r s e c t i o n of j ( l 5 j 5 n ) M n ) such t h a t no two of them b e l o n g t o d i s t i n c t columns ( o f M1, t h e same M V t h e n

...,

P r o o f : The formula i s t r u e f o r j=1. Assume now 2 2 j 2 n. Let E, be a column o f M, f o r v = l , . . . , j - 1 and l e t C1, Cr be t h e columns o f M j : Observe t h a t t h e C 1 , C r a r e p a i r w i s e d i s j o i n t and t h a t t h e i r union i s t h e whole s e t of p o i n t s . Hence

...,

and t h u s a

j-1

=

r

...,

a.. T h i s p r o v e s t h e theorem.

J

A t t h e b e g i n n i n g of s e c t i o n 3 we r e q u i r e n > l f o r a s t r u c t u r e T ( t , q , r , n ) . But o b v i o u s l y t h e s t r u c t u r e s T ( t , q , r ) of s e c t i o n 2 c a n be c o n s i d e r e d

a s s t r u c t u r e s T ( t , q , r , l ) , s i n c e ( i i ) p l a y s no r61e i n c a s e n=l. The s t r u c t u r e s T ( t , q , r , Z ) have been s t u d i e d e x t e n s i v e l y i n t h e l i t e r a t u r e i n c a s e q = r . See f o r i n s t a n c e t h e r e s u l t s i n C e c c h e r i n i [ 3 1 ,

W.Benz

22

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 Mn is given by MI,

...,

Proof. If b is a point, denote by [b]i the column of Mi through b. Ct of M1.We now like to define Consider t distinct columns C1, sets D1, Dt.Those sets (but not their cardinalities) will depend on certain points al,a2,. Put D1 := C1. In case Du(l 1, of Theorem 2 , we g e t a s t r u c t u r e T ( 2 , v n Y 1 , Y , n ) by t h e d e r i v a t i o n proce s s . T h i s s t r u c t u r e can be d e s c r i b e d as follows: Take t h e K n , K = = GF(Y), and d e f i n e the columns of M i t o be t h e h y p e r p l a n e s X i = = const ( i = l , n ) . The b l o c k s a r e given by t h e l i n e s

...,

D.

For i n t e g e r s t , X , r such t h a t 2

5

t 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. F o r given points a,b write a b iff they are in the same colMp. This is an equivalence relation and since umns of M1, # (C1n n C ,) = X,Ci a column of Mi (i=l,...,n) the equivalence hrl be a block classes contain exactly X points. Let H := {hi, 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

...

...,

...,

...,

...,

...,

.

...,

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

...,

...

...,

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 q i n @(t,X,r) such that 9(il) = jl,...,T(it) = jt.

W.Benz

28

R e m a r k s . 1 ) With t h e examples @ ( 2 2 , 3 ) ,4 ( 3 , 2 , 4 ) one c a n c o n s t r u c t s t r u c t u r e s T ( 2 , 2 ~ 3 " - ~ , 3 , n ) ( p u t r2 = = r n =: S3) , = r n =: S 4 ) i n case n > 1. T(3,2.4"-l,4,n) (put r2 =

...

...

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 g e o m e t r i e s T ( t , X , r , l ) and b e c a u s e o f t h e n o n - e x i s t e n c e o f T ( 3 , X , A + 2 ) , X odd, ( s . H e i s e [ 5 ] ) , t h e r e do n o t e x i s t T ( 3 , A . ( X + 2 ) " - l , A + 2 , n ) f o r A odd a n d n E N a c c o r d i n g t o Theorem 6 . 3 ) By a p p l y i n g 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 d o e s n o t e x i s t a T ( 3 , 1 0 , 1 3 ) ) . T h 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 A2 < (A-l)(r-t+2). n- 1 4 ) A s f a r as t h e number o f b l o c k s o f a T ( t , A r , r , n ) is c o n c e r n e d , Theorem 5 , 6 l e a d t o a new p r o o f o f Theorem 4 : A s h a r p l y (r-t+r) = t - t r a n s i t i v e p e r m u t a t i o n s e t on a r - s e t c o n t a i n s r . ( r - I ) = t ! ( r ) many e l e m e n t s . Hence

...

f

b e c a u s e of # O ( t , X,r) = A t . T h i s remark d o e s n o t c o n c e r n 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 i s d e t e r m i n e d by t h e p r o o f of Theorem 4.

References

W.

Benz, V o r l e s u n g e n iiber Geometrie d e r A l g e b r e n . S p r i n g e r - V e r l a g ,

Berlin-New York 1973. K . A . Bush, O r t h o g o n a l a r r a y s o f i n d e x u n i t y . Ann. Math. S t a t . 23 ( 1 9 5 2 ) , 426-434. 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 , 4 0 ( 1 9 6 6 ) , 218-221.

H.R. H a l d e r , W. H e i s e , K o m b i n a t o r i k . H a n s e r V e r l a g , Munchen Wien 1976.

-

W.

H e i s e , E s g i b t k e i n e n o p t i m a l e n (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 , L a g u e r r e und Minkowski-m-Strukturen.

Rend.

1st. Mat. Univ. T r i e s t e I V ( 1 9 7 2 ) . W. H e i s e , P . Q u a t t r o c c h i , S u r v e y on S h a r p l y k - T r a n s i t i v e S e t s o f P e r m u t a t i o n s and Minkowski-m-Structures. A t t i Sem. Mat. F i s . Univ. Modena 27 ( 1 9 7 8 ) , 51-57.

t8l

P . Q u a t t r o c c h i , O n a t h e o r e m of P e d r i n i c o n c e r n i n g t h e non-exis t e n c e o f c e r t a i n f i n i t e M i n k o w s k i - m - s t r u c t u r e s . J o u r n . Geom. 13 ( 1 9 7 9 1 , 108-112.

On a Test of Dominance [ 91 H. -J. Smaga, Dreidimensionale reelle Kettengeometrien. Journ. Geom. 8 (1976), 61-73.

[ l o ] H. Schaeffer, Das von Staudtsche Theorem in der Geometrie der Algebren. J. reine angew. Math. 267 (1974), 133-142.

29

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

ON n-FOLD

31

B L O C K I N G SETS

Albrecht Beutelspacher and Franco Eugeni Fachbereich

Mathematik der Universitat Mainz F e d e r a l R e p u b l i c o f Germany I s t i t u t o Matematica Applicata Facolta' Ingegneria L'Aquila , 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 bounds f o r t h e number o f c o m p o n e n t s i n a n nfold blocking set i n p r o j e c t i v e and affine spaces. 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 any element of 9 ' ( a n y " l i n e " o r "block") 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 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 sets of P . Any B = { B , ,Bz B, 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 While blocking sets have 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 .

,..., 1

.

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 of o r d e r q h a s a n n - f o l d b l o c k i n g set, 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 order 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 sets i n i m p r o v e s t h i s r e s u l t . Later 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 p r o j e c t i v e and a f f i n e spaces. Let B be a projective or affine any t-dimensional space. A set B of 2 i s s a i d a t-blocking set i f I-B. subspace of H c o n t a i n s a t l e a s t a p o i n t of B and a p o i n t of A s e t B = { B, B,} of n mutually disjoint t-blocking sets is s a i d an n - r x d t-blocking set of 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 maximal number n of components of an n-fold t-blocking s e t i n PG(r,q) or AG(r,q). We shall prove 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 number. 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 prove the following fact:Given a positive particular, we shall 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 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 Mason [ l o ] ,

set"

in

a

A . Beutelspacher and F. Eugeni

32

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, L e t S b e a n S ( 2 , k , v ) n-fold blocking set, then n s k - 2 .

S t e i n e r system. I f

S

admits

an

,...,

a n n-fold b l o c k i n g set of S .Consider PROOF. D e n o t e b y { B, B,} 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 with a t l e a s t one point of Bi , i t r e s u l t s IB,I 5 r , where 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 I B i l r=(v-l)/(k-1) = 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 one 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 t w o p o i n t s o f Bi i s t o t a l l y cannot c o n t a i n e d i n B i . S i n c e B i h a s a t l e a s t t w o p o i n t s , {B, , , . , , B n } .,n I b e a n n - f o l d b l o c k i n g s e t , S o , IBiI 2 r + l f o r any 1 E 1 , 2 , ( T h i s i s a l s o a c o n s e q u e n c e o f Theorem 1 i n 171). T h e r e f o r e ,

..

.

n

v =

E I Bz, I( r + l ) n . i -1

On t h e o t h e r h a n d , we h a v e v - 1

= r(k-1).

Toghether we get

n(r+l)sv=r(k-l)+l Hence n s k - l - ( k - Z ) / ( r + l )
2 ) , then i t h a s a l s o a f u n c t i o n s bp ( n , t , q ) and t-blocking set. Consequently, t h e r e e x i s t b a ( n , t , q ) such t h a t PG(r,q) ( o r AG(r,q)) c o n t a i n s an n-fold t-block i n g s e t i f a n d o n l y i f r < b p ( n , t , q ) (0: r < b , ( n , t , q ) ) (112.2). Clearly, bp(n'.t,q) 1x1. This contradiction proves Lemma 2.0 LEMMA 3 . Let L be a line parallel to HI and denote by 5 a normal 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 M u (HI is contained in a maximal clique through L. 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

PROOF. Clearly, 5'

=

l f l l = f(0)

- (d-l)(c+l) + x.

n

Therefore, m u [HI is contained in a maximal clique. Since kl u {HI, Lemma 3 is proved.0 LEMMA 4 . If

f11nf12

=

fll

and

fi2

L

€

are different maximal cliques, then

(Hl.

PROOF. Because /)Il secting lines L1

f12

and €

M1

and

are maximal cliques, there exist interL2 € 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 From hypothesis ( 3 ) we obtain therefore

If11CJfi21 Ifill

flll

H

in

h1nfi2.

5 h(L,H) + ItL,HII 5 n+c+2, and /112 1 =

-k

On the other hand

I

L

+

I"1uf12

+

IA1n M 2 1

Q

n+c+2

+ e.

we have IM2

->

2(n -

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

together we get a con radiction 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 suppose 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

Embedding Finite Linear Spaces

43

for 3 . Because there exists no claw of order d+l, we have f(0) = 0. From Lemma 1 we set d y ) = nd + x d, and f(y)(y-1) x - d l+c). y=l y=2 so I d d f(y) 2 nd + x - d f(y (y-1 f(1) = nd + x - d y=2 y=2 > nd + x - d + x d(l+c) = nd - d(c+2) + 2x. Put S = {L1, L d f , and define I?,’ as the set of all lines L E S-{L. 1 which are parallel to H ahd intersect every line of SILill Obviously, the sets I?; are distinct; therefore,

if

-

-

-

-

...,

-

-

lfil-[HII

+...+

(fid-IHII

1fi;I

+...+

IfiiI

Ihl-IL1ll +...+ Ihd-ILdll = f(1) + d. Assume that there is another maximal clique fld+l. Then f(1) + d + n - (d-l)(c+l) + x -< f(1) + d + Ihd+l-IH)I 5 Ihl-IHII +...+ [Ad+ 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. Since f(1) > nd-d(c+2)+2x, we conclude n 2 (2d-l)(c+l)-2x. Hence implies e 5 0. But in view of d 2 2 and h(L1,L2) condition ( 6 ) -> 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 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 I; are parallel classes of S. x ’ = x+z. 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 satisfied (note that x‘ replaces x and x is replaced by x+z). Obviously, the conditions (l), (21, (7) of 1.1 are fulfilled. Now, let L be a parallel of H with k = ntl-d’. Since L intersects kL(d-l) parallels of H and H kas nd+x+z para lel lines, we get

c

A . Beutelspacher and K.Metsch

44

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 e l 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 € fi 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 c Z i q u e is a maximal set fi of mutually parallel lines with ] M I t(L) for at least one line L E f i . A clique M is called norrnaZ, if I f i l = n. By we denote the set of all cliques of S . For p € p and 18 € j we define p M , if p 1 L for at least one line L of M . we put For h E

-

51(h)

= IF)'

Z,cm

=

I

fi' € j r mnh' = @ I ,

G1(M) u { M I , 4 p I ) = Ip For every normal clique M we define

I

p E p, p

1.

MI.

Now we can dFfine the incidence structure S ' = ( P d , L the following way: pI'L * P I G for all p € p r L € L ; p I' 5 ( f i ) p E 4(M) for all p 6 p , G ( f l E 2; MI'L L E A for all M E j, L E L ; M I' * fi E G ( M ' ) for all ill E /3, 5 ( f i ) 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.

-

Q

~(4')

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 E A. clique f i l with

Embedding Finite Linear Spaces

45

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 fl 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 . ~ , 1.~ t(L'). f l ~ Now condition of Theorem 2 . 1 gives M 3 = M . contraaicting IhI < t(L) 5 I f l j I . O

(a)

1

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 fl 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 n+l n+l v-1 = (kL.-l) = (n+l)n di. 1 i=l i=l In view of (b) and Lemma 1 we conclude

2

n+l

n+ 1 I{fi

=

€

13,

Li E M I 1

i=l Together, our assertion fol1ows.U

=

1

di.

i=l

LEMMA 3. Let fl be a normal clique, and denote by L a line with = n+l-d, and l e t t denote the number of points p L 4 h. Put on L with % h . Then there are exactly 1-t cliques which are disjoint to fl and contain L. In particular, we have 0 5 t 5 1. PROOF. L has kL-t points p with p M . Therefore, there are exactly m := I f l l - (kL-t) = d + t - 1 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 m . for i f j . Obviously, for all i. Therefore, by Lemma l(b), hi I I { M ' I h ' E /3, M n f l ' $. @, L E f l ' l l = m. Now, Lemma l(a) shows 1 { h ' I h ' E 13, h n h ' = @, L E h ' l l = d - m = 1 - t.0

kb

Q

...,

LEMMA 4 . Denote by fl a normal clique. Then (a) l G 2 ( M ) l 5 1. In other words: There is at most one point p with P 1. f i . (b) l G ( h ) I = n+l. PROOF. (a) Assume that there are two distinct points p and q with h. Let L be the line which passes through p and g , and dep,q fine t as in the preceding lemma. Then t 2 2 , contradicting our Lemma 3.

+

A . Beutelspacher and K. Metsch

46

M = iL1,

(b) If

...,Ln)

and

k L , = n+l-d. 1

I G 2 ( M ) I = 1 Ip

I p

€ p

M}I = v

Q

then n

- 1 i=l

(n+l-di).

+

n'+n+l.O

From Lemma 1 we get

and the assertion follows in view of

v

=

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: (i) There is a unique clique in G ; , ( A ) containing L. If G 2 ( h ) f 0, then L is not incident with the point of 4 ( M I . (ii) NO clique of G ( M ) contains L, I G , ( M ) I = 1 an& L is incident with the point 4 (&). (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

02

LEMMA 6. Let

M1

and

M2

4(A1) = 5 ( A 2 )

(a)

be two distinct normal cliques. Then A 1 n M2 =

0

0.

(b) 14(M1) 5 ( M 2 ) I = 1 0 Mln M 2 9 0. PROOF. (a) One direction is obvious. Suppose therefore M1n M 2 = 0. Then M € 51(fll); hence, by Lemma 5(b), cl(hl) c Gl(M2). SirnilarZy, we have G1(h2) c G l ( f l l ) , hence equality. 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 L. Since 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 Pi have 5,(fll) = 5 , ( M 2 ) . (b) In view of (a), one direction is obvious, Let us suppose A 1 n M 2 f 0. Then fil and f12 intersect in a line L.Define T = r M I ki € ,E, M nMl = 0 fl n M 2 } . G ( M 2 ) , we have Since M1, M 2 [ 5(M1)

.

+

I5(A1) n 5 ( A 2 ) I = IM E

F

= rM E

i~ I

=

I

MnMl

=

MnM1 =

ia

= MnM2)1

$11

-

+ 152(M1)n52(M2)1

li-I + 142(fil)n52(M2)1

- li-l + 1 5 2 ( M 1 ) n 5 2 ( M 2 ) 1 .

IG1(fl1)l

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 ( 4 ) Because no clique of G1(hl) contains two lines of M 2 , we have' 171 = IA2-ILfI = n-1, and so

T

.

l5(fl1)n5(fi2)l since

IG1(M1)I + 1

=

=

151(f11)l - ( T I = n - (n-1) = IG(M1)I = n+l.

IT1(M1)I

=

1,

41

Embedding Finite Linear Spaces

152(fi1)I

=

1

G2(M1)

and

= 52(fi2).

G 2 ( f i , ) = S 2 ( f i 2 ) , no line of fi2 is incident with the point fi 1. As in case 1, this implies I T ( = I f i -ILlI = n-1. Since I 1= 1 we now have I G , ( A , ) I = n-1, and t6is imp1 es

15(fi1)nG(f12)l

=

lGl(fil)l - I T 1

+ lG2(fi1)nG2(fi2)l

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 incident2wi&h the point of G ( f i l l . In view of L € f i l L', and so 171 = ( M 2 -{L,L')y = n-2. This shows again

= 1.

Case 3 .

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

-

f i 2 which is we have L $.

I T 1 = 1.

Since IG2(fii)I 5 1, we have handled all cases. Thus. Lemma 6 is 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 is a linear space with n2+n+l lines (Lemma 2(c)) and Lemma 7, S 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 P 1 t - - - r P n + 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 b > n', this shows that there are at least two 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 S , and G(p,) = tpl,...,pn+ll = L, contradicting 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 I 0 1 throughspaces are the projective planes, we will assume A out. 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 is a line which has at least one parallel line,LtEen 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 (kL,-l)g 1. (n-a)a parallels. On the other hand, L intersects exactly k L m n = (n+l-a)n lines. Hence, 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 8 e k t e by 4 the set of all lines parallel to L. Then (rp-kL) = (v-kL )a [ M I (n+l-a) 5 x k X = -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) IMI 2 L - ,= na- + ( 2 - g - 1 )+~ n+l-Z n+l-Z n+l-Z Our hypothesis yields n+l-Z > gZ(Z-5) + 1 - a 1. (a-1)3(Z-g-l), It follows therefore I M I 2 ng + (2-2-1)s. b = 1 + kL*n + I M l 5 n2+n+l + (a-g-l)~, i.e. z 5 (2-g-1)a.m Suppose b 5 n’+n+l and assume that following conditions: (1) n > i ( ~ 2 - 1 ) (~’+~-2g+2 + ) T1 z ( f i - 1 ) z I

THEOREM 3.2.

S

satisfies the

49

Embedding Finite Linear Spaces

+ 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 + k2 ( ~ - l ) z , or ( 2 ) n > 2(Z-1)(Z2-Z+l)

n 5 2(Z-l)(Z2-Z+1) + 2(Z-l)z. (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, 4 = g and x = 0. Then d 2 d 2 4, and our hypothesis ( 1 ) shows PROOF.

a

-

> (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 in a projective plane of brder n, contradicting 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 1 -

then

S

+ 3.4. If b > n’+n+l, then n 5 7(a2-l)(a2+z-2a+2 f-OROLLARX -_ Ta(a l)(a-g-l)g. PROOF. Since b > n2+n+l, there is no line of degree n+l. Assume 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). 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 1. ~ ( ~ - 1 1 2 n > $ ( ~ ’ - 1( )~ ‘ + ~ - 2 a + 21.) L2 ( ~ 2 - 1 ) -> -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 ) ( ~ ’ + ~ - 2 g + 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

(%‘-a)

-

A . Beutelspacher and K. Metsch

50

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 ba and (b) If

=

(n-c+a)t +

bc = n2+n+l - c

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

-

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

b = n2+n+l, then (c-a)’t2

-

(c-a)[(n+l)(c-a)+n]t

+

cn(n+l-a) = 0,

or JD t = t to 2(c-a)

‘

where n+l 2(c-a) and

D

=

[ ( ~ - a + 1 ) ~ - 4 c ]+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 1 n > zc(c-1)(c2+3c-1), then S is embeddable in a projective plane P of order n. Moreover, 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 ) ~

Embedding Finite Linear Spaces

51

is a perfect square. PROOF. First we show that S is embeddable in a projective plane of order n. From our hypothesis we get 1 + zc(c-l)(c-2). 1 n > 7(c2-l)(c2+c) (*) 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 (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,. so

n 5 c2-2c+l = (c-l)2I contradicting ( * ) . I NOW (1) and (2) follow immediately from ( * ) . Hence S is embeddable in a projective plane P of order n. In particular, b 5 n2+n+l. Let L and n be as above. We distinguish two cases. II. C a s e 1 . All long lines are contained in Then t 5 1 and s o t = 1. From Lemma 3 . 5 we get b = b + b = n’+n+l - c. a c L in P which are not Hence there are exactly c lines L1, lines of S. Since b = n2 + 1 III , 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 L1,. ,LC‘ 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. Adding 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.

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 Baersubplane of P, then P-c meets the conditions of 3.7(c). c, COROLLARY 3 . 8 . Let a and c be positive integers with 2 5 a 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: ~) - 2 n > T1( C Z - ~ ) ( C ‘ + C - ~ ~++ c3z’+4(c-a+1)2 (1) and 1 n > z(c2 -1)(c2+c-2a+2). (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 = x 2 + 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 f(t) = [(t-a)(c-a)+l-a]t - [t(c-a)+l-cl(t-c) (3) n+l-a n+l-c Obviously, f(t) is a polynomial of second degree with negative coefficient in t‘, which has its maximum in t = (n+l)(c-a) + n 2(c-a) From (2) we get (4) f(0) = - c(c-l) > -1, and n+l-c n a(a-1) > c-a-1. f(-) c-a = f(n+l) = c-a - n+l-a First we show that S is embeddable in a projective plane of order n. If z 5 0, this follows from ( 2 ) and Corollary 3.3. Therefore, we may assume z > 0. Then

.

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

53

Embedding Finite Linear Spaces

Z(C*-C3+3C2+C-4) 1 + 1 < 51 -(c‘-c3 +3c2+c-4)+1-2c 2

c3-c2+4

c3-c2

Thus I

and c3-c2+4 2 = f(t) c 5 f ( t ) - c < 4(c,-cd)(c-n+l)’ (5) In view of ( l ) , this implies 1 1 n > -(c2-1)(c2+c-2a+?) + Zc(c-1)z. 2 On the other hand, Theorem 3.2(b) yields

-

-

c.

n 5 2(c-l)(c2-c+l) + z(c-1)~. 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 constitute a parallel class of s , every line L of degree n+l-d is contained in exactly d parallel classes nl(L),...,IId(L). Furthermore, 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 + -n 2 L . 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 with h . = n. Therefore the lines of S is a j l € {l,...,n+l{ through p . form a parallel ciass n ’ with exactly n elements. deAotes the number of long lines in n ‘ , then If s ’ 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 c-1, and now we are in case ( b ) of Corollary 3.8. Case 2 .

Consider

is a divisor of

Every parallel class of S has at most n elements. a short line L of S . ThenC n’+n+l-c 5 b = 1 + k n + ( I I I ~ ( L , )5~ -n2+n+l-c. ~) i=l

C

A . Beutelspacher and K. Metsch

54 b = n’+n+l-c

Hence

and

...

Ini(L)I = n (i E [l, ,cl) for every short line L. In particular we have f(t) = 0. By (4) we obtain t < A. c-a Consider now o long line L of S . We get a n2+n+l-c = b = 1 + k n + ( Ini(~)I-l)5 n’+n+l-a. i=l 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 n we have (t-a)(c-a)+l-a < n+l-a On the other ha.nd, (6)

.

-

=,

we get from (2) that (t-a)(c-a)+l-a 2 -a(c-a)+l-a > -c(c-1 +l-c = -c2+1> -(n+l-a), so, I(t-a)(c-a)+l-al < n+l-a. NOW, ( 7 ) implies t-a)(c-a)+l-a = 0, in particular -a v = (n+l-a)(n-c+a), and t = a(c-a)-1 c-a Toaether with [ (c-a)(n+l)+n] + c(c-1) (n+l-a) 0 = f(t) = t 2 (c-al2 - t.(c--a) (n+l-a)(n+l-c) 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. n f a-1, we obtain a(c-a + 2(a-l)(c-a) ( ~ - 1 = ) ~0 , and so c = a + Ja-1. Therefore, there is a positive integer x satisfying a = x2+l, c = x’+x+l, v = (n-x2)(n-x), b = n’+n+l (x2+x+l), ba = x’+x+l)(n-x), and bc = n2+n+l - (x2+x+1) ba. Moreover. if n is a parallel class corresponding to a point of P outside S , then one of the following possibilities occurs: Since

-

-

n contains x2 long and n-x2 short lines; V n+3.-a , and n consists of lomg lines only.

(I) 1x1 = n, and (11)

1 !1

=

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 (81 n-l , b4 = n2+n-3 - (2t-3jft-4) n-3 and b = n2+n-3 + f(t) with f(t) = (2t-5)t - (2t-3)(t-4). n-1 n-3

Emhediiiug Finite Linear Spaces

55

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 From n-1

f(t) 5 3. 1 (2t-5)t and f(t) 2 f(4)

=

n 2 5

t 1. 4; consequently,

we get

12 n -l > 0

or

f(t)

2 f(n+l)

=

2

-

1-l n

1.

with s E (1,2,3}. By (l), we obtain 3n+2r JD with D = (9-4s)n2 + (16s-36)n + 52-12s. t = 4 Now we distinguish three cases.

Hence

f(t)

=

s

C a s e 1 . s = 3. Then D = -3n’ + 12n + 16 C a s e 2 . s = 2. Then D = n’ - 4n

we get

+ 28

=

2

0, and therefore

(n-2)’

+

24. Since

n = 5. C

is a perfect square,

n = 7.

C a s e 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 spaces, Arch. Math. 44 (1985), 557-568. [2] Bose, R.C.: Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963), 389-419. [ 3 ] Bruck, R.H.: Finite nets 11. Uniqueness and imbedding, Pacific J. Math. 13 (19631, 421-457. [ 4 ] Hall, M.: Projective planes, Trans. Amer. Math. SOC. 54 (19431, 229-211.

A . Beutelspacher and K. Metsch

56

[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 fundamen-

tal characterization theorem of R.C. Bose on partial geometries, Rend. Sc. fis. mat. e nat. 59 (1975), 86-90. Lincei

-

[7] Vanstone, S . A . , The extendability of (r,l)-designs, in: Proc. third Manitoba conference on numerical math. 1973, 409-418. [ 8 ] De Witte, P., On the complement of a triangle in a projective

plane, to appear.

Annals of Discrete Mathematics 30 (1986) 57-68 0 Elsevier Science Publishers B.V. (North-Holland)

57

VERONESE QUADRUPLES Alessandro B i c h a r a D i p a r t i m e n t o d i Matematica 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 conics i n a p r o j e c t i v e plane i s generalized s t a r t i n g f r o m Buekenhout o v a l s . T h i s l e a d s t o t h e d e f i n i t i o n o f a Veronese quadruple which i s c o m p l e t e l y c h a r a c t e r i z e d as a p r o p e r 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 r o p e r 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 s e t , whose elements a r e c a l l e d p o i n t s , L i s a p r o p e r 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 c o n t a i n s b o t h 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 r o p e r subspace o f

2,

H consist o f col-

if

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 r e 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 r o p e r PLS

5

c o n t a i n i n g p r o p e r 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 l a n e o f o r d e r g r e a t e r t h a n 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 ] P,

of lines i n

P. F o r a l i n e 1 i n

we d e f i n e

al

= t

Such a

!I,s

nl

1

: s E i31.

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 l a n e

A . Biclzara

58

(three pairs [l,sl

s,,

s,,

s,

1,

[l,s,],

[ l , s , ] a r e 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

a r e c o n c u r r e n t i n P ) . Such a p l a n e i s isomorphic t o t h e dual p l a n e

o f P. Def ine

PI =

Next, f o r p

a

P

E

In

1

: 1 E DI,

3 define

t[l,sl,

pE1,

PES,

1, S E 01

I f a Buekenhout o v a l [ 2 ] B ( p ) i s d e f i n e d on t h e p e n c i l F ( p 1 o f l i n e s i n P t h r o u g h p, t h e n t h e s t r u c t u r e o f l i n e a r space lows: t h e t h r e e p a i r s [ l , ,sl

1,

[lp,sz

as f o l P [ l 3 , s 3 1a r e c o l l i n e a r i f e i t h e r an i n -

1,

( B ( p ) ) can be g i v e n t o

a

v o l u t i o n o f B ( p ) i n t e r c h a n g e s li ans s ( i = 1,2,3), o r 1 , = l 2 = 1 ). Therefore, i 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). 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

I n what f o l l o w s ,

o v a l B ( p ) i s g i v e n . 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 b e l o n g i n g t o some elment i n P,u

P 2 . Hence

the 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 r o p e r 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 ass o c i a t e d w i t h t h e f a m i l y i B ( p ) : p~ 71 o f Buekenhout o v a l s .

I n o r d e r t o c h a r a c t e r i z e t h e Veronese space o f a p l a n e a s s o c i a t e d w i t h a c o l l e c t i o n o f Buekenhout o v a l s , 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 o w i n g c o n d i t i o n s , (i) The p a i r S = ( P , L ) i s a p r o p e r i r r e d u c i b l e PLS. ( i i)

P

and

P,

a r e two d i s j o i n t f a m i l i e s o f p r o p e r subspaces such t h r o u g h

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 ,

(1.1)

F o r any n E P , , and any a E P , e i t h e r

(1.2)

Any two d i s t i n c t subspaces i n common.

P

i

ann =

(i= 1,2)

0

and

o r any l i n e i n a meets n . have p r e c i s e l y one p o i n t i n

Veronese Quadruples

59

Through any p o i n t t h e r e a r e a t l e a s t t h r e e elements o f P I u P ,

(1.3)

most two elements o f P I

.

I f a p o i n t p i s on two elements o f

(1.4)

and a t

P,, t h e n p i s on e x a c t l y one element

o f PI. Any t h r e e elements i n Y',

(1.5)

meeting t h e same element i n P , have a common

point. Any t h r e e elements o f P I meeting t h e same element i n P, meet e v e r y e l e -

(1.6)

ment i n 7 , i n c o l l i n e a r p o i n t s .

P , i n c o l l i n e a r p o i n t s then

I f t h r e e elements i n 7 , meet a subspace i n

(1.7)

t h e r e e x i s t s an element i n

P, h a v i n g 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 ' , ?',

,PI2) i s a bijection f : P

+

p ' such t h a t

( 1 .a)

Both f and f - ' map l i n e s o n t o l i n e s .

(1.9)

Both f and f - ' p r e s e r v . t h e two c o l l e c t i o n s o f p r o p e r subspaces. It i s easy t o check t h a t t h e Veronese space ( P , L , P ,

.P,

) o f a projective

( 7 , f l ) a s s o c i a t e d w i t h a f a m i l y i B ( p ) : p E S ) o f Buekenhout o v a l s i s

plane P

a Veronese quadruple. Furthermore, i f

P can be c o o r d i n a t i s e d by a (commutative)

f i e l d K and each B ( p ) i s a s s o c i a t e d w i t h a c o n i c i n P , t h e n ! p z L P I $, 1 i s i s o morphic t o t h a t p a r t o f t h e c u b i c s u r f a c e Mt i n PG(5,K),

representing the conics

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. - If Q = (P,L,P,,P,

i s a Veronese quadruple, t h e n t h e r e e x i s t s a p r o -

j e c t i v e p l a n e o f o r d e r g r e a t e r t h e n 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 o v a 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 t h r o u g h p. Furthermore, Q i s i s o m o r p h i c t o t h e Veronese space o f P a s s o c i a t e d 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. PLS and

?1

Let Q and

=

( ? , / . , P I ,?,)

be a quadruple i n which ( P , L ) i s an i r r e d u c b l e

f', a r e two f a m i l i e s o f p r o p e r subspaces such t h a t t h r o u g h any

S t h e r e i s a t l e a s t one subspace o f P I U ,y' line of -

f i l s axioms ( 1 . 1 )

,..., ( 1 . 6 ) .

Suppose moreover t h a t Q

If S i s f i n i t e , then also (1.7) holds.

ul-

A . Bichara

60 2. SOME PROPERTIES

OF VERONESE QUADRUPLES

L e t Q = (P,.L, PI,?',)

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 ment i n

P , passesand by A t h e s e t o f a l l p o i n t s

2 t h r o u g h which e x a c t l y one e l e i n 2 t h r o u g h 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 r e non-empty and P = A uV. __ Proof.

By (1.3),

2

t h r o u g h any p o i n t p i n

a t most two elements pass of

P I . I f p IE A,

i.e. t h r o u g h p a t most one element passes o f P I

exist i n 8,

c o n t a i n i n g it; t h e r e f o r e ,

two elements

t h r o u g h p p r e c i s e l y one subspace passes

o f P , ( s e e ( 1 . 4 ) ) and p e V . Consequently, P Next, A #

, then

A uV.

=

,

0

w i l l be proved. Take q E P and v , a subspace i n P , t h r o u g h q,

( b y t h e p r e v i o u s argument such a subspace e x i s t s i n P I ) . S i n c e ( P , L ) i s a prcjpe r e r PLS and

of P,

V,

i s a subspace a p o i n t q, e x i s t s i n

t h r o u g h q 2 . Obviously,

~ l ,

f

P\

vl. Let

and by (1.2) t h e p o i n t

IT,

n2

be an element belongs t o

n,na,

A; t h u s , A # 0.

ai

Finally, V #

0 will

P2, i

passes (see ( 1 . 3 ) ) .

E

Assume

= 1,2,

a, =

az

and l e t

be proved. Through b o t h q

CI

If a , #

be an element i n

a2,

and q 2 a t l e a s t one element then

C I , ~CI,

i s a p o i n t i n V.

P2 t h r o u g h a p o i n t q o f f

a , ; thus

a n a , E V and t h e statement i s proved.

The n e x 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

t h r o u g h 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.

If n € P , ; then I n n V I ~ l .

P r o o f . Assume

TI

c o n t a i n s 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 , t h r o u g h p , two d i s t i n c t subspaces a , and a : o f

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

ly, t w o subspaces a,and

subspaces

a., a ' . , 1

1

P2 pass which share j u s t t h e

a; e x i s t s i n P,meeting

i = 1,2,

o f P , share a p o i n t b y (1.5).

belongs t o a, na; so t h a t a , n a ~ > t p , , p , ] ;

TI.

Similar-

a t p, and n o t skew w i t h

IT. The

Therefore, p I =

a c o n t r a d i c t i o n , s i n c e p 1 # p 2 . The

statement f o l l o w s .

V I . If T E P , , a E P , , then e i t h e r P r o o f . Assume __

nna =

0 o r l n n a l 2 3.

a n n # 0; t h u s , a p o i n t p e x i s t s i n

a t h r o u g h p and q a p o i n t i n a n o t on

1 (since

a

anv.

L e t 1 be a l i n e i n

i s a p r o p e r subspace i t i s n o t

Veronese Quadruples coincident with

1;

61

q e x i s t s ) and q ' a p o i n t on 1 . By ( 1 , l )

hence,

-

S and - o b v i o u s l y t h r o u g h q and q ' meets n a t a p o i n t p ' E spaces

and

TI

If

EP,,

TI

then e i t h e r

a€$,,

P r o o f . By pr0p.s V and V I ,

if

. The

p 1 and p , o f A l i e on n n a

nna =

0 or

111

nna

By prop.

nanA.

I V , through p

ses o t h e r t h a n n. Moreover, meet

a

( a t l e a s t a t pi,

hence, p l ,

sub-

T,

resp.

EL.

l i n e 1 i n L t h r o u g h them i s c o n t a i n e d i n n n a ,

i'

i = 1,2,3,

nn = p , .

TI

1 , t a k e a p o i n t p, i n

n o =

p r e c i s e l y one subspace

The t h r e e subspaces

and b y axiom (1.6) meet

p , a r e c o l l i n e a r and

p,,

2 3.

# 0 t h e n a t l e a s t two d i s t i n c t p o i n t s

Tna

as two subspaces meet a t a subspace. To p r o v e t h a t TI

p ' # p. The

o f ( P , L ) meet a t a subspace which c o n t a i n s t h e l i n e t h r o u g h 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 ,

VII.

the l i n e

nna

n A c l . If n

T

T., 1

TI

i

o f $,pas-

i = 1,2,3

i n c o l l i n e a r points;

n a =

T I ~ G ~ tAh e, n 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 then,

by prop. V,

nna

nV

=

q, whence

nna =

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 ,

all

nna,

1 U L q ) . The subspace n n a o f ( $ , L )

q €1 o t h e r w i s e any l i n e i n

nn

(I

join-

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 .

3. THE PLANE (P2,B)

Let TI

be an element i n P 1 . By prop. V I I ,

TI

e v e r y subspace a o f , ' Y

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

veeting

P 2 i s defin-

ed B(n) = { a € $ , : ( 1 n n ~ L 1 .

VIII.

If

elements i n Proof. -

Ti

E

P I , t h e n B ( T ) i s d i s t i n c t f r o m P , and c o n t a i n s a t l e a s t two

P,; hence, By prop.

P, passes which meets a p o i n t p, i n

TI\

B ( n ) i s a p r o p e r subset of

IV,

t h r o u g h any p o i n t p

n at a line 1 i n

1 t h e r e i s a subspace

O f course, ~ , E B ( T I ) and s i n c e a , € B ( n ) ,

,in

P2. TI

a t l e a s t one element

a2

other than

a1

P , meets

TI

Through

IB(TI)I,Z.

P 2 . Thus, any sub-

=

i n a l i n e i n L , so t h a t , by axiom (1.5);

i n V t h r o u g h which a l l t h e elements i n P , pass. Therefore, axiom ( 1 , 4 ) ,

B(n).

o f P, ( p , € a , and p,BaI).

To p r o v e 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 ) space o f

(I,

L ( s e e prop. V I I ) ; t h u s ,

of

t h r o u g h q p r e c i s e l y one element

n' E

PI

a point q exists

s i n c e I B ( T I ) ~ by ~ ~ ,

passes.

TI'

contains t h e

A . Bichara

62

p o i n t q which i s on a l l elements i n P 2 ; hence,

i s met by e v e r y subspace o f

P , a t a l i n e i n L t h r o u g h q (see prop. V I I ) . Next, l e t q ,

and q z be two p o i n t s

o t h e r t h a n q and n o n - c o l l i n e a r w i t h i t . By prop.V,

q , and q , b e l o n g t o A

in

9 '

and t h r o u g h t h e n t h e subspaces and

TIi

#

IT,

. Through

TI,

ses, The subspace u meets

and n ,

TI,

the point q' =

meets

p r e c i s e l y one element a EP? pas-

n,nn,

~

l

'

(since

' 1 . Thus, by axiom (1.11, t h e l i n e 1

Consequently, q , i s on 1 , and s i n c e

IT,.

l a r argument, q, E elements i n ?,, TI';

TI

, TI, #

TI,

on

a t a p o i n t . Since 1, i s c o n t a i n e d i n n, i t passes t h r o u g h t h e p o i n t

11'

IT'^

q1 =

P 1 pass; moreover,

of

a t a l i n e 1 , i n L and i s n o t skew w i t h

TI,

no e l e m e i t i n 8 , i s d i s j o i n t f r o m a

, resp.

. The subspace

D

TI

D.

By a s i m i -

a E P , passes t h r o u g h q, t h e p o i n t on a l l t h e

t h u s i t c o n t a i n s q, q l ,

hence, I q,q , , q , ~ c u n

q , l i e s on

I,CU,

and q 2 . These t h r e e p o i n t s a r e a l s o on

a c o n t r a d i c t i o n s i n c e these p o i n t s a r e n o n - c o l -

I,

l i n e a r (see prop. V I I ) . The statement f o l l o w s . I f n , , n 2 ~ P I yIT, # TI^

IX.

, then

P r o o f . Through t h e p o i n t p

~

prop. I V ) . O b v i o u s l y , l i n e 1, E L and meets n 2 Since

IB(n,)nB(n21I

= n,nn,

a t a p o i n t pi E n 2 . Since p '

TI^ and TI^

a u n i q u e element

E

P,

passes (see

U E B ( T , 1 n B ( n , 1. Take D ' ~ B ( T I , ) ~ B ( T I , D' ) ; meets a t a l i n e 1 , L. ~ 1

IT,

1.

=

i s contained i n

El

, and

a'

and by axiom ( 1 . 1 )

1 ,C n,, p ' E n, whence p ' E TI^

meet j u s t a t p, p ' = p. Therefore,

at a

TI,

Tin,.

a ' passes t h r o u g h p and i s

c o i n c i d e n t w i t h u t h e unique element t h r o u g h p i n P 2

.

Thus,

D

= B(n,)nB(n,)

and t h e statement i s proved. By prop.

IX,

B = { B ( n ) : n c P , } o f p r o p e r subsets o f P 2 i s d e f i n -

the family

ed.

X. The p a i r (F'2yB) i s a p r o j e c t i v e plane. ~

P r o o f . By pr0p.s

V I I I and I X ,

0 i s a p r o p e r 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 P 2 .

Next, we prove t h a t two d i s t i n c t elements a u n i q u e element o f

B. Set p = a , n a ,

P I passes (see prop. I V ) . Obviously, u , , XI.

TI

€ P I implies I n n V l

a, and

a2

i n P z are contained i n

; t h r o u g h p e x a c t l y one element a, E

TI

iri

B ( T I ) and t h e statement f o l l o w s .

1 . Furthermore, t h e l i n e s i n

TI

which a r e 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 ~ )a r e p r e c i s e l y t h e l i n e s i n t h e pencil (in

TI)

with centre a t the point v =

nnV.

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 t h e y belong t o t h e same element i n B(TI).

__ Proof. L e t

a,

and

a2

be two d i s t i n c t elements o f

B(n

1 ( t h e y do e x i s t by

Verotiese Quadruples prop. V I I I ) and s e t v = a l n sume v

a

lines 1

(see axiom ( 1 . 2 ) ) ; o f course, v belongs t o V . A s -

a2

n; t h r o u g h v a u n i q u e element

,

i = 1,2,

= a,

63

n n ' and 1, = a, n

and belong t o a, and

n'

P,

of

IT'

a r e d i s t i n c t , as l a , n a , resp. Since

az,

passes o t h e r t h a n

1

=

I T .Thus,

1

1 and

the

1.1 2 2 , 1

a , and a, belong t o B ( n ) ,

the

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 t h e r w i s e v would be c o i n c i d e n t w i t h q,

i m p o s s i b l e as v a n . Hence, ~ n n '

= q,,

c o n t a i n s 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 ( s e e axiom ( 1 . 2 ) ) . v belongs t o

IT

and by p r o p . V

By axiom ( 1 . 5 )

meets n

n

~i

Therefore

nV = [ v ) .

any element i n B ( n ) c o n t a i n s t h e p o i n t v = a , n

; thus, i t

a2

a t a l i n e i n L t h r o u g h v (see prop. VII: S i n c e t h r o u g h e v e r y p o i n t i n

t h e r e pass a t l e a s t one element i n

are t h e sections o f

n

P,,

by t h e elements i n

hence i n B ( I I ) , t h e l i n e s i n n , which are p r e c i s e l y t h e l i n e s through

B(n),

v. The statement f o l l o w s . XII.

(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 as

n

E

P , and v

=

i s a p r o p e r subspace of

i~

n V. A l i n e 1 does e x i s t i n n n o t t h r o u g h v,

( a , L ) . Since

(a , L )

i s irreducible,

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 c o n t a i n e d i n least three l i n e s e x i s t i n

IT

t h r o u g h v. By prop. X I ,

111 3 3 . The n

.

Thus,

at

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

C Z EP

, and La be t h e s e t o f a l l l i n e s i n L on

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 plane ( p ,

,o)

a;

consider t h e p e n c i l

through t h e p o i n t a

E

P 2 ; ob-

v i ously , F(a) = iB(n) €5: a n

I f l c L a , a correspondence i

1

il( B ( v 1 )

(4.1)

\

= B(n')

1

= B(n)

~i

EL 1

: F(a) w

+

F(a) i s d e f i n e d as f o l l o w s

n n n '

n

VE

€1,

n

#

n',

I.

By axiom ( l . l )any , element i n P I m e e t i n g 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 b e l o n g t o A and (see p r o p . 111) t h r o u g h

each of them e x a c t l y two elements o f P , pass. T h e r e f o r e , 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 r e 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 ; F(a) i f

n n s '

f u r t h e r m o r e , i i n t e r c h a n g e s t h e l i n e s B ( n ) and B ( n ' ) , 1 ~ 1. Thus, t h e n e x t statement has been proved.

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 ) 1 an i n v o l u t i o n .

XIII.

If

a E

P2 t h e n t h e f a m i l y

6

e(a) =

.

1'

+

* #

11')

of

F ( a ) d e f i n e d by (4.1) i s

1 E L a } o f i n v o l u t i o n s o f F ( a ) i s de-

fined. I f a ~ 7 ', ~t h e n IF( a l l

XIV.

24. Furthermore,

the pair (F(a),e(a)) i s a

Buekenhout o v a l . Proof. -

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 1 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

By prop. X I I I ,

t i o n . Since 111 '3,

+

Buekenhout o v a l , i .e. t h a t (21 ( i l e v e r y 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,

# n'

i , j = 1,2 j' 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 t e r c h a n g i n g B( n,)

and

B(n,)

and

From prop. X I 1 1 ( i )f o l l o w s . Thus, B(n;),

the points

nl

mi

w i t h B(n:).

B(n:)

and B ( n ; ) =

and (B(n:),B(n:)),

),B(n2))

then

( i i ) w i l l be proved.

= n 2 and n I l =

11,

.A

I f B(n, ) = B(n, 1

unique l i n e 1 e x i s t s i n

#

through

nV and n', n V, b o t h on a s i n c e B ( n , ) , B ( n : ; E F ( a ) . I f t h i s occurs

t h e n 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:), t h e n n 1 n;

a

nI2; t h e two p o i n t s

TT,

n n 2 and

and

i i s the 1 and B ( n , ' ) w i t h B(n;).

L a t h r o u g h b o t h of them. Again,

i n t e r c h a n g i n g B ( n , ) w i t h B(n,)

e(a)

n2

i n A belong t o a and a r e d i s t i n c t ;

n ' , nn',

hence, t h e r e i s a unique l i n e 1 i n unique element i n

#

A s i m i l a r argument proves ( i i ) i n t h e r e m a i n i n g cases. The statement f o l 1ows.

--

-.

.

Next, l e t ( F ' , L , P , , P collection {(F(a),

O(a)):

be t h e Veronese space o f a E p 2 1 0 f Buekenhout o v a l s .

Consider t h e mapping [B(n),

p2,8) a s s o c i a t e d w i t h t h e

@

: P

+

P

B( n')] o f d i s t i n c t l i n e s i n Li(i.e.

a s s o c i a t i n g w i t h e v e r y unordered p a i r n f n ' ) t h e p o i n t n n n ' E P and w i t h

the p a i r [B(n),B(n)] o f coincident lines i n D the point

XV

. The mapping

Proof. -

(4.2)

-

0 i s an isomorphism between

By pr0p.s 111, X I and axiom (1,2), @ i s one-to-one

and o n t o .

5

TI

nV in P

.

-

(rJ,P1,F2)

and ( P , L , P l , p 2 ) .

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

L o n an element i n P , ;

L on an element o f

mapped by @ o n t o a l i n e i n

f u r h t e r m o r e , t h e i n v e r s e image under @ o f any l i n e i n

8, i s a l i n e i n

By axiom (1.61,

, is

L

on an element i n

d,.

t a k i n g i n t o account prop. X I ;

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

(4.4)

P , a r e mapped

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 1 . From axiom ( 1 . 7 )

by 0 onto

and prop.

X I the

n e x t statement f o l l o w s . Three c o l l i n e a r p o i n t s on an element i n P , a r e mapped b y

(4.5)

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.

@-' onto

The statement f o l l o w s f r o m ( 4 . 2 ) -

(4.5). From pr0p.s X, X I I , X I V ,

XV, prop. I f o l l o w s .

5. THE PROOF OF PROP. I 1

remark t h a t a l l p r e v i o u s r e s u l t s b u t ( 4 . 5 ) were proved w i t h o u t

Firstly,

t h e h e l p o f axiom ( 1 . 7 ) . Thus, w i t h t h e same n o t a t i o n as b e f o r e , t h e n e x t propos i t i o n can be s t a t e d . XVI.

Under t h e assumptions i n prop.

I 1 f o r t h e mapping

@

:

P

* P(4.21,

( 4 . 3 ) and ( 4 . 4 ) h o l d . Next, any space

-=

4

n

B(

fl

E

n =

E?,

determines b o t h t h e l i n e B(.

b e l o n g i n g t o B and t h e sub-

b,in ( P , L ) d e f i n e d b y ( s e e s e c t . 1 ) : n

B( n )

= ([B(n),B(n')]:

Consider t h e mapping 6 :13

V'E

PI].

o f t h e dual p l a n e o f ( 2 , ,/I) o n t o ? d e f i n e d

+

by (5.1 1

6 ( B ( n ' ) ) = [B(n),B(n')

1

.

C l e a r l y ( s e e s e c t . 11, (5.2)

$

i s an isomorphism betwecn t h e dual p l a n e o f ( P 2 , D ) and t h e sub-

space % o f (P , L ) . D e f i n e a mapping a ' : ii

(5.3)

@'([B(n),B(n')

1)

n by

= @([B(a),B(n')

1);

o b v i o u s l y , 0' i s one-to-one and o n t o . S i n c e @ 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 bijection.

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 .

A . Bichara

66

p l a n e ( ' P , , B ) ; i f i t i s o f o r d e r q, t h e n 101 = q z t q t 1; hence (see ( 5 . 4 . ) ) ;

lfll

(5.5) By (5.2), (5.6) in

(5.31,

0'

J,

q

= q2 t

1.

t

and (5.4)

maps t h r e e c o n c u r r e n t l i n e s o f B

onto t h r e e c o l l i n e a r p o i n t s

TI.

Furthermore, t a k i n g i n t o account prop. X I , a l i n e i n point v

n n

=

passes

through

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

the

h'zJ

t h r o u g h a p o i n t on B ( n ) . Consequently, (5.7)

There

are

precisely q

t

1 lines i n

through v =

n V and on

each o f them q t 1 p o i n t s l i e . Next, l e t 1 be a l i n e i n t o v b y a l i n e , by ( 5 . 7 ) 111 (5.8)

Any l i n e i n The q

t

q

t

5

1~

n o t t h r o u g h v. S i n c e e v e r y p o i n t on 1 i s j o i n e d

q + 1; hence

c o n s i s t s o f q t 1 p o i n t s a t most.

1 lines i n

(P z , B ) a l l have s i z e q

ped by O ' J I o n t o a l i n e i n n ( s e e (4.4) q

t

q

t

t 1 and each o f them i s map-

and (5.8));

t h e r e f o r e , on

1 l i n e s l i e each o f them having s i z e q t 1 . Since

*

11

at least

i s a subspace of

( B , L ) , by (5.51,

(5,9)

1~

i 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 and t h e mapping 0 'J, i s an i s o -

morphism between t h e d u a l p l a n e o f L e t pi,

i = 1,2,3, @-'(pi)

( P 2 , B )and

n.

be t h r e e p o i n t s i n n. C l e a r l y , = ( e ' l - ' ( p . 1)

Since b o t h JI and

$ I $

= (J,dJ-l(O')

')(Pi)

= J,(O'J,)-'(pi)

a r e isomorphisms t h e t k r e e p o i n t s

Q-l(p.1 are c o l 1

l i n e a r iff t h e p o i n t s p . a r e 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 r e v i o u s argument and prop. X V I ,

P L,?,, $:

under t h e assumptions i n prop. 11,

+ ? 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 =

F2)

(d,

and Q = ( P , L ,P,, P J . Since f o r t h e Veronese space Q axiom ( 1 . 7 ) h o l d s ,

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 r o d u z i o n e a l l a geometria p r o i e t t i v a d e g l i i p e r s p a z i , Pisa,

E. S p o e r r i (19071. [2]

F. Buekenhout, Etude i n t r i n s e q u e des o v a l e s , Rend. d i Mat. V (1966) 333-393

Veronese Quadruples [3]

G.

Tallini,

Spazi p a r z i a l i d i r e t t e ,

Sem. Gem. Comb. Univ. Roma 14 ( 1 9 7 9 ) .

61

s p a z i p o l a r i . Geometrie subimmerse,

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 69-84 0 Elsevier Science Publishers B.V. (North-Holland)

S-PARTITIONS OF

69

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 [ 2 6 ] and Scarselli [17] mainly investigated the following question: 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 investigate "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 r u c t u r e s 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 n o t a t i o n . 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 t h e s e t o f a l l P Sylow p-subgroups of G and HK/K i s denoted by A. If H l l K = then K X H denotes the s e m i d i r e c t product of K by H. I f G i s a permutation group on a s e t R and r G f i then G

r

denotes t h e g l o b a l s t a b i l i z e r o f ?i i n G. A s e t R i s a G-set i f t h e r e 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 GR instead o f v(G). L e t G be a group and S a subgroup of G with SzG. A s e 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 s a i d t o be a (keguLatr) S-pwrLLtiion o f G i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : (i) (ii) (iii)

S H n s K = S f o r each H , K e Z with H#K;

f o r each g H

4

t

G t h e r e e x i s t s H t C such t h a t g f SH;

C i m p l i e s d-'Hs t C f o r each o E S.

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 t e r e s t e d i n t h e f o l l o w i n g special class o f S-partitions: a S - p a r t i t i o n C o f a group G i s s a i d t o be a L i n g e n b a g S . - p a t , t i L i o ~ w l t h respect t o t h e subgroup T o f G i f t h e f o l l o w i n g hold:

(j) T 6 S < NG(T) < G; (jj) . 16 P 0 a p o i n t : JL 6

.then C

=

{T()L) :

aE

8- { p } } LJ

Linyenbety S - p a h R i L o n o d G d R h ~ ~ e n p e otot T ( p ) and

M. Biliotti

I2

L e t 5 be a q u a s i - t r a n s l a t i o n 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 t h e group o f a l l t h e automorphisms o f 5 f i x i n g every p o i n t o f

and every l i n e p a r a l l e l t o

4.

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 t h e f a m i l y 8 = { H ( d : k c O } . Moreover, 9 i s a complete c l a s s o f conjugate subgroups o f G = c H ( h ) : h e @ > so t h a t 5 may always be represented i n t h e form [G,Gp,H(p)l,

2. PROPERTIES AND EXAMPLES LEMMA 2.1.

each x,y

t

where P E ~ E O .

OF LINGENBERG S-PARTITIONS

d e t e m i n e u Lingenbetlg S - p a k L i t i o n , t h e n TXnTY= 6vtl

LeR (G,S,T)

G w i t h TX#TY. W

N

Paood. Consider t h e 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 s o 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 a r e d i s t i n c t and b o t h are f i x e d by

?

then ?(P) = P and t h e r e f o r e

?

= I.

Lingenberg S - p a r t i t i o n s may be d i v i d e d i n t o two classes according t o t h e f o l l o w i n g definition:

a Lingenbetlg S-pwc.tLtion SnTX

ench

d o t

=

bay t h a t

xt

it i~ a6 t y p e

C ad G w i t h kenpect t o t h e bubgtloup T LA 06 t y p e 1 4 6 ld t h e S - p a h t i L i o n 0 not 06 t y p e I,we 4haU

G with TX#T. 11.

For Lingenberg S - p a r t i t i o n s o f type I we have:

d e t e m i n e n a Lingenbekg S-pakLLtihion 06 .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) d e t e m i n e n a Lingenbekg S-pahtLtion 0 6 t y p e It h e n t h e duUowing hold: ( I ) 7 u c . 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, (G,S,T)

PROPOSITION 2.2.

121 N G ( T ) n T X = 6ok each TK#T, and 131 [NG(T):S]

= ITI-1.

R o o d . Assume (G,S,T)

x

t

determines a Lingenberg S - p a r t i t i o n o f t y p e I and l e t

NG(T) n T y w i t h TY#T. Then

?

f i x e s both t h e 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 G i t f o l l o w s t h a t x = 1 and ( 2 ) holds. Now l e t

w t G w i t h ?#T.

Then

and so z = 1 and

7

Z E

x. Therefore,

x c S and from SnTY=,

z 6 T and NG(T)w,z = NG(T)w f o r some

NG(Tw) and hence z t NG(Tw)nT. But, by (21, N G ( T W ) n T = < l >

a c t s 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 t h e converse. L e 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

through R, which i s d i s t i n c t from NG(T), i s f i x e d by

on 0

-

{NG(T)},

z t. T,

z.

R 6 a. Then t h e l i n e o f 0

Since

?

acts semiregularly

i t f o l l o w s t h a t z = 1 and ( 1 ) i s proved. Now l e t lGl=g, ING(T)I=n,

I S ( = b and ( T I = t . I n G,

t h e r e e x i s t g/n - 1 d i s t i n c t complexes o f t h e form STX with

Tx#T and each complex contains e x a c t l y n t elements since S n T X = c1>. I f we take

73

S-Partitions of Groups and Steiner Systems SNG(T) = NG(T) c o n t a i n s n elements o f G and (G,S,T)

account o f t h e f a c t tha:

mines a S - p a r t i t i o n of G, then we must have ( n t - n ) ( g / n

- 1)

deter-

n = g so t h a t ( 3 )

t

now f o l l o w s .

16 (G,S,T) d e t e h m i n u a Lingenbefig S-pahtition 06 t y p e It h e n h a h u o l w a b d e SReinek nyotem w i t h paharneteu [ w , k l whehe w = [G:S1 + 1

COROLLARY 2.3. [G,S,TI

and k

=

ITI.

P J L U O ~ .This i s an immediate consequence o f (1) and ( 3 ) o f P r o p o s i t i o n 2.2.

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 eT(t) n

PROPOSITION 2.4. d o h each k e 06

0 id and o n d y

46

it 6 i x e 4 &Why &ne

06

0. A non-idenLicCLe automohpkcnm

0 6 i x i n g 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 t h e r e 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 . L i k e wise, T(n) = a - ’ T ( n ) a f i x e s e v e r y p o i n 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) = , 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 i m p l i e s ~ ( 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 t h e c o n t r a r y , q t 0 then t h e r e l a t i o n a ( Q ) = Q can be obtained by u s i n g t h e 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 f o l l o w s . Now we s h a l l g i v e some examples o f Lingenberg S - p a r t i t i o n s . We assume the reader i s acquainted w i t h t h e s t r u c t u r e o f groups SL(2,q); PSU(3,qz), q = ph p a prime; S Z ( ~ ~ ~ ’ R ’ )( 3; 2 n f ’ ) , n > l , and a l s o w i t h the elementary

,

p r o p e r t i e s 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]r e s p e c t i v e l y . EXAMPLE I. G

2

SL(2,q),

q = ph , q>2. L e t P c S (G) and assume T = S = P; then i t i s P

an easy e x e r c i s e t o show t h a t (G,S,T) I and t h a t [G,S,T],

determines a Lingenberg S - p a r t i t i o n o f type

the completion o f [G,S,T],

i s t h e a f f i n e plane over GF(q).

This i s t h e c l a s s i c a l example which i n s p i r e d t h e work o f Zappa [24]. I t a l s o exp l a i n s the d e f i n i t i o n o f t h e p a r a l l e l i s m i n [G,S,T] EXAMPLE 11. G

7

as given i n s e c t i o n 1.

S z ( q ) , q = Z Z n f ’ , ~ 2 1 .L e t P E S2(G) and l e t Z(P) be t h e c e n t r e 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 { N G ( T ) then G NG(T)nTX = 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

M . Biliotti

74 g =

h,.t:

= h2RY w i t h b 1 , n 2 e S, R l , . t t 2 E T and hence

n;'n,

=

d1.t;1 1 x .

If. t , f l , t n f l

and G i s regarded as a c t i n g i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree

q z + l then the element .t$(.t;'JX, being t h e product o f two i n v o l u t i o n s w i t h o u t common f i x e d p o i n t s , f i x e s an even number o f p o i n t s . 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 e x a c t l y one p o i n t which i s a c o n t r a d i c t i o n . As we have p r e v i o u s l y shown, we cannot have R =1 f o r only one 4=1,2 and so R =1 for 4=1,2 and 4

~ 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 t h e 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 t h e f o l -

lowing r e l a t i o n holds:

4'

1 ( I S T ' ~- ~ IS^)

This proves the a s s e r t i o n .

t

i- I

The completion o f the A-structure [G,S,T]

I N ~ ( T )=I I G ~ .

i s a r e s o l v a b l e S t e i n e r system w i t h

parameters (q(q2-q+I ) , q ) . EYAMPLE 111. G

2

PSU(3,qz), q = 2 h , h > l . L e t Pe S2(G). I t i s w e l l known t h a t NG(P) =

C, where C il c y c l i c o f order ( q 2 - I ) / d with d = ( 3 , q + l ) . Denote by Cl t h e subgroup o f C of order ( q t i ) / d and s e t T = Z(P), S = P X C,. Then (G,S,T) determines = P X

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), N G ( T ) n T X = , 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 r o p e r t i e s o f G, we have t h a t M = = SL(2,q) and M'ING(P) = or Z ( P ) X C,, w i t h 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 , q t ] / d ) = I and so, i f S n M # < l > then S n M = Z(P) = T. But, as we

I. (M,T,T) determines a Lingenberg S - p a r t i t i o n o f type I and

have seen i n Example

hence T X T Y n T = c l > . I t f o l l o w s t h a t T X T Y n S = and t h e r e f o r e S T X n S T Y = S. The t h e s 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 Examp l e 11. i s a r e s o l v a b l e S t e i n e r system w i t h

The completion o f the A-structure [G,S,T] parameters ( q ( q 3 - q 2 + I ) , q ) .

I n t h e case q=Zh, with h even, t h i s S t e i n e r system has

been already obtained by Schulz [la]. EXAMPLE I V . G

2

R(q), q=

32n+I

,

~ 2 1 .We s h a l l make use o f t h e r e p r e s e n t a t i o n o f G

i n P G ( 6 , q ) due t o T i t s [22],§5. L e t xI,x2,. ..,x7 be a coordinate system f o r P G ( 6 , q ) 3n+I Furthermore, l e t I be t h e hyperplane o f and l e t o c A u t ( G F ( 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 x=x,/x7,

y=x2/x7,

z=x3/x,,

u=xs/x7,

u=xs/x7,

w=xb/x7

i s a non-homogeneous coordinate system for A . F i n a l l y , s e t and denote by

r -

( m ) : ( I ,O,O,O,O,O,O) { ( m ) } the s e t o f p o i n t s o f A whose coordinates s a t i s f y t h e

equations

(1)

- X Z t yo p y a - za f xy' xzo - x o + i y

u = xzy

p + 3

u

t

=

1o =

yz -

X'CJ'3 x2y2

-

- zz

X ~ a f 4

75

S-Partitions of Groups and Steiner Systems Then G

PGL(6,q)r

2

and G a c t s on

r

i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n

o f degree q 3 + l . L e t P be t h e unique Sylow 3-subgroup o f G l y i n g i n G(,+.

By u s i n g

t h e r e s u l t s o f T i t s [ 2 2 ] , § 5 , about t h e r e p r e s e n t a t i o n o f t h e 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 n o t hard t o prove t h a t t h e p r o j e c t i v i t i e s l y i n g i n Z(P) are e x a c t l y those o f t h e form t c : (XI,xZ,x3,X4,x-,xb,X-)

+

+ ( X I, X L , X 3 + C X 7 , - C X I+ & + , c x ~ + X ~ - C ' X ~ ,C'X1-2CX3+X6-CZX7

c

,X7),

E

GF(r().

According t o T i t s [ 2 2 ] , § 5 , we have a l s o t h a t

: ,xZ,xJ,X4,x5,X6,x7) (X5,xb,X3,x2,xl ?-x7,-X6) 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 n o t l i e i n G

fA

(a).

Therefore,

uZ(P)W i s the c e n t r e 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 , d c GF(q), c + O , d#O, then

'""tCi does dd

n o 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 w i t h Frobenius k e r n e l Z(P) (see [23],111.4),

we can suppose, w i t h o u t

loss o f g e n e r a l i t y ,

d=l. We then have

dCdI : (X,,X2,X3,X,,Xj,X6,XI) ( X I + CX, + coxE,x 2 - cx j, ( J + ZC) x g X-'C - ( C + c P ) x 6 + x ,, - x + ( I - c)x, -c"xE,x 2 - 2cx3+( I -C+C~)X,+C'X 6-X7, X,-(2*2C)X3+CX,,'CuXSf

( It2CtC"c')X6-X7

2CX3'CoXg-c2Xgfx7)

A s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t d c d Jpossesses the eigenvalue I whose

eigenspace i s generated by t h e v e c t o r ( 0 , I ,l/'Z,-cu~*,O,l/c,l).

Now suppose d c d J

l i e s i n a Sylow 3-subgroup o f G, then t h e f o l l o w i n g hold: ili.tc does dI n o t have any eigenvalue d i f f e r e n t from I , f o r we are i n charac-

-

t e r i s t i c 3;

-

LK

C

wx

I

must f i x a p o i n t o f

r

- {

(m)}.

From t h a t which we have proved p r e v i o u s l y , we can i n f e r t h a t t h e f i x e d p o i n t o f

d2dl

on

r -

{ (a)] must have non-homogeneous coordinates ( O , l , I /2, - c O - ~ ,0, I /c)

.

But these coordinates do n o t s a t i s f y (l), a c o n t r a d i c t i o n . Now we may argue as i n the previous examples t o show t h a t i f we s e 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 The completion o f t h e A-structure

[G,S,T]

I.

i s a r e s o l v a b l e S t e i n e r system w i t h

parameters ( q ( q 3 - q '+ I ) ,q 1. EXAMPLE V. G = SL(n,q),

q = ph , n>3. L e t K = GF(q), V = K" and U a 1-dimensional

subspace o f V. Denote by T ( g , p ) t h e t r a n s v e c t i o n ptHomK(V,K) w i t h p(2) = T(U) =

0,p#O.

y

-f

l-u(l)awhere g c

b of u(b)=gj.

For a f i x e d non-zero v e c t o r

{I, T(b,u) : O # U E HomK(V,K),

V and

U, s e 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 h e 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. I t i s indeed enough t o observe t h a t t h e A - s t r u c t u r e which i s obtained from t h e a f f i n e space A associated t o V by removing t h e o r i g i n 0

i s a q u a s i - t r a n s l a t i o n A - s t r u c t u r e w i t h respect t o the p e n c i l 0 o f the l i n e s

M.Biliotti

16 through

0 (disregarding

the point

which a r e conjugated t o T(U).

0)and

t o t h e f a m i l y 3 o f t h e subgroups o f G

Furthermore, a t r a n s v e c t i o n 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 n o t semiregular on 8 . The

a s s e r t i o n now f o l l o w s from P r o p o s i t i o n s 1.2 and 2.2.

3. FURTHER RESULTS ON LINGENBERG S-PARTITIONS OF TYPE I We w i l l r e q u i r e t h e 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 ; S z ( q ) , q = p2 n f 1 p=2, @ I ; S U O , ~ ~,) q - p

i,,

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

conditioMn: (I1 /TI - I (21

TnZ(G)

0 4 G. 16

T

a nomat dubgtoup

06

NG(P) b ~ ~ q 4 u 2 g

[NG(P):Tl and =

,

then T = Z(P).

Pmod.

We s h a l l i n v e s t i g a t e t h e v a r i o u s cases separately.

L e t G = sL(z,q),

q s 4 . Assume q i s odd. Then I Z ( G ) ( = 2, ] P I = q , ING(P)I = q ( q - 1 )

= N (P)/Z(G) i s a Frobenius group w i t h Frobenius k e r n e l p. Since T Q N, then G by [ l l ] , V , Satz 8.16, we have t h a t e i t h e r 7 < or h P. I n t h e f i r s t case, i t

and

r

f o l l o w s t h a t T < PZ(G). But, T n Z ( G ) = and hence T < P, which i m p l i e s T = since P i s a minimal normal subgroup o f NG(P). I n t h e l a t t e r case, c o n d i t i o n (1) y i e l d s T = P. Since P i s elementary abelian, t h e p r o o f i s achieved. The case q even i s s i m i l a r . L e t G = S z ( q ) . N (P) i s a Frobenius group w i t h Frobenius k e r n e l P, moreover G

/ N G ( P ) / = q 2 ( q - l ) , lP1 = q z ,

I Z ( 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) = = s i n c e 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

IT[-I.

So, i n t h e former

case, i t f o l l o w s T = Z(P) from c o n d i t i o n (1). I n

the l a t t e r case we have P = T X Z(P). However, P/Z(P) i s a b e l i a n and hence P must be a b e l i a n which

is a c o n t r a d i c t i o n .

L e t 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’ G

C i s c y c l i c o f order q2-I and c o n t a i n s Z(G).

-

Moreover, I Z ( P ) )

= q.

N = NG(P)/Z(P)Z(G) i s a Frobenius group with Frobenius k e r n e l

plements isomorphic t o clearly

p

c. Since io N, we must have e i t h e r

i s a minimal normal subgroup o f

and hence

7=

and

and Frobenius com
Z(P) i m p l i e s T = P ' and c o n d i t i o n (1) i s n o t s a t i s f i e d . So T = Z(P). We canX E PI n o t have TfiZ(P) = -1> s i n c e if

der 2 q z (see [23],111.2)

-

Z(P) then i t s c e n t r a l i z e r i n NG(P) has or-

and hence I T ( > q , c o n t r a r y t o T S P I . I n t h e l a t t e r ca-

se, we cannot have T n P ' = , since for each t Z(P) < PI (see [23],

[23],111.2)

xE

P - P' we have o ( x ) = 9 and

Theorem). Nevertheless, T n P ' # i m p l i e s I T 1 2 q'

x3 6 (see

and again c o n d i t i o n (1) i s n o t s a t i s f i e d . This completes the p r o o f .

The f o l l o w i n g 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 t h e 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

even o t d m t h e n one 0 6 t h e doLLowing h u t & : G = O(G)T und T iA a Fhobeniun cornpLement; (u) h ( 6 . 1 ) G 2 SL(2,q), 9.2 , h62; T = S = P w i t h P t S2(G); ( b . 2 ) G 2 SZ(~),q - 2 2ntJ a21; T = Z(P), S = P w i t h P6S2(G); i, ( 6 . 3 ) G 2 PSU(3,qz), q = 2 , h22; T = Z(P) w i t h . P G S2(G) and S = ( q + l l / d , &eke d=13,4+11.

type I.1 6 T ha^

= P X

CI with

IC1I =

Phood. I n t h e 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 = N G ( T ) t 0 then E = Ne(?) and, by P r o p o s i t i o n 2.2, ? a c t s semiJl = r e g u l a r l y on 0 - { a } . Then by [ l o ] , Theorem 2, e i t h e r t h e 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 a c t s on 0 i n i t s u s u a l doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree q + l , q 2 + 1 , q 3 t 1 , q 3 + 1 r e s p e c t i v e l y . Then, i t is w e l l known t h a t h = Nc(F) with P"eS2(G) and hence Nc(?) = N c ( B ) . By t a k i n g account o f P r o p o s i t i o n 2.2, we see t h a t 7 s a t i s f i e s c o n d i t i o n s (1) and ( 2 ) o f Lemma 3.1. Therefore ? = Z ( P ) . When E = SL(Z,q),

e

Sz(q)

E

or

PSU(3,qz) we o b t a i n (b.1) - (b.3) i n view o f P r o p o s i t i o n 2.2,(3).

= SU(3,q2) w i t h 31qtJ we have

171

= IZ(p)l =

If

q and hence I ~ l ( = q - l Since . 1Z(G)I=3,

t h i s i m p l i e s t h a t 3 Iq- I by P r o p o s i t i o n 2.4, a c o n t r a d i c t i o n . Therefore, t h e case

G"

= SU(3,q2), w i t h 31q+l, cannot occur.

We p o i n t o u t t h a t Examples

I, I 1 and I11 o f s e c t i o n

2 show t h a t t h e cases (b.11,

(b.2) and (b.3) a c t u a l l y occur. On the c o n t r a r y , 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 t h e case ( a ) .

M. Biliotti

78

I n succession, we g i v e some r e s u l t s and examples concerning t h i s case.

LeA (G,S,T)

PROPOSITION 3.3.

d&tehmine u Lingenbehg S - p a d t i o n ud t y p e I und

M-

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 then Rhe ~oUoiu4ngh o l d : h 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 ,

dume [ T I 2 3.

16

[G,S,Tl,

A-n&ucRuhe

( I )G = M X m,hLJ;

( 2 1 IZ(G)I = lZ(M)I = 4,

I T 1 = q + I , S = T,

NG(T) = TZ(M).

P m a 6 . By P r o p o s i t i o n 2.4, Z(E) i s t h e k e r n e l o f t h e r e p r e s e n t a t i o n o f E on 0. Therefore, Go = E / Z ( c ) and a c t s on 0 as a Frobenius group by our assumptions.

c’

Denote by

G

the Frobenius k e r n e l o f

By P r o p o s i t i o n s 2.2 and 2.4,

IT], I i l ) = I

Moreover,

since

= G/Z(G) and l e t M

we have t h a t IZ(G)I

I /TI-1

i G such t h a t M/Z(G) =

M.

and hence ( l T l , l Z ( G ) l ) = l .

i s contained i n a Frobenius complement o f

c.

There-

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 = G 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 P r o p o s i t i o n 2.2. Since, fore, T n M =

r

as we have p r e v i o u s l y seen, IZ(G)I

ii

Note t h a t since

group of M with P

I

R - 1 , i t f o l l o w s t h a t / Z ( G ) I d-I and S = T .

i s n i l p o t e n t so i s M (see [11], V.a.7).

$

Z(G) and l e t N = PT. Since (G,S,T)

L e t P be a Sylow p-sub-

determines a Lingenberg

S-partitionof type I, t h e f o l l o w i n g r e l a t i o n holds

It‘

(2)

- $1

!nlRc - I )

+

Rc

c

y1

,

= I N ( and c = ( P n Z ( G ) ( . From (2), i t f o l l o w s t h a t t - l j c since n>tc and

where

hence c=X-I and Z(G) < P. This y i e l d s M = P. Consider the commutators o f t h e 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 c o n t a i n s e x a c t l y t-J n o n - i d e n t i c a l d i s t i n c t commutators o f t h e form [ x , g ] .

Moreover, i f T 9 # T6 then T T g n T T 6 = T and hence, t h e R-7 commutators

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‘. by s e t t i n g [M,T].

(GI

Since

= 6 there e x i s t a t l e a s t ( X - I ) ( i - 7 ) t I

Since [ M , T ]

IITX

: X E G}

2 M and I M I = m ( X - I ) , i t f o l l o w s t h a t [ M , T l

= M.

t h e r e e x i s t s a c h a r a c t e r i s t i c a b e l i a n subgroup A o f M such t h a t A group AT contains e x a c t l y a = [ A : Z ( G ) n A ]

I

=

121,

then

d i s t i n c t commutators l y i n g i n

6

Now suppose Z(G). Then the

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)tJ

t i o n y i e l d s Z(G) < A , 13.4(b),

,

( t - I ) a . The l a t t e r r e l a b u t t h i s c o n t r a d i c t s a r e s u l t o f Zassenhaus [11],III, Sat2

where a>?. From t h i s i t f o l l o w s t h a t A = [ A , T ]

and ( A ( =

since IZ(G)) > I. Therefore, a c h a r a c t e r i s t i c a b e l i a n subgroup o f M i s

c e n t r a l i n G. I n conclusion we have proved t h a t :

(I) ( I M I , I T I ) = I , (11) [ M , T l = M, (111)

T c e n t r a l i z e s every c h a r a c t e r i s t i c a b e l i a n subgroup o f M.

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

79

S-Purtitiorisof G r o i q s arid Steirier Systems

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 h , h21, and let l M l phtn. Since a h Frobenius complement of G/Z(G) has order . t = p h + I , it follows that p +llp"-l. From so this, we have that ph+ I I pn+ph=ph (pn-h+1 ) and hence ph+ 1 I pn-h- Ph =Ph (Pn-2h- I ph+ 1 1 p n - 2 h - I . Let b E P such that bhSn< Ib+ lih. By iterating the above procedure, it . completes the proof. is easy to prove that b must be even and P ' - ~ ~ - I = O This

.

A Lingenberg S-partition of type I satisfying conditions ( 1 ) and (2) of Proposition 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 projective plane over GF(qz), q=ph , 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 parallel 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? parallel 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: - G = 1 . Moreover,

i s solvable, i t f o l l o w s t h a t

a c t s r e g u l a r l y on

R i n v a r i a n t and a c t s semiregularly on s1 -

LR

R. Moreover, ?R =

[ k } . From t h i s , we i n f e r t h a t

i s elemen-

7

leaves

?ITn

is a

Frobenius group. Now s e t F = , 5, = S n F , No = NG(T)nF, I S o \ = h a , = n o , ( T I = t and I F ( =

IN,(

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 f o l l o w i n g r e l a t i o n holds: (3)

(,t~o

S

[N:S] = . t - 1 .

- I ) t no 2 6. t-1. On t h e o t h e r hand we have t h a t

- bo)(d/na

From t h i s , i t f o l l o w s t h a t n o / b o t

i t i s n o t d i f f i c u l t t o see t h a t ifwe s e t R = : x t F 1 } u {NFl(T1)

then C I = { T f

[N,:S,]

S

Therefore, n o / b o = t-f and ( 3 ) h o l d s as an e q u a l i t y . Using t h i s ,

respect t o the subgroup TI.

But

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 w i t h

-R-R

-R

F I = I- T

and hence, by P r o p o s i t i o n 3.3,

XI i s a

b p e c i d lingenbekg S - p a t L i L i u n . From a geometrical p o i n t o f view, t h e previous r e s u l t can be expressed as f o l l o w s .

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 tt 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

cuntaim a bubbpace w h i c h i~ a b p e c i d

ud G. Then t h e A-bRhuctwle [G,S,T]

A-btkuc-

tuke.

Pkoo6. L e t [FI,S1,T1]

be the s p e c i a l A-structure r e l a t e d t o the s p e c i a l S , - p a r t i -

t i o n & o f Fi described above. I f S I X i s a p o i n t o f [F,,Sl,T1] the map from t h e s e t o f p o i n t s o f [FL,SI,T1] defined as f o l l o w s

n

-

: SIX

-t

sx

and

x = Kx,

l e t q be

i n t o t h e s e t o f p o i n t s o f [G,S,T]

.

I t i s s t r a i g h t f o r w a r d t o show t h a t 11 i s w e l l d e f i n e d and g i v e s an embedding o f [FI,Sl,TIl

i n t o [G,S,Tl.

4 . LINGENBERG 5-PARTITIONS OF DOUBLY TRANSITIVE PERMUTATION GROUPS determines a Lingenberg S - p a r t i t i o n . Since i n Examples I - V vie have

Assume ( G , I , T ) that: (a)

t h r gnoup

cicib

Z-tcanoiLivek?y un t h e pen&

06 f i n e d

0

06

.the A - ~ i h u c . t u c e

[G,S,Tl. then the n a t u r a l question a r i s e s whether i t i s p o s s i b l e 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 - p a r t i t i o n s s a t i s f y i n g c o n d i t i o n ( a ) . I n the f o l l o w i i i g , 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 i v e n provided t h a t t h e 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 f o l l o w s f r o m t h a t o f F i n i t e

simple groups. THEOREM 4.1.

&Lion

Annwrie

(G,S,l) deXehmAneb a Lingenbehg S-pamLCi.on C

( u l . 16 C 0 ad t y p e It h e n une

06

t h e ,joXCawing h d h :

b a t ~ A 6 y ~ ncung

81

S-Partitionsof Groups and Steiner Systems

q = p c l , ph22; T = S = P l ~ i h i hP E S (C); P 2n+ I ntl; T = Z(P), S = P wi2h P 6 S 2 ( G ) ; ( 3 1 G 2 PSU(3,qz), q=2', h t 2 ; T = Z(P) wiRh PE S2(G), S = P A C I , wh&he I C I = ( q + l ) / d ,d=(3,y+ll; ( 4 ) G R ( q ) , q = 3 2 n C 7 , n 2 i ; T = Z(P), S = P w a h PES3(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.

(I]

G

(21

G

SL(Z,q),

Sz(q), Q =

2

I. By P r o p o s i t i o n

Pk006. Assume (G,S,T) -,@

t l i e group G

2.7,

I=

a LinyenbErg 5 - p a r t i t i o n o f type - determines s a t i s f i e s the following condition:

* G/Z(E)

(h) f o r each h e @ , t h e s t a b i l i z e r o f s e m i r e g u l a r l y on

o-

5

6'

in

c o n t a i n s a normal subgroup which a c t s

{t}.

From t h e 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

(see [ 4 ] ) i s a c t u a l l y a theorem (see

h a ? t h a t t h e so c a l l e d "Hering conjecture"

[19],p.302)

c0

asserting that i f

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 t h e f o l l o w i n g holds:

c0

(j) c o n t a i n s a r e g u l a r normal subgroup, (jj) Eo PSL(Z,(i), q L 4 , Sz(q), PSU(3,q"), q > 2 , or R ( y ) , 4'3, -1

aiid

Go

a c t s on 0 i n

i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n . We s h a l l i n v e s t i g a t e these cases separately.

( j ) . Let

C~c.14

hence

Go =

E0

be t h e r e g u l a r rlorrnal subgroup o f

cO. We have t h a t Go

" G Z ( e ) i s a Frobenius group. If( T I = 2 then

then by P r o p o s i t i o n 3.3,

-,o

we must have I N

I

=

EB =

and

=

= SL Z(G) f o r some Ty with TX# fTy.

I t f o l l o w s t h a t T X T Y n Z ( G ) # < l >since TXnTY= by Lemma 2.1. So t h e r e ex-1

sists 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)=

5, b u 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) = . Moreover, by P r o p o s i t i o n 2.2, we have: (2) ' 3 does n o t a c t semiregularly on Q - {NG(T)x]. By a w e l l known r e s u l t o f O'Nan [ 1 5 ] , Theorem A, c o n d i t i o n s (1) and ( 2 ) i m p l y 6 PrL(n,q) with n23. So t h e t h e s i s f o l l o w s from P r o p o s i t i o n 2.4. PSL(n,q) 5 = S then z

E

As a f i n a l remark, we note t h a t t h e 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 r e q u i r e d o n l y 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.

REFERENCES.

[l] A l p e r i n , 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, Proc. Am. Math. SOC. 17 (1966), 515-519. [ 2 ] Andre, J., Uber P a r a l l e l s t r u k t u r e n , T e i l I : Gundbegriffe, Math. Z. 76 (1961),

85-102.

S-Partitions of Groups and Steirier Systems

[ 31

Andr6, J.,

83

Uber P a r a l l e l s t r u k t u r e n , T e i l I 1 : T r a n s l a t i o n s s t r u k t u r e n , Math. Z.

76 (1961), 155-163. [41 Aschbacher, M., F-sets and permutation groups, J. Algebra 30 (1974), 400-416. [5] Bose, R.C., On t h e 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 s e r i e s o f balanced Kirkman arrangements, in:The Golden Jub. Comm., C a l c u t t a 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 g r u p p i f i n i t i , Boll. Un. Mat. I t . (6) 3-8 (1984), 665-685. [ 71 Burkhardt, R., Uber d i e Zerlegungszahlen der u n i t a r e n Gruppen PSU(3,2 2 f 1 , 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 t h e known f i n i t e simple groups, B u l l . Am. Math. Soc. 78 (19721, 68-71. 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, [9] Griess, R.L.,Jr., 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

(19721%622-629. [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 K o l l i n e a t i o n e n , whelche e i n e perspect 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 r a n s l a t i o n 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 t h e one-point s t a b i l i z e r o f a doubly-trans 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 g r u p p i non a b e l i a n i d ' o r d i n e 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 i n c e i , Rend C1. S c i . F i s . Mat. Nat. ( 8 ; 62 (1977), 300-304. [18] Schulz, R.H., Zur Geometrie der PSU(3,q ) , i n : B e i t r a g e 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 p o i n t s , i n : Geometry - von S t a u d t ' s P o i n t o f View, Proc. NATO Adv. Study I n s t . Bad Windsheim, 1980 ( 0 . R e i d e l P.C., Oordrecht, 19811, 275-311. [20] Suzuki, M., On a c l a s s o f doubly t r a n s i t i v e groups, Ann. Math. 75 (19621, 104-145. [ 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, i n : Sem. Bourbaki, 13e annCe, 210 (1960/61), 1-18. 1231 ~- Ward, H.N., On Reels s e r i e s o f simple groups, Trans. Am. Math. SOC. 121 (19661, 62-89. 1241 ZaDOa. G.. S u a l i sDazi a e n e r a l i quasi d i t r a s l a z i o n e , Le Matematiche (Catan i a ) i9 (i9647, 127-143: 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) 1251 . . Zappa, G., 74 (19661, 1-14. [26] Zappa, G., P a r t i z i o n i g e n e r a l i z z a t e n e i gruppi, i n : C o l l . I n t . Teorie Comb. 1973 (Acc. Naz. L i n c e i , Roma 1976), 433-437. i

,

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Annals of Discrete Mathematics 30 (1986) 85-98 0 Elsevier Science Publishers B.V. (North-Holland)

COLLINEATION GROUPS

85

STRONGLY

IRREDUCIBLE ON AN OVAL

Mauro Biliotti

Gabor Korchmaros

Dipartimento di Matematica Universitl degli Studi di Lecce via Arnesano, 73100-LECCE Italy

Istituto di Matematica Universitl degli Studi della Basilicata via N. Sauro, 85, 85100-POTENZA Italy

In recent years, Hering has written several papers concerning the composition series 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 o n 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 is

R

which is an oval in a proper subplane of

4.t/Lony4y u z n e d u c ~ 6 4 eon f i e o v a l n.

since it fixes the knot K of

n.

We say that r

Clearly r is not strongly irreducible on

TI

0.

Our main result states that if r has even order then r contains some involutorial perspectivities, i.e. elations. The subgroup



generated by all involutorial e-

lations is essentially determined. If r has a fixed line then



is the semidi-

rect product of O() with a subgroup of order two generated by an elation. If r has no fixed line then r acts a s 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



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

M . Biliotti and G.Korclirnaros

86

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

group cannot i n v o l v e PSU(3,q‘). H

or

a c t i n g t r a n s i t i v e l y on R a r e 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 )

Sz(q)c z c A u t S z ( q ) . The groups a r e always 2 - t r a n s i t i v e on

t h e former case,

IT

i s a desarguesian p l a n e o f o r d e r q and

R . Furthermore, i n

i s a c o n i c . For t h e

l a t t e r , we may o n l y a s s e r t t h a t , a t t h e p r e s e n t 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 d u a l Luneburg p l a n e of o r d e r q z (see ~ l l ~ , [ 1 3 ] , [ 1 4 ] ) .

2. NOTATION AND PRELIMINARY RESULTS F a i r l y s t a n d a r d 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 l a n e o f even o r d e r n c o n t a i n i n g an

o v a l R . Here an o v a l i s d e f i n e d 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 linear. The f o l l o w i n g elementary r e s u l t s a r e used i n t h e p r o o f s . 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 t a n g e n t o f a. The t a n g e n t s a r e c o n c u r r e n t ; t h e i r common p o i n t K i s c a l l e d t h e k n o t o f 0. Each l i n e t h r o u g h K is a tangent o f n. A l i n e o f

IT

is

an

e x t e r n a l l i n e o r a secant l i n e o f R a c c o r d i n g

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 c h o r d 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

o n t o 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 h e n 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 o n l y 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 r e p r e s e n t a t i o n . 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 c e n t e 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 c e n t e r . The a x i s o f any e l a t i o n i s a t a n g e n t 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 subp l a n e o f o r d e r hi, c a l l e d t h e Baer-subplane o f f .

BAER INVOLUTIONS MAPPING R ONTO ITSELF

87

Collineation Groups F d e n o t e t h e Buen-4ubpLune

u{

f. Thm

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

?noo[.

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 and a l l p o i n t s

Q

consists of P

Q f o r which t h e l i n e PQ b e l o n g s t o F. Other t h a n t h e t a n g e n t o f

a t P, t h e r e a r e e x a c t l y J E l i n e s t h r o u g h P b e l o n g i n g t o F. T h e r e f o r e ,

R

I R F I=,'ii+l.

T h i s proves ( 1 ) . There i s a unique l i n e r t h r o u g h R b e l o n g i n g t o F. F' Moreover, f does n o t f i x R and so r i s n o t a t a n g e n t o f R . L e t ( R , S } = r n a . Then a l L e t R be any p o i n t o f R- R

so SEn

-

T h e r e f o r e , 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' 1Q . 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 F proves ( 2 ) . R

-

$.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 b e l o n g i n g t o F which i s

an e x t e r n a l l i n e o f R and f(P)+P,

F'

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 ) Hence f g ( P ) = P w it h P B F. T h i s i m -

g(P)+P, i t f o l l o w s t h a t f ( P ) = g ( P ) .

plies that f g i s the identity collineation o f

?mw,L L e t F ( r e s p .

TI

which is a c o n t r a d i c t i o n .

GI be t h e Baer-subplane o f f ( r e s p . 9). Since f g = g f , t h e n f l e a -

ves G i n v a r i a n t . L e t f ' 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 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. A c c o r d i n g 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 b o t h 4itbplunea F und G , ua

F

nG

c4 u 4 u b p l m e

OL

onden 4fi uz

M.Biliotti and G. Korchmaros

88

We prove that the former possibility cannot occur. Suppose that H = F n G is a subplane of order ' / A .

H such that It n It

n

n 0 is an oval of H by (1) of Prop. 1 , Choose a line t of H I=O. Applying ( 2 ) of Prop.1 to G, f' and nG, we can infer that

So 0 =H

H

n 1=2. Similarly. It G

n a F 1=2. This yields I t

=4. A contradiction, since n

So we may assume that

nlilt

n

Q

F I+lt n nG I+lt n nH I=

is an oval.

(ii) holds. In this case nF n aG=tr n a ] . The lines through C

which are secants of either n or F

G

belong to F fl G. Since R is an oval, such li-

nes are pairwise distinct. Thus

Suppose there is a point P E Q

-

( n fl r) fixed by fg. Then P # n F U

QG.

Set Q=f(P)=

g(P). Again, Q # OF U n The line t joining P and Q meets R - ( a U R 1 in two poG' F G 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 F

G

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.

denote the involutorial collineation induced by g on G. By way of

P m m F . Let g '

contradiction, assume that g ' is either an elation or the identity. Choose an external line r of n in the subplane G such that r is fixed by g'. Applying ( 2 ) of Prop.

G

1 to G , g2 and nG, it follows that r meets n

- nG in two points P and 8. As g lea-

ves r invariant, then g2 fixes P and Q. On the other hand g 2 fixes G pointwise. Since P , Q $ G , tions.

it follows that g* is the identity collineation, contrary to our assump-

Collineation Groups

89

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 g e n e r a l i z e d q u a t e r n i o n 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 o c c u r . 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 g e n e r a l i z e d q u a t e r n i o n group. Then t h e c o l l i n e a t i o n group an elementary A b e l i a n subgroup tions i n

f

7

5

induced by S on F a d m i t s

o f o r d e r 4 . By Prop. 4 , each o f t h e t h r e e i n v o l u -

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 . B u 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 involutions of

7.

and any two F F i n a l l y , t h e statement c o n c e r n i n g t h e o r d e r o f S f o l l o w s f r o m [ 2 ]

4. '1.10.

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

o r c h o r d of a we have t h a t e i t h e r

~ = o0r

Y on a . As r l e a v e s i n v a r i a n t no p o i n t

1 ~ 1 2 3 .I f 1~123,Ly f i x e s a quadrangle

s i n c e t h e k n o 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 f o r m a subplane

TI'

r

P

leaves

and p = n 11 ill i s a suboval o f 0. Since

r

i n v a r i a n t . As

Y

is a normal subgroup o f r , t h e n

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 i m p o s s i b l e . Thus P

i s empty. As Y is an elementary A b e l i a n 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 I n + l . Now we s h a l l p r o v e 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 I E i = n ( n - 1 ) / 2 and ( n + l , n ( n - l ) / Z ) = I ,

t h e n 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 r o m t h e k n o 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 xes r . But t h e n , by ( 2 ) o f Prop. 1 ,

r

E.

As 'Y is a normal subgroup o f

r,

then

r fi-

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 ) , t h e n 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 :

M . Biliotti and G. Korchmaros

90

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 a c c o r d i n g 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 1,

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.

r c o n t a i n s no B a e r - i n v o l u t i o n . 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

c e n t e r belongs t o r . B u t t h e n 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 c e n t e r s 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 involutions 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 o r d e r .

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 s e m i d i r e c t p r o d u c t o f O() by a group o f o r d e r 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 a p p l y He-

r i n g ' s main theorem on bewegend groups [6]. As t h e k n o t K of n cannot be t h e c e n t e 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 t h e d u a l i t y , w i t h t h a t conside-

r e d 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 =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() has o r d e r 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, set of all fixed points of

Z() on

Z() R.

has some f i x e d p o i n t s on P. L e t p be t h e

r leaves

p i n v a r i a n t . As

r

leaves i n v a r i -

a n t no p o i n t o r chord o f n t h e n 1~123.But as we have shown i n t h e p r o o f o f Prop, 6

1 ~ 1 i~m3p 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 impossible. A s i m i l a r argument shows t h a t t h e

r 5

Aut .

centralizer

o f i n

r i s t r i v i a l . Therefore,

Collineation Groups

91

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 of

we have seen in the proofs

A . As

o f Prop. 6 and Theorem 6 , the following statement holds:

Now we shall prove

B 4 4 . In our situation,

tions in





has exactly one class of involutions. So all involu-

are elations.

Given any point P t S , let

a

denote an involutorial elation of

be the center of the unique Sylow 2-subgroup L of



A

fixing P. Let z(Z)

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



with the same fixed point P

on S commute because they have the same axis PK. Thus, < A > acts on S as the corresponding simple group acts on the set o f its Sylow 2-subgroups. This completes our proof. Assume ~SL(2,q), q=2' U iK1

and q24. By a result of Hering [7], Theoren 2.8.c, D U S U

is a desarguesian subplane

r is strongly irreducible on

n'

of order q o f n and

n, then we must have S=n

.

S is

a

Hence,

suboval of n. Since n'=n

.

Moreover, n

is a conic. Therefore, we have

Assume either c a > - S z ( q ) , q=Za q>4, or zPSU(3,q2), q=2' q24. The present state o f

M.Biliotti and G.Korchmuros

92

our knowledge does n o t a l l o w u s t o determine t h e u n d e r l y i n g p l a n e rical

For a geomet-

TI.

approach t o t h i s e s s e n t i a l q u e s t i o n , 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

o f D as w e l l as t h a t o f has c l a s s [XI,

...,x

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 l a n e

] when I r I l U l belongs t o t h e i n t e g e r s e t I x l ,

...,x

K

1 f o r any

l i n e r i n t h e plane. R e s u l t s about s e t s w i t h p r e s c r i b e d c l a s s a r e g i v e n i n [ E l , ~ 1 7 1 , [IE

P/LUO[. 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 b e l o n g i n g t o A. We

s h a l l denote t h e i r c e n t e r s by R . ( i = 1 , 2 ) and t h e l i n e through them by r . We want t o 1

determine

1 r n D I.

We a l r e a d y 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 t h e p r o o f o f Theorem B, < A > does n o t l e a v e r i n v a r i a n t . Hence D g r n D. Assume f i r s t a 1 a 2 = a 2 a , . An argument s i m i l a r t o t h a t used i n t h e p r o o f o f Prop. 8 shows t h a t r i s a t a n g e n t 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 cent 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 h e Sylow 2-siibgroup o f < A > c o n t a i n i n g 011a2.

Assume now a l a 2 f a 2 a 1 . Then r i s n o t a t a n g e n t o f R. By [ 1 4 ] , Lemma 5.1, r i s t h e unique f i x e d l i n e o f u 1 a 2 which does n o t pass t h r o u g h K. L e t A be any d i h e d r a l subgroup o f < A > which c o n t a i n s u 1 a 2 . Then A a l s o Thus, t h e c e n t e r s o f i n v o l u t i o n s o f with a E A ,

A

l e a v e s r i n v a r i a n t and f i x e s K .

b e l o n g t o r , a l s o . If A i s such t h a t no



u B A , leaves r i n v a r i a n t then r n 0 consists o f the centers of the invo-

l u t i o n s i n A and

Irn Dl=lAl/Z.

We p o i n t o u 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 . 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 b o t h o f cases under c o n s i d e r a t i o n . Assume zsZ(q), q=Za and q>4. Then < A > admits e x a c t l y t h r e e c o n i u g a t e c l a s s e s 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 c o n t a i n e d i n any o t h e r d i h e d r a l subgroup. They have o r d e r s 2(q-1) of .

o r 2 ( q V G + l ) . Those o f o r d e r 2(q-1)

a r e maximal subgroups

Moreover, each o f those o f o r d e r 2 ( q + / q + l ) i s c o n t a i n e d i n e x a c t l y one ma-

93

Collineation Groups x i m a l subgroup o f o r d e 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 ; l D I = q - l

or l r n D l = q + d z + l .

Assume =PSU(3,q2), q=2a and 924. Then f o r any two d i s t i n c t Sylow 2-subgroups 6,

we have < Z ( 6 , ) , Z ( z z ) > = S L ( 2 , q )

( s e e [I.?],

Satz 4 . 3 . v i i ) .

El,

T h e r e f o r e , each d i h e d r a l

subgroup o f i s c o n t a i n e d i n some subgroups o f < A > i s o m o r p h i c 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 c o n i u g a t e d c l a s s e s 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 c o n t a i n e d i n any o t h e r d i h e d r a l subgroup o f SL(2,q). groups have o r d e r s 2(q-1)

These

o r 2 ( q + l ) a c c o r d i n g t o whether I r n S 1 = 2 o r 0. I n PSU(3,q')

t h e subgroups i s o m o r p h i c t o SL(2,q) a r e c o n i u g a t e . 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 t h e s e f a c t s we can i n f e r t h a t < a l , a 2 > i s c o n t a i n e d i n a d i h e d r a l subgroup A o f o r d e r 2 ( q - I )

o r 2 ( q + l ) a c c o r d i n g 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 c o n s i d e r t h e subgroup Y o f < A > which l e a v e s

rnS

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 r o d u c t o f A w i t h a c y c l i c group o f o r d e r 3. T h e r e f o r e 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 t h o s e o f A . We may assume t h a t r and

S a r e d i s j o i n t . L e t M=SL(2,8) be t h e subgroup o f c o n t a i n i n g A , Denote by

the

0

subgroup o f o r d e 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 r d e r 9 o f A

t h e n 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 c o n t a i n e d

i n O X M t h e n 0x5 i s t h e u n i q u e Sylow 3-subgroup o f < A > which c o n t a i n s f. L e t R de-

n o t e 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 c e n t e r s l i e i n r . For any P , , P ~ E R w i t h p,6p2,

we have t h a t cp,,p,>is

c o n t a i n e d i n a d i h e d r a l subgroup o f o r d e r 2.3'.

Thus

R i s a f u l l c l a s s o f c o n j u g a t e i n v o l u t i o n s i n < A > and Glauberman's theorem (see [ 9 ] B B and I/=3 Moreover, O x A has

.

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 be any d i h e d r a l subgroup o f < A > o f o r d e r 2(q-1) o r 2 ( q + l ) .

I n o r d e r t o p r o v e t h a t each i n v o l u t i o n O E A , w i t h 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 t h e n 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 b o t h 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 c e n t e r s , s i n c e t h e y map ?i o n t o i t s e l f . T h e r e f o r e , t h e y must have t h e same a x i s t . But then, t h e y b o t h cannot l e a v e r i n v a r i a n t s i n c e t , b e i n g a t a n g e n t o f a , i s d i s t i n c t f r o m 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:,=

M. Biliotti and G. Korchmaros

94 =Z , t h e r e

I f u g h t h e n =SL(2,q),

Z(Z)>.

(see [9], 11.8.27).

(2,q) now

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 , ) ,

as A i s a maximal subgroup o f < z ( Z , ) , z ( Z ) > = s L

Hence, < o , A >

#.

c o n t a i n s two commuting i n v o l u t i o n s . Assume

Then N(Z,) c o n t a i n s two d i s t i n c t c y c l i c subgroups

0 and o 2 o f o r d e r ( q + l ) / d w i t h d=(3,q+l)

which c e n t r a l i z e < Z ( Z , ) , Z ( Z ) >

Z ( z 2 ) > , r e s p e c t i v e l y (see [ 1 2 ] , S a t z 4 . 3 . v i i ) . i n v a r i a n t , since the centers o f a,

0,

and

a,

These c y c l i c subgroups b o t h l e a v e r l i e on r . We p r o v e t h a t

two commuting i n v o l u t i o n s by showing t h a t In V ~ ~ , @ ~ < Y = Zand ,O k,o/Z(z,)

0

over,

and

0,

=2

,

Z(Z,)1>2. By [121, Satz 4 . 3 . v -

z,.

More-

a r e F r o b e n i u s complements o f Y . Since 1Z11=q2, IBI=l@,l=iq+ll/d,

<e,o,>=!.

where ~ 2 1 s i n c e each element o f

Z(Z,).

admits

i s a Frobenius group w i t h F r o b e n i u s k e r n e l

i t i s n o t d i f f i c u l t t o show t h a t Y

and < Z ( Z , ) ,

I f <EI),@~>=Z,O

z,-Z(Z,)

Since t h e q-1 i n v o l u t i o n s i n Z ( Z , )

w i t h z,

is a IA,Bl d i h e d r a l group w i t h c y c l i c stem o f o r d e r q-1 o r ( q * - l ) / d a c c o r d i n g t o whether ? 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

I n t h e l a t t e r , we have a g a i n Ir n DI=q-I s i n c e

contains exactly q-I involuiA,B) 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 o f o r d e r 2(q-1). Conversely, i f I r f l O l = q - l t h e n r i s f i x e d by a d i h e d r a l group H o f o r d e r 2(q-1) which, o f course,does n o t f i x any o t h e r l i n e . B u t , H i n t e r c h a n g e s 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

lr flSl=2.

T h i s completes t h e p r o o f o f P r o p o s i t i o n 10.

5 . COLLINEATION GROUPS

OF EVEN ORDER WHlCH ARE TRANSITIVE ON AN OVAL

95

ColIimeation Groups

P/iou[.

Clearly r is strongly irreducib e on 0. So we can apply Theorem B. Since r

acts transitively on

R,

every point o f R is fixed by an involution of A. But then,

by Gleason’s lemma (see [2], 4.3.15).



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 involution 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 zPsu(3,q2) holds. Since S=R, we have n=q3 with q=2a, a 2 2 . 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 S U{K}I=(q +l)(q+l).

U

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 ] 1 2 . 2 . 1 ,

this implies (q+1)l(q3+l)q+l, a contradiction.

Notice that the possibility +‘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, 1968).

P., Finite Geometries (Springer Verlag, Berlin-Heidelberg-New York,

M.Biliotti and G.Korchmaros

96

C e n t r a l elements i n c o r e - f r e e groups, J . Algebra 4 (1966) 403-

[ 3 ] Glauberman, G., 420. [ 4 ] H e r i n g , 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 ] H e r i n g , 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 l a n e s 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. Pur e Math. 37 (1980) 473-477. [ 6 ] H e r i n g C., Hon. T.G.

On B e w e g l i c h k e i t i n a f f i n e planes, i n : F i n i t e geometries, Proc. Conf. Ostrom, Wash. S t a t . Univ. 1981, L e c t . Notes Pure Appl. Math. 82 (1983)

197-209. [ 7 ] Hering, C.,

On p r o j e c t i v e planes o f t y p e V I ,

i n : C o l l o q . i n t . T e o r i e 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 geometries o v e r f i n i t e f i e l d s (Clarendon Press,

Oxford, 1979). [ 9 J Huppert, B.,

E n d l i c h e Gruppen 1 ( S p r i n g e r V e r l a g , Berlin-Heidelberg-New York,

1967). [ l o ] Huppert, B. and Blackburn, N.,

F i n i t e Groups 111 ( S p r i n g e r Verlag, B e r l i n - H e i -

delberg-New York, 1982). S y m p l e c t i c groups, symmetric d e s i g n s and l i n e o v a l s , J . Algebra

[ l l ] K a n t o r , W.M.,

33 (1975) 43-58. [ 1 2 ] Klemm, M.,

f 2f C h a r a k t e r i s i e r u n g d e 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. Le o v a l i d i l i n e a d e l p i a n o d i Luneburg d ' o r d i n e 2

[ 1 3 ] Korchmaros, G.,

2r

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 s o m o r f o a 1 gruppo semp l i c e S Z ( ~ ~ A) t, t i Accad. Naz. L i n c e i , Memorie, C 1 . S c 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 ( S p r i n g e r V e r l a g , Berlin-Heidelberg-New York,

1980). 1151 S t r o t h , G.,

On Chevalley-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 geometrie d i G a l o i s , Relazione n.30

1 s t . Matem. Univ. N a p o l i (1973).

Collineation Groups

97

[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, Rendiconti, C1. Fis. Mat. Nat. (8) 40 (1960) 812-818 and 1020-1025.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 99-104 0 Elsevier Science Publishers B.V. (North-Holland)

ori

SETS

99

OF PLUCKER CLASS TWO IJ

PG(~,Q)

-2)

P a o l a R i o n d i and r J i c o l a Melone Dipartimento d i Matematica e A p p l i c a z i o n i "R. Cacc i o p p o l i" Universitd d i Napoli ITALY

I n t h i s paper t h e concept of t h e Plucker c l a s s o f a k-set 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 of P l u c k e r c l a s s t w o i n 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 m o s t the 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 geometries.The t h e o r y o f k - s e t s , i . 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 s u b s p a c e s 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 b5], DG], [?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 a n d u s e f u l t o o l i n s u c h c h a r a c t e r i z a t i o n s . A k-set

in

I
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 2 line N N , let ,ti (i=1,2) be the set o f the lines through N. and 1 1 f: + E 2 be any bijection. It differen$ from k, moreover let is easy to prove that K= { snf(s); s E g l } u { N1, N2 ] is a thick (q+Z)-set and Nl, N2 are nuclei of K . So the number of the is equal to the number of the (q+2)-sets with nuclei N1,N2 bijections from 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.

El

El

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 )

R. Capodaglio di COCCO

118

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 ) v m € G F ( 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 Px3' The line X 3 r ( X 3 ) intersects K at X 3 and at an Other point, say X4. 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 ) a s the improper Let D be the only point of the axis y and C as the point ( 0 , O ) . 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. 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 ) v m E G F ( 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 w i t h 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 B 1 , B2t

. . .1 0 , .

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 4 . Proof: If n=qtl, we have nothing to prove. Suppose n=q; if C ~t is point different from Bl,B2,. . , B n r we have N,B1 ,B2,...,BnrA} , where A is a convenient point of the 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 , . . , B n , 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 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 instead of P ~ ~ . A f t ear B. Segre's sake of brevity we write P, 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, P I C , Pf are represented respectively by the equations x1 =m2x0, x ~ = ~ x2=mOx1 ~ x ~ with , m,-=mom 2 If we consider all the points C of K, C=FN,Al,A2,A3, we obtain that m2 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 P=P by the above arguments we have that the only point of K 3 $3 ' , vhich is on the line PI P3 is a so on the 1 ne A1A3 . If this point is distinct from A2, then the line A1A2 intersects K in at least four points; otherwise the point A4, the intersection of K with the line P2P3, is certainly distinct from A 1 and lies on the line AIA2. THEOREM 5 : Suppose K is a minimal thick

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 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 r 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 i n g 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 ) h o l d s f o r t h o s e b E X \ Ai d The number d w i l l be c a l l e d t h e such t h a t A . u t b l i s an i n j e c t i v e subset o f N

...

.

1

dimension o f I r ( X ) . We g i v e now t h e de i n i t i o n o f

3-geometries.

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 v e n i n a s l i g h t l y d i f f e r e n t f o r m ) . An 9-geometry o f r a n k 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 -

r

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

... 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 ) r r AEA w i t h t h e r e s t r i c t i o n t h a t axiom ( 3 ) h o l d s f o r t h o s e b € X \ Ai such t h a t i n t o A = A,u

A . U t b l E 3. 1

Ai a r e c a l l e d t h e f l a t s o f r a n k i o f t h e geometry G ( X I . r = A. v i b l f o r t h e s e t Ai+l mentioned i n axiom ( 3 ) . We s h a l l w r i t e A i+l 1 S A m a t r o i d M ( X ) i s a geometry Gr(X) = (S,Y,X,A) w i t h 9 = 2 and X = S, and The elements

r

AiE

2

c o n v e r s e l y . A p e r m u t a t i o n geometry Pr(X) = (X,A), w i t h XcN , i s a geometry Gr(X) = 2 = (S,3,X,A) w i t h S = N and 3 = IF c N2 : F i s a subpermutation o f N l , and convers e l y . P a r t i a l a p p l i c a t i o n ( r e s p . c o a p p l i c a t i o n ) geometries can be e a s i l y c h a r a c t e -

P.V, Ceccherini and N. Venanzangeli

128

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) 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 r ced from [ l ] , [5], [ 3 ] where examples o f permutation geometries and o f i n j e c t i o n geometries are given. We now g i v e some o t h e r examples. EXAMPLE 2.6.

Free 9-geometries.

Gr(X) = (S,9,X,A)

The f r e e geometry

d e f i n e d by assuming X = S, 9 a s i m p l i c i a 1 complex o f S, A

i

is

= t A E 9 : ( A ( = il,

ral.

O 1 and

conversely ( c f . Theorem 5 . 3 ) . EXAMPLE 2.9. metry w i t h A =

Truncations o f an 9-geometry. I f Gr(X) = (S,9,X,A) A,U

...u A r

(k)

Then Grlk) = (S,9,X,A

i s an 9-geo-

and i f 1 6 k < r , we can consider A ( k ) = A , u . . UAk. . . ) 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

Gr’

EXAMPLE 2.10.

Cotruncation o f an 9-geometry. If Gr(X) = (S,S,X,A)

i s an

9-geometry and i f Ah€ Ah (06 h c r ) i s i n c l u d e d i n some Are Ar, then we can consider =

2i = g,i(A,,)

=

(S,g,X,A) w i t h

G (A,) r

=

tAhti€Ah+i

i, U... ~ f i

c_

Ahti,

i=O,l,.

c f . 5.1 ( a ) .

= Gr,

EXAMPLE 2.11.

B i t r u n c a t i o n s o f an 9-geometry.

metry and i f A h € Ah, w i t h O < h c k c r and A h c A sider

-

-

A = A,

Ji

=

..,r-h].

Then G r ( A h 1 = ~i s -an~9-geometry , o f rank r - h , Note t h a t

: Ah

i.(A ) i h

=

{Ahti

E Ahti

: AhCAh+i,

k i=O,

u... uA;(-~i s an 9-geometry o f rank k-h.

EXAMPLE 2.12.

I n t e r v a l s o f an 9-geometry.

I f Gr = (S,Y,X,A)

i s an 9-geo-

f o r some A k € A k then we can con-

...,k-h}.

I f Gr

=

Then G = (S,Y,X,#)

(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 =

{Ahti

E Ahti

: A h L AhtiCAk,

o f rank k-h on Ah.

i=O

,...,k - h l .

Then i i , u . . . u f i k-h

with

A ) = h’ k gives a m a t r o i d

129

On a Generalization of Injection Geometries

EXAMPLE 2.13.

Restrictions of

J-geometries.

l e t Gr(X) = (S,9,X,A)

be an 9-

geometry and l e t be S ' such t h a t A € S ' C X f o r some A E d Define A' = 1 1 1' LJ Z A ' , r ' = max t i : A i E A ' l . Then = tAEA: AcSIl, X I = u ,A, 9' = A€ A A' € 4 ' G r , ( X ' ) = (S'J',X',A') i s an 3-geometry o f rank r ' on X ' .

A s p e c i a l i n t e r e s t i n g case i s obtained when S 1 = X ' = U A f o r some f i x e d AE B Bc A such t h a t D n A l # 0; i n t h i s case r ' = max l i : d i n B f 01, A' = t A E A : A C E f o r some B C B } and 9' =

A€ D

2A.

3. D I R E C T SUMS OF 9-GEOMETRIES By s t a r t i n g from two g i v e n 9-geometries, 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 described below. DEFINITION 3.1. G",,

=

Q

of

9-geometries.

Let GIr,

= ( S ' , 9 ' , X 1 , d ' )and

( S " , ~ " , X " , ~ " ) be two 9-geometries; we can suppose w i t h o u t l o s s o f genera-

0. Assume:

l i t y t h a t S'n S" =

A. = [A!, 1

D i r e c t sum

1

uA" : A ! , € A ! , , A!,,€ i" 1 1 1

A = A, u.. . u A

r = r'

r'

t

A'.',,, i ' t 1

i " =

i},

r".

w i l l be c a l l e d t h e d i r e c t sum G '

Then (S,9,X,A)

r'

8

GI,, o f G,;

and G I , , . It i s

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 i s an 9-geometry

o f rank r = r ' t r " . DEFINITION 3.3. = (S',9',X',A')

F u l l d i r e c t sum 6 o f i n j e c t i o n geometries. L e t G,;

and G'',

= (S",?',X",A")

dimension d. A c t u a l l y X ' c - S'

=

be two i n j e c t i o n geometries o f t h e

N o d and X " c - S" = NtId. We can suppose w i t h o u 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" = N =

N'UN",

-

s

A. = ( A ' u A;, 1 i' A = A, u...uAr, Then (S,?,X,A)

d

= N

-

, x

=

x'ux", T = r 1 E 2

: A i l € A ; , , A;,

r

~ame

0 . Assume:

: I i s injective),

E A;,,, i ' t i"= il,

= r ' t r".

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,,.

P.V , Ceccherini and N. Venanrangeli

130

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 di

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. P r o o 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;,, c o n s i d r r" d ered as 9-geometries ( c f . Theorem 3.2). A c t u a l l y S = S ' U S " = N ' u N1IdcNd = 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. F o r 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 p e r m u t a t i o n

( c o n s i d e r e d as i n j e c t i o n geometries o f dimension d.2) i t i s a p e r m u t a t i o n geometry.

o f dimension two, i . e . REMARK 3.6.

t e d meaning -

L e t GAI and G",,

o f (1 1.

geometries

i s an i n j e c t i o n geometry 0

be two " p e r m u t a t i o n geometries" i n t h e r e s t r i c -

Then t h e f u l l d i r e c t sum GAI

G;I,

i s a p e r m u t a t i o n geometry

( i n o u r meaning), b u t i t i s n o t a " p e r m u t a t i o n geometry" i n t h e meaning o f [l], 2 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 ,

4. REGULAR 9-GEOMETRIES

A geometry Gr (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 n c l u d e d i n

.

Every s t a r 9-geametry i s r e g u l a r . The f o l l o w i n g G2(X) = (S,S,X,A) some A E A r r i s not regular : A, = tA,

X = S = ta,b,c,d),

A

9

=

(A,,

2

[A

=

2

t b ) , {c}, t d l ,

:[

all,

A;,

1

=

[A

= Ia,bl,

1

A ' = Ia,cl, 1

A" = ta,dl.}, 1

A = A, u A u A 1 2'

= Ia,b,c}},

A1,

A

A", 1

{b,cl,

A21. We n o t e t h a t A;' i s n o t i n c l u d e d

in A

2' The same example shows t h a t i f G (X) = ( S , S , X , A ) i s a geometry, t h e n 9 i s r 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 . I t 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 r o p e r t y (191).

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 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 [ ~ } 1 E . PROPOSITION 4.1. 1

1

1

P r o o f . Suppose G ( X I r e g u l a r , A. E Ai,

1

i< r . L e t A . c A E dr. We have r 1 1 r A.cA s i n c e i < r ( c f . Prop. 5.1. ( e ) ) . If b E A r \ A t h e n b E X \ A i and i r i' A . W b I E 9 , s i n c e A.u:bl C_ A r E 9 and 7 i s a s i m p l i c i a 1 complex. 1

1

On a Generalization of Injection Geometries Conversely, i f f o r each A.E Ai, 1

A.U 1

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

I b I E 9, t h e n A . i s i n c l u d e d i n some 1

times.

131

ArE

A : t h i s f o l l o w s by a p p l y i n g ( 3 ) r - i

r

0

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 lArl

1 i f f Ar

=

=

[XI.

( 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 P r o o f . I f A = ( X I , t h e n 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 r 1 1 r AEA s o t h a t G 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 A = ( A I . I f X E X = u A, t h e n XE A. f o r some A. E A , , so t h a t X E A . c A AE A 1 1 1 1 r r r by r e g u l a r i t y . T h e r e f o r e X c Ar, i.e. X = A r

.

5. FIRST PROPERTIES OF .?'-GEOMETRIES PROPOSITION 5.1.

( a ) IA,1

L e t Gr

=

(S,S,X,A)

1 and t h e element A,EA,

=

be an 3-geometry. 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. t h e n t h e r e e x i s t s a k k h k k). c h a i n A h C Ah+l c...c A w i t h A E A ( s = h , h t l , k s s ( c ) Each A. E A . ( O < i s r ) i s i n c l u d e d i n a c h a i n A,C A C . . C A. w i t h

...,

1

A

S

E

AS (s=O,l,

1

1

...,i ) .

1

( d ) I f Gr i s r e g u l a r , each A.E A . ( O < i < r ) i s i n c l u d e d i n a c h a i n 1

A, c...c Ai

C...C

( e ) I f Aie where Ai

=

A, v { b }

=

1

w i t h A E A ( s = O ,...,r ) . r s s A i , A . € A . ( w i t h i , j = 0,1, A

J

...,r ) ,

J

and i f A . L A . , 1 J

then i C j ,

A . i f and o n l y i f i = j . J P r o o f . ( 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,, AEA 1 t h e n A,, C - A:. Then A, = A;; o t h e r w i s e i f bEA:\ A,, then from ( 3 ) i t f o l l o w s t h a t A

C A; w i t h AIEA1, contradicting (2). 1C A w i t h Ah+l~Ahtl. I f h t l = k, From (31, Ah v I b } = A ( b ) Let b E A k \ A h' htl - k t h e n Ahtl = Ak by (3). I f h t l < k , t h e n Ahtl C A and i t e r a t i o n o f t h e same a r g u k ment l e a d s t o a c h a i n A h C ... C Ak-l C "A, A,, w i t h A S e A ( h < s < k ) and A ; ( € A k .

S

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, required. ( c ) Take h = 0 and k

=

and Ahc

r e g u l a r i t y . By ( c ) we have a c h a i n A, C...C

s=O,

Ar-lC

...,r ) .

Ar,

a c h a i n as

i i n (b).

( d ) If i = r, ( d ) f o l l o w s f r o m ( c ) . I f i c r , t h e n

Ai

... C A ~ - ~ kC iAs

C...C

s o t h a t we g e t a c h a i n

A.C

i

A

r

f o r some A

r

A. and b y ( b ) we have a c h a i n 1

A,C

...C

A.

1

C...C

Ar

(AS€ A

S'

E

A

r

by

P. V. Ceccherini and N . Venanzangeli

132

( e ) We have i c j by ( 2 ) . From Ai pose now A.

A ! w i t h Ai,

C

1 -

1

we g e t a c h a i n

A.

1

=

A,C

A . i t f o l l o w s o b v i o u s l y t h a t i = j. SupJ I f i = O , t h e n A, = A: by ( a ) . I f i f O , t h e n by ( c )

A;EAi.

...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

v { b ] = A! where b E A . \ A . So A . C A! i m p l i e s A. = A ! . Ai-l 1 1 1-11 1 1 1

0

From 5.1 ( e l f o l l o w s COROLLARY 5.2. then AinA;

E A .

J

THEOREM 5.3.

L e t Gr be an 9-geometry.

I f Ai,

A; €Ai

w i t h Ai

# A;(O c i s r ) ,

with O s j c i - 1 . L e t Gr

=

(S,3',X,A)

be an

%geometry.

I f x i s i n c l u d e d i n some i n c l u d e d i n a unique A E A 1 1' A . ( j > O ) , t h e n x i s i n c l u d e d i n some Ai E Ai f o r e v e r y O < i c j . I n p a r t i c u l a r i f J G i s r e g u l a r , t h e n each x i s i n c l u d e d i n some A. E A . f o r e v e r y 0 < i c r . ( a ) Each X E X \ A,is

r

1

1

( b ) I f 0 < i < j < r , each A . E A . i s t h e u n i o n o f t h e elements o f A . which J J 1 are included i n A j' ( c ) I f Ai E A i ( O 6 i < r ) i s i n c l u d e d i n two d i s t i n c t element A ! A'.' E Aitl, ltl'

ltl

then A. = A ' n A:' = n A. 1 it1 it1 Ai 5 A €Aitl ( d ) 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 : ( d l ) f o r a l l 0s i < r, each A . E Ai i s i n c l u d e d i n two d i s t i n c t elements o f 1

Aitl; ( d ) f o r a l l 0 c i < r , each A . E A . i s i n c l u d e d i n two d i s t i n c t elements o f 2 1 1 ( 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 o f A. c o n t a i n i n g A i ' J P r o o f . ( a ) Suppose x E X \ A , . Since X = U A, we have X E A . f o r some A.EA AE A J J j' Then x E A , 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 1 - j 1 1 x E A . and O < i < j , t h e r e i s a c h a i n A,CA C . . . C Aic c A. ( A S € A s , O t s < j ) , J 1 J w i t h x E A1 = A, v {XI, so t h a t x E A cA 1 i' ( b ) I t 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 CA This f o l l o w s from ( a ) .

...

j'

5 Aitl

= 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 n J J A! n Uyt1 by P r o p o s i t i o n 5.1 ( e ) . i it1 ( d ) ( d l ) * ( d 2 ) . From ( d i t f o l l o w s t h a t each A.E A . ( O < i < r ) i s i n c l u d e d 1 1 1

( c ) Ai

j = i and A

=

On a Generalization of Injection Geometries

133

and Ar-l i s i n c l u d e d i n two d i s t i n c t elements o f A Ar-1 r' ( d 2 ) * ( d 1. From ( d 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 = ments o f Ar. I f x ' EA; \ A:; and x" EA; \ ,A; t h e n Ai 2 A;+1 n AYtl w i t h A! 1+ I = A. v t x ' l and A! = A. v t x " 1 , A ' 1 l+l 1 it 1 # AYtl. i n a chain A.C 1

. e C

L

1 2 1 2 ( d 3 ) . From ( d 1 we have A. c A. "Aitl w i t h Ai+l # Ai+l and f r o m ( a ) 1 1 - 1+1 L we g e t Ai = A : nAit,. I f i t 1 = r t h e n ( d ) i s proved. I f i t 1 < r t h e n f r o m ( d ) 1 +1 3 1 11 12 2 21 and f r o m ( a ) we g e t A1 = Ait2 nAi+2 and Ai+2 = A. nA:f2 so t h a t it1 1t 2 hk A. = /--\ Ait2, and so on. i l ~ h k, c 2 (dl)

=,

(d4) * ( d

3

1. Obviously.

(dl) * ( d 4 ) . 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 ) . 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 including j-1 A.. By t h e p r e v i o u s 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 r e 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 h e n t h e r e e x i s t s Aj-, E Aj-l w i t h J J J A.C A . 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 r e j o i n a b l e i f t h e y a r e i n h k k' h k A; c l u d e d i n some A E A . I f A h and A k a r e j o i n a b l e , we d e f i n e Ah v A k = A n EA A2AhU A t h e n Ah v A = A E A f o r some c >, max [ h , k l . k k c c

P R O P O S I T I O N 5.4.

I f A,,€

Ah, Ak E Ak a r e j o i n a b l e w i t h Ah v A

A nA = A . E A . t h e n h+k>Ii+c. h k 1 1 Proof. I n d u c t i o n on k - i 2 0 . I f k = i , t h e n A =

A

h

v Ai

=

Ah;

k

= A . C_

i

A

h

and A

c

= Ac E Ac and

k =

A v A = h k

i t follows htk = c t i . E A Then w i t h Ai 5 Ak-lCAk. k-1 k-1 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 , , t h e n

ifk & i + l , l e t be A

Ac, = Ah v Ak-l x

c_

) v Ix) = A v Ix) = = A v Ak = Ah v (A v {XI) = (Ah v A c h k- 1 k- 1 C' I n c o n c l u s i o n c ' t l > c . By t h e i n d u c t i o n h y p o t h e s i s h t k - 1 2 i + c ' , so

A so ~ t-h a~t A

~

Acl+l' that h

t

k %i t c'

THEOREM 5.5.

t

1> i

t

L e t Gr(X)

c. =

0

(S,S,X,A)

be a geometry o f r a n k r 6 3 . Then Gr(X)

satisfies the following condition: ( * ) i f Ah€

and A

k

$,

AkC

Aky A h n A k = A . w i t h h

are j o i n a b l e ,

1

t

k s i t r and i f AhU A E 7, t h e n Ah k

P.V. Ceccherini and N . Venanzangeli

134

Proof. We can suppose w i t h o u 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.

If r

=

-

2, t h e n 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

k

a r e n o t i n i n c l u s i o n . I n t h i s case, t h e p o s s i b l e values o f h, k and i a r e : 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' 1 1 1 1 1 1' and i f x E A ' \ A i t i s easy t o prove t h a t A v ( X Ii s an element o f A i n c l u d i n g 1 1 1 2 A1 and A;.

If r

= 3 then a c t u a l l y h t k -

i -, 4 t h e r e e x i s t s an 9-geometry (resp. a p e r m u t a t i o n

THEOREM 5.8.

geometry) Gr(X) o f r a n k r, such t h a t c o n d i t i o n ( * ) i s n o t s a t i s f i e d . P r o o 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 ( r e s p . a p e r m u t a t i o n geometry) G;-l =

(S',S',X',A')

o f r a n k r - 1 2 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 ; i n o t h e r

A ' E A ' such t h a t : A ' n A ' = A!, h t k - i s r - 1, k k h k i A' UA' E 7' and A' and A ' a r e n o t j o i n a b l e . h k h k be an F g e o m e t r y ( r e s p . a p e r m u t a t i o n geometry) of L e t GI' = (S",7",X",d") 1 r a n k 1 w i t h A: = (A:). Then we c l a i m t h a t Gr = GAq1 b G; = (A,9,X,A) ( r e s p . words, t h e r e e x i s t

-

b G;) i s an 9-geometry ( r e s p . a p e r m u t a t i o n geometry) o f rank r such Gr = G ' r-1 t h a t c o n d i t i o n ( * ) i s n o t s a t i s f i e d . Indeed

Ah = A ' U A t E A h h'

A = A' U A: E A a r e such t h a t : k k k

A h n A k = (A; n A ; ( ) u A :

= A;UA:

€Ai,

5

k ( A ' U A " ) E A t h e n A,',uA'

h

t

k - i s r

-

I 2,

l e t f denote

t h e s i z e f u n c t i o n o f M. F o r any f l a t Y o f r a n k 2, we have t h a t lIl(O,Y)l

P

I n o t h e r words, t h e p o s e t

= ( Y I = f ( 2 ) = f(d(0,Y)).

has t h e same s i z e f u n c t i o n f t h a n 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 h o l d s . So c o n d i t i o n ( 1 ) o f Prop. 2.3 h o l d s too, and t h e r e s u l t ( b ) - ( c ) i s proved.

0

4. F-GEODETIC GRAPHS a l l graphs w i l l be f i n i t e w i t h o u t l o o p s o r m u l t i p l e edges,

I n what f o l l o w s ,

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

p =

p(t) =

G'

(V,;)

=

i s o b v i o u s l y t h e Hasse diagram o f a poset

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 p a t h f r o m x

(V, 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 i n o m i a l t y p e , s a y C and CA, ll are isomorphic a s coalgebras:

c

(14)

n

bi

a c----t

qi

!/Ai

.f b i .

Proof. T r i v i a l . Here a r e some examples: i i i 1) C o a l g e b r a o f p o l y n o m i a 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 . I n t h e l f o l l o w l n g J we s h a l l d e n o t e t h i s coalgebra w i t h CN. i 2 ) C o a l g e b r a 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 11 1 1 rl of f o r m a l power s e r i e s . [i] ! 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 rl J J [jlq! [i-jlq! 2

:= l + q + q + . . . + q c o e f f i c i e n t s ) and n i = [ i J algebra of formal e u l e r i l n series.

i-1

.

C*

n

i s s a i d t o be t h e

C o a l g e b r a s 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 . poset) t h a t s a t i s f i e s t h e following f u r t h e r conditions: 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 c o n d 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 a y 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 e m e n t . A f t e r [ 1 2 ] , t h e s e p o s e t s a r e s a i d t o be 1.f. p o s e t s of full b i n o m i a 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 i n o m i a l t y p e o f i n f i n i t e l e n g t h one c a n a s s o c i a t e a c o a l g e b r a of f u l l b i n o m i a l t y p e C,=(K[X],A,,E) - t h e s o - c a l l e d maximally 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 ( 7 - by d e 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 and assuming n i = B i . Thus, e a c h s t r u c t u r e c o n s t a n t hf g i v e s t h e number h l = Bi/B,Bi-j o f e l e m e n t s o f r a n k j i n a n y i n t e r v a l o f 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 c o r r e s p o n d r e s p e c t i v e l y t o t h e following posets: a) t h e l a t t i c e of 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 of a l l f i n i t e - d i m e n s i o n a l subspaces of a v e c t o r space of dimension w o v e r GF(q)

.

164

L. Cerlienco, G. Nicoletti and F. Bras

12.

I n 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 i n o m i a 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 s e q u e n c e s 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 the p a r t i c u l a r c a s e of t h e coalgebra of polynomials). 2.1. Let u s b e g i n by g e n e r a l i z i n g t h e n o t i o n o f p o l y n o m i a l s e q u e n c e o f b i n o m i a 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 i v e n i n some i n t r i n s i c way, i t i s c l e a r t h a t we may a r b i t r a r i l y c h o o s e 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~J r, c i o c c u r r i n g i n A V . = . L T J v~. @ v and E ( ~ . ) = E How1 i s h f h i A g 'but a usefuf t o o l . Thus, e v e r , t h e chosen b a s i s (vi) i t 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;) c a n n o t be compared t o each o t h e r by means o f t h e map v i - v i . 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 . ,E) be a c o a l g e b r a o f ( f u l l ) b i n o m i a l t y p e and l e t ( b i ) L e t C,=(V,a b e 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 - b a s i s of C i f t h e t h e map rl f : C -c (15) 0

b.

1

-

rl

b:

i s a n automorphism o f c o a l g e b r a s , t h a t i s

.

A,b!

(16)

= j=O

1

.

I'![J q

b!gbi-j. 3

C o n s i d e r t h e isomorphism $I:

c -cN n

bi-

ni!/i!

x

i

from C n t o 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 C N . We s h a l l s a y t h a t a s e q u e n c e p i ( x ) of p o l y n o m i a 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 such t h a t p i ( x ) = $ ( b i ) . I t i s simple t o prove t h a t : (b:) i n C, P r o p . 6 . A polynomial sequence p i ( x j d ( [ x ] , 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 : 1 ) degfpil = i; 2 1 p o ( x l = I; 3) p i l o ) = 0 f o r every i f 0 ;

;EN,

i s q-nornial i f and

0 The i n t e r e s t i n q-nomial s e q u e n c e s o f p o l y n o m i a 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 - a n a l o g o f umbra1 c a l c u l u s a l o n g t h e l i n e s f o l l o w e d b y Rota and o t h e r s [17] , [18] ( s e e a l s o [ 8 ] , 191 ,[14). 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 e q u e n c e s . L e t C, be a c o a l g e b r a o f f u Z Z binomiaZ t y p e a n d Zet be a morphism of coaZgebra8. T h e n t h e r e p r e s e n t a t i v e m a t r i x i s c o m p l e t e l y d e t e r m i n e d by f"(bl):

Prop. 7 . f:C,,-C, M(f)

165

Polynomial Sequences and Incidence Coalgebras

w h e r e 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 t 0 r+s .[ 1 = j g o 151 <slflt-j>; < 0 / f / t > =t6 n rl which imply (18). i If a = .l Zd 0 a.b , B = .1)Z 1 Bib i EC,,* the element 1.3Z0 ( a1. / n1.! ) B iE C is ~ said 1 to be the c o m p o s i t i o n of a and Prop. (191

8.

The map

13

and denoted by

a0B.

--

A

Aut(C,l

C;

f ' f*(bl) i s a n i s o m o r p h i s m o f t h e g r o u p A u t l C , ) o f - t h e a u t o m o r p h i s m s of t h e of t h e e l e m e n t s 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 g r o u p ( C G , o ) a=ZaibicC; s u c h t h a t ao=O#al. Proof. Because of (18), map M(fog)=M(f)xM(g) it follows:

(fog)'

1 (b )

=

19) is a bijection. Moreover, from

i$o 1, w i t h a r e g u l a r p o i n t x . 1 D e f i n e PI a s t h e s e t P \ x I n B' t h e r e a r e t w o 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 of B which are m t i n c i d e n t 11 with x, the elements of type ( b ) are t h e hyperbolic 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). I n t h e even c a s e t h e g e n e r a l i z e d quadrangle P ( W ( q ) , x ) , x a p o i n t 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 q u a d r a n g ! 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-conc 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 one 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 a r e 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 7 . R-REGULARITY OF POINTS AND LINES 1. DEFINITIONS

Consider a generalized quadrangle S

(P,B,I)

of 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 .

M. de Soete and J.A. Thas

174

I n what f o l l o w s w e a l w a y s assume t h a t t h e g e n e r a l zed q u a d r a n g l e 2 o f o r d e r (s,s+2) c o n t a i n s a s p r e a d R ( I R - s t l ) ) .

S = (P,B,I)

1.

= {z E P 1 I z -. x , z # x , zx 4 R u { X I I* For a p a i r o f d i s t i n c t p o i n t s x,y we d e n o t e t h e set x n yl* as

F o r x E P, we d e f i n e x

-y

.

I stl. b u t xy 4 R , t h e r e h o l d s I { x , y ) 1. II x E A } . So f o r a More g e n e r a l l y , f o r A C P we d e f i n e A'* 9Ix l*l* p a i r o f 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, 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 1.1. = xy and s o ~ { ~ , y } ' *= ~s*t l~. b u t xy 4 R , t h e n c l e a r l y I x , y l Ix,y}'*.

If x f y or x

1.

-

.

-

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 p r o v i d e d x -. y 1.1, and xy 4 R , o r x f y and I I x , y } I = s t l . A p o i n t x i s R-regular

p r o v i d e d ( x , y ) i s R - r e g u l a r 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 follows :

P'

I x i j E P II i = l Y . . . , s t 2 , j = l, . . . , s t 2 ,

B' = I L 1 , . . . , L s t 2 , I'

I n Li

((PI

17

Xijxji

Mj

M1,...

,Mst2}

X B ' ) U (€3'

X

Rji

c B \ R , and

P I ) ) , w i t h Li f L

x i j i f i f j , Li = Rij

and i # j } ,

E R for 1 6

i,

J

Mi -f- M j y

- xjj iyy and

x !j

j . Mi,

G st2.

11.

We d e n o t e t h e set { L 1y...,Lst2! ( r e s p . !MI,.. * ,Mst2)) by { L i y L j 1 o r {Mi,rCl1* (resp. (MiyM.Ill 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 h e n by d e f i n i t i o n t h e p a i r ( L 1 , L 2 ) R-regular i f f ( L l a L 2 ) b e l o n g s t o a R-grid. I n such a c a s e t h e r e e x i s t s a unique R E R for which L1 R L2. A l i n e L E B \ R i s

is

- -

weak R-reguZar i f f ( L,M) i s R - r e g u l a r f o r a l l M E B \ R w i t h L .f- M 1 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 - r e g u l a r and f o r a l l M E B \ p a i r (L,M) i s r e g u l a r .

R , L -f- M y w i t h I{L,Mll

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-regularity f o r l i n e s i s not t h e dual of R-regularity f o r points. 2 . EXAMPLES

2.1.

Theorem. C o n s i d e r P ( W ( q ) , x )

(P',B1,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 ) i n B ' ( s e e 1.2.(d)) Then e ac h p o i n t Each l i n e of B ' \ R i s R-regular i f f 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 o b v i o u s i s R-reguZar.

t h a t t h e set ( ( x , y l

11

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).

175

R-Regularit). of the Generalized Quadrangle P(Wls1. 1-11 L e t y,z E I", y % ' z . T h e n i t f o l l o w s t h a t y i s re&ular

'd(g). Let

ill

'-I x1

i [y,z)"

= {? a t l e a s t two d i s t i n c t values, any two rows o f

(C,)

X

Let

A

are d i s t i n c t .

be a nonempty s e t , and l e t

d i s j o i n t s e t s o f nonempty subsets o f

Lo,L1,

...,L t

(with

X. The elements o f

tzl)

3 :=(X;Lo,L1 ,...,L t )

poi n t s and t h e elements o f ,..,, Lk Zines. Then 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)

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 ,

(S,)

each c l a s s

Li

partitions the point set

w i l l be c a l l e d

X

u,,,,,

be m u t u a l l y

X.

C o n d i t i o n (S2) j u s t i f i e s t h e t e r m p a r a l l e l c l a s s f o r each o f t h e l i n e s

Li.

The

n o t i o n o f a seminet g e n e r a l i z e s such well-known s t r u c t u r e s l i k e a f f i n e p l a n e s , n e t s and (more g e n e r a l l y ) t h e 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

3

in

c a l l e d a transi.arsa2 o f t h e seminet

5

e x a c t l y one p o i n t . I f p a r a l l e l class

Li

5

r:=

then

r

has a t r a n s v e r s a l c o n s i s t i n g o f

contains e x a c t l y

r

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 versals o f

i f i t i n t e r s e c t s each l i n e o f

(X;Lo,L1

3

is

p o i n t s , t h e n each

l i n e s ( a n d hence each f u r t h e r t r a n s v e r s a l r

p o i n t s ) . I f T1,T

,...,Lt;T1,T2 ,..., Tv)

*,...,TV

are trans-

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 r a n s v e r s a l c o n s i s t s o f

r

p o i n t s ) . B o n i s o l i and Deza r41 p o i n t e d o u t t h a t t h e r e 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 t h o g o n a l p e r m u t a t i o n a r r a y s and o t h e r mathematical s t r u c t u r e s . F o r i n s t a n c e , t h e y proved t h a t each s e t o f p e r m u t a t i o n a r r a y s i s e q u i v a l e n t t o a 1-design w i t h number

r

and

ttl

mutually orthogonal v x r

t

v

treatments, r e p l i c a t i o n

m u t u a l l y o r t h o g o n a l r e s o l u t i o n s (see S e c t i o n 5 ) . Moreover,

i t was shown t h a t any o f these a r e e q u i v a l e n t t o a t r a n s v e r s a l ( t t 1 , r ) - s c m i n e t

with

v

t r a n s v e r s a l s . T h e r e f o r e many o f t h e 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 c o m b i n a t o r i a l designs w i t h m u t u a l l y ortogonal resolutions.

Oil

Pennutatioii Arrays

187

2. AN EQUIVALENCE AND A CONSTRUCTION METHOD

J :=

Let 1 2 {lo,l

o,...,lL).

(X;Lo,L1,..

. ,Lt;T1,T2,.

.. ,Tv)

Lo:=

be a t r a n s v e r s a l seminet w i t h

( I n f a c t , t h r o u g h o u t t h i s paper t h e s e t o f p a r a l l e l c l a s s e s , t h e Lo o f any t r a n s v e r s a l seminet a r 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

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 k d e f i n e t m u t u a l l y o r t h o g o n a l v x r p e r m u t a t i o n a r r a y s Ak = ( a . . ) , k=1,2, ...,t, 1J i n t h e f o l l o w i n g way: F o r i E I 1 , 2 ,... ? v 1 , j E I 1 , 2 ,...,r l , k < I1,2 ,...,t ) l e t Ti n l;, and l e t

be t h e unique p o i n t . c o n t a i n e d i n

x

x. L e t

be t h e u n i q u e p o i n t w i t h

y

t h r o u g h y. F i n a l l y , d e f i n e one can conclude t h a t

a:j:=

1

y c T 1 n 1, and l e t

be t h e 1;

L k - l i n e through

be t h e

Lo-line

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

&J):=IA1,A2, . . . ,A t )

i s , i n fact, a s e t o f mutually

orthogonal permutation arrays. The c o n d i t i o n s (C,)

o f S e c t i o n 1 can be paraphrased i n terms o f

and (C,)

t r a n s v e r s a l seminets as f o l l o w s : There i s no p o i n t o f t h e seminet w h i c h i s c o n t a i n e d i n a l l t r a n s v e r s a l s ,

(D1)

...,

. = ni=1,2, v Ti 1 , a n d v.2.)

1.‘.

line (D2)

0.

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 .

I n t h e c o n s t r u c t i o n procedure f o r

Ti

i

J?-(r) only

T

for

j

X have been T h e r e f o r e one can always

those p o i n t s o f Ti.

r e s t r i c t o n e s e l f t o t h e reduced t r a n s v e r s a l seminet

( X ’ ;LA,Li,.

d e f i n e d by

XI:=

ui=1,2,...,v

f o r each

iz j .

used which a r e c o n t a i n e d i n one o f t h e t r a n s v e r s a l s Tv)

I1 I t 2

(This implies, i n particular,

,

. ,LC;T1,T2,. . . ,

L k’ : = ( 1 n X ’ j l c L k I .

Ti’

I n t h e 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 t h o g o n a l p e r m u t a t i o n a r r a y s f r o m t r a n s v e r s a l seminets can be r e v e r s e d : L e t k p e r m u t a t i o n a r r a y s Ak = ( a . . ) , k=1,2, 1J

+

r ; , and l e t

& be ...,t .

a set of Define

be t h e e q u i v a l e n c e r e l a t i o n on

Y

t

Y:=

with

mutually orthogonal v x r {1,2. v) x {l,Z,

...,

...,

( i , j ) $ ( i ’ , j ’ ) i f and

j i Fi i ,(A). D e f i n e t h e p o i n t s e t X o f t h e seminet as t h e set o f equivalence classes o f 9 , i . e . X:= Y / $ = “ ( i , j ) l l b 1 (i,j)t:YI. For c = C 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 l k : = 1 2 k 1 2 ,...,1 L I . { I ( i , j ) l d j a . . = c l , and d e f i n e L o : = {lo,lo ,..., lor}, L k : = Ilk,lk

o n l y if j = j ’

and

~

1J

Finally, l e t (X;Lo,L1,..

T.:= 1

.,Lt;Tl,T2,,

= A .Summarizing,

’,r

(i,j)l$

..

1

j = 1 , 2 , . ..,rI, i = 1 , 2

,...,v .

Then

J(R):=

,Tv) i s a (reduced) t r a n s v e r s a l seminet w i t h one o b t a i n s

&(T(R))

M. Deza arid T. Ihririger

188

The existence of a s e t

2.1. PROPOSITION.

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. T h i s e q u i v a l e n c e was a l r e a d y observed i n 141. One can show even a l i t t l e more.

Let

. . ,Lt;T1,T2,..

= (X;Lo,L1..

.,Tv)

be reduced t r a n s v e r s a l seminets w i t h

.

and ‘U = (Y;Mo,M1,. ..,Mt;U1,U2,. .,Uv) 1 2 ,...,1 L I and Mo 1 2 ,..., Lo = ilo,lo Imo,mo

m i l , and assume R(’3’)= A(%). D e f i n e a mapping $ : X 4 Y as f o l l o w s . F o r X E Ti n 1: l e t $ ( x ) be t h e unique element c o n t a i n e d i n Ui nm:. Then 0 i s an and U , i . e .

isomorphism o f

0 satisfies

onto p a r a l l e l l i n e s ( i n f a c t , for all 2.2.

PROPOSITION.

If

y

&(J) =

T. nT. = 0

= Ui

$Ti

and

i i $lo = mo

u

are reduced transversa2 seminets with are isomorphic.

.

.

= ( X;Lo,Ll,. .,Lt;T1,T2,. .,Tv) corresponds t o a o f m u t u a l l y o r t h o g o n a l Zatin rectanglos e x a c t l y i f

IA1,A 2,...,Atl

for a l l

J

and

and

The t r a n s v e r s a l seminet

1

$Li = Mi,

i ) . This y i e l d s

J , ( J )= L ( U ) then

set

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,j, i z j . The a r r a y s

A1,A 2,...,At

form a s e t o f m u t u a l l y

o r t h o g o n a l Zatin squares i f and o n l y i f tT1,T2,,..,Tv\,

(X;Lo,L1,, ..,Lt,Lt+l), w i t h Lt+l:= I n o t h e r words, t h e e q u i v a l e n c e o f P r o p o s i t i o n 2.1

i s a net.

s p e c i a l i z e s t o t h e c l a s s i c a l correspondence o f m u t u a l l y o r t h o g o n a l l a t i n squares w i t h nets. The f o l l o w i n g theorem p r o v i d e s 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 t h o gonal p e r m u t a t i o n a r r a y s v i a seminets, u s i n g groups. 2.3.

THEOREM.

G he a f i n i t e group with neutral eZement e . Let

Let

t and

be p o s i t i v e integers, and Zet So,S1,.. .,St and F1,F2,.. .,F be nontrivial S subgroups of G such that the foZZowing conditions are s a t i s f i e d f c r aZl i , j E S

10,1,

..., tl,

k,l

E

11,2

,..., ~

4 Si n S j

(2)

i j

(2)

S. O F = {el,

(3)

k * 1 =$ Fk z F,,

(41

l F k I = CG:Sil.

i

= (el,

k

Then there e x i s t s a s e t o f S r : = I F I and v:= -./GI . 1 r

Proof. i.e.

1 :

F o r each

Li = {Sig

1

t

i E (O,l,

geG1. Then

mutualZy orthogonal v x r p e r m t a t i o n arrays, with

...,t l

let

(G;Lo,L l,...,Lt)

Li

c o n s i s t o f the r i g h t cosets o f i s a (tt1)-seminet:

Si,

C o n d i t i o n (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 c o s e t

Fkh i s a t r a n s v e r s a l o f (G;Lo,L1, ..., L t ) : I t has t o be Fk n F k h l = 1 f o r a l l g,hsG. As a consequence o f ( 2 ) one o b t a i n s

o f one o f t h e subgroups show that

lSig

1S.g n F k h l i 1. Assumption ( 4 ) then i m p l i e s u f C F k S i f = G, and hence 1

lSig n F k h l

2

1. Each t r a n s v e r s a l has

d i s t i n c t r i g h t c o s e t s . Thus t h e r e a r e form

Fkh, w i t h

(D1) and

k t t1,2,

...,s l

and

r = lFll

elements, and each

v = :*IGI

Fk

IGI

has

d i s t i n c t transversals o f the

h e G. Finally, the n o n t r i v i a l i t y conditions

( 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. The seminet

seminet, i . e .

n

,,...,

L t ) o f t h e above p r o o f i s , i n f a c t , a transZation (G;Lo.L 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 represeritation o f

G

each maoping

XH xg,

g r G , maps e v e r y l i n e

o n t o a p a r a l l e l l i n e . On t h e o t h e r hand, each t r a n s l a t i o n seminet can be o b t a i n e d i n t h i s way f r o m a group

G

and subgroups

So,S l....,St

satisfying condition

(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 i n s t a n c e ,

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 n e t s , 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 ( s e e e.g.

rll,

1151. C221, [31 and 1201). Marchi r181 uses

s i m i l a r i d e a s 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 l o o p s . P r o b a b l y one can f o r m u l a t e an analogue o f Theorem 2.3 u s i n g l o o p s 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 g e n e r a l i z a 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 l a s s e s o f examples f o r Theorem 2.3. C f . Huppert 1141 and W i e l a n d t 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.

Let

G

be a n o n s o l v a b l e t r a n s i t i v e p e r m u t a t i o n group o f p r i m e

v:= p 2 , r:=-I,G I 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 P d = r (mod p ) . Then one can c o n s t r u c t a s e t o f t : = 1 mutually orthogonal

degree p . L e t and

i-

v * r p e r m u t a t i o n a r r a y s : Assume tl,Z, . . . , p 1

define

FkzF,

k z

for

So,S l,...,St,

1

Fk

since

G

t o o p e r a t e on

t o be t h e s t a b i l i z e r o f

6

*,.... a P I .

ial,a ak

in

G, i . e .

F o r each

k

Fk:= Ga

6

Then

k' i s d o u b l y t r a n s i t i v e ( c f . Theorem 11.7 o f r 2 4 1 ) . L e t

be t h e Sylow p-subgroups o f

G

(with

t ' 1 1 because

G

i s non-

s o l v a b l e ) . O b v i o u s l y , these subgroups s a t i s f y t h e assumptions o f Theorem 2.3. Hence t h e r e a r e show P

of

t'

m u t u a l l y o r t h o g o n a l v x r p e r m u t a t i o n a r r a y s . I t remains t o r G has e x a c t l y Sylow p-subgroups. L e t

t ' = t or, equivalently, t h a t

be a Sylow p-siibgroup o f

P

is

P

i t s e l f . Hence

a

G. The o n l y Sylow p-subgroup o f t h e n o r m a l i s e r

NG(P)

i s s o l v a b l e and t h u s o f o r d e r

pad'

NG(P)

with

( c f . r141, Satz 1 1 . 3 . 6 ) . T h e r e f o r e t h e number n o f Sylow p-subgroups G r satisfies n=[G:N ( P ) l = p . d ' = a T . From n = l (mod p ) one o b t a i n s d ' = r (mod p ) , G r i . e . d = d ' and n =

d"p-1

a.

The n o n s o l v a b l e t r a n s i t i v e p e r m u t a t i o n groups o f p r i m e degree have been comp l e t e l y determined, due t o t h e c l a s s i f i c a t i o n o f f i n i t e s i m p l e qroups ( s e e

M. Deza Q

190

I I T. ~ Ihringer

C o r o l l a r y 4.2 o f F e i t C111). N o t i c e t h a t solvabZe t r a n s i t i v e p e r m u t a t i o n groups o f p r i m e degree o cannot be used i n t h e 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 i m p l y 2.5. EXAMPLE.

t = 0.

F o r each i n t e g e r

mr2

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 ( 2 - 1 ) - 1 9 v : = (2m+1)2 and r : = m 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 G = PSL(2,q), w i t h q = 2 ,

orthogonal v x r permutation arrays, w i t h

t:= 2

as a p e r m u t a t i o n 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

o f t h e p r o j e c t i v e l i n e o v e r t h e q-element f i e l d . F o r each define

Fk

t o be t h e s t a b i l i z e r o f

ak

in

G, i . e .

be t h e ( m u t u a l l y c o n j u g a t e ) c y c l i c subgroups o f

=

=-,

t'

o f conjugates o f

ttl, i.e.

k

t

..,aq+l) . . ,q+11

{al,a2,.

{1,2,.

Fk:= G L e t So,S l,...,Stl ak' q + l . By t h e r e s u l t s

o f order

G

t h e s e subgroups s a t i s f y t h e assumptions o f Theorem 2.3.

i n 1141, pp. 191-193, t h e number

points

one o b t a i n s

So

For

t ' + l= C G : N G ( S o ) l = w

t ' = t.

i n o r d e r t o conPSL(2,q) s t r u c t designs w i t h m u t u a l l y o r t h o g o n a l r e s o l u t i o n s . F o r i n s t a n c e , f o r each q E N o t i c e t h a t Hartman [121 used some o f t h e groups

{19,31,431 t i o n s and

there e x i s t s a design w i t h

v=qtl

t r e a t m e n t s , r =?

replica-

t+l=q mutually orthogonal resolutions.

3. BOUNDS FOR THE NUMBER

OF MUTUALLY ORTHOGONAL PERMUTATION ARRAYS

.

o f m u t u a l l y orthogonal p e r m u t a t i o n a r r a y s i s c a l l e d {A1,A2,. . ,At) maxima2 i f t h e r e e x i s t s no p e r m u t a t i o n a r r a y which i s o r t h o g o n a l t o a l l Attl Ak, k = 1,2 ,..., t. A t r a n s v e r s a l seminet (X;L0,L1, Lt;T1,T2 ,Tv) i s c a l l e d L-mo.ximaZ i f t h e r e 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 l a s s Lt+l such t h a t (X;Lo,L1,

A set

...,

..

.,Lt,Lttl;T1,T2,. obvious. 3.1.

LEMMA.

. .,Tv)

i s a g a i n a t r a n s v e r s a l seminet. The f o l l o w i n g lemma i s

of mutually orthogond permutation arraps is maximal

A set

i f mid only if the associated transversal seminet

3.2. PROPOSITIOM.

,...

J (A)is

L-maximal.

of rnintualZy orthogoxu2 permutation arrays of

Eaeh s e t

Example 2 . 5 is maximal.

Proof.

Let

G = PSL(2,2m)

a s s o c i a t e d t r a n s v e r s a l seminet

be t h e group used f o r t h e c o n s t r u c t i o n of

7

has

G

as p o i n t s e t . The subgroups

...,St

a r e e x a c t l y t h e l i n e s t h r o u g h t h e n e u t r a l element

groups

F1,F2,.

. .,Fq+l

e

are e x a c t l y the transversals through

of

e

A.The SO'S1,

G, and t h e sub( c f . the proof

191

Oir Permu tation Arrays

G. Hence

o f Theorem 2 . 3 ) . By Satz 11.8.5 o f Huppert L141, t h e s e subgroups c o v e r 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.

J 2 J(&).

By Lemma 3 . 1 t h e p r o o f i s complete, s i n c e P r o p o s i t i o n 2.2 y i e l d s

0

A c t u a l l y , t h e a s s e r t i o n o f Propos 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

F(&)

structure

of

a:L e t

gonal p e r m u t a t i o n a r r a y s w i t h

,..., B t , I

3

B1,B2

F(JJ )

= F(&).

S e c t i o n 2 shows t h a t t h e t r a n s v e r s a l seminets p o i n t s e t s and t h e same t r a n s v e r s a l s

n

y @ ) As .

number

Y

lines o f

through

e

7113)

7

G. T h e r e f o r e each l i n e o f

T(&)t h r o u g h

x

have t h e same o f the proof

contains e x a c t l y

J(&)

t h a t t h e same i s t r u e f o r

a consequence, t h e r e i s a p o i n t

t'+l of lines o f

and

pairwise d i s j o i n t transversals

0

=

7 7(A)i m p l i e s

p o i n t s , and

for

u s e s oF1s

S E S ~ w, i t h

')'(a)

The t r a n s v e r s a l seminet

n:= I S I = CG:F1l

o f Proposition 3.2 contains F1s,

be a s e t o f m u t u a l l y o r t h o -

The c o n s t r u c t i o n procedure o f

x

of

J ( 3 ) such

and a l s o

t h a t the

c a n n o t exceed t h e number

( 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

t+l o f

y(2)) .

of

T h e r e f o r e P r o p o s i t i o n 3.2 can be improved as f o l l o w s .

3.3. PROPOSITION.

Get

a ={A1,A

2,...,Atl

be one of t h e s e t s ofmutuaZZy

3

orthogonal permutation arrays of Exnmple 2 . 5 . Let

= IB1,B2,.

of mutually orthogonal permutation arrays with F ( B ) = F(&).

. . ,Bt, Then

be a s e t

1 t'

5

C..

The n e x t lemma g i v e s an upper bound f o r t h e number o f m u t u a l l y o r t h o g o n a l p e r m u t a t i o n a r r a y s 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 r a y s . 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 s e v e r a 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] I

pemutatioil arrays, and l e t

..., r l

j o 4

equidistant with of i t s g e n e r a t o r

z e r o e n t r i e s and any non - z e r o codeword o f

can be i n t e r p r e t e d as a l i n e i n a g e n e r a t o r m a t r i x o f

HAW(r,q)

HAMl(r,q)

which i s

o b t a i n a b l e from t h e o r i g i n a l g e n e r a t o r m a t r i x by a p p l y i n g o n l y elementary o p e r a t i o n s on t h e l i n e s . The group o f a l l l i n e a r code - a u t a n o r p h i s m s o f tIAtP(r,q)

. Since

i s t h e g e n e r a l l i n e a r g r o u p GLr(F)

b y d e f i n i t i o n any t w o o f t h e l i n e a r

, i cZn , a r e l i n e a r l y independent ( i n f a c t i t i s forms nilml(r,q) d(HAM(r,q)) = 3 ) t h e g r o u p o f a l l l i n e a r c o d e - a u t a n o r p h i s m s o f t h e Hamming code HAM(r,q)

i s isanorphic 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 g e a n e t r y 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

- linear

code

- ismorphisms".

REFERENCES

[11 Heise, W. and Q u a t t r o c c h i , P. , I n f o r m a t i o n s - und C o d i e r u n g s t h e o r i e ( S p r i n g e r , B e r l i n - H e i d e l b e r g - N e w York -Tokyo, 1983).

121 MacWilliams, F. J . and Sloane, N. J . A., The t h e o r y o f e r r o r - c o r r e c t i n g codes ( N o r t h - H o l l a n d , Amsterdam

- New

York - O x f o r d , 1977).

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 225-242

225

0 Elsevier Science Publishers B.V.(North-Holland)

ONTHE CROSSING NUMBER OF GENERALIZED PETERSEN GRAPHS S . Fiorini

Department of Mathematics, University of Malta

ABSTRACT la,, a*,

The Generalized Pcte:-scfiGmnn P ( n , k ) is defined to De the qraph on 2n vertices !abel led an,bl,b2 ,...,bn} and edges ta.b.,a.a. 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. 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-I1 '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, PRELIMINARlES

.

,...,

...,

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. In our case, the only in the 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. A common point of two arcs other than the image of a 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. rhe number of crossings in an optimal drawing of a graph C is denoted by v(G); the number of crossings in a drawing U of C is denoted by vp(G).

S.Fiorini

226

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 standard. Some g o d drawing 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 . By some ad hoe 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 h o p e f u l l y 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. If two g r a p h s C and H are homeomorphic, t h e n 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 ,

THEORW 1

then

and where w

'n, k fvn-n f a , k-klo ' n,k

denotes v(P(n,k.) f

.

P B Let H be obtained from P ( n , k ) by d e l e t i n g k s u c c e s s i v e spokes and l e t 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 c a s e u 4 2. Then H is homeomorphic t o 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 f o l l o w s from 'Theorem 1. / /

-

If C is a graph and X 5 V(G)oE(G) t h e n the subgraph induced by X is denoted by u(>.

THEORM 2

If v3= PROOF

(The Decomposition Theorem) Let 0 be an optimal drawing of a 0. graph G and l e t E(C) = XWYJZ, XnY = YnZ = ZlrX 0 , then v ( G ) = vo(uY> + vaCd,z>

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>

+

= vyu(aY> +

w p z > v

- va*

0aCuZ>, s i n c e v 0(x>

0

The followiw c o r o l l a r y r e a d i l y follows by i n d u c t i o n on k:

COROLLARY

2

Let D be an optimal 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).

, of E(C) t h e n

YinY. = 0 ( i d j ) is a d e c m p o s i t i o n J

v(G) f

&

v <XUYi>.

/I

The Crossing Number of Generalized Petersen Graphs

THEOREM 3 (The Deletion 'heorem) Let u be the least number of edges of a graph G whose deletion fran G results in a planar subgraph H of C. Then u (GI b a. PROOF Assuming on the contrary that w < a, then deleting the (at most)v edges being intersected results in a planar subgraph of G I contradicting the minimlity of a. / / We often make use of this simple conclusion in conjunction with Euler's polyhedral formula as in the following:

THEOREM 4

w(9,3)

I

2

PROOF The graph of Figure 1 (i) shows that 2 is an upper bound for v(9,3). 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); (the subgraph is

obtained by deleting an edge from each of the three triangles of P(9,3). ) 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

G

5(b

Thus, ~ ( 9 ~ 3 + )a

5

- a) 6 2(18 - a).

r4/31 = 2. / /

(ii) Fig.1.

227

S.Fiorini

228

THEOREM 5 (lhe Contraction Theorem) Let 0 be a grawing of a graph C and l e t e E E(G) make 0 contribution t o 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. Then wv ( C ) p wDi

PROOF (i)

(ii)

Let wv

fEE(G) such that f is adjacent t o e

If f d E(Ge) ( t . g uw is missing i n 0' ;

E

uv,

E ( G ) ) , then any crossing involving f i n 1)

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'.

Since a l l o t h e r crossings a r e unaffected, i n a l l cases

v

If is an g p t i m l drawing i n which e then v(G) 5 w(G 1.

COROLLARY 1

t o v,(G),

Pi3OOF

E

E ( G ) niakes 0 contribution

By t h e Contraction Theorem, v(G)

uD(C)

a u u , ( G e ) 9 "(Gel. / /

Repeated use of the Contraction Theorem y i e l d s t h e following:

...,

COROLLARY -2

Let <e e > be a sequence of edses of G each of whicb makes 0 contribution to vD!h) i n iome drawing U . I f w e d e f i n e recursively G = G,

Oo = 0, Gi = (G1-'Iei,

Vi 0

t h e drawinp; of Gi induced by Ui-',

uOO(G ) 2

=>

-

4(14 a ) & 2(24 v 2 u 2 4,

- a)

and t h e s t a t e m e n t is v a l i d i n t h i s case.

W e now c o n s i d e r an optimal 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 h e n by t h e Decomposition t h e i ' t h set of three s u c c e s s i v e chords [heorem w i t h CC, = C and Yi ( i = 1,2, ...,K), we conclude t h a t v(Gk) k , s i n c e ( X U Y i ) = 1. It follows t h a t i n t h i s c a s e v ( G x, 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 itself. 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 s u c c e s s i v e edges of C, we o b t a i n Ck-l f o r which Lhe inductive hypothesis implies:

a contradiction.

/I

The same a r q c n e n t , o n l y s l i T h t l y modified, 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 f a i l s i n i t s 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 w i t h k = 3 for t h i s case.

229

230

S.Fiorini

THEOREM 7 If Ck denotes t h e derived graph P1(3k+h,3), then f o r h = 1 , k 3 3 and for h = 2 , k & 2 , v(G,) = k + h. F u r t h e r , t h e r e exists a n optimal drawing i n which t h e ( j k + h ) - c i r c u i t C does n o t i n t e r s e c t itself. That v(Gk) c k + h follows from t h e drawings of Fig. 3.

To e s t a b l i s h t h e r e v e r s e i n e q u a l i t y we proceed by induction and n o t e t h a t for ( h , k ) = ( 1 , 3 ) or ( 2 , 2 ) t h e g i r t h is 4 and (n,m) = (10,201 and (8,16) r e s p e c t i v e l y . 'he Deletion Theorem, then y i e l d s : PROOF

f

= 12

a and f

10

L 2(20

- a)

- a)

4(12 respectively;

-

- a,

r e s p e c t i v e l y , so that

and 4(10

- a ) c 2(16 - a),

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 c o n t r i b u t i o n t o vI) i n some drawing 0 .

.men C

is p l a n a r l y embedded and a l l chords e i t h e r l i e i n I n t ( C ) or i n Ext(C).

Case ( i )

if a l l a d j a c e n t chords l i e i n d i f f e r e n t r e g i o n s , then two d i s t i n c t

sub-cases a r i s e none of which

i3

optimal;

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 r e g i o n , then two f u r t h e r sub-cases, according as ai-2ai+, l i e s i n the same or i n d i f f e r e n t r e g i o n s a s t h e s e , arise. In a l l cases that l o c a t e

ai,lai+2, a re-drawing is p o s s i b l e which both does not i n c r e a s e which some chord i n t e r s e c t s C.

v

and i n

i4e conclude t h a t i n all. cases t h e r e e x i s t s a n optimal 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 s u c c e s s i v e edges, we o b t a i n a homeomorph of Ck-l for which t h e i n d u c t i v e hypothesis implies: k + h

-

1

v(Ck-.,) f v(Gk)

-

1 5 k

+

h

-1-

1,

a contradiction. The drawings of Figure 3 are then seen t o be both optimal and i n which C does not i n t e r s e c t itself. / I

The Crossing Number of Generalized Petersen Graphs

Fig.3.

23 1

S. Fiorini

232 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 t h e lower bound a l s o holds, we consider two cases for a minimal

counterexample: Case ( i ) If t h e r e e x i s t s an optiml drawing i n which no spoke is i n t e r s e c t e d , then t h e Contraction Theorem implies that

v(3k + 1,3) 2 v'(3k + 1,3) = k + 1 (By Theorem 71, for k ? 3. That v(7,3) =

3 follows from

t h e work of Exoo, Harary and Kabell.

Case ( i i ) If some spoke is i n t e r s e c t e d , then d e l e t i n g t h a t spoke and two successive spokes, w e o b t a i n a homeormorph of P(3k 2,3) whose crossing number is k, by t h e rninimality of k. But then,

-

~ ( 3 +k 1,3) 2 v(3k a contradiction.

- 2, 3) + 1

//

Fig . 4 .

= k + 1,

233

The Czossing Number of Generalized Petersen Graphs

The remaining two cases: v(3k + h,3) = k + h (h exactly the same way once we prove that u(d,3)

= 0,2) are established in

4 = u(12,3).

4'

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). 'Thereafterthe 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 'heoremwith H = X. If not, then H must intersect itself in exactly two of its edges to yield a 2-looped drawing of itself. A contradiction is obtained for each pair of edges. 'To this end 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. He 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 ~ 1 4 , 5 , t~h~a t i n c l u d e s 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 c h a i n 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 n o t i n t e r s e c t e d i n some optimal 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 ) , ( 1 1 , 1 2 ) , (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 r o s s i n g s . If on the other nand, H i n t e r s e c t s i t s e l f , then one of t h e crossed edges must l i e i n t n e segnent . But each non-spoke i n t h e first is s e p a r a t e d from each non-spoke i n t h e second by t h e p a r a l l e l edges ( 2 ~ l l 1 1 ) , ( 5 1 , dexcept ~) for ( l l l , d t ) and ( 5 ' , 2 ' ) which are i n t u r n s e p a r a t e d by (d,c)) and ( 3 * , 1 2 ' ) . W e conclude that t h e c h a i n < 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 ) . But 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 ! ) , ( 7 1 , 1 0 1 ) i n t h e correspondinr: drawing of P ( 1 2 , 3 ) .

...,

' 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 = e n o t e c h a t L fl U z (5',d1),,(4',7')and t h e spoke (4,4') which < 5 ' , b 1 , ...,4>. W we ignore. Lf g ( t I ' , S ' ) , t n e n g cannot cross any edge i n t h e sub-segnent , s i n c e t h e y are separated by p a r a l l e l c h o r d s ( 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 since these edges are separated by (11,4') and either (3,4) o r (5,5'), which bound no edge in . 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 If e, say, is (5I,2l), then {e,gl &! and {f,h} C_ 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 excluding , 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) o r (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

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 f o l l o w i n g 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 s u c c e s s i v e l y 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 t h 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 , t h e n it is a l s o va1.i.d for t + l . 'The same cannot be s a i d for even t. ( I t 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 e conclude that v ( 4 k , 4 ) 2k. b i p a r t i t e ~ y a p h s : c f . [ 2 ] 1. W

References: 1.

2.

G. EXOO, F. Harary, J. Kabell, Ine C r o s s i n s numbers o f some Generalized P e t e r s e n 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? Graph l'heory.(F. r k r a r y , ed.)

.i--n

3.

Iy.

Watkins, A 'Theorern on hit Colourings

..., J.C.T.(B) &

(1969) 152-104.

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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. In the Desarguesian plane of even square order, this gives an example of an arc achieving the upper bound for complete arcs other than ovals. 1.

INTRODUCTION

Our aim is to demonstrate the existence of complete (q2 - q + 1)-arcs in a 2 2 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 intersection of two Hermitian curves in PG(2,q2) . The importance of these arcs, not observed in [S], is Segre's result that for q e en, a complete m-arc in 1 . Thus, this example of a PG(2,q) with m < q + 2 satisfies m 5 q - Jq +

complete arc attains the upper bound f o r q even As a by-product of the investiis the disjoint union of gation, it is shown that a Hermitian curve in PG 2,q') q + l of these arcs. 2.

NOTATION

Let

n

= n(q

2

)

be a cyclic projective plane of order 9'.

identify its points with the elements i of

ZV, v = q4

+

q2

+

cyclic group is generated by the automorphism u with o(i) = i [ 3 ] , 5 4 . 2 . The lines are obtained from a perfect difference set lo =

j = O,l,.. ., v

Ido,dl,., . dq2} as the sets u J ( l o ) , Let b = q2 + q + 1

and k = q

2

- q

are relatively prime, Zv= Zb x Zk. i = (1,s)

where i

In this notation u(i) = (r + 1, s

+

+

For

1;

i

-

l),

+

1, i

E

Zv,

1.

then v = bk.

Since b

and k

in Zv, we write

r(mod b), i L s(mod k)

taken modulo b and the second modulo k . to any arithmetical operation in Zv.

One can so that the

1,

.

where the sum of the first component is The notation extends in a natural way

J.C. Fischer, J . W.P.Hirschfeld and J.A. Thar

244

By the multiplier theorem of Hall [2], q 3 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) If we define r ( q 3 - 1) E 0 (mod b)

:

r

E

3

Zb1; this is because q r - r

=

.

= I(r,s) : r

B

E

Zbl for s

O,l, ..., k

=

-

1,

then o(Bs) = BS+l and the q2 - q + 1 Baer subplanes Bs partition Il. A l i n e of a Baer subplane Bs is a line of Il meeting Bs in q t 1 points. Similarly define = {(r,s) : s

K

whence o(Kr) = K

r+l It will turn out that

3.

iZk 1 for r = 0,1,. . . , b - 1 ,

also partition Il. Thus i = ( r , s ) = Bs n K r is a complete (q2 - q + 1)-arc.

and the Kr Kr

,

COMPLETE k-ARCS LEMMA 3 . 1 :

$(i)

E

=

(1,

k

-

s)

Proof:

i = (r,s) i n

For each

izb

Zk, we have that

x

of i

, 3

q s + s = s ( q + l)k

LEMMA 3.2:

=

fixes the first component r

It was noted in 52 that

Now, for each s in iZk

whence

izv

.

q3s F - s s k

For any l i n e

IK

n

~~1

.t

-

3

0 (mod k) ,

s (mod k )

.

0

BS , w i t h

of the Baer subpZane odd if

(r,s)

E

(r,s)

=

Bs n Kr,

K

is even

if ( r , s ) #

p..

Proof: By lemma 3.1, the involution $J fixes exactly one point of Kr namely the point (r,O) where it meets B o ; the other points of Kr are interwhich implies changed in pairs. If K is a line of Bo it is fixed by $ , that the number of points of .t n Kr outside Bo is even. Thus the parity of 1.t n Krl 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 3

245

Complete Arcs in Planes of Square Order Let

LEMMA 3 . 3 :

be an automorphism group that a c t s regularly on t h e

S

p o i n t s of some p r o j e c t i v e p h n e

n(n)

of order

n,

and suppose t h a t

VO,V1, ...,Vt are the orbits of t h e p o i n t s under t h e a c t i o n of a subgroup s . If .t i s a Zine of n(n) and A . = 19.. n v.1 , then 1

G

of

3

A.(A.

j=1

J

-

1) = I G I

1.

-

J

n2 + n elements y of S \ { l } there corresponds Q of 9. for which y ( P ) = Q ; in fact, P = y - 1 ( 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 Proof:

To each of the

a unique pair of points P ,

in two ways. First, each y

IJI

whence

= IGI - 1 .

other than the identity gives a unique pair

(P%Q),

Second, 9. is a disjoint union o f the sets 9. n V. , 3

and to each pair (P,Q) , P # Q , in 9. n V . there is a unique y in G such J that y ( P ) = Q ; hence IJI = 1 4 . (A. - 1) and so I J I = A . (A. - 1 ) . 0 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. For

THEOREM 3.4:

k

=

q2

lie i n

-

q + 1 in

B

q

n(q2) .

are t h e

q + 1

2,

>

each o r b i t

is a complete k-arc w i t h

Kr

Furthennore, the l i n e s through tangents t o

Kr

at

(r,s)

Bs n Kr = ( r , s ) t h a 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 C I . + 1 = 12 n K I ; J rj for the remaining orbits, set @ . = n K 1 , j = q + 1, q + 2 , ... , b - 1 . 1 j' By lemma 3.2 both a . and B . are even. I

3

By definition,

By lemma 3.3, b-1

1

j =q+1

Bj (Bj - 1)

+

j=1

( a . + 1) a J. = q J

whence subtraction yields b-1 j =O

2

- 9,

J.C. Fischer, J. W.P.Hirschfeld and J.A. Thas

246

B. I

Consequently

E

{0,2}

for

j 2 q

(rj,s) Krj

E

so t h a t

II n Kr j

a t the point

1.

II o f t h e s u b p l a n e B s ,

Summarily, f o r any l i n e (i)

t

aj

= 0,

(!2 n

K

rj

1

either

= 1 and

is t a n g e n t t o

II

i = (r.,s) 1

or (ii)

.t n K,.

1

Bs = 0 and II meets

n

in

Krj

or

0

points.

2

i s a l i n e of e x a c t l y one o f t h e s u b p l a n e s Bs, it f o l l o w s i n more t h a n two p o i n t s ; t h a t i s , Krj is a (q2 q + 1 ) - a r c . From ( i ) it i s c l e a r t h a t , f o r each p o i n t ( r j , s ) o f K,. , I at 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 a n g e n t s o f K r j t h i s point.

S i n c e each l i n e of

il

t h a t no l i n e meets

Krj

-

For q

a s i m p l e c o u n t i n g argument suffices t o show t h a t t h e k - a r c

2 4 ,

Kr

i s complete. Assume t h e c o n t r a r y . Then t h e r e i s a p o i n t P through which p a s s q 2 - q t 1 t a n g e n t s o f Kr , one from each of i t s p o i n t s . S i n c e P E K r , for

some

r’ # r ,

q 2 - q + l

i t f o l l o w s t h a t through each p o i n t o f

tangents of

t h e g r o u p g e n e r a t e d by

(because

K br a )

.

Since

Krl

l i n e s i s counted more t h a n t w i c e , whence tangents.

But a s

i ( q 2 - q + 1)’ When

L

Kr

(q

q = 3,

t

has e x a c t l y l)(q2 - q

+

and

Krl

is i t s e l f a k - a r c , none of t h e s e t a n g e n t 2 - q + l )2 has a t l e a s t ;(q

Kr

(q + l ) ( q 1)

,

- q

t

t a n g e n t s , we have

1)

a contradiction for

it must f i r s t b e observed t h a t

p l a n e of o r d e r 9 , Bruck [l].

there are

Krl

are o r b i t s u n d e r t h e a c t i o n o f

Kr

Then t h e o n l y 7 - a r c o f

q

2

4.

i s t h e unique c y c l i c

PG(2.9)

whose automorphism

PG(2,9)

group c o n t a i n s an element o f o r d e r 7 i s a complete arc, [ 3 ] , 514.7. q = 2

The c a s e

i s a genuine e x c e p t i o n : 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

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

-

PG(2,q),

Jq + 1 .

So, f o r

q

247

Complete Arcs ira Planes of Square Order an even s q u a r e , 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 S e g r e ' s theorem cannot be improved i n t h i s c a s e . §1 0 . 4 , t h e comparable theorem s t a t e s t h a t a complete m-arc i n is e i t h e r a c o n i c , t h a t i s a (q + 1 ) - a r c , o r

m

odd

q ,

q

odd,

This r e s u l t

However, t h e e x i s t e n c e o f a

iq + 1

i s n o t t h e b e s t bound f o r a l l

514.7.

[3],

LINES IN lI(q2)

4.

Any l i n e of

THEOREM 4 . 1 : (i)

q

+

(ii)

d ( k - 1) = j = 1,2,..

('12

.,

(d,O)

A line

one p o i n t .

Zb;

Each o f i t s l i n e s

Bo.

.

Since

of

j

Zk\{O}

i s f i x e d by

$,

r

Zb

p a i r e d with t h e same element

of

j

r. # D

I t remains t o show t h a t

3

t h a n twice among t h e p o i n t s o f

L

.

f o r by t h e

i = (0,s)

, s # 0,

k - 1 differences

that lie in l i n e of

=

Kt

Bs ,

common wi t h i t .

in

(I

q

+

D

generates 1

for

.

of t he p a r t i t i o n i n exact l y

interchanges

( r j , j ) and j

and

k - j

and t h a t no

r.

I

in

Zb\D

are

can ap p ear more

T h i s f o l l o w s from t h e fact t h a t t h e p o i n t s of

Zv : each of t h e

k - 1 differences

Bo

-

j))

.0

g iv e n i n t h e theorem i s e s s e n t i a l l y t h e

is a k - a r c whose t a n g e n t s are t h e l i n e s through

Kr

Zb

.

t ( ( r j , j ) - (rj, k

(r,s)

The d e s c r i p t i o n of p o i n t s of t y p e ( i ) and ( i i ) shows t h a t any

i s a ta n g e n t t o t h o s e

Bo

the other os(Bo)

Bs.

ok

must o c c u r e x a c t l y once, and t h e s e are accounted

The d e s c r i p t i o n of a l i n e of a l t e r n a t i v e proof t h a t

Bs

it f o l l o w s t h a t both

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

of t h e form

for

o ccu r s a s t h e second component o f

Lemma 3 . 1 shows t h a t

L. L

therefore contains

meets any o t h e r subplane

Bo

of

II

since

q,

i s an element o f a p e r f e c t d i f f e r e n c e s e t

d

Thus each element

e x a c t l y one p o i n t o f

(rj, k - j)

i s an element o f a p e r f e c t

d

is i t s e l f a c y c l i c p l a n e of o r d e r

where

L

, where

3

a c y c l i c group f o r elements

(d,O)

pairs of p o i n t s of the form ( r j , j ) and ( r j , k - j ) $ ( k - 1) , w i t h the r . d i s t i n c t elements of Zb\D .

Bo

Proof:

for

D

c o n s i s t s of

Bo

1 p o i n t s of the form

difference set

a

-

q

shows t h a t

PG(2,9)

odd, [ 3 ] ,

PC(2,q),

q - Jq/4 + 7 / 4 .

5

has been s l i g h t l y improved by t h e t h i r d a u t h o r . complete 8 - a r c i n

q

For

0

or

2

points.

Kt

t h a t meet i t i n a p o i n t o f

Bo;

it meets

The proof i s completed by n o t i n g t h a t

which e i t h e r c o i n c i d e s with

Bo

o r h as 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 HERMITIAN CURVES

5.

The only known cyclic planes are the Desarguesian ones and, in this section, we restrict our attention to P G ( 2 , q 2 )

.

Lo of Bo and define are incident exactly when i + j (mod v)

It is convenient to distinguish one line

.

R . = o-’(Ro) 1

element of

Then

Lo.

i

and R .

1

In particular, L

theorem 4.1; now, D

B0

= { i = (d,O) : d

H = {

(d/2, s) : d

Hermitian curve and is t h e d i s j o i n t union of the

liii H is odd or even, whence

n

q

6

+

D, s

.

‘b

Zk} i s a

E

1 compZete k-arcs

Bs is a conic or a l i n e of

tI is a disjoint union o f

is an

as in (i) of

DI

E

0 is a distinguished, perfect difference set f o r

iil The s e t

THEOREM 5 . 2 :

n

Kd/2 ’

Bs according a s

k subconics or

q

k sublines

accordingly.

Define the correlations $ : i a . and p : ( r , s ) c-f 2 (r,-s) . 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 Pro0f:

Then

+-f

$

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. In Bo in Zb. So

the points are

Bo

(r,O)

while the lines are R

is self-polar with respect to

self-conjugate points of the polarity

@

p

(r,O)

and meets H

induced on

B0

by

p

’

.

both with r in the q + 1 These self-

conjugate points form a subconic when q is odd and a subline when q is even. there exists s ‘ such that bs’ t s (mod k) since b and k are Given s , coprime. Thus H n Bs = 0bs’ (13 n Bo) = I(d/2, s) : d 6 D} is a conic or a line of Bs according as q is odd o r even, and the last part of (ii) follows. 0 THEOREM 5 . 2 : q

The tangents t o any complete

( q 2 - q + 1)-arc

in

2 PG(2,q )

,

even, form a dual Hermitian arc. Proof:

See Thas [ 6 ] .

THEOREM 5.3: The tangents t o any o f t h e complete 2 PG(2,q ) form a dual H e m i t i a n curve i f and only i f q

(q2 - q + 1)-arcs

Kr

in

i s even.

Let q be even and consider the arc K O , where D has been Proof: chosen s o 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

249

Complete Arcs in Planes of Square Order to

II II

with

(d, 0 )

to

d

namely t h e l i n e s o f

in

E

D = 2D

determined by

p

c o i n c i d e s with t h e s e t o f t a n g e n t s t o Now l e t

q

b e odd.

have t h e form (0,s)

,

takes

KO i s {I. : j = ( d , s ) , d 6 U , J these lines a r e the self-conjugate

Thus t h e s e t of t a n g e n t s t o

H.

KO

P

not i n

and so i s odd, t h i s number i s never

1

+

do n o t form a d u a l H e r m i t i a n a r c . 0

Kr

(q2 - q

Each of t h e

Tb'EOREM 5 . 4 :

,

to

H.

- q

q'

Hence t h e t a n g e n t s t o

q + 1 .

(0,O)

(0,O)

S i n c e t h e number o f t a n g e n t s from a p o i n t

h a s t h e p a r i t y of

Kr

to

which t a k e s

t h e set of t a n g e n t s t o

9,

l i n e s of t h e p o l a r i t y

Kr

containing

Bo

obs' ,

Since

D .

(d,-s) ' Zk}. From t h e assumption t h a t

(d,O)

s

(0,O) ,

at

KO

+

i s t h e intersection of t u o

Kr

1)-arcs

Wermitian curves.

Proof:

First, let

contained i n a p(H)

=

H

q

be even.

H,

Hermitian c u r v e

and l e t

H*

by t h e t a n g e n t s t o Now, l e t

q

Kr

Then a s i n theorem 5 . 1 , t h e a r c

which d e t e r m i n e s a p o l a r i t y

Then, a s i n theorem 5 .

Hrl

n Hr2

d2

= Kt

D

in

,

so

Zb,

Let

n Hr21

(b)

If

H

iZk such t h a t

f o r any

k

is a perfect

+

r2,

r1 # r 2 .

Also

s i n c e t h e r e e x i s t unique

dl

0 , q

even, and l e t

ti.

then

q > 2 ,

Suppose m

theorem 1 0 . 3 . 3 , c o r o l l a r y 2 ) , (q2 + 2 ) - a r c .

E

be a Hemitian curve in PG(2,q')

m = 4 if q = 2 . 2 m = q - q + 1 and (i)

=

rl = i d

+

(m+1)-'zw in H , 2 m = q - q + l if q > 2 ;

Pro0f:

D, s

d l - d 2 E 2 ( r 2 - r l ) (nod b ) .

If there is no (a)

(ii)

IHrl

t = ad

where such t h a t

be an m-izric contained in

(i)

E

s c izk

we have t h a t

THEOREM 5.5: K

D,

In f a c t , s i n c e t h e r e e x i s t s r ' kr' Hr = o (Ho). Since D

i s a l s o a Hermitian c u r v e .

and

E

Hence Hr = { ( d / 2 + r , s ) : d

difference set in

is

Kr

Let

b e t h e d u a l H e r m i t i a n c u r v e o f theorem 5 . 3 t h a t is formed Then p(Kr) = H* n h , whence K = p(H*) n H .

be even o r odd.

is a Hermitian c u r v e .

,

.

.

H = Ho = { ( d / 2 , s ) : d

k r ' : r (mod b)

p

>

K

Now, count t h e p a i r s

q'

then -

K

q + 1

.

is compLete. Then by S e g r e ' s theorem ( [ 3 ] ,

is c o n t a i n e d i n an o v a l (P,Q)

such t h a t

P

E

t h a t is a

0, K,

Q

E

0 ,

P

# 9 and

250

J. C.Fischer. J. W.P. Hirschfeld 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

Hence 3m

2q2

5

+

4,

and 3q2

2 ( q 2 + 2 - m)

m

.C

-

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 ' whence q2

-

3q

-

1

5 0 ;

-

q

+ 1 5

2(q

that is, q = 2 .

+

l),

U

For q odd, the points of Bo together with the q2 + q + 1 Remark: 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

in PG(3,q),

of the q2 + q + 1 quadric surfaces through a twisted cubic T 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, G e m .

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 S Q S ( o ) H A V I N G A PRESCRIBED PQS I N COMMON" Mario G i o n f r i d d o ' , Angelo L i z z i o ' , Maria Corinna Marino'

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,ul , 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 a n y t w o o f t h e m i n t e r s e c t i n u quadruples, which occuring i n each of t h e S Q S l v l s

.

1.

Introduction

A p a r t i a l quadruple system v

is a f i n i t e set having sets c f

elements and

such t h a t every

P

an element

of

s

. If

(PQS)

3-subset of

(P,sll

and

is a p a i r

{ x ,y , z } c p

are

t h a t every

,

s

her

then

DMB

, then

3-subset (P,s)

/PI = v

of

p

a r e two

(P,s2)

I

=(I,

if

.

If

If 2 . (P,s) is a

= v (v-1) (~-2)/24

s

,

they n s

1 s

1

2

=@

if and

(P,sll

s

such

PQS

i s c o n t a i n e d i n e x a c t l y one element o f (SQS)

is t h e o r d e r and i t i s well-known t h a t an v :2

I n what f o l l o w s an 19

PQSs

(DMB)

i s s a i d a S t e i n e r quadruple system

re e x i s t s i f and o n l y i f

4-sub-

is contained i n an element of

l s l l = 1s21

P

i s c o n t a i n e d i n a t most

P

and o n l y i f i t i s c o n t a i n e d i n an element of lP,s2)

where

is a family of

s

a r e s a i d t o be d i s j o i n t and m u t u a l l y balanced and any t r i p l e

,

(P,s)

SQSlv)

or

.

The n u t

SQS(vl

the-

4 (mod. 6 ) .

w i l l b e denoted by

(&,a)

. We

have

.

On o f t h e m o s t 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

SQSs i s t h e

determination of the parameter: 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 }

.

" L a v o r o 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 (1983).

' 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 Catania, Italy. ' 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 Messina, I t a l y .

M. Gionfriddo. A . Lizzio and M.C. Marino

252

I n [2]

J. Doyen h a s 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

systems. 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

.

SQSS

for

D(v,ul

2 . Known r e s u l t s Let

be a

(P,s)

d(x) = r

degree

X,YEP

,

X # Y

9

if

FQS

x

. We w i l l

say t h a t an element

belongs t o e x a c t l y

we w i l l i n d i c a t e b y k

tained in exactly

quadruples of

quadruples of

r

(x,ylr s

. We

x,.:. P

a pair

has

. If

s

{x,y~P c

have

con-

.

dlx) =41s1 X E P

The d e g r e e - s e t o f a where

ve d e g r e e (he),

are t h e elements o f

,

hi

for

..., ( hs II,.

2

If An

.

xly,..

(of

K

1x1 )

i s a factor [I] o f

Fi

F . n F . = @ f o r every 1

1

K

3

I < h < IxI-1 ,

F

i,j =1,2,

on K

elements of

ri

c

DS = ( h

w i l l write

1x1

i s called a partiaZ

)

1

P I J

X

F = [ Fl,--.,Fh~

is a family (on

,

ha-

P

b e t h e c o m p l e t e g r a p h on

1x1 X

...

. If

f r = IPI

+...

is a f i n i t e set, l e t

X

I-factorization

where

r

where

P

, we

i =1,2, . . . , p

DS = [ d ( x ) , d l y ) , . . . ]

is the s e t

IP,s)

PQS

.

,

X) a n d , f u r t h e r ,

i #j

.

I t is

h = 1x1-1

.

If

1-factorization (of

On

x. A partial

1-factorization

embedded i n an

1-factorization

if

Y s X

Let XnY=@

G = I GI

, and e v e r y X

. If

and

Y

F={Fl

,..., G u - l }

an

,

then

{1,Z,...,u-Z}

{xl,x2,yl,ye} C X A Y

Fa\ = { F ; , F ; , ..., Fe} h F = {F .,F 1 on l’** k

on a s e t

X

,

i f and o n l y

.

F9‘: E Fg: is contained i n a F .€ F z 3 b e two f i n i t e s e t s such t h a t 1x1 = I Y I = u

,..., F

V-

1

1

i s an

I - f a c t o r i z a t i o n on

I - f a c t o r i z a t i o n on IF,G,cl)

such t h a t

Y

,

a

is

Y

X

and

,

a p e r m u t a t i o n on

i n d i c a t e s t h e s e t of the quadruples

253

On the Maximum Number of SQS(vI I t i s well-known

with

X n Y =0

,

I n 1-31, [4],

,

(X,A)

and

IQ,q) = [ X u Y ] IA,B,F,G,rxI

then

q=AuEwr(F,G,cO

morphism, a l l

that, if

is an

[6]

SQS(2v)

fY,B)

a r e two

, where

Q =XuY

having

m=8,12,14,15

(i.e.

m~ 1 5 q) u a d r u -

These r e s u l t s a r e t h e f o l l o w i n g :

1,2,:,4 1,2,5,6 1,3,5,7

1,2,3,5 1,2,4,6 1,3,4,7

1,4,6,7

1,5,6,7

2,3,5,8 2,4,6,6 3,4,7,9

2,3,4,8 2,5,6,8 3,5, 7,9

5,6,7,9 3,4,8,0 3,5,9,0

4,6,7,9 3,4,9,0 3,5,8,C

4,6,9,0

4,6,8,0

5,6,8,0

5,6,9,0

and

.

pies.

92

,

M. Gionfriddo has constructed, t o within iso-

DMB P Q S

97

SQSlv)

1,4,5,6 1,4,?,8 1,5,7,9 1,6,8,9 2,4,5,7 2,6,7,8 2,6,5,9 2,4,8,9 3,4,6,8 3,5,&,7

1,4,5,7 1,4,6,6 1,5,6,9 1,7,8,9 2,6,8,9 2,4,5,5 2,4,7,8 3,6,5,7 3,5,7,9

3,4,5,9 3,7,8,9

3,4,8,9 3,6,7,8

3,4,5,&

1,2,3,4 1,2,5,6 1,2,7,6 1,3,5,7 1,4,6,7 1,3,6,6 2,3,5,8

1,2,3,5 1,2,4,7 1,2,6,8 1,3,4,6 1,5,6,7 1,3,7,8 2,3,4,a

2,4,5,7 2,4,6,8 3,4,5,6 3,4,7,8 5,6,7,8

2,4,5,6 2,5,7,8 3,4,5,7 3,5,6,8 4,6,7,8

M . Gionfriddo. A . Lizrio and M.C. Marino

254

1,2,3,4 1,2,5,6 1,3,5,7 1,4,6,7 2,3,5,8 2,4,6,8 3,4,7,8 5,6, 7,9 5,6,8,0 5, 7,8,A 5,9,O,A 4,7,9,A 4,8,0, A 4,6,9,0

41

42

7,3,4,5 1,3,6,7 1,3,8,9 1,4,6,8 1,5,7,8 1,4,7,9 1,5,6,9 2,3,4,6 2,3,5,8 2,3, 7 , 9 2,4,5,9 2,5,6, 7 2,6,8,9 2,4,7,8

1,3,4,6 1,3,5,8 1,3,7,9 1,4,5,9 1,5,6,7 1,6,8,9 1,4,7,8 2,3,4,5 2,3,7,6 2, 3,8, 9 2,4,6,8 2 , 5 , 7,8 2,4, 7,9 2,5,6,9

3. The v a l u e o f

1,2,3,4

1,2,3,5 1,2,4,6 1,3,4,7 1,5,6,7 2,3,4,8 2,5,6,8 3,5,7,8 4,6, 7 , 9 4,7,8,A 4,6,8,0 5,6,9,0 5,7,9,A 5,8,0, A 4,9,0, A

1,3,5,7 1,4,6,7 2,3,5,8 2,4,6,8 3,4,?,9 3,4,8,0 3,6,9,0

1,2,3,5 1,2,4,6 1,3,4,7 2,5,6,7 2,3,4,8 2,5,6,8 3,4,9,0 3,6,8,0 3,5,7,8

3, 6 , 7 , 8 5,6,8,0 5,6,7,9 4,5,9,0 4,5,7,8

3,6, 7,9 4,6,7,8 5,6,9,0 4,5,8,0 4,5,7,9

4,

9"

1,2,5,6

1,2,3,4 1,2,5,6

1,2,7,8 1,3,5,7 I, 4 , 7 , 6 I, 3 , 6 , 8 1,4,5,8 2,3,5,8 2,4,5,7 2,4,6,8 2,3,6,7 3,4,5,6 3,4,7,8 5 , 6 , 7,8

~ ( v , q ~ - m lf o r some c l a s s e s o f

1,2,3,5 1,2,4,7 1,2,6,8 2,4,5,8 2,5,6, 7 2,3, 7,8 2,3,4,6 1,3,4,8 1,4,5,6 1,5,7,8 1,3,6,7 3,4,5, 7 4,6,7,8 3,5,6,8

SQSfvl

We prove t h e f o l l o w i n g theorems.

THEOREM 3 . 1 . L e t

(P,sII

,..., ( P , s h )

be

h

D M B PQS

.

If t h e r e

On the Maximum Number of SQSlv) e x i s t an

I x , ~ ) ~i n

, then

IP,s.) 2

It f o l l o w s

Proof.

{ r , y l ~ P such t h a t i t i s

.

h(2k-1

in

ix,ylk

,

(P,s.l 3

..., h l .

f o r every , j ~ I 1 , 2 ,

{ x , ~ , a ~ ~ , a ~ ~ } a { ~ , ~ , a . ~, {~x ,, ya, a~ k~l 3 } aa k.2 .} ~ s i ' let F i IIallJa12}a{a21,a22},.. {akl,ak211 It f o l l o w s

If be the

is a partial

.,Fkl

1

1-factorization of

torization on

I-factors

Fi elements, it follows

2k-1

IF1 = k i 2 k - l

We have

THEOREM 3 . 2 . L e t

(in the case

on

KZk

,.

Since the set o f the

can (at most) be an h 2k-1

Dlv,qv-k'(2k-1)l

Proof.

Let

be two

r e s p e c t i v e l y , and l e t

Y

be an

(Q,ql

F

1x1 = I Y I

=2k

I-factorizations

of

be a p e r m u t a t i o n on

CY

containing

SQSlvl

]-fa5

necessarily.

Y

G

on

h =2k-I

A).

and

X

KZk

1-factorization of

the set A = ~ all,al2,aZlJaz2,.. .,ak1"ak2 l is exactly an

.

.,

1-factors F ={F , F 2 , . ,

that

and a p a i r

i€{l,,,.,hl

255

,

I'(F,G,al

. SQS(vl

containing the family

lF,G,aI

.

I t is

If 1

2

...

1

2k-1

...

2

2k-1

[a .+iE a +i a ti 1 2

Zk-1

i

for where ''*

a

2k-1

z 2k-I 1

+i

, then the quadruples of khe families TfF,G,a.l, i =0,1,2, ..., 2lk-1) , form 2k-1 DMB PQS f P , s ) , I P , s I ,... 1

=1,2,.

..,2k-2

2 (k-2 ) l

('9

D(v,q-k

...

3

2

,

all embeddable in an

(Zk-1)) L 2 k - 1

THEOREM 3.3.

If

SQS(v)

.

2k i2

or

Hence

.,

k€N

i s such t h a t

4 (mod. 6 )

,

then

M. Cionfriddo. A. Lizzio and M.C. Marino

256

Proof. I f

is such t h a t

k€N

p o s s i b l e t o c o n s t r u c t an

or

of o r d e r

Zk

s ~ ~ l 2 k lw i t h

( ~ ~ , q b e~ two l

two

SQS

2k - 2

I-factorizations of

. Let

on

and

Q,

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 2 -k ( 2 k - I ) q u a d r u p l e s i n common. Hence

and

lQ1,ql)

~ = @~ , a n dQ l e t ~ F

Q

K2k

(mod. 6 ) , t h e n i t i s

4

and

G

be

r e s p e c t i v e l y . From

Q2

having

SQS(4k.J

q4k

2

D(4k,qqk-k

TI-IEOREM 3 . 4 .

v -2

For e v e r y

6'11

4 (mod.

or

Proof. Let

. Further, . ., 2 k - l b e

,

k'2

. w =min { v E N : u , 4 k ,

let

F =IF 1

let

i i=I,.

..,2 k - 1

1x1 =

tions

F'=fFII

with

or

F

on a s e t

embedded i n

4 (mod. G ) }

X ' n Y ' = @ and

G' ={GI}

z i=l,...,w

.

If

.

F'

ip

then

i s embedded i n

G

SQS(wl

and

a

a bijection

I - f a c t o r i z a t i o n on

{r,ylEGE

{ g

G'

-1

(x),m

. Further,

i s a permutation of

-1

From Theorem 3 . 2 i t f o l l o w s

Y

l-factoriza-

Y

,v

~2

such t h a t

X' +Yfy

Y'

such t h a t

(y)lEFI

if

,

fX',ql),

{l,Z, ...,w-

11

, IY',q2)

a r e two

,.

, t h e n we c a n c o g

SQS(2wl = [.x'uY'] Iq7,qZ,F',G',al

s t r u c t an

and

X

I X ' I = ~ = m i nC V E N : u >4k,8

Let

lY'I = / X ' I

on

X C X ' , l X f 1 , 2 1 X I = 4 k >-8 ,

such t h a t

is a set, containing

Y'

the

,

X'

IYI = 2 k > d

and

1 two I - f a c t o r i z a t i o n s o f K e k i i=I,. 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

G = { G

.

>2k-l

D(2w,qZM-k'!2k-l))

b e two f i n i t e s e t s w i t h

Y

X n Y =@

and

,

I t follows

,

and

X

k€N

(2k-1)) Z2k-1

containing T(F',G',al 2 D(2w,qgM-k ( 2 k - I l l 2 2 k - 1

COROLLARY, F r o m t h e s a m e h y p o t h e s e s of T h e o r e m 3 . 4 it f o ~ l o w s D(2v,qw-k

2

(2k-1)) >2k-l

,

f o r every

v 'w

, v

:2

or

..

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 o f

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 s c h e m e :

.

257

On the Maximum Number of SQS(v)

k

-

w> 4k

2

q2u-k

VLW, v - 2

i2k-1)

o r 4 (mod. 6)

-

2

8

3

14

928

4

16

5

20

6

26

7

28

- 112 q4@- 225 q s 2 - 396 q50'- 637

8

32

964 960

..

........ ........

q16 1 2

- 45

qs2

-

*.

... ...

I t is easy t o see that v

4 . The v a l u e o f

2

(2k-1))

= + m

.

-t+m

for

D(v,qv-rn)

D(v,qv-k

Zim

m=8,14,15

a n d the value o f

D( 8, qg-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

THEOREM 4 . 1 .

s

i

for

DMB P Q S

(for

and

m =8,14,15

.

D18,q8-121

, > a

D(v,qv-m)

(

a quadruple

Let

i I,. ~

(P,s..J

. .,Jil)

such t h a t

b

be

h

i

=1,2

t h e r e exist t h r e e e l e m e n t s

.....

x,y,z

h ) . If

E P

and

( ~ , ~ ) ~ , ( x , ~ ) ~ , { x= , by E, s~ . } , t h e n

h < 2 . Proof.

and

h=3

hence

From Theorem 2 . 1 i t i s

,

then

.

h < 3

.

If

{~,14,~,~},{2,y,a,b}Es 3 .

I t follows

{z,z,c,~~€s~,

(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 m =8,14,15

quadruples.

DMB PQS

with

M. Gionfriddo, A . Lizzio and M.C. Marino

258

P r o o f . I t i s e a s y t o s e e t h a t i n t h e u n i q u e p a i r s of

with

rn = 1 5

and

rn = 8

DS = [17/2, (617]

and

,

,

and i n t h e p a i r s o f

,

(x,y12

(x,zJ2

,

DS = [(714, (612, (4)41 and a q u a d r u p l e

DS = [ ( 7 / 2 , ( 6 1 3 , ( 4 ) 6 ] , (see

b ={z,y,zl

DS = [(7)8]

, since

and

. If

it is

h =3

.

then

{1,2,3,x}~s w i t h z ~ { 6 , 7 , 8 } But, x = 6 3 implies {2,2,4,81, {3,2,5,71 E s [resp. {1,2,4,6), 3 w i t h { 1 , 4 , 6 , y } ~ s ~ [ { 1 , 3 , 8 , y l ~ s ~ ] and y @ { l , Z x = 8

it follows

. Therefore,

y e {1,2,.,.,81

DS = [ ( 6 1 8 ]

T h e i r d e g r e e - s e t is

P r o o f . I n t h e p a i r s of

or

DS = [ ( 6 ) 4 , ( 4 ) &

( X , Y ) ~

,

(x,z)

with

G = { G ,G ,G 1 1 2 3

X = {1,4,5,8)

. It

I

with

has degree-set

I

F3

E s3]

{1,2,5,8)

,... ' 8 1 . From

,

with

m=l2

m = 12

quadruples.

h a v i n g DS=[f6)6,(4)31 such t h a t :

z,y,z

(see 5 2 ) . Therefore,

Consider t h e l a s t p a i r of DS = [ ( 6 j 8 ]

. Let

1 - f a c t o r i z a t i o n s of

F={F

K4

respectively:

I

x=7]

{ 1 , 5 , 8 , y l ~ s ~ and

b =Ix,y,z)

(Theor. 4 . 1 ) .

Y = {2,3,6,7)

[resp.

the

.

DMB PQS

be t h e f o l l o w i n g and

Fl

F =

D M B PQS

2 ) . The-

with

t h e r e e x i s t t h r e e elements

h =2

m =12

3 '

h =2

and a q u a d r u p l e

2

f o r them i t i s PQS

it is

There e x i s t t h r e e

THEOREM 4 . 3 .

..

{1,3,5,6),(2,3,4,7)~s

§

. Consider

h =2

m =14

such

x,y,zEP

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 case

m = 14

D M B PQS w i t h

t h e r e e x i s t ( i n every c a s e ) a t l e a s t t h r e e elements that

D M B PQS

I

I

G2

I

G3

DMB

F F 1, 1, 2' 3

on

259

On the Maximum Number of SQS(v) Further, l e t

=( If

=

1 2 3

1(618]

)

1 2 3 3

, we

(P,T(F,G,a

3

.

Further,

r(F,G,a

J

1

)

2 3 1

can v e r i f y t h a t

I )

are three

and

,

D M B PQS

) 1

with

*

l) , m=12

and

it follows t h a t it is not possi-

m

with

7 2 ~ 3 D M B PQS

(P,I'(F,G,a

{1,2,3}

a r e t h e two f a m i l i e s i n d i -

r(f,G,ci2)

cated i n 5 2. Since it is ble t o construct

1 2 3

)

3 1 2 '

,..., 8 1

,

(P,r(F,G,a21)

DS

1 2 3

P={1,2

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

a i

=12

and

DS = [ ( 6 j 8 ]

. Hen-

c e , i t follows t h e statement.. THEOREM 4.4. Proof.

and

Let

D(8,q8-121 (X,A)

( i= 1 , 2 , 3 )

(Y,8)

let

I - f a c t o r i z a t i o n s of

K4

b e two

F = f F , F ,F 1 1 2 3

on

X

, where

SQS(4)

and

Y

, ,

SQS(8)

W e have :

.

X={1,4,5,8}

G = { G ,G G 1 1 2' 3

and l e t

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 .

191 t h a t t h e p a i r

i n an

.

. Further,

Y={2,3,6,7}

t h e two

and

= 3

a

be

i

I t i s known

260

M. Gionfriddo, A . Lizzio and M.C. Manno We can see immediately that:

,

q1nq2=qlnq3 =q2nq3

for every

i,j€{I,Z,31,

have

Di8,q8-12) = 3

v = 2

Proof.

n+2

,

u=5-2

Since for

=2

. From

Theorem 3.1, in the case

k =2

, we

. D(v,qv-8) = D ( v , q - 1 4 ) =D(u,q - 1 5 ) = 2 v v

THEOREM 4 . 5 . We h a v e every

3

i # j .

D(8,q8-12) >3

Hence

lqinq.l

n

v =2

, n+2

to construct at least two

u=7.2

,

, and

v =5.2

SQS(vl

quadruples in common (see [ 6 ] ,

n

n

,

with

nL2

[ 7 ] , [13]),

.

v =7.en

qv-8

or

, for

it is possible q -14 V

or

4,115

the statement follows

from Theorem 4.2, directly..

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111 C. Berge, Graphes e t h y p e r g r a p h e s , 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 , Proc. Amer. Math. SOC., 32 (1972), 409-416.

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Theory, 20 (B) (1976), 265-282.

26 1

On the Maximum Number of SQSlvl

[ 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 v ~ 5 . 2 and ~

n z 3

, with

qv-Zl

and

qv-25

S(3,4,vl

,

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 ) , 183-191.

1121 A . Lizzio, S. Milici, O n 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 , Rendiconti 1st. Mat. Un. T r i e s t e , (to a p p e a r ) . 1131 G . L O F a r o , On t h e s e t order 47. 1141 A .

v =7.2n

with

Jlvl n22

f o r S t e i n e r quadruple systems o f

, Ars Combinatoria, 1 7 ( 1 9 8 4 ) ,

39-

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 .

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 263-268 0 Elsevier Science Publishers B.V. (North-Holland)

263

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 structureand 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 ~E ~V, V w E W, k E K. Clearly f is bilinear iff y=l.

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 ( * ) . A group

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) = X O ( ~ * Z ) . Moreover xn = x+...+x (n times) holds and so xp=ol since K has chaPacteristic p . At last for the commutator of x and y we have [x,yl = x-I= y-l- x'y = 2f (x,y), and so G' 5 W 5 Z(G) is valid.

A . Herzer

264

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 least K=GF (p)) Defining p+l f(x,y) := [x,yl 2 for p odd, 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 2 and class 5 2 and a partition n of G, such that = S ( G , n ) holds.

s

s

s

Concerning the proof the implications a)*b) and c)-a) follow from the preceding propositions. The implication b)-*c) follows from the fact, fulfilling b) can be considered as an affine two-sided linearthat ly fibered incidence group1, which then is represented by an affine kinematic algebra A= K (3 M as defined at the end of chapter 1.

s

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(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 (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 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 , (The analogic thing holds for the lines and their parallelism Ilr.)

.

s

s.

s

s

REMARK. Since the parallel structures 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 which is affinely embedded and possesses a paralparallel structure for different points x l y l zof the lelism //r on R compatible with 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).

s

s,

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 lo= ' x VxEM, see [ 5 1 or [91. cal K-algebra A = K @ M with , REFERENCES

[I] Andrg,

Math.Z.76 (1961) 85-102 and 155-163. [ 2 ] Biliotti, M., Sulle strutture di traslazione, Boll.U.M.I.(5) 14 A (1978) 667-677. [3] Biliotti, M. and Herzer, A . , Zur Geometrie der Translationsstrukturen mit eigentlichen Dilatationen, Abh.Math.Sem.Hamburg (1984) 1-27. J . , Uber Parallelstrukturen I.II.,

268

A . Iferzer

[ 4 ] Biliotti, M. and Scarselli, A., Sulle strutture di traslazione

[5]

[6]

171 181 [91

dotatedi dilatazioni proprie, Atti Acc.Naz.Lincei, Rend. C1.Sci.Fis.Mat.Nat. (8) 6 7 ( 1 9 7 9 ) 75-80. BriScker, L., Kinematische Raume, Geom.Ded. 1 (1973) 2 4 1 - 2 6 8 . Herzer, A., Endliche translationstransitive postaffine Riume, Abh. Math.Sem.Hamburg 48 ( 1 979) 25-33. Herzer, A., Endliche nichtkommutative Gruppen mit Partition ll und fixpunktfreiem WAutomorphismus, Arch.Math.34 (1980) 385-392, Herzer, A., Varietg di Segre generalizzate, Rend.Mat.(Roma) to appear 1986. Karzel, H., Kinematic Spaces, Symposia Mathematica XI ( 1 9 7 3 ) 4 1 3 439 (Istituto Nazionale di Alta Matematica).

Annals of Discrete Mathematics 30 (1986) 269-274

269

0 Elsevier Science Publishers B.V. (North-Holland)

Sharply 3 - t r a n s i t i v e groups 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 H e i n r i c h W e f e l s c h e i d

The s e t J: = { y E G ) y 2 = i d , y + i d } o f i n v o l u t i o n s of a g r o u p 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 s e t 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 g r o u p G . F o r example t h e c h a r a c t e r i s t i c 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 a s p r o p e r t i e s of J In t h i s paper we a r e looking f o r

2

.

s h a r p l y 3 - t r a n s i t i v e permutation-

g r o u p s 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 :

K c Z 2 and r e s u l t s on g r o u p s G w i t h G = J 3 a r e g i v e n . Also a c l a s s of examples of g r o u p s w i t h G = J 3 a r e 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 r o u p s G s u c h 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 r o u p s 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 r o u p s G which a r e 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 . G = J 2 n G SPGL(2,K)

fi

F o r u n d e r s t a n d i n g 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 K T - f i e l d and t h e b a s i c r e p r e s e n t a t i o n theorem: Definition 1:

F ( + , . , a ) i s c a l l e d a KT-field

i f t h e f o l l o w i n g axioms

are valid: FB 1

FB 2

( 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) F o r e a c h p a i r of e l e m e n t s a , b F F t h e r e e x i s t s a n e l e m e n t - x f o r each x E F. d a I b E F such t h a t a + ( b + x ) = ( a i b ) +d a,b ( F * , - ) i s a g r o u p ( w i t h i d e n t i t y 1; F*=F { O } )

FB 3 -

a.(b+c) = a-bta-c

KT

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 :

and

0.a = o

for a l l a,b,cCF

270

M.Hille and H Wefehrheid

-

O ( l + U(x) = 1

F:=

f o r all x E F \ { 0 , 1 } .

x

--f

an e l e m e n t n o t

FUI-1.

The t r a n s f o r m a t i o n s of t y p e a a n d B of a:

-

L e t F be a KT-field,

R e p r e s e n t a t i o n - T h e o r e m 2: i n F a n d d e n o t e by

a ( l + x)

8: x

a+m-x

+

F

onto

F:

a+cs(b+m-x)

w i t h a , b , m E F , m $ O a n d a ( - ) = - , o ( n ) = 0 , o ( 0 ) =-, f o r m a g r o u p 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'. C o n v e r s e l y each s h a r p l y 3 - t r a n s i t i v e group i s isomorphic a s a permutation g r o u p t o t h e g r o u p s T (F) of a u n i q u e l y d e t e r m i n e d K T - f i e l d . 3 In the following let

G

r e p r e s e n t e d i n t h e form x

+

a

8,:

x

+

-b + n o ( b + x )

case of

,

3:

-1

6(a) = b

Lemma 4:

If

*

F*.

and

S

b E F and

-

I f t h a t i s t h e case

=

x1,x2

- b _ + ~

).

6

6 E G with

2

*

is called a

id

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

and G

n E S := { s E F*lo(s).s=lI

then

A transformation

pseudo-involution with

are:

G

a E F ( i f char F = 2 , then a * O )

- ;o) I z E F * ) .

n = z.o(z

a s being

.a has t h e o n l y f i x e d p o i n t ;in 2 t h e n .a h a s t h e two f i x e d p o i n t s and p o s s e s s e s f i x e d p o i n t s x 1 , x 2 i f and o n l y i f

*

char F whereas

Definition

x

char F = 2

n E R := { z . a ( z with

T 3 ( F ) The i n v o l u t i o n s of

a0:

I n case of a-2-I

-

be a s h a r p l y 3 - t r a n s i t i v e g r o u p

F

a,b E

6(b) = a 6 then 6

c o n t a i n s a pseudo-involution

4

J

2

Proof: L e t be 6 E G a p s e u d o - i n v o l u t i o n w i t h S ( 0 ) = m a n d 6(-)

Then 6 h a s t h e form 6 ( x ) = a ( m - x ) = o(m) - a ( x ) t h e r e e x i s t two i n v o l u t i o n s 6 , m =

6(0) =

0 = 6(m) =

1 . C a s e : L e t be

,8 ,

E J such t h a t 6 =

e102co,

*

8, (-1

B,B2(-)

=b

B,

B,

(m)

Then 66(z) = 6 S B 2 ( 0 )

= B(0) = z

w i t h m E F*\S.

Bl 8,.

= 0.

Suppose

Thenwehave:

= B2(0)

(0) =

02(4

* 0,-.

= 6R,B2B2(0) = 6 8 , ( 0 ) = 6 B 2 ( m )

=

I~~B,B,(-)

=

Sharply 3-Transitive Groups =

PI(-)

=

z. * u ( m ) * m - z = 6 6 ( z )

-

= z

27 1

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 .

6,

2 . Case:

(a) =

* B1 h a s t h e 1 B 2 i n t e r c h a n g e s 0 a n d OJ. T h e r e f o r e (3, h a s t h e form @ , ( x )= n.o[x) w i t h n E S . * u ( m ) -r?(x) = 6 ( 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.

Then w e g e t B 2 ( 0 ) = B, (-1 f i x e d p o i n t s 0 and

m

=$

=

=.

0 = B2(m) = @ ( 0 )

=

-

QD.

B, ( x )

x.

.

3 . Case: @ ,

*

B2(0)

=

= 0

(m)

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 1 and B2.

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 g r o u p . Then G = J2 i f and o n l y i f G S P G L ( 2 , K ) and I K I > 2 . Proof: “4‘ PGL(2,K) S J

2

if

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 . 2 w*11 Let be G = J Because of lemma 4 t h e g r o u p G c a n n o t c o n t a i n any

.

pseudo-invo1ution.Therefore F * \ S = #. A K T - f i e l d w i t h F* = S i s a commutative f i e l d ( c f . [ 3 1 , S a t z 1 . 6 ) . Now w e t u r n t o g r o u p s G w i t h G = J

3

.

(F,+,*u) i s c a l l e d 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 . D e f i n i t i o n 6: A KT-field

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 . group.

(F,+) i s a n a b e l i a n

(cf. [ I ] , 5.6).

Theorem 7 : L e t be (F,+,.,u) a K T - f i e l d w i t h I F / > 2 a n d l e t b e G t h e induced s h a r p l y 3 - t r a n s i t i v e group. G~:=

I yEGly(a)

= a

Then

~ f o r ca 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 : = =

CsEF*ls-U(s) = I}.

M . HiNe and ti Wefelscheid

212

Proof:

W e conduct t h e demonstration i n s e v e r a l s t e p s :

L e t b e F a p l a n a r K T - f i e l d a n d S - S = F*.

A:

1 . Gm -

cJ' 1 0

Proof:

The e l e m e n t s of t h e g r o u p Gmlo:={yEGI y ( 0 ) = 0 , y ( m ) h a v e t h e form:

=m)

= {LIEGI p : = - , k x w i t h k E F * ] . .

Gm10

Because of S - S = F* t h e r e e x i s t n 1 , n 2 E S s u c h t h a t nl' n 2 = k . Then ( 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 ) , 1 2 1 2 2 1 p,(x) = u ( n 2 x ) a n d B 1 , B , - E J .

'V ( x ) = n n

A

2*

f o r each a E F .

Gm,acJ2

P r o o f : L e t b e a : x-, a t x . Then G -1

b e c a u s e of

3.

J

ci

c1-I

=

J.

a

= ci G

ci

-1

c c1 J

2

2 a-l

--J

=I0

Gm= { y E G l y ( x ) = c + m-x, c , m E F , m t O } c J

2

Proof: L e t be T:= { T E G I T ( x ) = ~ = , x = ~ } . Then w e h a v e

S i n c e F i s p l a n a r , t h e s e 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 h a v e Hence T c J

2

T

.

B . L e t be G a c J 2

Then Gm = y G a y

-1

= a 2 a 1 E J2 w i t h a i ( x )

= ai-x

f o r some a E F . 2 -1

c y J y

= J

2

f o r some y E G w i t h y ( a ) =

L e t now b e c 1 E Gm w i t h a ( x ) = c + m x , m + 1 a n d c1 =

B,(x)=-b.+n *o(bi+x),i = 1,2. i

a n d a 2-al = b .

i

B2Bl

EJ

m.

2

with

Because of B i E J w e h a v e n

i

ES.

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 - .

B 2 B 1 (-1

= - b 2 + n 2 u ( b 2 + ( - b +n o ( b l + - ) ) )

m

= a(-)

=,

= -b +n. o ( b -b ) 2 2 2 1 b2 = b l = : b . +. a ( x ) = -b

=)

The s e c o n d f i x e d p o i n t of

=

1

(n2n1-'b

t

ci

i s -b.

1

t

n2n1 - 2 . x )

We h a v e

Sharply 3-Transitive Groups

213

S i n c e e a c h 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 w h e r e a s f o l l o w s t h a t (F,+) i s a n a b e l i a n g r o u p .

* 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 c h o s e n .

Remark:

I f Ga c J2 t h e n G c J3. Wether t h e c o n v e r s e 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 e x a m p l e s of s h a r p l y 3 - t r a n s i t i v e g r o u p s G w i t h 3 b u t G t. J2. W e 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

G = J

s h a r p l y 3 - t r a n s i t i v e g r o u p s 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*,.) s u c h t h a t :

2

EF*laEF*}cA

(1)

Q:=

(2)

T h e r e e x i s t s a monomorphism

(3)

i (x) E

{a

xA f o r all

T

E

TI

F*/A

TI:

+

Aut(F,+,')

(F*/A) a n d a l l x E F*.

D e f i n e Oob = 0, and f o r a + O , a o b = a - a ( b ) where a cp

Then ( F , + , o , o ) i s a K T - f i e l d . coupled Dickson-nearfield,

(To b e e x a c t :

(F!+#o)

cp

= a(a.A).

is a strongly

which i n a d d i t i o n i s p l a n a r ( c f . [ 5 ] ) )

( 1 ) i n d u c e s [F*:A] = 2111 where I d e n o t e s a n a p p r o p r i a t e i n d e x 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 i n d e x sets s u c h K T - f i e l d f o r which t h e i n d u c e d g r o u p G s a t i s f i e s G = J 3 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 b e 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

r1

e x t e n s i o n of K . C o n s i d e r t h e f o l l o w i n g automorphisms a 1 , a 2E AutKL

al:

{

X

+

1-Xl

x2

+

x2

k E K

+

k

a2:

+

1-x2

x

+

x1

k E K

--*

k

u 1 and a2 a r e i n v o l u t a r y a n d s a t i s f y u 1 u 2 = a 2 a 1 . Take F:= L ( t l f t 2 ) where t h e t i are t r a n s c e n d e n t a l o v e r L. L e t b e 7i

E A u t 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 a 1 f o r h E L [ t ,

/t21:

:

214

M . Hille arid H. Wefelscheid

and define

h, := h2

T , (-)

I

ri(hl) for -r;(hZJ

h, A l s o let be gradi€ = gradi$-:=

f = -hlr h2

h ,h EL[tl,t21 1 2

gradihl

- gradih2 the degree function

of the polynomials h l r h 2EL[tlrt21with respect. to ti. Now we define for € , g E F : grad f grad2f T2 fog:= f m f w ( g ) with fQ:= Then Fr+,oru)is a planar KT-field (to be exact: (F,+,oru)is a planar KT-nearfield; o(a) = a-1 the inverse with respect to ( - 1 1 . We note thatA = Kern cp = {fEF*l gradifZO mod 2 for i = I r 2 } c S and [F* : A] = 4 = 211[. The 3 other cosets are: t,*A, tZ-Ar t,-t2-A The Dickson-group r:= If EAut F / fEF*] consists of the 4 automorcp -c2, T ~ T ~ } . Fany O ~ T €rand any z E Itl,t2,tl.t21 phisms: r = {id, ~~r 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 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 Mathematische 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)

275

OY T I E G E N E R A L I Z E D CIIROMATIC NUMBER F l o r i c a Kramer and I l o r s t K r a m e r Computer T e c h n i q u e R e s e a r c h 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 r e l a t i v e t o d i s t a n c e p,den o t e d b y 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 a n y t w o v e r t i c e s 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. W e q i v e upper '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

bounds f o r t h e c h r o m a t i c nur.ber

o f S . A n t o n u c c 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 ] o f C.Oerqe. We r e s t r i c t o u r s e l v e s t o s i m p l e g r a p h s , c o n n e c t e d u n d i r e c t e d f i n i t e g r a p h s which h a v e no l o o p s o r mul-

i.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 s e 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 L g ( x ) ; x ~ V t3h e maximum

t i p l e edqes. L e t G = ( V , E )

d e g r e e o f t h e v e r t i c e s of G , by d G ( x , y ) t h e d i s t a n c e b e t w e e n 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 i s c o n s i r i e r e d t o be t h e number o f e d g e s w h i c h f o r m t h e p a t h ) . d e n o t e s t h e c o m p l e t e g r a p h w i t h n v e r t i c e s and K t h e complete n m*n b i p a r t i t e qraph. it

I n 1969 w e h a v e 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

follow-

in? c o l o r i n g problem of a graph: DEFINITION 1. L e t p h e 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-color 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) f: V

9 {1,2,...,k)

such that

2

=>

is a function

v x , y € V with 1 ~ d C ( x , v ) p f(x) # f(y). The s m a l l e s t i n t e g e r k f o r w h i c h t h e r e e x i s t s an a r ' r n i s s i h l e k-colo-

F. Kramer and H. Kramer

216

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 c h r o m a t i c number r e l a t i v e t o d i s t a n c e p of t h e q r a p h G . DEFINITION 2 .

L e t G=(V,E) b e a g i v e n g r a p h and p an i n t e g e r , p

W e s h a l l d e n o t e by G P = ( V , E

set

17

2 1.

t h e g r a p h , which h a s t h e same v e r t e x P a s t h e g r a p h G whereas t h e edge set E is d e f i n e d by: )

5Pp.

( x , y ) E Lp i f and o n l y i f 1 s d G ( x l y )

There e x i s t s t h e n t h e f o l l o w i n q r e l a t i o n between t h e c h r o m a t i c numb e r r ( p , C ) and t h e o r d i n a r y c h r o m a t i c number g(C) : )$(PIC)

In t h e papers

= $-(l,CP) =

[ill,

K(C,P). [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 c h r o m a t i c number

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

g r a p h s G f o r which we have r ( p , G ) = p + l . The problem of c o l o r i n q a g r a p h r e l a t i v e t o a d i s t a n c e s w a s recons i d e r e d i n 1 9 7 5 by P.Speranza [ l G ] u n d e r t h e name of L s - c o l o r i n g of a g r a p h ; t h e same oroblem was a l s o s t u d i e d by M.Gionfriddo [6], [7] and [f311S.Antonucci [11 and by G.Wegner L171. 2 . UPPER BOUNDS FOR THE CHROMaTIC NUMBER r ( 3 , G ) G.Weqner 1111 proved t h e f o l l o w i n g theorem: T!IEOREY

1. L e t G=(V,F) be a s i m p l e p l a n a r g r a p h s u c h t h a t

W e have t h e n :

s(2,G)

-

A

( C ) f 3.

f 8.

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 g r a p h C, w i t h maxi-

mal d e q r e e 3 such t h a t w e have $ - ( 2 , G ) = 8 o r else t o prove &-(2,G)f7. 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 c h r o m a t i c number ) ( " ( 3 , G )of b i p a r t i t e g r a p h s . L e t C=(A,B;E) be a q i v e n b i p a r t i t e graph. t h e g r a p h , which h a s t h e v e r t e x s e t A and t h e cdqe s e t EA is d e f i n e d by:

W e s h a l l d e n o t e b y Gp=(A,EA)

( x , y ) E E A i f and o n l y if d G ( x l y ) = 2 . The g r a p h GA c a n b e o b t a i n e d from t h e g r a p h C, by t h e e l i m i n a t i o n o f 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 operations: 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 ( i t 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

struct in

GA

a d j a c e n t t o y are i n A ) t h e n w e con-

t h e c i r c u i t x1x2x3x1.

h) I f a verte:: y o f B h a s d e q r e e 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 o f 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 r e c e d i n g two o p e r a t i o n s a p p e a r mult i p l e e d g e s 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

011the Generalized Chromatic Number

211

of t h e s e e d g e s 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 g r a p h G B = ( B , E R )

with t h e v e r t e x set 8

and t h e e d g e set EB d e f i n e d by: ( x , y ) E E B i f and o n l y i f d c ( x , y ) = 2 . An upper hound f o r t h e c h r o m a t i c number

o r d i n a r y c h r o m a t i c numbers

and

$-(C,)

r ( 3 , G ) i n f u n c t i o n of t h e

r(CR)

is qiven i n t h e follo-

winq: TlICOREM 2 .

L e t G = ( A , B ; E ) b e a b i p a r t i t e q r a p h . Then

(1) Proof.

T(3,G)

5 pA) + K(CB).

b e i n g t h e o r d i n a r y c h r o m a t i c number of t h e g r a p h

$(GA)

t h a t ( x , y ) E EA w i t h )-'(CB)

,...,

CA

3 Ll,? r ( G A ) ) of i t s v e r t i c e s such 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

t h e r e is a c o l o r i n g f l :

A

c o l o r s o f t h e v e r t i c e s of t h e g r a p h GB f 2 : B +

..

[bp(GA)+l,

.

such t h a t ( x , y ) 6 EB i m p l i e s f2 ( x ) # f 2 (y) 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

...,

f(x) =

(2)

{

fl(x),

i f xEA

f2(x), if xEB i s t h e n 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 g r a p h G=(A,B;E) w i t h '$"(GA)

+

r ( G B ) colors. W e 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 ( d i s t a n c e 3 of t h e g r a p h

C,,

i.e.

t (C,) ) - c o l o r i n g r e l a t i v e t o t h a t x , y 6 A V B and 15 d G ( x , y ) 3

f-(GA)

6

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 , l e t it be x , b e l o n g s t o A and t h e o t h e r , y , b e l o n g s

.

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 d G ( x , y ) = 2 it f o l l o w s from t h e b i p a r t i t e c h a r a c t e r o f C t h a t b o t h v e r t i c e s x and y a r e i n A , o r b o t h a r e i n R . C o n s i d e r now t h e c a s e s : a ) i f x , y E A and A G ( 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 f ( x ) = f1(x) # f p = f(y). b ) i f 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 (XI = f 2 ( x ) f f 2 ( y ) = f ( y ) . 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 i m p l e p l a n a r b i p a r t i t e g r a p h 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 r a p h s f o r which

X(3,C)=S.

P r o o f . 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 r a p h G and t h e c o n s t r u c t i o n way of t h e g r a p h s 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

and [41) w e have t ( G A ) 4 and t h e n immediately i n e q u a l i t y ( 3 ) .

5

L31

'rl:

Fig. 1 i n Theorem 3 we o b t a i n :

$(GB)

5

4.

B y Theorem 2 f o l l o w s

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 , c a n be s e e n from t h e f o l l o w i n g example: L e t G=(A,B;E) be t h e g r a p h o f 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 g r a p h w i t h 8 v e r t i c e s and d i a m e t e r 3 . I t r e s u l t s t h a t w e have = 8. I f w e renounce t o t h e p l a n a r i t y of x ( 3 r G ) = IAlJBl

be a b i p a r t i t e g r a p h such t h a t

THEOREM 4 . L e t C = ( A , B ; E )

(GI

f

G

3.

Then w e have: p(3,C)

(4)

5

14.

The o b t a i n e d bound i s 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 c u b i c q r a p h s 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

an(' c,., d e f i n e d above. From A ( C )

correspondinq qraphs

L,

3 results

i p m e d i a t e l y A ( C , ) f 6 and A(C,) L, 6 . A well-known t.!ieorem (see f o r i n n t a n c e [ l 5 ] v o l . 1 , S a t z I V . 2 . 1 ) asserts t h a t f o r any g r a p h G , t h e o r d i n a r y c h r o m a t i c number i s a t most one g r e a t e r t h a n t h e maximum d e g r e e A (GI. W e have t h e n ( G A ) 5 7 and f 7. Applying a g a i n Theorem 2 w e have Y ( ? , G ) 14. The f a c t t h a t t h i s bound i s s h a r p ,

F(GB)

5

can be s e e n from t h e example o f t h e well-known

Ileawood g r a p h (see F i q . 2 )

,

which i s a c u b i c b i p a r t i t e g r a p h w i t h 1 4 v e r t i c e s and o f d i a m e t e r 3 . W e ha-

r ( 3 , C ) = I A U B I = 14. 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 f o l l o w i n q theorem.

ve t h e n :

Fig.

2

THEORE'fi 5 . L e t C: be a s i m p l e b i p a r t i t e qraDh w i t h maxinum clegree

A

=

a (c).

Then w e have: ~

(5) P r o o f . From

A

(

3

f~

~

+ 1A

(

A -

1)).

= 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 i v e n v e r t e x

On the Generalized Chromatic Number

A (A-1)

x i n t h e q r a p h G i s a t most and

b(GB)

-

r ( C B )f

2 n(A-1)

and t h e r e b y

a(a-1)+1. Applying

COROLLARY 1. L e t G = ( A , B ; E )

279

A(c;,) A( A - 1 ) +1 a n d

.\ve h a v e t h e n

g(GA)5

= L

A(A-1)

respectively

now Theorem 2 w e o b t a i n i n e q u a l i t y ( 5 ) .

be a simple A - r e g u l a r

b i p a r t i t e graph of

d i a m e t e r 3. Then w e h a v e : ( A \

16)

+ !B1 5 2 ( 1 +

h ( a - 1).

Examples o f g r a p h s f o r w h i c h w e h a v e t h e e q u a l i t y s i q n i n ( 5 ) and

1) f o r

also in (6) are:

b

= 2 t h e c i r c u i t C6 o f

a=

lenqth 6; 2) f o r

3 t h e a b o v e d i s c u s s e d ifeawood

g r a p h and 3 ) f o r

A

= 4 t h e 4-regular

b i p a r t i t e q r a p h 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

implies again r ( 3 , C ) = I A U R \ = 26. An improvement of Theorem 5 and o f Theorem 4 c a n b e o b t a i n e d i f w e appl y Brooks theorem (see f o r i n s t a n c e [ 9 1 Theorem 1 2 . 3 ) : I f

G is A-colorable

A(C)=A t h e n

unless:

= 2 and G h a s a comnonent w h i c h

(i)

i s a n odd c y c l e , o r ( i i )A > 2 and

K A + l i s a component o f

Fis. 3

G.

w a s d i s c u s s e d i n [ _ 1 1 ] , 1 1 2 1 and L131.Let u s assume t h a t 7 2 . The g r a p h s CA and GB a r e c o n n e c t e d s i n c e C, i s c o n n e c t e d . I f

The c a s e A.2

A

now a t l e a s t o n e of GA, GB

-

s a y , GA

-

is not a K

t h e n by

g(GA)fA ( A - 1 ) and b y Theorem 2 w e h a v e t h e i n e q u a 5 2 A ( A - 1 ) + 1 . T h u s i f e q u a l i t y h o l d s i n Theorem 5 w e \ B \ = n(n- l ) + land 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

Brooks t h e o r e m , lity

K(3,G)

have ( A 1 = d i a m e t e r 3. bb(GB)5

I f n o n e o f GA,GB

n(n-1) and

is a K

by Theorem 2

A ( A -1) +1 t h e n )$'(GA) f a(n-1), a ( A - 1 ) . W e c a n now

x(3,G)f2

enounce : COROLLARY 2 . L e t C = ( A , B ; E )

A ( G ) =A >

be a simple b i p a r t i t e qraph such t h a t

2 and min ( I A I , \ B \ ) )9(3,G)

In p a r t i c u l a r , i f

A

>A(a-l)+l.

52 4 ( A -

= 3 and I A \ > 7 , \ R 1 > 7

1;(3,G)

5 -

12.

Then w e h a v e :

1). t h e n w e have:

280

F. Kramer and H. Kramer

3 . LOWER BOIJNDS FOR THE CHROMATIC NUMBER 2('$

,G)

F i r s t we s h a l l p r o v e a r e s u l t needed i n t h e s e q u e l . THEOREM 6 .

(i)

L e t G=(V,E) b e a s i m p l e g r a p h w i t h t h e p r o p e r t i e s :

t h e derjree o f e a c h v e r t e x i s a t l e a s t 2 ,

( i i ) t h e d i a m e t e r o f t h e g r a p h D(G)=2, ( i i i ) G d o e s 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 r a p h . P r o o f . 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 e a s t o n e c i r c u i t i n t h e g r a p h G. By ( i i ) and ( i i i ) r e s u l t s t h a t t h e g i r t h o f G i s 5 . (7 is t h e n a Moore q r a p h by a r e s u l t o f R . S i n g l e t o n l 1 4 1 which a s s e r t s

t h a t a simgle graph with diameter k

2

1 and g i r t h 2 k + l I s a l s o regu-

l a r and h e n c e 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 f o r 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 o f 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 h a v e : n

(7)

3

n 3- 4m2 But S . A n t o n u c c i d i d n ' t g i v e a n y example o f g r a p h s f o r which t h i s hound is a t t a i n e d . W e s h a l l 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 , without c i r c u i t s of

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 w h i c h w e h a v e

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 .

P r o o f . 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 h a v e n = 1 , m=O, '$'(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). I f D ( G ) = l , G is a c o m p l e t e g r a p h K w i t h n - 2 . The o n l y c o m p l e t e 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 r a p h K2, f o r graph Knl n which w e h a v e n=2, m=1, ) f ' ( 2 , K 2 ) = 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 h a v e 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 of

On rhe Generalized Chromaric Number

l e n g t h 3, G i s a ( n - 1 ) - s t a r w i t h n

2

28 I

3 , i . e . G=(V,E) w i t h V= Ca1*a2* i=1,2 n-1J W e h a v e t h e n m=n-1 and E= [ ( a i , h ) , =n. R e l a t i o n (El becomes n ( n 3 - 4 ( n - 1 ) 2 ) = n3 The o n l y sol 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 w h i c h 1 3 4 c o r r e s p o n d s b e c a u s e as w e h a v e s e e n a b o v e w e h a v e n 'I, 3. The c o n c l u s i o n is t h a t w e can n o t have b)

x = min

$=

,...,

.

.

1.

{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 e d q e s . Because D ( G ) = 2 w e h a v e 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 r o v e d . REMARK.

By a well-known r e s u l t o f A.J.Hoffman

and R . R . S i n q l e t o n

(101

a Moore g r a p h o f d i a m e t e r 2 h a s o n e 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 o n e Moore g r a p h of d i a m e t e r 2 ( i t is. n o t known w h e t h e 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 ) .

lower bound f o r 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 can a l s o b e deduced f o r g r a p h s which h a v e c i r c u i t s of l e n g t h 3 o r 4 .

A

TIIFOREM 9 . L e t C=(V,E) be a s i m p l e c o n n e c t e d 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 w h i c h w e d e n o t e by: (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 ;

( i i ) 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 w h i c h n o 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 ; 1 ( i i i ) c 4 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 o n e p a i r o f o p p o s i t e v e r t i c e s i n t h e c i r c u i t a r e 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 . If G doesn't

c o n t a i n a n y s u b a r a p h of t h e t y p e K t h e n t h e chroma2,3 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 3 n (9)

c

3 0 1 2 n +n ( 6 c 3 + 4 c 4 + 2 c 4 )- 4 m

I

**

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 t h e o r e m and f o r which w e h a v e t h e e q u a l i t y siqn in (9). P r o o f . The p r o o f o f t h i s t h e o r e m c a n 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 r o o f g i v e n by S . A n t o n u c c i f o r Theorem 7 . A s w e h a v e o b s e r v e d

282

F. Kramer and H. Kramer

above w e h a v e

Y(2,C)

=

s q u a r e of t h e q r a p h G.

$(llC2)

=

2

r(C;) ,

where c; 2 = ( V I E ) i s t h e 2

I f w e d e n o t e b y m2 t h e c a r d i n a l i t y of t h e

e d q e set E 2 , t h e n w e h a v e by a Theorem of C . B e r q e r(2,C= ;) g(G2) 7 =

(10)

2

( c 5 J l p.321)

.

n2-2m2 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 r a p h by t h e sum

2 (q(ii)).

xyz of lenq&'

G

i s qiven

If we introduce corresnondinq t o each path

2 i n C an e d q e ( x , z ) w e s h a l l o h t a i n a q r a p h C " = ( V , E " ) .

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 e d g e s 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 c o n n e c t e d i n O b v i o u s l y E2 C;

by a t l e a s t o n e p a t h of l e n q t h 2 . W e h a v e 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 cont 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 mult i p l i c i t y of t h e e d q e ( 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 e d q e ( 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 e d g e of t h i s c i r c u i t by one u n i t y , w e h a v e t o d e l e t e 3c3 e d q e s 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 e d g e s . 2 ) a and 13 a r e n o t a d j a c e n t i n C. Because G d o e s n o t c o n t a i n any s u b g r a u h 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 c a n h e c o n n e c t e d 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 c o n n e c t e d 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 obviously a simple edge of t h e graph G " . 2b) a a n d b a r e c o n n e c t e d 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 v e r t i c e s i n a c i r c u i t of l e n q t h 4 i n G . B e c a u s e 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 o n e d i a q o n a l , i n o r d e r t o obe d g e s from El' b e s i d e t h e t a i n t h e q r a p h C2 w e h a v e t o d e l e t e 2c:+c: 3c3 edctes a l r e a d y d e l e t e d . W e h a v e t h u s n g(xi) 1 (3c3+2cy+c4). m2 = m + i=1 I t r e s u l t s then:

t(

)-

283

On the Generalized Chromatic Number

2

- -2m-

.

1

( 3c3+2c>c4)

n

T h i s i n e q u a l i t y and. (10) v i e l c l s

As

r(2,C)

w h i c h [r]*

i s an i n t e g e r w e o b t a i n i m m e d i a t e l y t h e i n e q u a l i t y ( 9 ) i n 2 r. An e x a m p l e o f a n r a p h f o r w h i c h w e

denotes t h e smallest i n t e q e r

have t h e e q u a l i t y s i g n i n ( 9 ) i s

4 f o r which we 1 h a v e n = 7 , m = l l , c = 3 , c 4 = l , c0=2, 4 3 D ( G ) = 2 and )f(2,C)=n=7. I t i s e a s y t h e q r a p h from F i g .

to v e r i f y t h a t f o r t h i s qraph we have t h e r e l a t i o n s : Fiq.

i3

n

4

3

'i'

= [343/551*

0

1

n +n ( 6 c 3 + 4 c 4 + 2 c 4 )- 4 m

= 7 = $(2,C,).

ACKNOWLEDCYENT. The a u t h o r s w i s h 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 s e c o n d 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

A l e x a n d e r von H u m b o l d t - S t i f t u n q 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) A n t o n u c c 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

q r a f o , Boll.Un.Vat.Ita1.

[23

Eq

[q

Appe1,K. ,Haken,W., Amer.Vath.Soc.

( 5 ) 15-B

( 1 9 7 8 ) 20-31.

E v e r y p l a n a r map i s f o u r c o l o r a b l e , B u l l .

82 ( 1 9 7 6 ) 711-712.

Appe1,K. ,Ifaken,IJ.,

E v e r y 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 .

D i s c h a r g i n g , I l l i n o i s J.Math. Appe1,K. ,Iiaken,W. , K o c h , J . ,

2 1 ( 1 9 7 7 ) 429-490.

E v e r y p l a n a r map i s 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 . M a t h . ti51 R e r q e , C .

[6!

,

Craphes e t hypergraphes

Gionfriddo,M., Mat.Ita1.

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 , B o l l . U n .

( 5 ) 15-A

[7]

Gionfriddo,M.

[8]

Gionfriddo,M.,

2 1 ( 1 9 7 7 ) 491-567.

(Dunod, P a r i s , 1 9 7 0 ) .

( 1 9 7 8 ) 444-454.

, Alcuni

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

d ' u n q r a f o , Riv.Mat.Univ.Parma

(4)

6

( 1 9 8 0 ) 125-133.

Su un p r o b l e m a 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 ( 1 9 8 0 )

I;. Krarner and H . Kramer

284

151-160.

1 IIarary,F., Graph Theory

(Addison-Wesley Publ.Comp. ,Mass. 1969). - - Hoffman,A.J. ,Singleton,R.R., On Moore graphs with diameters 2 and 3, IBM J.Res.Deve1op. 4 (1960) 497-504. [l 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: (5)

Al#A2

a

AlnA2

= S;

wa.

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

( V x , y e A : A ~ - ~ # A y -= 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

D.Lenzi

294

property (2) hold, and:

= A1nA2

A1 # A 2

( 4 ' ) VA1,A2eC2:

= I.

Then I i s a s u b g r o u p o f G. can suppose

We

.

of Q

101 #

1. Now l e t A1.A2 different elements -1 -1 -1 -1 T h e n , f o r e v e r y i , j e I , i j eAlj nA2j But A , j - ' # A 2 J ,

PROOF.

.

h t ~ n c t( ~b y ( 2 ) and ( 4 ' ) )

A1j

N.2.

THE

THE

OF

CASE

-1

nAZj-'=I, therefore i j - l e I .

GEOMETRICAL

OF

SPACES

Q.E.D.

TYPE

"n-STEINER".

Let u s g i v e t h e f o l l o w i n g

DEFINITION

6.

shall

We

say

a

that

fl+ o f

subset

thr

power

? ( G ) o f 8

v (v-1) ,..., -=nl--in--Ian-2,n-3,n-5I 2

k€H

then there exist two

l-factoriza-

G.Lo Faro

300

e

If

is the i d e n t i t y permutation, it i s a routine matter t o see t h a t

[ Q U Q ' I[q,q',F,G',e]

and

((Qu&!'l,(q,@))

k

SQS(2v)s with e x a c t l y

a r e two

b l o c k s i n common. The statement f o l l o w s .

I t i s w e l l known 111 t h a t i f 32'2~

,

then

there

y

t i o n of

(a

a r e even p o s i t i v e i n t e g e r s and

1 - f a c t o r i z a t i o n of order

exists a

1 - f a c t o r i z a t i o n of o r d e r

y

1 and

2-factorization

c o n t a i n i n g a sub-

z

n

of o r d e r

is a

I-factoriza-

K ).

2

...,n+98,n+lOO,n+104,n+112) Proof. L e t

. If

v iv-1) n =, v> 16

THEOREM 3 . 2 . Let

(Q,q)

1 , then k E J ( 2 v )

be a n

SQS(v)

Let

= (T,z)

SQSS(16) ( ( , ? U R ' ) , (?,'$)) F={FIJFg,

...,Fv-2 1

SQS(8)

(R,rJ

as a subsystem.

((QuQ'), (q,0)) = ( P , p )

SQS(2vl

F"2,.. ,,F"}

be two

7

I-factorizations

.

Q and R r e s p e c t i v e l y w i t h F$GPi , f o r e v e r y i = l , 2 , , . . , 7

of

SQS(2vl

l y easy t o s e e t h a t t h e

SQS(l6)

Ir,r',F",F"',i]

IRuR'I

IQuQ'I

con-

.

F"={F"

and

..

.

containing an

It i s s t r a i g h t f o r w a r d t o s e e t h a t t h e tains the

k E {n-I,n-Z,n-3,n-5,n+l,n+Z,.

Iq,q',F,F',i]

=(P,s)

It i s equal-

contains the

.

=(T,t)

Obviously, we have:

= 28 k E J(16) =1(16)

Let

such t h a t If SQS(2vls

.

(i",al

SQS( 6 )

I t i s p o s s i b l e t o c o n s t r u c t two

(T,b)

and

lanbl = k ,

fp-zlua=p'

Is-tlUb=s'

and

with exactly

v (v-1 I

--28+k

, then (P,p'l

(P,s')

and

are two

b l o c k s i n common.

2

T h i s completes t h e proof of t h e theorem., LEMMA 3 . 3 . O _< h _ , n - r ,

w e remember t h a t t h e m a t r i x which a g r e e s

i n t h e f i r s t n-r rows and l a s t n-r columns w i t h

..

A'

is non-singular.0

LEMMA 3 . T4e elements w .€ K , i = 0,1,. ,s , are l i n e a r l y F-independent if und n-1 t n o n l y if the vectoras w . = (wi wq w! ) EK , i = 0,1,. ,s, are Zinearly

.. .

-1

..

K-independent . Proof. Let us examine t h e c o n d i t i o n S

1 k.w.

(7)

i =O

1-1

=

0,kiEK,

under t h e h y p o t h e s i s t h a t t h e e l e m e n t s wi a r e F-independent. I f w e suppose t h a t a t l e a s t one c o e f f i c i e n t k i , then w e can d e t e r m i n e k E K such t h a t h = kk 0

0'

f o r example ko, i s n o t z e r o ,

t r ( h o ) # 0 ( ' ) h o l d s . From ( 7 ) ,

we

obtain S

T: h.w. 1-1

= 0 , h . = kk

-

i=O

I

i'

s

j

R a i s i n g t h e l e f t s i d e o f t h i s e q u a t i o n t o t h e powers qJ we o b t a i n ? h: i =O

...,n-1.

j = 0,1,

particular

wi

=

0,

S

I f we add t h e s e n e x p r e s s i o n s , w e f i n d C ( t r ( h i ) ) w i = i =O

2;

in

S

Y ( t r ( h i ) ) w i = 0, t r ( h o ) # 0 , i=O i n contrast with the hypothesis.

It i s e v i d e n t how t h e second p a r t of t h e t h e s i s may be proved.! COROLLARY 4 . Y7ie n x n matrix

nm-singular

2

=

(%,El,... ,%-,),

if and onlg if { u o , ul,. . .

...

n-1

)t,iz li= ( u i :u uqi , u ~ - ~is} a b a s i s o f the vector F-space K.!

For any polynomial ( 2 ) , t h e s e t Z(L) = { x € K : L(x) =

01

is o b v i o u s l y a v e c t o r subspace of K . Moreover, i f Z(P) = { x E K : P ( x ) = 01

(8)

(l)

Z(P)

=

xo+ Z ( L ) , x0E Z ( P ) .

t r ( x ) = t r ( x ) = x + xq + F

...

n-1

+ xq

,v

XEK.

# d then

G. Menichetti

306

Given an a f f i n c polynomial ( l ) , l e t

A(P)

= G(L):

= A(lo,ll,...,ln-l).

If rank(A(L))

PROPOSITION 5.

...,w

Proof. Let {wo,wl,

r then d i n $ Z ( L )

=

=

n-r.

] b e 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 h e homogeneous l i n e a r s y s t e m

A(L)JI-=

(9)

0,11. =

.. . yn-l)t.

(yo y1

From Lemma 3 , i t f o l l o w s t h a t the v e c t o r s w.= (w. wq -1

1

1

.,. w:

n-1

,...,s ,

)t,i=O,l

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,

Let

(11)


oots in K if and onZy ?'f t h e equalions of t h e Zinear sistems (13) and y= b ' m e compatible.

A(L') -

Proof. If x E K i s acommon r o o t of b o t h P(x) and P'(x) t h e n x

4

O

(xo :x

=

n-1

... x:

) 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 t h e a s s e r t i o n .

Conversely, i f t h e e q u a t i o n s of b o t h systems a r e c o m p a t i b l e , we f i n d , by ( 1 4 ) , a common r o o t f o r t h e given polynomials.[ I t i s easy t o prove t h a t , when t h e c o n d i t i o n of t h e p r e v i o u s p r o p o s i t i o n i s

s a t i s f i e d , t h e 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 s u b v a r i e t y of K whose dimension i s n - r ' ,

(-Gi1:

,-;I

where

(-:

A(L)

r ' = rank

-

A(L)

=

)

b

-I - :. A(L')I b'

rank

--

Now we want t o use t h e p r e v i o u s r e s u l t s t o d i s c u s s t h e e q u a t i o n

m

xq

(15)

- 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 , t h e i n t e g e r s i m + j ,

i

= O,l,

...,k-1,

j = O , l , . . .,d-1,

a r e p a i r w i s e incongruent modulo n.

I n t h i s c a s e , t h e l i n e a r s y s t e m (13) becomes

'+m Y2m+ j

- 'm+j

'( k-1 ) m+ j

-

= bq'

...................

(16)

'i

-

(k-2 )m+j (k-1 )m+ j

=

'+(k-2)m bqJ

=

bq

j + (k-1 ) m

,

j = 0.1,

and t h u s i t s e q u a t i o n s a r e compatible i f and o n l y i f

...,d-1, im j k-1 j+im k-1 C bq = ( C bq )' = 0. i=O

i=O

From t h i s , we deduce t h a t (15) has some r o o t s i n K i f and o n l y i f (17)

k-1 im C bq = t r F , ( b ) = 0,

i =O d where F' = GF(q )

(')

C_

GF(qn)

(2),

The i n t e g e r s h d , h = 0 , 1 , k-1

modulo m and t h e r e f o r e

I: bq i =O

... ,k-1, im

and i m , i = 0,1,

k-1 =

C bq h=O

hd

.

...,k-1,

a r e congruent

Roots of Affine Polynomials m

-

L(x) = xq

309

x implies obviously

d Z ( L ) = GF(q ) , d = ( n , m ) .

(18)

T h e r e f o r e , w e can d e t e r m i n e a r o o t x E K of (15) u s i n g P r o p . 7 and supposing t h a t C

(17) i s s a t i s f i e d . From (16), by s u c c e s s i v e s u b s t i t u t i o n s , we f i n d

i-1 yim+l

j

hm

C bq

= yj + (

, i

)q

1,2

=

,...,k-1,

...,d-1,

j = 0,1,

h=O and by (17) yim+j =

A. 1

k-1

j

hm

C bq

(

)'

h=i

, X.EK,

i = 0,1,

J

...,k-1,

j = O,l,...,d-l

Let u s c o n s i d e r t h e p a r t i c u l a r s o l u t i o n =

'im+j

k-1

-

j

hm

C bq

(

, i

)q

=

O,l,

...,k-1,

j

=

...,d-1.

X = 0, j = 0,1, j From ( 1 4 ) w e o b t a i n

obtained f o r

n-1

x tr(v) 0

h

h

k-ld-1

E 24 vq n-h

=

h=O d- 1

Hence, s e t t i n g v = wq

, we

=

i=O

' j=o

qn-(im+j)

n-(im+j)

"4

'im+j

have

k-1 d-1 n-(im+j) n-(im+j)+d-1 x t r ( w ) = C C z;m+j wq 0 i=O j=O where t r ( w ) = t r ( v ) # 0. I f we o b s e r v e t h a t n-(im+j) = -

zSm+j

k-1 n+(h-i)m k-i-1 rm Cbq = C bq , h=i r=O

t h e n , s u b s t i t u t i n g i n t o t h e p r e v i o u s e q u a l i t y , one h a s

x tr(w) 0

k-1 k-i-1 rm d-1 n-im+(d-1-j) Z C bq C wq 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 = - C Z bq ( C w ' ) ~ i=O r=O s=o =

-

.

From h e r e , p u t t i n g d-1

a =

C

s W

~

s=o

and o b s e r v i n g

k-1 d-1 tr(w) =

c

i=o we deduce

I: wq j=O

im+j

im

k-1 =

c

i=O

a'

= trF,(a),

...,d-1,

0,1,

h=i

.

G.Menichetti

310 k-1 C i=O k = C h=1 k-1

xOtrFI(a) =

-

c

= -

k-i-1 rm n-im C bq aq r=O r m hm h-1 C b q aq

r=O h-1 ~

hm

rm b

q aq

.

h = l r=O T h e r e f o r e : The equation (15) has r o o t s 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 r o o t s i s the a f f i n e subvariety ( 8 ) in which Z ( L ) i s given by (18) and x

=--

k-1 h-1 rm hm C C b q a' trF,(a) h=l r=O

,

t r F l ( a ) # 0.

I f ( k , p ) = 1 ( p = c h a r K ) then t r F l ( l ) = k # 0 and t h e r e f o r e , we can s e t a = l . The p r e v i o u s r e s u l t a l l o w s u s t o determine t h e r o o t s of a second d e g r e e e q u a t i o n i n a f i e l d K of c h a r 2 . I n f a c t , f o r q = 2 , m = 1, w e f i n d t h e w e l l known c o n d i t i o n t r ( b ) = 0 i n o r d e r t h a t t h e e q u a t i o n X'

+ x

t

b = 0

h a s a r o o t in K = GF(2").

Moreover, from (18) and ( 1 9 ) , we deduce t h a t t h e r o o t s

of t h e above e q u a t i o n a r e

n-1 x

C

=--

h-1

C b

2r 2h a

and

xo+ 1 ,

t r ( a ) h=l r=O where a E K i s a f i x e d element w i t h 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 M e n i c h e t t i G . , On a g e n e r a l i z a t i o n of Kantor's l i k e a b l e planes, Geom. D e d i c a t a , 1 7 (1985) 253-277.

[ 3 ] Dickson, L . E . ,

Linear Groups w i t h an e x p o s i t i o n o f t h e Galois fieZd theory

(Teubner, L e i p z i g . R e p r i n t Dover, New York, 1958).

Annals of Discrete Mathematics 30 (1986) 31 1-330 0 Elsevier Science Publishers B.V. (North-Holland)

O n the parameter

n(v,t

31 I

for Steiner t r i p l e systems ( " )

-13)

Salvatore Milici ("") Abstract. L e t D ( v , k l ([l], [ 8 ] ) b e 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 b e c o n s t r u c t e d i n s u c h a way t h a t an3 t w o of t h e m h a v e 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 STS(v), Let t v =vlv-11/6 I n t h i s p a p e r 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 PnrtiaZ T r i p l e System

a finite non-empty set and

(PTS)

(P,P) where

is a collection of

P

2-subset of

called blocks, such that any

P

is

P

3-subset of

,

P

is contained in at

.

P

most one block of

is a pair

Using graph theoretic terminology, we will say that an element

of

x of

P

P

has d e g r e e

if x

d(x) =h

. Clearly

belongs to exactly

. We will

d ( x ) =31PI X E P

of a

PTS

(P,P) the

. If i = I , . .. , s

are the elements of degree

h

,

i

for

,

where

then we will write Two balanced

PTSs

A set of

s

r

1

Ih I

i~

(P,PII if

(DMB)

in a block of

p

Pl

PTSs

+... =h

and

call the d e g r e e - s e t

.

DS = I d ( x ) , d ( y i , . . ]

n-uple

,

where

elements of i , we will write DS =

there are

+ r = IPI S

r

.

If

blocks

h

P

(DSl

..

x,y,.

having

r . = I , for some

i

,

i '

IP,P21

are said d i s j o i n t and m u t u a Z l g

= 0 and a 2-subset o f P is contained 1 2 if and only if it is contained in a block of p2 P n P

(P,P1),iP,P2i,.

. ., ( P , P s l

is said to be a set of

( A )

Lavoro eseguito nell'ambito del GNSAGA (CNR) e con contributo finanziario MPI (1983).

(*$ 1 4 ,

if

if

Let

3

Let

UAli) i=l

Y = P -

i =I

lyl =

.

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

u A(i,{l,2,3})

obtain

RGM

I '

=5

1x1 = 4

.

Further, it follows

YGM

3

. Then

otherwise

326

S. Milid

1x1 = 4 , w e

If

I

=i:lI

and hence If

obtain

2 x b

3 y b

4 6 y

5 6 b

2 a y

3 x a

4 5 x

4 7 a

5 7 y

,

we o b t a i n

. This

=2

and

Y={4,5,61

,

Y={4,5,6,7}

l a b

lA(7,f4,7,a}lnA(a,14,7,a})l IXI=5

,

X={a,b,x,y}

6 7 x

is impossible.

.

X=I7,a,b,z,yl

It

3

follows

Ali)

7 E

,

(i,j , k l C A ( 7 1

otherwise

with

i,j

E {1,2,31

i=l

k ~ 1 4 , 5 , 6 l or

and

i,j

j

E {{1,2,31

If

z E A ( ~ J and

-{ill

z =b

and h e n c e

,

with

' 2

IA(7,{7,i:,yllnA(k,{7,k,y}il

r i l y w e have

i

1x1

#8

. It E

P

,

=2

1x1

y E{7,a,x,y}

R

such t h a t l R n M

with

1x1

#6

i24

.

4

s

i24

for

1x1

Further, it follows 6 E M u M

,

and hence

f 7

,

YEM

and hence

.

3

I

=2.

Y =P-(A(I)U

Lemma 4 . 2 w e o b t a i n

since otherwise

bEA(l)nAI2JnA(a)

n ~ l b , t 3 , 4 , b I ) n ~ ~'2 I

with

,

. From

=6,7,8

.

with

X = A ( l l u A ( 2 ) u A f a l -{1,2,a}

If follows

follows

I

,

E {l,2,31

we o b t a i n

R = { 1 , 2 , a } E PI

{3,4,bI

Then n e c e s s a -

,

IAii,{i,b,y}InAIb,{i,h,y}iI

uA(2)uA(a)l.

.

y E {a,b,z,yl

Now, 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 Let

and h e n c e

k E{1,2,31

with

z # A ( j )

z E {a,bl

and

and

E {4,5,61

, with Y={3,4), 3 IA(3,{3,4,bI) n

since otherwise

{3,4,63

E

IA(3,{3,4,61)nA(4,{3,4,6}1nMi)

PI ( 2

, for

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 . Proof.

There is no

(P;P ,P ,P ,P 1

2

Suppose t h a t t h e r e e x i s t s a

DS= [(5j3,(3j8]

, Let

M

3

={1,2,.,.,8}

3

with

4

DS =

(P;P ,P ,P ,P 1

,

M

5

2

3

4

[ (S13,

I

= { a , b , c } and

(3Jg]

.

with P=M

3

u M

5 .

327

Parameter D(v. t,-13) for Steiner Triple Systems

Since

n =I1 Let

with

ct,B,y

it follows t h a t

{a,b,c} $P,

,

i n3

i=l

(x,ylC_P

,

1

for

x , y E ia,b,cl

.

w i t h o u t l o s s o f g e n e r a l i t y . Then

.

~{4,5,6,7,8}

Since u = B = y = 4

,

(A(i,{1,2,3}j

, f r o m Lemma 2 . 1 we o b t a i n

‘ 3

;

1 2 3

1 l I 2 2 3 4

l a b 1 2 2 3 3

4 a 4 4 b

. Necessarily

{ 1 , 2 , 4 ) E P2

A t f i r s t , suppose

c c b a c

2 a b a 3 4 a

1 1 1 2 2 3

4 3 c b c b c

2 4 3 4 3 4

a b c c b a

...

1 1 1 2 2 3

I

...

2 4 3 4 b a

1 2 b 1 4 3

c a b 3 a c

...

...

l e a 2 3 a 2 4 c 3 b c

...

I

1

...

or

-

ii)

p3 1 2 3

1 2 4

l a b

l u c

1 2 2 3 3

1 2 2 3

4 a 4 4 b

...

c c b a c

3 3 c 4

...

Pan

p4

#@

and

b a b c

...

PI

2 4 b 4 3 a

... ...

a 3 c c b c

1 1 1 2

f0

A

. Then

2 4 3 3

1 2 c

b a c a

1 1 2 2 3

2 4 c 3 4 b

... I *.. I

PI n F 5 # @ and

I n case i ) we h a v e have

1 1 I 2 2 3

PI n a

3 4 4 b b . * * * * *

F3 # 0

(P; P

a b 3 a c

1

. In

case i i ) we

,P 2 ,P 3 ,P 4 )

cannot

exist.

Now s u p p o s e

{I, 2 , 4 }

4P

j

f o r every

j =2,3,4

. Necessarily

S.Milici

328

i) 1 l 1 2 2 3 3

2 a 4 a 4 4 b

3 b c c b a c

p4

pZ

p3

1 2 a 1 4 3

1 2 b 1 4 a

l 2 2 3

1 2 2 3

b 4 3 a

c c b c

...

4 c 4 b

2 b a 3 b c

2 3 4 3

c 0 b 4

...

...

...

...

a , .

3 c a 3

7 1 1 2

. I .

_ I

or I

I

I

ii) 1 l 1 2 2 3 3

2 3 a b 4 a 4 4

e c b a b c

p2

p3

1 Z a 1 4 b 1 3 c

1 2 b 1 3 4

2 4 c

p4

2 4 a 2 3 c 3 a b

1 1 1 2

l e a

2 3 b 3 4 a

... ...

...

2 4 3 3

e a b 4

2 b a 3 a c

... ...

... ...

P n P # @ i n c a s e i ) and

I t follows t h a t

i i ) . Then a

I

(P;P ,Pz,P 1

1

4

3

, P 4)

P l n P 2 # @ in case

c a n n o t e x i s t s and t h e p r o o f i s comple-

te.

Lemma 4.6. or

DS=

1413,(319]

=

RcM

or

1 5 , 4 , ( 3 1 1 0 -I 3

.

.

Suppose t h a t t h e r e e x i s t s a

Pro0 f DS

(P;P ,P ,P ,P I 1 2 3 4

T h e r e i a no

Let

2 . 1 we o b t a i n

{2,2,3}

€PI

I3

UA(i,{1,2,3}) i=l

(XI = 4 + k

and

1x1 # 4 ,

otherwise

(YI = 5

(YI =4

Let

with

1

,P

2’

P ,Pql 3

1

10

with

e v e r y case e x i s t s a b l o c k

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

clearly and

(p;P

. In

DS = [ ( 4 ) 3 , ( 3 ) 9 ]

DS = [5,4, ( 3 1

vith

=

YAMi=@

Y={4,5,6,7}~M 3

for

=4,5

.

3

Let

Y

=P- U A ( i l , i=l

for

(Y(= 5 - k

and

1x1

k =0,1

. Then

l P , l 224

. Since

i=4,5 m=13

.

I t follows t h a t

we h a v e

1x1

=5

. ,

we o b t a i n t h a t ( x , y ) C P 1

Parameter D f v , tv-13)for Steiner Triple Systems

with

.

xJy E Y

Let

2 = {z E

I t follows t h a t

Z nM with

3

IZI =3

and hence

zl f z 2

=a . zl

E

Z

M

,

otherwise

IA(4,{4,5,z

1

~

.M A t~ t h i s

3

={1,2

PI w i t h

{ 4 , 5 , z I l , {6,7,z

2

I5

{4,5,z1))

p o i n t we h a v e

,..., 9 ) ,

E

2

1

E

3'

M

5

.

s , y E }'l

P

1

, with

. Observe

lAfl,{l,zl,y}l nafzl,{l,zlJyII

DS = ( 1 4 )

and hence

{zJy,x}

1 ) nA(zl,

O t h e r w i s e we o b t a i n

= {8,9]EM3

Let

.., 7 1 : 7

P - {1,2,3,.

329

I

that =O

3 nA(i,{lJ2,311 =

=@,

i=l

.

13J9]

4 M

4

and

={a,b,c]

U Mi '

P =

i=3

Then

or

I n c a s e i ) we h a v e ii),

l ~ ( l , { ~ , O , ~ } ) n ~ f u J { ~ ,5u2, ~ , }In l l case

.

l ~ f ~ , { l , 8 , a } ) n A f u J { l J 8 , 0 } 5) l2

( P ; P ,P , P , P ) 1 2 3 4

Then a

can-

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 proof., Lemma 4 . 7 .

Proof.

T h e r e is no

3

.P 4

with

DS=[(3)13]

.

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 f r o m Theorem 2 . 1 o f [ S ] .

Theorem 4 . 1 . Proof.

(P;Pl,P2,P

D(v,t -13) = 3

for every

V

A p p l y i n g Lemmas 4 . 1 , 4 . 2 ,

obtain that a

I F ; PI, P2,.

..,P

)

e x i s t , Then, s i n c e t h e e x i s t e n c e of o f them i n t e r s e c t i n

tu-13

4.4,

4.5,

4.6 and 4.7, we

m=13

and

s > 3

4.3,

with s

.

v z 1 5

STS(vls

blocks (these

,

t -13 V

cannot

s u c h t h a t a n y two blocks occurring,

330

S.Milici

moreover, in each of the (P;PIJP2,...,PSl v L15

every

.

S T S f v l s ) implies the existence of a

, from Theorem 3.1 we obtain D ( v , t V - 1 3 1

=3

for

REFERENCES

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 , Proc. Amer. Math. SOC., 32 (1972), 409-416.

L2-1

J . Doyen and R . M . Wilson, E m b e d d ings of S t e i n e r t r i p l e s y s t e m s , Discrete Math., 5 (1972), 229-239.

[3]

S . Milici and G. Quattrocchi, Some r e s u l t s on t h e maximum numb 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 i n t h e same b l o c k - s e t , preprint.

14.1

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 DMB P T S e o n e l e m e n t i d i g r a d o 2 , Le Mate matiche (to appear). S. Milici and

Z ' e s i s t e n z a di t r e

[5]

G . Quattrocchi, S u l m a s s im o numero d i D M B P T S a v e n t i 1 2 b l o c c h i e i m m e r g i b i l i i n u n STS , Riv. Mat. Univ. Parma (to

appear). 161

G. Quattrocchi, SuZ p a r a m e t r o D ( 1 3 , 1 4 1 di S t e i n e r , Le Matematiche (to appear).

1.71 G . Quattrocchi,

SuZ p a r a m e t r o

D(v,tv-lOl

p e r S i s t e m i d i Terne

,

19 ' v < 33

per

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. 181

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 Discrete Math., 7 (1980), 115-128.

A.

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 i p a r t i m e n t o d i Matematica V i a Campi, 213/B 41 100 MODENA ( ITALY)

.

We g i v e a new s i m p l e c o n s t r u c t i o n o f p a i r s o f d o u b l y d i a g o n a l o r t h o g o n a 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 d o u b l y d i a g o n a l o r t h o g o n a l L a t i n squares o f o r d e r n, DDOLS(n), i s a p a i r o f o r t h o g o n a l L a t i n squares o f o r d e 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 r a n s v e r s a l b o t h on t h e f r o n t d i a g o n a l D1 and on t h e back d i a g o n a l D2 The r e a d e r i s r e f e r r e d t o t h e monograph [I] by J.Denes and A.D.Keedwel1 for t h e d e f i n i t i o n s which a r e n o t g i v e n here. W.D.Wallis and L.Zhu proved t h e The problem was posed by K . H e i n r i c h and e x i s t e n c e o f 4 DDOLS(12) i n [ Z ] A.J.W.Hilton i n [ 3 ] .

.

.

Let

Q

be a L a t i n square o f o r d e r n based on t h e s e t be t r a n s v e r s a l s o f

S, T

t o t h e element o f of

r)

. We

Inoccupying t h e c e l l

In occupying t h e c e l l

In={O,l

form a permutation

(k,i)

(h,i)

,..., n-1)

and l e t

Inas f o l l o w s :

on

o f S we a s s o c i a t e t h e element

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). We denote by Q(S,T) t h e L a t i n square o b t a i n e d 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 element a S a T ( s ) . O b v i o u s l y we have: (a) i f

U

i s a transversal o f Q then

U

i s also a transversal o f

Q(S,T);

( b ) i f R i s a L a t i n square which i s o r t h o g o n a l t o Q t h e n R i s a l s o o r t h o g o n a l t o Q(S,T). Let

Q

be a L a t i n square and l e t h be a symbol; we denote by

o b t a i n e d by r e p l a c i n g each e n t r y

s

THEOREM. For an even p o s i t i v e i n t e g e r k l e t l e t T1, T2 be two common t r a n s v e r s a l s o f A

Q

(h,s).

A, B be a p a i r o f DDOLS(k) and 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 i a g o n a l exists a pair o f

Qh t h e copy o f

o f Q w i t h the ordered p a i r

.

D1

and

D2

DDOLS(3k).

P r o o f . Consider t h e two o r t h o g o n a l L a t i n squares o f o r d e r

3k

, then

there

C. Pellegrino and P. Luncellotti

332

O f course 8 possesses a tranSversa1 on t h e f r o n t d i a g o n a l , w h i l e t h e back diagonal i s a transversal o f B S t a r t i n g f r o m A and B we f o r m t h e f o l l o w i n g L a t i n squares o f o r d e r 3k

.

From ( a ) and ( b ) i t f o l l o w s i m m e d i a t e l y t h a t t h e square

i

on t h e f r o n t d i a g o n a l w h i l e we have :

Aij

( c ) each subsquare having (d)

A”

,D,

T,,T,

and

of

,D,

s t i l l has a t r a n s v e r s a l

has a t r a n s v e r s a l on t h e back d i a g o n a l . I n a d d i t i o n

Bij

and

6

of

i s a doubly d i a g o n a l L a t i n square

as 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 ;

are orthogonal.

Since t h e square i s obtained from subsquares, we have f o r j=1,2, ...,k :

Hj

the set

o f the entries o f the

D,

transversals

, T,

A,,

of

H’. o f t h e e n t r i e s o f t h e J

D, o f

A,,

T, o f

A,,

and

7i by s u i t a b l y renaming symbols i n t h e j - t h column o f

which l i e on t h e

A,, and D, o f A, coincides w h i t t h e s e t ( k t j ) - t h column o f l y i n g on t h e t r a n s v e r s a l s of

a

D, o f

A,,

;

the set

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 j 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,, c o i n c i d e s w h i t t h e s e t

Kj

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

D, o f each

, T,

A,, j=1,2,

of

...,k

A,,

and D, o f

exchange i n

A,,

?i the

’li

elements o f

same row; s i m i l a r l y we exchange t h e elements o f Ki same row o f

A”

(property

d i s t i n c t c e l l s ) : from

A^

(e)

(c) and

l y i n g on t h e t r a n s v e r s a l s

. H

j and

implies t h a t the elemints o f (f)

matrix i s a L a t i n square. F u r t h e r e a s i l y shows.

and

K:

ij

Hj

appearing on

appearing on t h e and

K! occupy

J

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

8

i s d o u b l y d i a g o n a l as t h e c o n s t r u c t i o n

333

Doubly Diagonal Orthogonal Latin Squares Observing t h a t

has p r o p e r t i e s which a r e analogous t o ( e ) and ( f ) we can

exchange elements i n

c

B

as we d i d i n d e r i v i n g

A”

from

A

d o u b l y d i a g o n a l L a t i n square ( d ) . Hence

8

and

8

which i s o r t h o g o n a l t o

are p a i r o f

and t h u s o b t a i n

R

a

because o f p r o p e r t y

DDOLS(3k).

EXAMPLE. S i n c e 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 h y p o t h e s i s o f t h e p r e v i o u s 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). 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 s u p p o r t e d by MPI. REFERENCES J.Denes and A.D.Keedwel1, New York, 1974).

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

W.D.Wallis and L.Zhu, Four p a i r w i s e o r t h o g o n a l d i a g o n a l L a t i n squares o f s i d e 12, U t i l . Math. 21 (1982) 205-207. K . H e i n r i c h and A . J . H i l t o n , Doub1.y d i a g o n a l o r t h o g o n a l L a t i n squares, D i s c r . Math. 46 (1983) 173-182.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 335-338 0 Elsevier Science Publishers B.V. (North-Holland)

335

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 imento d i Ma tema ti ca V i a Campi, 213/B 41 100 MOOENA ( ITALY)

I n t h i s paper we pi-ov t h a t t h e upper bound, g i v e n by Mandeli and Lee and Federer 131 f o r t h e number t o f orthogonal squares, a l s o h o l d s f o r F1(n;xl), F2(n;;x2), ... , Ft(n;ht) x 1 ,ml 1 9 t h e number t o f o r t h o g o n a l F1(n;x1,1,x1,2 F*("12,1 ,A*,2'* *. J ~ , ~ ~ , Ft(n;ht,l,xt,2 L ,A t,mt 1 squares.

,...,

...

1

-

,...

DEFINITIONS AND PRELIMINARY RESULTS

Hedayat and Seiden, i n c o n n e c t i o n w i t h some r e s u l t s b y o t h e r a u t h o r s , 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 b y 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 l u m b e r o f t i m e s i n each row and i n each column. They c a l l such squares frequency-square o r s h o r t l y F-squares. More p r e c i s e l y t h e y g i v e t h e f o l l o w i n g d e f i n i t i o n : DEFINITION 1.

...,am 1 .

Let

L

We say t h a t

write briefly

A

k.':J 1

F =

m a t r i x d e f i n e d on a

i f f o r each

times

Ak

nxn

F-square o f t y p e

,..., A,,,),

F(n;xl,x2

appears p r e c i s e l y

i s an

F

a

b

,..., A),

(n;xl,x2

k=1,2,.

A = Ial,a2,

m-set

..,m

and we

t h e element

ak

of

i n each row and i n each column o f

ik 2 1)

F

.

A ~ = X ~. .== A. = A t h e n m i s determined u n i q u e l y by n and h , m hence we s i m p l y w r i t e F(n;A) Note t h a t an F ( n ; l ) square i s s i m p l y a l a t i n I t i s easy t o prove t h a t an F(n;Al,x 2,...,hm) square of o r d e r n square e x i s t s m

In particular i f

.

.

i f and o n l y i f & X i = n

I n [l; Hedayat and Seiden a l s o e x t e n d t o F-squares t h e c o n c e p t o f o r t h o g o n a l i t y o f l a t i n squares t h r o u g h 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

and

,...,p s )

F2(n;p1,p2

i s orthogonal t o the p a i r each

(ui,v.)

J=1,2,

J

F 2 , and w r i t e of

UxV

Ai

Let

F-square o f t y p e

be a

(n;xi ,1

i s a set o f

i p j , i,j=1,2

on a F1

appears

r

)

square on a

s-set

r-set

U = {u,,u2,.,.,ur}

,...,v S I ,

V = {vl,v2

we say t h a t

1.F2

, i f upon s u p e r i m p o s i t i o n o f

xipj

times, f o r each

i=1,2,

F,

F1 on and f o r

...,r

5

...,s .

OEFINITION 3.

, ,.,Ft

square

,..., t.

t

m.-set,

, A ~,2,.

1

.

i=1,2

"i ,mi )

,...,t.

on t h e s e t

Fi

be on

We say t h a t

F,,F2,

F o r each

Ai

mutually (pairwise) orthogonal

.

i, l e t

F-squares i f

FiLFj

,

C. Pellegrino and N. A . Malara

336

Hedayat, Raghavarao and Seiden proved t h a t the maximal number o f mutually I n [2] orthogonal F(n;x) squares i s (n-1) 2/(m-1) , where m=n/A. I n [3] Mandeli, Lee and Federer proved t h a t the maximal number t o f mutually orthogonal , Ft(n;xt) squares (where f o r each i = l y 2a...,t Fi i s F1(n;;X1), F2(n;;x2), 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

...

2.

ON THE MAXIMAL NUMBER OF MUTUALLY ORTHOGONAL F-SQUARES

I n analogy t o THEOREM. Let

we prove the f o l l o w i n g

[3]

...,h Z a m 2 ) , ... ,

F l ( ~ ; ~ l , l y h l , 2 a . . . a ; x l , m l ), F2(n;;x2,1,h2,2y

Ft(n;;xtal,xt,2,...,A

)

be

t

mutually orthogonal

tamt

i = l a 2 ,...,t Fi i s defined on the mi-set 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

-t5

Ai

F-squares, where f o r each mi and n = xi,j Then t h e

&

.

.

(n-1) 2

1=

Proof. From Fh(n;;xh,lyhh,2y...yx ) we d e f i n e a n2 xmh m a t r i x Mh = [a:ja,], hamh where ah -1 i f the k - t h symbol o f Ah occurs i n the c e l l ( i , j ) ( i a j = l , 2 , ij,kn ) o f Fh and 0 otherwise. L e t M = [MlIM21 . I M t ] . By the property o f the

...,

..

(n-1) 2t 1 and

F-squares, the number o f l i n e a r l y independent rows i n M i s a t most so we o b t a i n

Now, we can w r i t e the product o f t h e transpose o f

M’M

=

L2Jm xm L~ 2 1

L2N2

M with

...

M i n t h i s manner:

2 Jm2xmtLt

:~ I

where

Li =

’

[ U ~ , ~ J( i = l y 2 y , . . a t )

,...,

i s a diagonal m a t r i x o f order

mi w i t h

urlr= i

f o r each r=1,2 mi, Ni = [nk,s] i s a diagonal m a t r i x o f order mi i nr,r = n f o r each r=1,2, ...ami i s a m a t r i x o f s i z e mixm and Jmixmj j ( i ,j=1,2,. ,t) w i t h the element 1 everywhere.

with

..

Maximal Number of Mutually Orthogonal F-Squares

337

Let

Om2xml

*..

L2

- J1

...

Lt

where

i s the matrix o f size

Omixm

j

everywhere.

As

A.

.#O,

-

JmtxmlL1

The e i g e n v a l u e s o f t

t- (mi-1) GT

and

t-1

are

M

.

(iyj=lyZ

1 . l

i s i n v e r t i b l e and t h e m a t r i x

A

1 ,J

as t h e m a t r i x

,..., t )

m.xm.

tn,

Jmtxm2L2

n

and

0

*

*

w i t h t h e element

M'M

*

0

has t h e same rank

Nt

j! .

with respective m u l t i p l i c i t i e s

1

,

Then t

1 +

(mi-1)

= rank(M) = rank(M'M) =

t rank(#) Hence

5 m i n { (n-1)

t

mi

-

t

1=

When

...=

XiYl=Xiy2=

and Federer

[3]

.

1.

i,m.

...

i

(i=1,2,

..., t )

. we have t h e r e s u l t b y Mandeli, Lee

Furthermore, t h e p r e v i o u s 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 t h o g o n a l

A~,,,~),

=A 1

< (n-1) 2

F-squares

Ft(n;;h t,l,~t,2,,.,y~t,mt)

F1(n;;xl ,l,;xl

y2

,...,A,,,,~),

a complete s e t i f

F2(n;;x2,1yx2,2y...y

338

C. Pellegrino and N . A . Malara

where

n=hiYl+xiy2t

...+x. ,mi 1

(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 o r t h o g o n a l F-squares design: a g e n e r a l L z a t i o n o f l a t i n square and o r t h o g o n a 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. W.T.Federer, On t h e c o n s t r u c t i o n o f o r t h o g o n a l Fsquares o f o r d e r n f r o m an o r t h o g o n a 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 . Plann. I n f e r e n c e 5 (1981) 267-272.

131 J.P.Mandoli ,F.C.H.Leey

Annals of Discrete Mathematics 30 (1986) 339-346 0 Elsevier Science Publishers B.V. (North-Holland)

339

CARTESIAN PRODUCTS OF GRAPHS AND THEIR CROSSING NUMBERS Giustina Pica

+

D i p a r t i m e n t o 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 N a p o l i , Naples, I t a l y Tomat P i s a n s k i

++

Oddelek za Matematik0,Univerza v L j u b l j a n i L j u b l j a n a , Yugoslavia A l d o G.S.Ventre

+

I s t i t u t o d i Matematica,Facolta 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 N a p o l i , Naples, I t a l y

Kainen and White have determined e x a c 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 r e p e a t e d C a r t e s i a n p r o d u c t s 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 g e n e r a l i z e d y i e l d i n g e x a c t c r o s s i n g numbers and bounds f o r v a r i o u s f a m i l i e s o f graphs.

INTRODUCTION

I n t h i s paper graph embeddings and i m n e r s i o n s a r e s t u d i e d . I n o r d e r t o keep i t s h o r t we a d o p t s t a n d a r d 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 t h e o r y t h a t can be found, say i n [ 2,3,4,5,13] . U s u a l l y o n l y normal imnersions o f graphs i n t o s u r f a c e s a r e considered, i . e . imnersions i n which no two edges c r o s s 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 a d i a c e n t do n o t c r o s s . We r e q u i r e i n a d d i t i o n t h e i m n e r s i o n t o be a 2 - c e l l immersion which means t h a t t h e complement o f t h e immersed graph i s a d i s j o i n t u n i o n 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 r o m t h e immersed graph i n o r d e 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 s u r f a c e . 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 i m n e r s i o n o r embedding a l l f a c e s 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 t h e r w i s e 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 h e 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 r e p e a t here f o r convenience ( s e e 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 s u r f a c e 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 f a c e s o f t h e embedding i n t o s+l s e t s F1yF2y,..,FS,R i n such a way t h a t t h e boundary o f each s e t Fi, l(i(s, i . e . t h e u n i o n o f boundar i e s of faces b e l o n g i n g t o F i s an even 2 - f a c t o r o f G, i . e . a spanning subgraph i’ o f G c o n s i s t i n g o f c y c l e s o f even l e n g t h s ; f u r t h e r m o r e , 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 a c e s 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 a c e s and may be empty. I f k=s we a r e dea-

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 t h e embedding i s a l s o m i nimal, y i e l d i n g t h e genus o r n o n o r i e n t a b l e genus o f G (depending on t h e o r i e n t a b i l i t y t y p e o f S), see [Ill L e t G I 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 h e 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 t h e v e r t e x s e t s o f G and G w h i c h induces a b i j e c t i o n 1 2 o f a l l s sets o f nonresidual 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 -2K2 i n t o t h e sphere. The o u t e r f a c e i s hexagonal and t h e r e a r e two r e ~ i d u a 1 ~ ' ~ f a c e Ps a. r t ( b ) of F i g u r e 1 shows an (l,O)-embedding of K i n t o t h e p r o j e c t i v e plane. There i s one hexagonal f a c e and t h r e e r e s i d u a l 3 9 3 f a c e s . The two ( 1 ,D)-embeddings agree, which i s shown by an a p p r o p r i a t e numbering o f v e r t i c e s i n b o t h graphs.

(b)

Figure 1 An i m n e r s i o n o f a g r a p h G i n t o a s u r f a c e S i s s a i d t o be an (s,k,c,e)-immersion if 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 a n (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 i n i m a l ) . H i s s a i d t o be a reduced graph o f t h e (s,k,c,e)-immersion o f G. L e t G have an (s,k,c,e) i n t o S ' . We say t h a t t h e two immer-immersion i n t o S and an (s,k,c',e')-immersion s i o n s agree, i f t h e c o r r e s p o n d i n g (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. F i g u r e 2 r e p r e s e n t s p l a n a r (1,0,3,2)-immersion of K , The reduced graph and i t s (l,O)-embedding i s d e p i c t e d o n , F i g u r e l ( a ) . Note 3 y 3 t h a t t h e immersions o f K on F i g u r e s 2 and l ( b 1 agree. 3,3 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 i g u r e 3 shows an (1,1,4,4)-immersion of K i n t o the sphere. I f t h e edges 1-6, 2-7, 3-8, and 4-5 a r e removed a ( 3 , 3 ) - e m f j d d i n g w h i c h 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 r e 3 s u l t s ; see F i g u r e 3(b). P a r t ( c ) o f F i g u r e 3 r e p r e s e n t s t h e well-known genus i n t o t h e t o r u s which i s a (4,4)-embedding. embedding o f K 494

34 I

Cartesiati Products of Graphs 2

1

6

3

4

5 Figure 2

N o t e t h a t i m m e r s i o n s on F i g u r e s 3 ( a ) and 3 ( c ) a g r e e as (1,1,4,4)i m m e r s i o n s , as t h e y have f a c e s 1-2-3-4 a n d 5-6-7-8 i n comnon.

(C)

Figure 3

and (l,l,O,O)-

342

G. Pica, T. Pisanski and A.G.S. Ventre

Example 4. F i g u r e 4 ( a ) r e p r e s e n t s a (2,2,8m - 8,4)-immersion o f t h e C a r t e s i a n product C x C 4 i n t o t h e sphere f o r t h e case m = 3. P a r t s ( b ) and ( c ) o f F i g u r e 2m analogous t o p a r t s ( b ) and ( c ) o f F i g u r e 3. Namely, by removing 4 are 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 l a n a r , r e s 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 d e p i c t e d by Figure 4(b). F i n a l l y , Figure 4(c) represents the f a m i l i a r 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 1,1,8m - 8,4) - and ( 1 , I ,O,O)- immersions (and n o t as (2,2,p,q)immersions f o r any P and 4). R e c e n t l y Beineke and Ringeisen have shown [ I ] t h a t c r ( C x C ) = 2m. T h i s means m 4 t h a t t h e immersion o f F i g u r e 4 ( a ) i s f a r from o p t i m a l .

Cartesian Products of Graphs

343

Figure 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 s u r f a c e s t h e f o l l o w i n g two c o m b i n a t o r i a l i n v a r i a n t s a r e handy. d (G) = q k Jk(G) = q

-

g(p g(p

-

2(1 - k))/(g 2 + k)/(g

-

-

2)

2)

Here p denotes t h e number o f v e r t i c e s , q denotes t h e number o f edges, and g denot e s t h e g i r t h o f G.In b o t h cases k i s a n o n n e g a t i v e 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 t h e ( o r i e n t a b l e ) genus and i n t h e second case t h e n o n o r i e n t a b l e genus o f some s u r f a c e . 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 t h e y r e f e r t o t h e graph and t h e s u r f a c e , 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 s u r f a c e i s i n v o l ved. They were i n t r o d u c e d b y Kainen. The o r i e n t a b l e v e r s i o n was i n t r o d u c e d i n [ 6 ] w h i l e t h e n o n o r i e n t a b l e one was d e f i n e d i n [ 8 ] and l a t e r used by Kainen and White [ 9 1 E u l e r d e f i c i e n c y t e l l s us t h e number o f s u p e r f l u o u s edges which o b s t r u c t t h e embedding o f a graph i n t o t h e s u r f a c e . The P o l l c w i n g lemma shows how E u l e r def i c i e n c i e s serve as l o w e r bounds f o r c r o s s i n g numbers c r ( G ) and Cr (G). k k Lemma 5.

.

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 n o n o r i e n t a b l e case i s e s s e n t i a l l y 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 t h e same; see a l s o [8,9] need t h e f o l l o w i n g lemna.

.

Lemma 6, 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 a d m i t s a n o r i e n t a b l e (s,s,c,e)-immersion, t h e n c r (G), 0

PG(n,q) n-1

I

,

n 23

,

= (rR-2Rtl)/(r-l)

= r-2

and

a

2

+ (r-1) /(R-r)

k > , r , so

by (2.6) it is

9 is a projective plane of order

a Galois space

.

whence we have

R-s):

k - 1 2 ER(r-2)+l]/(R-r)

(2.6)

+ VR2 0

s(R+1)]

v = (k-l)r+l , whence we obtain:

(see also (2.5) and dividing it by

in

.

by (2.3) we have: 3

(2.4)

Being

, ,

t

q = k-1

is a

k

=

r

,

and every plane JG

.

It follows that (S,zrzq.% is n-1 (q t1)-cap in PG(n,q) (being

n-2

= R-s+l =

qi + 1 = q

qi -

i=O

n-1

+ 1). It is known (see [5])

that in

i=O

n-1 PG(n,q) , n > 4 , ( q +l)-caps don't exist, whence 2 (q tl)-cap in PG(3,q) So Theorem 1 i s proved.

n

=

3

and

is a

.

3.

-

Let

CAPS

H

IN

be a

(S,z.q)

h-cap in a planar space

n E 9 , n = IJGAH(L2

,

$7 n H

is a

n-arc in

( S , y , p ) satisfying (1.21. Given

. We

easily prove that:

351

Ovoids and Caps in Planar Spaces n s r + 1 ,

(3.1)

H

.

P,Q E H , P # Q

Let

a

n = r+l

(3.2)

in at most

H

has no tangent lines

By (3.1) each of the and

h < R + 1

,

(3

.

odd

planes through

s

P

points different from

r-1

r

3

, whence h

=

meet

PQ

I t 1 1 1 (r-l)s+2,

that is (see (1.5)): (3.3) the equality holding iff:

=

~ 9I , JdnHI22

J 7 E

,

I n n H I = r+l

so that, by (3.2): h = R+l

(3.4) @

H

0

has no tangent lines

H

F)

is of type ( O , r + l ) with respect to planes

Let us now suppose:

we remark rha-c, by ( 3 . 2 ) , (3.5) is fulfilled, if r

P f Q , by (3.5) and (3.11, each of the at most

points different from

r-2

P

s

is even. F o r any

planes through

and

(3

h l R-s+l

,

, whence

h

meets

PQ =

P,QEH , H

in

IH( S ( r - 2 ) s + 2

,

that is (see (1.5)): (3.6)

the equality holds iff

u n ~ @ [nnH122 , a

lnnql = r

,

that is: (3.7)

h

=

R-s+l e H

If (3.5) holds and a

=

s

,

there are just s

through

P , not in

fi

tangents to

H

z p , in a line. So

is an ovoid, by ( Z . l ) ,

h So Theorem 2 is proved.

=

/ H I = R-s+l

R-stl

F)

through

Jd

PQ

,J 7 nH

is

P , A s the number of planes through PO at

tangents by (3.7) is a subspace z

of such

=

s

with respect to planes.

, for any plane

R-s+l

r-arc having only one tangent at

is

H

h

is of class [O,l,r]

H

P H

. The

P of

( S a

set theoretical union meeting every plane

is an ovoid. Conversely, if

and (3.5) holds (see sect. Z ) , i.e.:

is an ovoid

.

G. Tallini

352 4. -

SETS

Let

OF

(0,n)

TYPE

s a t i s f y i n g (1.2). Let

S'

r'

be

(lS'l-l)/(k-l)

,

P

e n t from

t h e number S'

If w e s e t

( S S

lines

through

through

P

meets

SinH

JslnHI =

rl(n-1)

S' = S

,

or

VJGEY,

(4.3) is of t y p e

1

=

, H

(O,r(n-l)+l)

1nnHI

,

r e d u c e s t o a p o i n t , so i t is n o t

that

n = k-1

if

prime

n E pis

every

points differ-

of

(S,Y)

, H

.A

,

1

,

r(n-l)tl

n = k

. Therefore:

.

is t h e complement o f a s u b s p a c e

meeting any l i n e n o t b e lo n g in g t o

subspace

If

.

1

lo

=

2Snlk-1

(S,Y)

.

S'

with respect t o planes.

(4.4) We remark

+

+

( H I = R(n-1)

H

n-1

in

and

S ' = J'C r e s p e c t i v e l y i n (4.11, w e h a v e :

(4.2)

whence

a point in

0

S'nH f

such t h a t

of

(S,y,,fl

whence:

(4.1)

n

(s,2',.9)

IN

LINES

be a subspace o f

P E S ' n H , every l i n e i n

If

TO

RESPECT

be a s e t of type ( 0 , n ) with r e s p e c t t o l i n e s i n a planar space

H

=

WITH

i n a point.

S'

(S,y,.!?)

planar space

S'

in

We d e n o t e s u c h a proper , i f

is c a l l e d

t h e l i n e a r c l o s u r e o f w h a t e v e r 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,.!?)

type

exists,

(O,k-l)l

then

i s t h e complement o f a p r i m e

(S,2?,9)

in

=

PG(d,q)

.

H

PG(d,q)

of

&

H

I t follows 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 p r i m e . Proof point. the

in

S' Let

line 3t

= S-H

is a s u b s p a c e o f

P,Q€S'

PQ

meets

is p r o j e c t i v e ,

,

T E H

and

being

i.e.

. The

plane

proper.

(S,2',9)

PQ = S'nn

meeting any l i n e n o t i n

(S,a

Each o f

i n a point, so

(S,y,Y, =

PG(d,q)

joining

J'C

,

r = k with

the

S'

in a

meets

S'

in

l i n e s through

T

P,Q,T

r

and e v e r y p l a n e i n q = k-1

.

(S,zfl

Thus t h e t h e o r e m

is p r o v e d . Let

n

b e a n i n t e g e r s a t i s f y i n g ( 4 . 4 ) and

(see (4.3))

/nnHI

= r(n-l)+l

.

If

3t

P € n -H

E 9 such t h a t

,

3t

n H # 0 , i.e.

t h e number o f l i n e s t h r o i i g h

P

353

Ovoids and Caps in Planar Spaces in 3c ,

n-secant of 3cnH

m = (r-l)/n

hence

must

number of planes through (4.2),(1.5) and being

therefore It is

r-m

is

be an integer. If

1

meeting

s-1

n < r - m I r-1

,

whence

k = r

In fact, if

r-m r

n = r-1 , but by (4.41,

so that every plane is projective. Then and

H , the

v = /Hl/(r(n-l)+l) , that is (by

(being m = (r-l)/n

the equalities hold iff

,

H

e

.

n+l > r ; by (4.4), k 2 n + l

have

u = (3cnHI/n , so that:

r-1 = mn) we have:

divides

.

r-m>n

,

is a set of type

.

So it is

r>k2n+l

,

(S,Y,YJ is a Galois

(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 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 Technion. Israel Inst. Tech, Sci. Publ. 6 (1954/5), 58-63.

Is]

G. Tallini,

Sulle

Q plans isomorphes,

d

points,

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.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 355-362 0 Elsevier Science Publishers B.V. (North-Holland)

355

(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 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 o f ( k , n ; f ) - a r c s a r e d i s c u s s e d i n 52 a n d 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 o f 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 . E.

1 . I n t h e p a p e r s of B a r n a b e i [ 2 ] and d ' A g o s t i n i [ 4 ] [ 5 ] a n a c c o u n t h a s been g i v e n of some r e s u l t s c o n c e r n i n g w e i g h t e d ( 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 a r e concerned w i t h e x t e n d i n g 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 w e i g h t e d ( 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 p a p e 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 w e i g h t e d a r c 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 a r e c o n c e r n e d w i t h a s e t K of k > 0 p o i n t s i n P G ( 2 , q ) t o e a c h 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 w e i g h t and such t h a t t h e t o t a l w e i g h t of t h e p o i n t s on any l i n e d o e s n o t exceed a g i v e n n a t u r a l number n , i . e . f o r e a c h l i n e R of PG(2,q) w e have C f (P) 5 n PER

A l i n e h a v i n g t o t a l w e i g h t i is c a l l e d a n i - s e c a n t o f K . Points 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 w e i g h t z e r o . F o l l o w i n g 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 o f w e i g h t j f o r j 3

=

O,l,...,w.

F o l l o w i n g t h e n o t a t i o n and t e r m i n o l o g y o r i g i n a t i n g i n [ I 3 1 l e t t i d e n o t e 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 call t h e t i t h e c h a r a c t e r s o f K a n d , i f e x a c t l y u of them a r e non-zero w e say t h a t the a r c K has u characters. I f t h e v a l u e s of i f o r which t i i s non-zero a r e m l < < n t h e n 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 s u b s e q u e n t l y 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 i v e n t o ( k , n ) a r c s h a v i n g e x a c t l y two c h a r a c t e r s . In p a r t i c u l a r t h e connection between such a r c s and H e r m i t i a n c u r v e s h a s been e x p l o r e d 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 d i m e n s i o n s d i s c u s s e d i n [ 1 5 ] . A good bibliography i s included i n [16].

...

B.J. Wilson

350 I t i s proved i n [ 5 ] t h a t f o r a 0 < m < n , it i s n e c e s s a r y t h a t

( k , n ; f ) - a r c K of t y p e ( m , n ) , when

w S n - m

(i)

and q 5 0 mod (n - m)

.

(ii)

Let W = C f(P) = C f ( P ) , and t h e n i t may e a s i l y b e s e e n [51 PEK P E E (2,q)

that m(qt 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 f o r 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 I n order t o have a n a r c have g = 2h and t h e n ( i )r e q u i r e s t h a t w 5 2 . which i s n o t s i m p l y a ( k , n ) - a r c w e t h u s must have w = 2 so t h a t (iii) gives

( n - 2) ( q +1 ) 5 W 2 (n- 2) ( q +1 ) + 2.

(iv)

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 a r c s have p o i n t s I t may e a s i l y be shown t h a t

having p o s s i b l e w e i g h t s 0 , l and 2; w i t h t h i s i n mind w e c a n 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 a r e g e n e r a l i s a t i o n s of theorems i n [ 5 ] and which c a n 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) w i t h n > m > 0 of minimal weight FI=m(q+ 1 ) having some p o i n t s of weight w = n - m , some p o i n t s of weight a for exactly one value of a s a t i s f y i n g both 1 2 u S w - 1 and ( w , d = I and a t l e a s t one point of weight 0. l a ) Suppose there i s exactly one p o i n t of weight 0 . Then a = w - 1 and the p o i n t s 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 i n g l e points of weight 0. ( b ) Suppose that 1 , > 1 . Then K c o n s i s t s of a q / w c o l l i n e a r p o i n t s each of weight w and the q 2 p o i n t s not co2lincar w i t h them, each of weight a. Further, n=aq+w. THEOREM 2.

Let K be a (k,n;f)-arc of type h , n l w i t h n > m > 0 of maximal weight W = ( n - w ) l q + 1 ) + W having some p o i n t s of weight 0 , some p o i n t s of weight a. f o r exactly one value of a s a t i s f y i n g both 1 s a ~ w 1- and ( w , d = 1 and a t l e a s t one point of weight w . l a ) Suppose there i s exactly one point of weight w. Then a = 1 and the p o i n t s of weight 0 form a I w q + w - q - l,w)-arc of which the ( w - l ) - s e c a n t s are concurrent i n t h e single point of weight w. Suppose k w > l . Then K c o n s i s t s of a q / w + 1 c o l l i n e a r p o i n t s each of weight Ibl w and t h e q2 poznts not c o l t i n e a r w i t h 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 w i t h 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 a r c 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 a r e 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 n u c l e u s of t h e a r c . 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 gives a (n,,q+n,-q,n,)-arc, known as a maximal

357

( k *ri;f)-Arcs und Cups in Finite Projective Spaces

(k,n,)-arc. 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 n o s u c h a r c s c a n b e 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 o f i n t e r e s t o n l y if n-m>3. 2 . I t f o l l o w s 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 need t o c o n s i d e r f u r t h e r i n a d i s c u s s i o n of maximal and minimal (k,n;f)-arcs the cases

and

R, > 0;

II, > 0; 1, > 0;

In particular we discuss (v).

(1,

(vi)

> 0.

I n t h a t case

( n - 3 ) ( q +1 ) 5 W 5 ( n - 2 ) q t n . F o l l o w i n g t h e n o t a t i o n of

(vii)

[ 5 ] w e u s e t h e symbol v!

t o denote the

number of i - s e c a n t s which p a s s t h r o u g h a p o i n t of w e i g h t j . t h e arguments i n [ 5 ] t h e v a l u e s of v:

and

vA-~,

j = 0,1,2

Using

are fixed

i n d e p e n d e n t l y of t h e p o i n t u n d e r c o n s i d e r a t i o n . F o 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 vo = q + l n-3

vo = 0 n (viii)

I t f o l l o w s i m m e d i a t e l y t h a t no p o i n t o f w e i g h t 0 l i e s 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

Solving ( i x ) gives

tn = ( n - 3 ) q / 3 tn-3

= (3q2+6q-nq+3)/3

Now l e t a b e a n - s e c a n t on which t h e r e are no p o i n t s of w e i g h t 0 so w e suppose t h a t on a a r e a p o i n t s o f w e i g h t 1 and 6 p o i n t s of w e i g h t 2. Counting p o i n t s o f u and w e i g h t s o f p o i n t s on a g i v e s a + B = q + l (xi) c c + B = n solving (xi) gives a = 2 ( q + 1) - n = n- (q+l)

(xii)

B.J . Wilson

358

C o u n t i n g i n c i d e n c e s between p o i n t s of w e i g h t 2 and n - s e c a n 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

(xiii)

S i m i l a r l y c o u n t i n g i n c i d e n c e s between p o i n t s of w e i g h t 1 and n-secants gives R,vk =

tnCl

whence, u s i n g ( v i i i ) , ( x ) and ( x i i ) w e have

L1

=

( n - 3) ( 2 q + 2 - n )

(xiv)

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 2q2+ (11-3n)q+n2-6n+11-2Ro= 0 It i s t h u s n e c e s s a r y t h a t (n-q),

-

(xv)

(48-16R0)

s h o u l d be a s q u a r e . W h i l s t it i s n o t t h e o n l y c a s e f o r i n v e s t i g a t i o n a n o b v i o u s 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 ( x v ) a r e t h e n n = 2 q + 1 and n = q + 5. By c o u n t i n g t h e p o i n t s of a n ( 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 w e i g h t 0 i t may e a s i l y be s e e n t h a t f o r n = 2 q + 1 i t i s i m p o s s i b l e f o r t h e t h r e e p o i n t s of w e i g h t 0 t o be c o l l i n e a r . A s i m p l e example may be found i n P G ( 2 , 3 ) . Assign t h e weight 0 t o and ( O , O , I ) , t h e w e i g h t 1 t o t h e p o i n t s t h e p o i n t s ~1,0,0),(0,1,0) ~ 2 , 1 , 1 ~ , ~ 1 , 2 , 1 ) , ( 1 , 1 , and 2 ) ( l , l , l ) and t h e w e i g h t 2 t o a l l o t h e r points. T h 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 w i t h 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 - s e c a n 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 a r c would, i f q > 3, v i o l a t e t h e L u n e l l i - S c e 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 . However i t was shown by H i l l and Mason [ 8 ] t h a t c o u n t e r e x a m p l e s t o t h i s c o n j e c t u r e c a n be found f o r a n 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 w e i g h t 0 a r e c o l l i n e a r t h e n i t may be shown t h a t t h e p o i n t s of w e i g h t 2 form a c o m p l e t e ( 2 q + 3 , 4 ) a r c of t h e t y p e ( 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 P G ( 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 o v e 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 :

359

Ik,n;fl-Arcs and Cops in Finite Projective Spaces 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 ) h a v e w e i g h t 0 and t h e t w e n t y two p o i n t s

Q, ( 0 , 0 , 1 ) Q l ( O t 1iO) Q, ( 0 , 1 ,a:)

Q3(0ilra 1 (1, a l l 1 ( 1 ,a,a)7 (l,a,a 1

(1 , a 2, a 3 ) ( 1 ,a;,a;) (1,a la 1 (1 ( 1 ,a 3 , a 4 1 (1,a3

(1,a5,a1

(1,aSIa:) ( ~ , a ~ ~ a ( 1 , a 6 ,1 ) ( 1 , a 6 ,a) ( 1 ,a67,a7) (1,a r l ) ( 1 ,a7,a) (I ,a7,a7)

have w e i g h t 2 . The r e m a i n i n g p o i n t s of P G ( 2 , 3 ) a r e a s s i g n e d w e i g h t 1 g i v i n g a n ( 8 8 , 1 4 ; f ) - a r c K O of t y p e ( 1 1 , 1 4 ) w i t h t h e p o i n t s of w e i g h t 2 f o r m i n g a c o m p l e t e ( 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 w e i g h t 0 . On v a r e 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 Qa,QlrQ2,Q3, w i t h c o o r d i n a t e s a s i n d i c a t e d a b o v e , of w e i g h 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 w e i g h t 1 which d o 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 t h r o u g h Q , and t h e n i n e l i n e s PiQ, a r e 1 1 - s e c a n t s meeting 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 . This configuration i s , i n P G ( 2 , 3 2 ) , d e t e r m i n e d c o m p l e t e l y by f i v e of t h e p o i n t s *.-

rQ,.

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 w e i g h t 1 form a $-arc, t h i s b e i n g t h e i r r e d u c i b l e c o n i c

xi

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 e a c h of which t h e r e i s a u n i q u e 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 a r e n o t on C may t h e n b e 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 , t h r o u g h e a c h o f 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 , t h r o u g h e a c h of which t h e r e a r e n o t t a n g e n t s t o C . A s s i g n i n g w e i g h t 0 t o e a c h i n t e r i o r p o i n t , w e i g h t 1 t o e a c h p o i n t of C and w e i g h t 2 t o e a c h 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 t y p e ( q + 1 , 2 q + 1 ) . By a s s i g n i n g d i f f e r e n t w e i g h 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 b e o b t a i n e d . F o r example a s s i g n i n g t h e w e i g h t 0 t o e a c h i n t e r i o r p o i n t o f C , t h e w e i g h t 1 t o e a c h e x t e r i o r p o i n t of C and t h e w e i g h t ( q + 1 ) / 2 t o e a c h p o i n t of c a maximal ( ( q 2 + 3 q t 2 ) / 2 , ( 3 q + 1 ) / 2 ; f ) - a r c of t y p e (q t 1,2q + 1 ) is obtained.

-

A s e c o n d i n f i n i t e c l a s s of

( k , n ; f ) - a r c s i s s u g g e s t e d 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 o f w e i g h t 0 i n t h e ( 1 0 , 7 ; f ) - a r c of 12 a s a s u b p l a n e . G e n e r a l l y l e t n o b e a s u b p l a n e 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 d o n o t c o n t a i n a n y p o i n t of n o . A s s i g n w e i g h t 0 t o p o i n t s of n o , w e i g h t 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 w e i g h t v t o t h e r e m a i n i n g p o i n t s where u / v = ( q - q:) / ( q q , q i ) i n i t s l o w e s t terms Then t h e r e i s formed a minimal ( ( q- 9 0 1 ( g + 9, + 1 ) I + 9,+ l ) u + ( q - q i - q , ) v ; f ) - a r c

-

-

(4

of t y p e ( ( q + q o ) u , ( q : + q o l+) u + ( q - q i - q , ) v ) .

B.J. Wilson

3 60

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. This proof required results listed by 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. However it may be shown using 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. 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)-archidi 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. [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?1 : ) , Rend. Mat. (Roma) Serie VII, 2 (19821, 371-376.

36 1

( k , n;fl-Arcs and Caps in Finite Projective Spaces L u n e l l i , L. a n d S c e , M . , Considerazione arithmetiche 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 ; n l q - a r c h i , 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 G a l o i s 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 c o n 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 c o n due c a r a t t e 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 a n o g r a f i c o f i n i t o c o n 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 c o n 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 Rend. Mat. e Appl. 26 (19671, 273-303. h e r m i t i a n e d i un S r , q . T a l l i n i S c a f a t i , M . , G r a p h i c C u r v e s on a Galois p l a n e , A t t i d e l convegno d i Geometria C o m b i n a t o r i a 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 - i n s i e m i 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 t w o - c h a r a c t e r k - s e t s a f f i n e s p a c e A G ( r , q ) , J . Geometry 22 (19841, 75-82.

-

i n an

T h a s , 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 . C o m b i n a t o r i a l Theory A 19 (19751, 228-232.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 30 (1986) 363-372 0 Elsevier Science Publishers B.V. (North-Holland)

363

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)-mtrixand 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)-mtricesand of certain corresponding carbinatorial structures. In particular, % is proved that a circulant (O,l)-mtrixC 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 the cyclic group of order n. n' 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 denote the transpose of C. T

... , cn-13

I. L e t c be a circulant matrix. If [co, cl, it follows [2] that the eigenvalues of c are n-1 x = c c,bJjr

r

j=o

is the first nm of C,

(1)

3 2ai where 0 5 r 5 n-1 and w = exp( 1.

n

Consider the circulant matrix A

=

C P. The first row of A is obtained frcm the

N . Zagaglia Salvi

364

first row of C by shifting it cyclically one position to the right. n-1 where 06 r 3 be a f i n i t e s e t a n d B w i t h I B I = b a s e t o f 3 - s u b s e t s of V. The e l e m e n t s of V a r e c a l l e d 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 or blocks.

I f any 2-subset o f V i s contained i n e x a c t l y one

l i n e , then t h e incidence s t r u c t u r e (V,B,E)

i s called a S t e i n e r

t r i p l e s y s t e m 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 S T S ( v ) 1 1 l i e s o n e x a c t l y r = ~ ( v - 1 )l i n e s a n d t h e r e a r e e x a c t l y b = - v ( v - I ) 6 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 i s n e c e s s a r y and s u f f i 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 s e 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 S T S ( 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 S T S ( 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 . = 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

The complement

a s u b s y s t e m S T S ( r ) a n d , v i c e v e r s a , t h e complement of a s u b s y s t e m 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 a n y v = 7 o r 1 5 + 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 l e a s t o n e h y p e r o v a l

[21,

14I

[7]. The s e t 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

o r 1 3 + 12n, n

E

INo.

are v = 9

,

374

H. Zeitler

1 . OVALS

1 . 1 DEFINITIONS Any non empty s u b s e t 0 c V i n a S T S ( v ) w i l l b e c a l l e d a n o v a l i f f there e x i s t s exactly one tangent a t each point of 0 ( t h i s tangent

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 ) a n d e a c h l i n e of S T S ( v ) h a s a t most t w o p o i n t s i n common w i t h 0 . Somet i m e s 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 a s 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 be c a l l e d t h e on-points,

t h e points on t h e tangents to 0 with

t h e points of contact deleted w i l l be c a l l e d t h e ex-points

and a l l

t h e remaining p o i n t s t h e in-points. 1 . 2 ENUMERATION THEOREMS 1 . 2 . 1 CLASSES OF POINTS

Each o v a l has e x a c t l y r = -1( v - l ) p o i n t s and t h e c o r r e s p o n d i n g 2 1 ) oints. counter oval exactly r + 1 = ~ ( v + l p Proof:

Each p o i n t o n t h e o v a l l i e s o n r l i n e s , a n d p r e c i s e l y o n e

of them i s a t a n g e n t . 1 . 2 . 2 CLASSES O F LINES

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 ( v - 1 ) t a n g e n t s , 1 1 exactly = 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) 6 24 external l i n e s .

(;)

Proof:

An e a s y c o u n t i n g a r g u m e n 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 o v a l - 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

follows.

1 . 4 THEOREM

For a n y v

E

HSTS t h e r e e x i s t s a n S T S ( v ) w i t h a t l e a s t o n e o v a l .

375

Ovals in Steincr Triple Systems

Proof:

When v E HSTS t h e r e e x i s t s a n S T S ( v ) w i t h a t l e a 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 o v a l . T h e s e o v a l s

are s p e c i a l ones, because 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 b e c a l l e d k n o t o v a l s . I f through each ex-point

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 ,

then t h e oval w i l l be c a l l e d a regular oval. 1 . 6 THEOREM

In a n S T S ( 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 a n S T S ( 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 INo,

a n d r i s o d d . 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 S T S ( 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 o v a l 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 contradiction. 1 . 7 THEOREM

For any r e g u l a r o v a L 0 i n at? S T S ( v ) t h e r e e x i s t s e x a c t l y o n e i n point. Proof:

L e t e be t h e number o f e x - p o i n t s a n d 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 a n d i n - p o i n t s t o g e t h e r a r e t h e p o i n t s o n the counter oval

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 i e s on e x a c t l y two t a n g e n t s a n d 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 are e x a c t l y t w o ex-points.

Consequently, e = r and i = 1 .

1 . 8 THEOREM

For a r e g u l a r o v a l t h e n u m b e r s 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 t h r o u g h 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 :

H. Zeitler

316

secants

tangents

in-point

1 1 ? ( r - 2 ) = T(v-5) 1 1 =p - 1 )

0

on- po i n t

r - 1 = ?1( v - 3 )

1

ex- po i n t

2

Tr

external lines 1 1 ~(r-2= ) T(v-5) 1 1 = x(v-1)

zr

0

i

The proof f o l l o w s immediately from t h e p r e c e d i n g 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 o n e r e g u l a r

oval. 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 k n o t 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 r e g a r d i n g 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: F o r 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 l e a s t one o v a l . I n [ 2 ] t h i s theorem i s proved by c o n t r a d i c t i o n . W e 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 a c c o u n t 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 e a s t one r e g u l a r o v a l . 2 . 1 THE POINTS

1 With v E RSTS, 101 = r = -2( v - l ) = 4 o r 6 + 6n, n E INo, and 1 161 = r + 1 = z ( v + l ) = 5 o r 7 + 6n, n E INo. T h e r e f o r e , t h e number o f p o i n t s o n t h e o v a l 0 is e v e n and t h e number o f p o i n t s o n 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 e d g e s

have d i f f e r e n t l e n g t h s . F u r t h e r m o r e , t h e r-gons a r e r o t a t e d w i t h

. The

...

v e r t i c e s Po,P, , ,Pr-, of 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 e x - p o i n 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 c o u n t t h e v e r t i c e s of each polygon c o u n t e r c l o c k w i s e . F i g u r e 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 . r e s p e c t t o e a c h o t h e r by a n a n g l e

f

t h e i n n e r polygon a r e t a k e n a s o n - p o i n t s ,

Ovals in Steiner Triple Systems

377

0

M

Figure 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 p a p e r w e r e s t r i c t 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 o f 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 h a v i n g no c o n n e c t i o n with an oval. L e t a r e g u l a r r-gon be g i v e n w i t h 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 l a s s 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 p o l y g o n . The two end-

H , Zeitler

378

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-subset o f t h e f i r s t class. ( b ) Second c l a s s o f s u b s e t s L e t i , j w i t h i <j b e t w o v e r t i c e s o f t h e g i v e n p o l y g o n . Then t h e

minimum o f j

- i,

r

+i-

j

is c a l l e d t h e " d i f f e r e n c e " of t h e t w o ver-

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 a r e p o s s i b l e : 1 1 1 1 , 2 , . . . , p - 1 = a ( " - 5 ) . (The d i f f e r e n c e ?r i s a l r e a d y e l i m i n a t e d w i t h t h e f i r s t c l a s s ! ) 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 , d 2 . d 3 ) i n s u c h a way t h a t e a c h o f t h e m e n t i o n e d d i f f e r e n c e s occ u r s a t m o s t o n c e 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 of t h i s kind are p o s s i b l e ? I n [ I ] ,

[21,

[31,

q u e s t i o n s a r e a n s w e r e d . Here w e g i v e t h e r e s u l t o n l y

[ 6 1 these

-

w i t h o u t any

proof.

r = 4 t 12n, n E IN (1,

5 n + l , 5n+21 , ( 2 ,

3n,

3n+2),

(31

5n,

3n-1,

3n+3),

(2n-1,

4n+2, 6 n + l )

5n+3), (4,

,

(2n, 2n+l, 4 n + l ) .

1 The d i f f e r e n c e 3 n + 1 = -r i s missing, any o t h e r d i f f e r e n c e o c c u r s 4 e x a c t l y once a s r e q u i r e d . A l t o g e t h e r w e h a v e 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 A G ( 2 , 3 ) . I t i s e a s y t o show t h a t i n t h i s system 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 s e t o f p o i n t s (x,y) with x,y

E

GF(3) and x 2 - y 2 = 1 ) . I n t h e 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 ex-points

QoIQ1f...fQ,-l

and M i s t h e i n - p o i n t o f a n o v a l 0 . Next, t h e e x t e r n a l l i n e s t o 0 a r e 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 ) a n d (Q,x) i n s t e a d o f Px a n d 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 l a s s d e t e r m i n e d by t h e p o l y g o n d i a m e t e r s

Ovals in Steiner Triple Systems

319

a r e 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 with x t

2 t h e number x modulo r .

{O, 1,.

. ., r - I

1. W e a l w a y s u n d e r s t a n d

( b ) The r e m a i n i n g e x t e r n a l l i n e s

( 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 x t {0,1,

..., r - I }

1 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 324 s u b s e t s o f p o i n t s . These l i n e s are e x t e r n a l l i n e s o f 0,too. Together 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 h a v e 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

is t h e set of a l l e x t e r n a l l i n e s . In t h e difference t r i p l e s 2 . 2 the difference 3n+ 1 is missing. 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+3n+l)) w i t h

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 connection with

x E {O,I,...,r-l} the tangents. 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 contair.inq t h e ex-point

-

f o r r = 16.

B

0

*

0

0

0

.,

M

, , , , , , , ,.

....0............... .........-0

----_ --- -- -------z

-0

01,-

o--

-0

0-

-

--+

-Q

-Q

o Q Figure 2

D

A

Q

H. Zeitler

3 80

Secants containing a n ex-point

( a ) First c l a s s o f p o l y g o n c h o r d s W e s t a r t w i t h t h e p o l y g o n v e r t i c e s A and B ( f r o m B t o A c l o c k w i s e ! ) 1 having t h e d i f f e r e n c e Tr. I n t h e set of polygon chords o r t h o g o n a l 1 t o t h e diameter through B we choose t h o s e T ( r - 4 ) c h o r d s which a r e 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 p o l y g o n c h o r d o f t h i s k i n d i s a p o l y q o n d i a m e t e r CA. The e n d p o i n t s o f t h e p o l y g o n c h o r d s

c h o o s e n i n t h i s way h a v e t h e d i f f e r e n c e s 2 , 4 , .

. .,-!-r-2, 2

respectively.

( b ) Second c l a s s o f p o l y g o n c h o r d s The p o l y g o n 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 . 1 I n t h e s e t o f p o l y g o n c h o r d s p a r a l l e l t o CD w e c h o o s e t h o s e -r 4 c h o r d s which a r e 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 1 t h e s e p o l y g o n c h o r d s h a v e d i f f e r e n c e s l 1 3 ~ . . . r ~ r - rle~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 e d g e o f t h e p o l y g o n . The e x - p o i n t

lying

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 p o l y g o n c h o r d , b o t h i n ( a ) a n d i n ( b ) , t o g e t h -

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-2) -4( r - 4 ) + s e c a n t s c o n t a i n i n g Q. By 1 . 8 w e h a v e f o u n d a l l s e c a n t s of t h i s kind.

ar

T h e r e 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 a s

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 1 and B i s e x a c t l y -r. 4 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 r o u g h a n y ex-point

(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 e x point (Qrx):

1 1 ( Q l x ) , ( P , x + ~ r - l ) , ( P r x 2+ - r + l ) } .

Furthermore,

1 (A,x) = ( P r x + , r ) ,

1 3 ( B , x ) = ( P , x + ~ r ) ,( C , X ) = ( P , x + - r ) . 4

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 ) a n d ( Q , x + + )

has

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 } .

38 1

Ovals in Steiner Triple Systems Secants containing t h e in-point 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 p o l y g o n form a n oval 1 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 secants

zr

c o n t a i n i n g M. By 1 . 8 w e h a v e f o u n d a l l t h e s e c a n t s o f t h i s k i n d : 1 namely, t h e y a r e { M , ( P , x ) , ( P , X + ~1 ,~ x) E { O , l , ...,r - 1 ) . Adding t h e numbers o f 1 1 o b t a i n r--(r-2) +-r = 2 2

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 1 1 -r(r-1) = - ( v - l ) ( v - 3 ) . 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 b e t w e e n t w o 1 p o i n t s o f s u c h a p a i r w a s d = -r 4 = 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 d o n e 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 of 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 a r e c o m p l e t e l y d e t e r m i n e d : 1 1 { ( 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 c o m p l e t e d . CONCLUDING REMARKS

Long a g o 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 o n e f o r a p o i n t s e t M i n a n S T S ( v ) c o n t a i n i n g no l i n e ? F o r w h i c h 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 p r o b l e m s t o h y p e r o v a l s a n d 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 of 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 h y p e r -

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 S T S ( 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 u p p e r bound i s a c h i e v e d i f M i s a r e g u l a r o v a l . 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 s e 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 . A n o t h e r 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 m e n t i o n some r e s u l t s : I n P G ( 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 ( 2d + l - 1 ) k n o t o v a l s a n d 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 a n d 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 neither knot

H. Zeitler

382 o v a l s n o r r e g u l a r ovals.

W e n o t i c e t h a t t h e r e a r e s t i l l many u n s o l v e d p r o b l e m s a b o u t o v a l s i n a n S T S ( v ) . (The t o t a l number of o v a l s 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 s y s t e m s S ( k , v ) w i t h k Z 3 t h e r e a r e o n l y a few r e s u l t s concerning o v a l s , hyperovals and similar n o t i o n s [ 4 ] . Extensions o f t h e ErdBs p r o b l e m s t o g e n e r a l s y s t e m s of t h i s k i n d a r e unknown, too.

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H i l t o n , A.J.W., On S t e i n e r and s i m i l a r t r i p l e s y s t e m s , Math. Scand. 24 ( 1 9 6 9 1 2 0 8 - 2 1 6 L e n z , H . and Z e i t l e r , H . , A r c s a n d O v a l s i n S t e i n e r T r i p l e S y s t e m s , C o m b i n a t o r i a Z 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 969 ( 1 9 8 2 ) 229-250 (Springer Verlag, Berlin, Heidelberg, N e w York) P e l t e s o n , R . , E i n e Losung d e r b e i d e n 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 R e s m i n i , 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 s y s t e m 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 49 (1981) 104-113 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 s e t h a v i n g 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 s y s t e m o n t h e s e t , Canad. M a t h . BUZZ. 1 2 ( 1 9 6 9 ) 7 7 7 - 7 7 8 Skolem, T . , Some r e m a r k s o n t h e t r i p l e s y s t e m s o f S t e i n e r , Math. S c a n d . 6 ( 1 9 5 8 ) 273-280 Z e i t l e r , H . a n d Lenz, H . , H y p e r o v a l e i n S t e i n e r - T r i p e l Systemen, M a t h . Sem. B e r . 3 2 ( 1 9 8 5 ) 1 9 - 4 9

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383

PARTICIPANTS

L.M.

V.

Abatangelo

Abatangelo

Bari Bari

S. Antonucci

Napoli

E.M.

Milano

Aragno M a r a u t a

R.

Artzy

Haifa

L.

Bader

Napoli

G.

Balconi

Pavia

A.

Rarlotti

Firenze

U.

Bartocci

Perugia

L.

Benbteau

Toulouse

W.

Benz

Hamburg

L.

Berardi

L I Aqui l a

C.

Bernasconi

Perugia

L.

Bertani

Heggio E m i l i a

A.

Beutelspacher

Mainz

A.

Bichara

Roma

M.

Biliotti

Lecce

P. R i o n d i

Napoli

P. B i s c a r i n i

Perugia

F.

Ronetti

Ferrara

A.

Bonisoli

Munchen

A.A. A.

Bruen

Caggegi

1. C a n d e l a

G.

C a n t a l u p i Tazzi

Western O n t a r i o Napol i

Bari Milano

Participants

384 R . C a p o d a g l i o d i Cocco

Bo 1ogna

M.

Bari

Capursi

B. C a s c i a r o

Bari

P.V.

Roma

Ceccherini

N.

Cera

Pescara

N.

Civolani

Milano

A.

Cossu

Bari

C.

C o t t i Ferrero

Parma

M. Crismale

Bari

F . De C l e r c k

Gent

E . Ded6

Milano

M.

De Finis

M.L. M. M.M.

D e Resmini

De S o e t e

Deza

C . D i Comite

Roma Roma Gent Paris

Bari

V.

Dicuonzo

Roma

L.

D i Terlizzi

Bari

B . DlOrgeyal

D i jon

J . Doyen

Bruxe 1l e s

F. Eugeni

L Aqui l a

G.

Faina

Pe r u g i a

L.

Faggiano

Bari

M.

Falcitelli

Bari

A.

Farinola

Bari

G.

Ferrero

Parma

0. F e r r i

L I Aqui l a

P. F i l i p

Miinchen

S.

Fiorini

Malta

M.

Funk

Potenza

M.

Gionfriddo

Ca t a n i a

M.

Grieco

Pavia

W. Heise A.

Herzer

Miinchen Mainz

Participarz ts R.

H i l l

Salford

J . Hirshfeld

Sussex

D.

Hughes

London

V.

Jha

Glasgow

H.

Kellerer

Miinchen

H. Karzel Keedwell

A.D. G.

Korchmaros

H.J.

Kroll

Miinchen Gui 1d f o r d Potenza Munchen

F . Kramer

Clu j-Napoca

H.

Kramer

Cluj-Napoca

P.

Lancellotti

Modena

B.

Larato

Bari

D.

Lenzi

Lecce

A.

Lizzio

Catania

G.

Lo F a r o

Messina

P.M.

Lo R e

Napol i

G.

Luisi

Bari

G.

Lunardon

Napoli

H.

Lunenburg

K a iserslautern

N.A.

Malara

Modena

M.

Marchi

Brescia

A.

Maturo

Pescara

F . Mazzocca

Case r t a

N.

Melone

Napoli

M.

Menghj-ni

Roma

G.

Menichetti

Bologna

G.

Micelli

Lecce

G.

Migliori

Roma

F. Milazzo

Catania

S. Milici

C a ta n ia

A.

Maniglia

Lecce

D.

Olanda

Napoli

A.

Palornbella

Bari

385

Participan Is

Pastore

A.M.

Bari

S.

Pellegrini

Bresc i a

C.

Pellegrino

Modena

G.

Pellegrino

Per u g ia

C. P e r e l l i Cippo

Brescia

M. P e r t i c h i n o

Bari

G. P i c a

Napoli

F. P i r a s

Cagl i a r i

Pornilio

A.I.

L.

Porcu

Rorna Milano

L. Puccio

Mess i n a

G.

C a t an i a

Quattrocchi

P. Quattrocchi

Modena

G.

Bari

Raguso

L. R e l l a

Bari

T . Roman

Buc a r e s t i

L.A.

Rosati

Firenze

H.G.

Samaga

Hamburg

M.

Scafati

Roma

R.

Scapellato

Parma

E. Schroder

Hamburg

R.

Schulz

Berlin

D.

Senato

Napoli

H.

Siernon

Ludwigsburg

C. Soma

Roma

R. S p a n i c c i a t i

Rorna

A.G. S p e r a

P a l e rmo

.

Stangarone

Bari

K.

Strarnbach

Erlangen

G.

Tallini

Roma

P..

Terrusi

Bari

R

J . Thas

Gent

M.

Ughi

Perugia

V. Vacirca

C a t a n i a.

Participants K.

Vedder

Gissen

A.

Venezia

Roma

F. Verroca

Bari

R.

Vincenti

P e rugi a

A.

Ventre

Napoli

H. Wefelscheid

Essen

B.

Wilson

London

N.

L a g a g l i a Salvi

M i 1a n o

H.

Zeitler

Bay r e u t h

E.

Zizioli

Brescia

R. Z u c c h e t t i

Pavia

387

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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. Z W E R M A N N 1981 x + 380 pages

390

VoI. 1 1 : Studies on Graphs and Discrete Programming edited by P. HANSEN 1981 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 viii 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-GREENand U. ZIMMERMA" 1984 viii 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 Vol. 28: Random Graphs '83 edited by M. KARONSKI and A. RUCIrjSKI 1986 approx. 5 10 pages Vol. 29: Matching Theory L. LOVASZ and M.D. PLUMMEK 1986 in preparation

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