NORTH-HOLLAND MATHEMATICS STUDIES 123 Annals of Discrete Mathematics( 30) General Editor: Peter L. HAMMER Rutgers' University, New Brunswick, NJ, U.S .A. Advisory Editors C. BERGE, Universite de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.-H. VAN LINT CaliforniaI nstitute of Technology, Pasadena, CA, U.S.A. G .4 .R OTA, Massachusetts Institute of Technology, Cambridge, MA, U. S.A. NORTH-HOLLAND- AMSTERDAM NEW YORK OXFORD .TOKYO COM BIN ATORICS '84 Proceedings of the International Conference on Finite Geometries and Combinatorial Structures Barl; ItalK 24-29 September, 1984 edited by A. BARLOlTI Universita di Firenze, Firenze, Italy M. BILIOITI Universita di Lecce, Lecce, Italy A. COSSU Universita di Bart Bari, Italy G. KORCHMAROS Universita delta Basilicata, Potenza, Italy G.TALLINI Universita 'La Sapienza: Rome, Italy 1986 NORTH-HOLLAND -AMSTERDAM 0 NEW YOAK OXFORD .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 Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U.S .A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VanderbiltAvenue NewYork, N.Y. 10017 U S A . Library of Congress Catalogingin-PublicationD ata International Conference on Finite Geometries and Combinatorial Structures (1984 : Bari, Italy) Combinatorics '84 : proceedings of the International Conferenco on Finite Geometries and Combinatorial Structures, Bari, Italy, 24-29 September 1984. (Annals of discrete mathematics ; 30) (North-Holland mathematics studies ; 123) Includes bibliographies. 1. Combinatorial geometry--Congresses. I. Barlotti, A. (Adriano), 1923- . 11. Title. 111. Title: Combinatorics eighty-four. IV. Series. V. Series: North-Holland mathematics studies ; 123 . QA167.158 1984 511l.6 85-3 1 121 ISBN 0-444-87962-5 PRINTED IN THE NETHERLANDS V PREFACE Every year, since 1980, an International Combinatoric Conference has been held in Italy: Trento, October '80; Rome, June '81 ; La Mendola, July '82; Rome, at the Istituto Nazionale di Aka Matematica, May '83. The International Conference Combinatorics '84, held in Giovanazm (Bari) in September '84 is part of the well established tradition of annual conferences of Combi- natorics in Italy. Like the previous ones, this Conference was really successful owing to the number of participants and the level of results. The present volume contains a large part of these scientific contributions. We are indebted to the University of Bari and to the Consiglio Nazionule delle Ricerche for fmancial support. We are pmfoundly grateful to the referees for their assistance. A. BARLOTTI M. BILIOTTI A. COSSU G. KORCHMAROS G. TALLINI vii Intervento di Apertura del Prof. G. Tallini al Convegno “COMBINATORICS 84” Vorrei dare, a nome del Comitato scientific0 e mio, ilbenvenuto aimoltipartecipanti che dall’Italia e dall‘estero sono qui convenuti per prendere parte a questo convegno. Esso si ricollega e fa seguito ai congressi internazionali di combinatoria tenuti a Roma nel giugno del 1981, a La Mendola nel luglio del 1982, a Roma presso I’Istituto Nazionale di Alta Matematica nel maggio del 1983. Questi incontri, ormai annuali in Italia e che spero possanc continuare, s’inquadrano nell’ampio sviluppo che la combinatoria va acquistando a livello internazionale. Come 8 noto il mondo modern0 si va indirizzando ed evolvendo sempre di pib verso la programmazione e l’informatica, al punto che un paese oggi B tanto pih progredito, im- portante e all’avanguardia quanto pib B avanzato n e b scienza dei computers. I1 ram0 della Matematica che B pi^ vicino a questi indirizzi e che ne I! la base teorica B proprio la combinatoria. Essa a1 gusto astratto del ricercatore, del matematico, associa appunto le applicazioni pib concrete. Cib spiega il prepotente affermarsi di questa scienza nel mondo e ne prova il fervore di studi e di ricerca che si effettuano in quest’ambito, le pubblica- zione dei molti periodici specializzati, i numerosi convegni internazionali a1 riguardo. Vorrei ringraziare gli Enti che hanno permesso la realizzazione di questo convegno, tutti i partecipanti, in particolare gli ospiti stranieri che numerosi hanno accolto il nostro invito e tra i quali sono presenti insigni scienziati. Concludo con I’augurio che questo convegno segni una tappa da ricordare nello sviluppo della nostra scienza. Annals of Discrete Mathematics 30(1986) 1-8 0 Elsevier Science Publishers B.V. (NorthHolland) 1 TRANSLATION PLANES WITH AN AUTOMORPHISM GROUP ISOMORPHIC TO SL (2,5) Vito Abatangelo Bambina Larato Universith di Bari Italia In this paper translation planes of odd order q , 2 51q -1 , are constructed. Their main interest consists in the fact that their translation complement contains a group isomorphic to SL(2,5) . At first these planes were obtained in other ways by 0. Prohaska in the case 51q+l ( [lo] ,1977) and by G. Pellegrino and G. Korchm5ros in the case 51q-1 ( [9] ,1982), but in both papers the Authors did not establish the previous group property. Moreover we show that Pellegrino and Korchmhros plane 2 is not a near-field plane of order 11 . 1. - AN AUTOMORPHISM GROUP OF THE AFFINE DESARGUESIAN PLANE OF 2 2 ORDER q , 51q -1 , ISOMORPHIC TO SL(2,5) 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 the form y = mx+b or x = c with m,b,c elements of K . The affine subplane no 2 V. Abatangelo and B. Larato is t h e po in t - se t o f an a f f i n e Baer subplane. Its Baer s u b l i n e a t i n f i n i t y has equa t ion (1 .2 ) as t h e l i n e y = xmtd i n t e r s e c t s (1.3) e i t h e r i n q p o i n t s o r i n t h e on ly p o i n t ( 0 , d ) accord ing as (1 .2) ho lds o r does n o t hold. The a f f i n e Baer subplanes with equa t ions of t h e form ( 1 . 3 ) c o n s t i t u t e a n e t on t h e a f f i n e p o i n t s o f n . The l i n e a t i n f i n i t y becomes t h e Miquelian i n v e r s i v e p l ane M(q) i f we t a k e a s c i r c l e s t h e sets (1.1) and ( 1 . 2 ) of elements o f K ~ { c o } . I n o r d e r t o cons ide r an automorphism group o f $C isomorphic t o SL(2 ,5) we d i s t i n g u i s h two c a s e s , accord ing a s 51q+l o r 51q-1 . CASE 5 ( q + l By assumption 5 1 q t l t h e r e e x i s t s an element a o f K such t h a t a5 = 1 (c f . [4]). W e p u t b = (a-aq)-' and c such t h a t c q + l t bq+l = 1. After t h i s l e t u s cons ide r t h e fo l lowing a f f i n e mappings o f a a : x ' = ax , yt = aqy f l : x ' = bx t cy , y ' = -cqx t bqy and t h e i d e n t i t y & : x ' = x , y ' = y . Note t h a t < a , f l> 2 S L ( 2 , 5 ) , ( c f . 141, p. 199) . Now our purpose is t o s tudy t h e a c t i o n o f < a , p > on t h e l i n e a t i n f i n i t y , i . e . t h e a c t i o n o f I' = < Translation Planes 3 PROPOSITION 3. - Any two circles of have two o r zero common points ac- cording as q E 3 (mod 4 ) o r q 5 1 (mod 4). PROOF - By Prop. 2 it is sufficient to prove that the circles C and C 1 2 satisfy our assert. STEP 1 - If C 1 and C 2 have any common point, its coordinates would satisfy both their equations which form a system which is equivalent to the following 1 =: 1 (1.6) 1 c q m -c mq = o , As { m = ch I h EGF(q) } is the set of solutions of the second equation of (1.61, the whole system (1.6) has any solution if and only if the equation cq+l h2 = 1 is solvable, i.e. if and only if cq+l is a square in F . STEP 2 - cq+l is a square of F or not according as q 5 3 (mod 4 ) o r q 1 (mod 4) . In order to prove this, let x and x be elements of F 1 2 such that a = x + tx * note that 1 2 ' 2 2 (1.7) aq+l = x 1 - sx2 E F q+l 5 and (a ) = 1 , so (1.8) '+'a = 1 4 2 2 (because in F there is no element of order 5 ) and therefore 5x + lox x s + 1 1 2 4 2 4 2 + x s = 0 , By means of (1.7) and (1.81, we get 16x1 - 12x + 1 = 0 , i.e. 2 1 2 -+ .r/;;= 8x1 - 3 : this proves that 5 is a 2 square in 2 F -.1 Now 2 2 -1 cq+l = c(a-aq)2 + 13la-a' = p(l+sx ) - 37 (4x2 s ) = (4x - 3 ) ( 4 x s) = 2 1 2 -1 -2 = -1 s (1 2 m 2 ( 4x2 ) q+l. and therefore c is a square in F if and only if -1 is not a square in F , i.e. if and only if q 3 (mod 4). PROPOSITION 4 . - Co is disjoint from any circle of % . PROOF - It is easy to check that C and C are disjoint. Our assertion 0 1 follows by means of Prop. 2. CASE 5)q-1 By assumption 51q-1 there exists an element a of F such that a5 = J. . -1 -1 Let b,c,d be elements of K such that b = (a-a ) , cd = -1-b . Now let us consider the following affine mappings of 7C -1 -1 y : x ' z a x , y' = (2+t)b x + ay 2 6 : X' = [-b+(Z+t)d]x + dy , y' = [2(2+t)b+c-(2+t) dJx + [b-(2+t)ay and note that<y,d >--SL(2,5) (cf.[4], p . 198). We shall now look at the action of< 7,6> on the line at infinity, i.e. the action of the group A = < Y , ~ > / < - E >on M(q). 4 V. Abatangelo and B. Larato PROPOSITION 5. - In M(q) the group A maps the circle Do : 2tmq+l-m+mq =O onto itself and leaves the set { D1e - 0 . ,D6 } invariant where D :4mq+1+m+mq=0 1 and 2 2i 2 -1 a-icd-ll .q+1 + C(2tt)-a 2i (1+2b2)(2bd)-3m+ D : C(4-t ) - 4a (1+2b )(2bd) - i+2 2i 2 + [(2-t) - a (1+2b )(2bd)-Y mq + 1 = 0 , (i = 0.1, ... ,4). PROOF - Long and easy calculations prove that y and 6 map D onto itself 0 and act on the set { D1 ,D2, . . . , D ~] as follows y : (D1)(D2D3D4D5Ds) and 6 : (D1D2)(D3D6)(D4)(D5) . PROPOSITION 6. - The group d acts on { Dl,D2, ... ,D6 } in the same way as PSL(2,5) acts in its usual two-transitive representation on six objects . PROOF - The proof is similar to that one of Prop. 2 , provided we identify D 1 with the symbol 00 , D2 with 0 , D3 with 1 , D4 with 2 , D5 with 3 and D6 with 4 and moreover y and 6 with 7' and 6 ' , respectively, where y ' : x' = x + 1 and 6' : x ' = 4x . Assume now 9 = { Do ,D1 , .. . ,D6 } ; by direct calculations and by Prop.6 we get the following PROPOSITION 7. - (i) Any two circles of 9 have two common points; (ii) no three circles of 9 have a common point. 2. - DERIVED AND R-DERIVED PLANES In this section we discuss briefly the processes of derivation and multiple derivation due to T.G. Ostrom and of R-derivation and multiple &derivation due to A.A. Bruen. The reader can find some details in Hughes-Piper c37, Ostrom c7,8] and Bruen Ll]. Let us assume q > 3 ; every circle C of M(q) is a derivation set, i.e. every pair of affine points lies in a unique Baer affine subplane, whose subline at infinity is C . From now on we will consider circles C with equation (1.2). An affine plane can be obtained by replacing lines whose equation is y = mx+b , for each m satisfying ( l . Z ) , with the q+l Baer affine subplanes, called also components, with equation y = xv + rxq g where g runs over A = { g E K 1 gq+' = 1 } , together with their translates. This construction is said derivation by C and the obtained plane is a trans- 2 lation plane of order q . Derivation can be repeated many times, when there is a set of circles C l , C a , C 3 , . . . ,C each disjoint from the other; in this case the process is said k. multiple derivation with respect to the circles C1 'C 2 ' * . * 'Ck About R-derivation, suppose that M(q) admits a chain of circles, i.e. a family % of circles satisfying the following properties: (i) any two circles of W have two common points; (ii) no three circles of % have a common point; (iii) % consists of (q+3)/2 circles. Translation Planes 5 Let I denote the point set covered by the circles of V . It follows that I contains exactly (q+l)(q+3)/4 points and each point of I belongs ex- actly to two circles of % . Hence any two points on a line with its ideal point on I belong exactly to two affine Baer subplanes whose sublines at infinity represent circles of V . The affine Baer subplanes with Baer sublines at infinity representing circles of % ' do not form a net on the affine points of 7c . However, for small q ( 5 1q S 13), several Authors have constructed a net N on the affine points of by taking a suitable half of such affine Baer subplanes; namely, 2 for each circle C of %? , (q+l)q /2 affine Baer subplanes with the same Baer subline representing C . If such a net N exists, we can obtain a trans- lation plane on the affine points of JZ by replacing the affine lines of R whose ideal pointslie on I with N . The resulting translation plane is said to be R-derived from 7c . Of course if there is a set of disjoint chains of circles on M(q) and the corresponding nets exist, then the above method can be repeated: the re- sulting translation plane is said to be multiple R-derived from 7c . 2 3. - A CLASS OF TRANSLATION PLANES OF ORDER q , 51q+l , CONTAINING IN THEIR TRANSLATION COMPLEMENT AN AUTOMORPHISM GROUP ISOMORPHIC TO SL(2,5) In this section we assume that q - 1 (mod 4 ) . By Propositions 3 and 4 we get three planes n . (j = 1,2,3): 7c by derivation from 7c with respect to the J 1 circle C ( s o 7c is the well-known Hall plane of order 81, cf. c23, 1 p. 225), a2 by derivation with respect to the circles C1'C2' 'C6 t R3 by derivation with respect to the circles Co 'cl* -.- ,C6 ; each of them contains in its translation complement the group<u,p >?SL(2,5) . Now we prove the following propositions PROPOSITION 8. - The group < a,P > leaves invariant each of the q + l components corresponding with the derivation set co * PROOF - The q + l components are the Baer subplanes B with equations g B : y = rx 9g where rqtl = -1 and g runs over A . A straigh forward cal- 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 components corresponding with the multiple derivation set C I U C U ... u c6 into 2 (q+1)/2 orbits each of length 12. PROOF - Let H, (i = 1,2, ... ,6) be the set consisting of the q+l compo- nents which corrispond with the derivation set . Then <a,p > acts on the Ci set { H1, ... ,H6 } in the same way as on the set { C 1, . . . ,C6 } . By Prcp. 2, < a,p>acts transitively on { H1, . .. ,H6} . The stabilizer of H in < a$ > 1 3 2 is< a , l >with A=papap , i.e. 2 A : x ' = c(2b -acq+l-aqb2)y , Y' = - ~ ~ ( 2 b ~ - a ~ c ~ + ~ -.a b ~ ) x