Combinatorics 86: Proceedings of the International Conference on Incidence Geometries and Combinatorial Structures, Passo Della Mendola, Trento,

CO MBINATOR ICS '86

ANNALS OF DISCRETE MATHEMATICS

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

Adviso ry Editors: C. BERGE, Universitede Paris, France M.A. HARRISON, Universityof California, Berkeley, CA, U.S.A. V. KLEE, Universityof Washington, Seattle, WA, U.S.A. J.-H. VAN LINT CaliforniaInstitute of Technology,Pasadena, CA, U.S.A. G.-C.ROTA, Massachusetts Institute of Technology,Cambridge, MA, U.S.A.

NORTH-HOLLAND-AMSTERDAM

NEW YORK

OXFORD *TOKYO

37

COMBINATORICS ’86 Proceedings of the International Conference on Incidence Geometries and Combinatorial Structures Passo della Mendola, Trento, ItalK 30 June -5 h I x 7986

Edited by

A. BARLOTI Universitadi Firenze, Firenze, Italy

M. MARCHI UniversitaCattolica del S. Cuore, Brescia, Italy

G.TALLlNl Universita ’La Sapienza’,Roma, Italy

1988 NORTH-HOLLAND -AMSTERDAM

0

NEW YORK

0

OXFORD 0 TOKYO

Elsevier Science Publishers B.V., 1988 AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 70369 1

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LIBRARY OF CONGESS L i b r a r y of C o n g r e s s Cataloging-in-Publication

Data

International C o n f e r e n c e o n I n c i d e n c e G e o m e t r i e s and C o n b i n a t o r i a l S t r u c t u r e s (1986 : Trento..Italy) C o n b i n a t o r i c s '86 : proceedings o f t h e International C o n f e r e n c e on I n c i d e n c e G e o m e t r i e s and Coabinatorial Structures, held at P a s s o della Mendola. Trento. Italy, 30 June-5 J u l y , 1986 / edlted by A. Barlotti. M. Marchi. G. Tallini. p. cm. -- (Annals o f d i s c r e t e m a t h e m a t i c s ; 37) B i b l i o g r a p h y : p. I S B N 0-444-70369-1 ( U . S . ) 1. C o n b i n a t o r i a l geometry--Congresses. I. Barlotti. A . (Adriano), 1923. 11. Marchi. M. 111. Tallini. G. (Gluseppe), 1930. IV. Title. V. Series. O A 1 6 7 . 1 5 9 1986 516'.12--dc19 87-32079 CIP

PRINTED IN THE NETHERLANDS

V

FdREWORD

This volume contains the proceedings of the Conference "COMBINATORICS '86" held at the Centro di cultura of the Universith Cattolica of Milano, in Passo della Mendola (Trent0 - Italy) (30 June - 5 July 1986). The Conference is the latest in a series of international meetings on the same subject held in Italy in the recent past: Roma, June 1981 (Conference in honour of €3. Segre); Passo della Mendola, July 1982; Roma, May 1983; Bari, September 1984. The Proceedings of the first and the last of these meetings are published in this Series "Annals of Discrete Mathematics" by North Holland Publ. C o . The participants in Combinatorics '86 numbered about 150 and represented more than a dozen nationalities. The subjects covered by this volume concern recent developments in combinatorial and incidence geometry in all its different aspects, and its links with foundations of geometry, graph theory, algebraic structures, and applications to coding theory and computer sciences.

We are indebted to the Universith Cattolica for the organization of the Conference and to the Italian Consiglio Nazionale delle Ricerche for financial support. We are also profoundly grateful to the referees for their assistance.

A. BARLOTTI M. MARCHI G. TALLINI

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vii

OPENING WELCOME Giuseppe Tallini

First of all I wish to heartily welcome the many participants, both from Italy and abroad, in "Combinatorics '86". This conference is another step in an already long series. I like to recall here the Rome conference in September ' 7 3 , organized by the Accademia dei Lincei, again in Rome that in memory of B. Segre, in June 1981, theaonferencesheld in July '82 at Passo della Mendola, i n May '83 in Rome at the Istituto Nazionale di Alta Matematica "F. Severi", in September '84 in Bari. I hope that our present conference will be scientifically fruitful and at the same time enjoyable and satisfying.1 also hope that this series will continue, say with aconference in Ravello (Naples) in 1988. Nowadays, these meetings and specialized seminars have the effective role of spreading ideas and promoting research, not only because of the information they present to the listeners during the formal talks, but mainly because of the opportunities they offer of discussions among the participants. The vitality of combinatorics and of the Foundations of Geometry is asserted on the one hand by the many specialized journals and a vast literature, and on the other hand by the numerousconferences which are recurrently held all over the world. This vitality resides, in my opinion, in the fact that today combinatorics has moved to the forefront of applied mathematics because of its important interactions with computer science and statistics. Moreover, the theoretical interest of our discipline lies in the fascination of the discrete with which it deals. The thrill of combinatorial problems stems from the fact that the configurations to be studied in relation to a given problem, though finitely many, actually come in so huge numbers that they behave as infinitely many both for humans and computers. That is, we are daily confronted by a finiteness which for us is really infinite. For example, given a set of ten objects, if we wanted to decide if there exists a sharply 2-transitive set of permutations (which already accounts for 90 elements) and wished to employ our computer, then 10F we would have to examine( cases and verify the required property for each of them. Such a check is impossible within the lifetime of mankind, even with the help of the most sophisticated computer. By the way, the number we were talking about considerably exceeds the number of atoms in the universe! For this reason, in order to solve combinatorial problems the geometric intuition of the researcher must intervene in a critical way, that is there is a quiddity man has while a computer does not (up to now). The problems which arise are of a varied nature and the suitable techniques to deal with them are to be devised for each situation. Indeed, one of the special features of combinatorics is the often sporadic nature of solutions which comes from its links with number theory. The branches of combinatorics are many and various. All of them are amply and appropriately represented in this conference whose scientific output, I hope, will reflect the fervour of the intense studies on these subjects and provide a new impulse to future developments.

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1x

CONTENTS

Foreword Opening Welcome G. T a l l i n i P a r t i c i p an t s

V

vii xiii

N e t of R a t i o n a l i t y i n a Minkowski P l a n e R. Artzy

1

A New Class of T r a n s l a t i o n P l a n e s R.D. Baker and G . L . E b e r t

7

Quasigroups and Groups A r i s i n g from Cubic S u r f a c e s L . Beneteau

21

Blocking S e t s i n t h e Large Mathieu Designs, I: The Case S(3.6.22) L. B e r a r d i

31

Blocking S e t s i n t h e P r o j e c t i v e Plane of Order Four L . B e r a r d i and F . Eugeni

43

K a l a h a r i and t h e Sequence "Sloane No. 377" D. B e t t e n

51

Enciphered Geometry. Some A p p l i c a t i o n s of Geometry t o Cryptography A. Beutelspacher

59

O n F i n i t e Grassmann Spaces P. Biondi

69

The Regular Subgroups of t h e S h a r p l y 3 - T r a n s i t i v e F i n i t e Permut a t i o n Groups A. Bonisoli

75

w.

Hyperovals i n Desarguesian P l a n e s of Even Order Cherowitzo

87

C i r c u l a r Block Designs from P l a n a r Near-Rings J . R . Clay

95

Extending t h e Concept of Decomposability f o r T r i p l e Systems C . J . Colbourn, E. Mendelsohn, and A. Rosa

137

T r a n s l a t i o n P a r t i a l Geometries F. D e C l e r c k , H. G e v a e r t , and J . A . Thas

117

O n Admissible S e t s w i t h Two I n t e r s e c t i o n Numbers i n a P r o j e c t i v e PLane M . J . D e Resmini

137

Contents

X

Commutative F i n i t e A-Hypergroups of Length TWO M. De Salvo

147

On S e t s of F i x e d P a r i t y i n S t e i n e r Systems V . Eugeni and S. I n n a m o r a t i

157

Blocking S e t s of Index Two F. Eugeni and E . Mayer

169

A S h o r t Proof

t h a t Ordered L i n e a r Spaces a r e L o c a l l y P r o j e c t i v e

R. Frank

177

Midpoints and M i d l i n e s i n a F i n i t e H y p e r b o l i c P l a n e C.W.L. Garner

181

Hall-Ryser Type Theorems f o r R e l a t i v e D i f f e r e n c e S e t s D . G h i n e l l i Smit

189

C o o r d i n a t i o n of G e n e r a l i z e d Quadrangles G . Hanssens and H. Van Maldeghem

195

C o n s t r u c t i o n of Some P l a n a r T r a n s l a t i o n Spaces A. Herzer

209

Regular S e t s i n Geometries J . D . Key

217

N.

Group P r e s e r v i n g E x t e n s i o n s of Skew P a r a b o l a P l a n e s Knarr

225

P r o d u c t s of I n v o l u t i o n s i n Orthogonal Groups F. Kniippel

231

Examples of Ovoidal MBbius P l a n e s of Hering C l a s s I11 H.-J. Kroll

249

A C o n s t r u c t i o n of P a i r s and T r i p l e s of k-Incomplete Orthogonal A r r a y s P. L a n c e l l o t t i and C . P e l l e g r i n o

25 1

R e l a t i v e I n f i n i t y i n P r o j e c t i v e De S i t t e r Spacetime and I t s R e l a t i o n t o Proper T i m e J.A. Lester

257

A f f i n e Hjelmslev Rings and P l a n e s J . W . Lorimer

265

I r r e d u c i b l e R e p r e s e n t a t i o n s of Hecke A l g e b r a s of Rank 2 Geometries S. L6we

277

A C h a r a c t e r i z a t i o n of Pappian A f f i n e Hjelmslev P l a n e s H. MSurer and W. N o l t e

28 1

Embedding L o c a l l y P r o j e c t i v e P l a n a r Spaces i n t o P r o j e c t i v e Spaces K . Metsch

293

On T o p o l o g i c a l I n c i d e n c e Groupoids R. Meyer, J . M i s f e l d , and E. Z i z i o l i

297

lsomorphisms of F i n i t e Hypergroupoids R . Migl i o r a t o

301

Contents

xi

Seminversive Planes D. Olanda

31 1

Geometric and Algebraic Methods in the Classification of Geometries Belonging to Lie Diagrams A. Pasini

315

The Thas-Fisher Generalized Quadrangles S.E. Payne

357

On Group Spaces Defined by Semidirect Products of Groups 3 . Pfalzgraf

367

On Permutation Properties for Finitely Generated Semigroups G. Pirillo

375

On k-Sets of Type (O,rn,n) in S with Three Exterior Hyperplanes R. Procesi Ciampi and R. Rota r y q

377

An Algorithm for L -colourations L. Puccio

385

A Blocking Set in S.

Rajola

PG(3,q),

q

35

39 I

A Characterization of all Abelian Groups whose Lattice of Precompact Group Topologies Represents a Projective Geometry D. Remus

395

Groups of Homologies in 4-Dimensional Stable Planes are Classical H.-P. Seidel

399

Polynomial Species and Connections among Bases of the Symmetric Polynomials D. Senato and A.M. Venezia

405

Set and Sequence Closure for Finite Permutation Groups

J . Siemons

413

P-Cyclic Hypergroups with Three Characteristic Elements S.H. Spartalis and T.N. Vougiouklis

421

Order and Uniform Structure in Projective Geometry H. Szambien

427

On Blocking Sets in Finite Projective and Affine Spaces G. Tallini

433

Symmetric Designs without Ovals and Extremal Self-Dual Codes V.D. ToncheB

45 1

Groups in Hypergroups T. Vougiouklis

459

The Perron-Frobenius Projection in the Theory of Graphs, Digraphs, Designs and Stochastic Processes K . E . Wolff

469

On the Non-Existence of Certain Difference Sets N. Zagaglia Salvi

479

Xii

Contents

On Complete 12-Arcs i n P r o j e c t i v e P l a n e s of Order 12 C. Z a n e l l a

485

Block D e s i g n s A d m i t t i n g F l a g T r a n s i t i v e Groups of Automorphisms P.-H. Zieschang

493

An Independence Theorem on t h e C o n d i t i o n s f o r I n c i d e n c e Loops E. Z i z i o l i

497

x iii

PARTICIPANTS

ABATANGELO L . Maria ABATANGELO V i t o ANTONUCCI S a l v a t o r e ARTMANN Benno ARTZY Rafael BADER Laura BAKER C a t h a r i n e Anne BARLOTTI Adriano BASILE A l e s s a n d r o BATTEN Lynn Margaret BENETEAU Lucien BENZ Walter BERARDI L u i g i a BERNARD1 Marco P a o l o BERTANI Laura BETTEN Dieter BEUTELSPACHER A l b r e c h t BICHARA A l e s s a n d r o B I L I O T T I Mauro BIONDI Paola BISCARINI P a o l a BONETTI F l a v i o BONISOLI A r r i g o BONNEAU BORZACCHINI Luigi B R I N I Andrea BROUWER A n d r i e s E v e r t BROWN J u l i a M. Nowlin BRYLAWSKI Tom CAGGEGI Andrea CAPODAGLIO R i t a CAPURSI Mauro CECCHERINI P i e r V i t t o r i o CHEROWITZO William C I V O L A N I Nino CLAY James R . CORSINI P i e r g i u l i o D'ANTONA O t t a v i o DE FINIS Massimo de RESMINI M a r i a l u i s a J . DE SALVO Mario DE SOETE Marjke

DE V I T O P a o l a D I M A R T I N 0 Lino

Bari - Italy Bari - Italy Napoli - I t a l y Darmstadt - Germany Haifa - I s r a e l Roma - I t a l y S a c k v i l l e - Canada Firenze - I t a l y Perugia - I t a l y Winnipeg, Manitoba - Canada Toulouse France Hamburg - Germany Italy L'Aquila Pavia - I t a l y Parma - I t a l y K i e l - Germany Munchen - Germany Roma - I t a l y Lecce - I t a l y Napoli - i t a l y Perugia - I t a l y Ferrara - I t a l y Modena - I t a l y L e Chesnay - F r a n c e Bari - I t a l y Bologna - I t a l y Amsterdam - Holland Downsview- O n t a r i o - Canada Chapel H i l l - N. C a r o l i n a - U.S.A. Napoli - i t a l y Bologna - I t a l y Bari - Italy Roma - I t a l y Denver - Colorado - U.S.A. Potenza - I t a l y Tucson - Arizona - U.S.A. Udine - I t a l y Pavia - I t a l y Roma - I t a l y Roma - I t a l y Messina - I t a l y Gent - Belgium Napoli - I t a l y Milano - I t a l y

-

x iv

Participants

D I C U O N Z O Vincenzo DODUNEKOV S t e f a n Manev DUBIKAJTIS L . EBERT Gary L . EUGENI F r a n c o EVANS David Mark FAINA G i o r g i o FERRERO Giovanni FIORI C a r l a FIORINI S t a n l e y FISHER J . C h r i s FRANK R o l f d i e t e r FUNK M a r t i n GARNER C y r i l GERBER P . Dean

Roma - I t a l y Sofia - Bulgaria A r c a v a c a t a d i Rende - Cosenza Newark Delaware - U.S.A. L'Aquila - I t a l y Tubingen - Germany Perugia - I t a l y Parma - I t a l y Italy Modena Msida - Malta Regina Canada Hamburg - Germany Potenza - I t a l y O t t a w a - Canada

GEVAERT H . G H I N E L L I Dina GIONFRIDDO Mario GRUNDHOFER Theo HANSSENS Guy HARTMANN P e t e r HEISE Werner H E R Z E R Armin HIRSCHFELD James I D E N Oddvar INNAMORATI S t e f a n o J H A Vikram JUNKERS W i lhelm KALHOFF F r a n z Bernhard KARZEL Helmut KAYA Rustem KEEDWELL Anthony Donald KEPPENS D i r k F . J . KERBY W i l l i a m KEY J e n n i f e r D . KNARR N o r b e r t KNEPPEL F r i e d e r KORCHMAROS Gabor KROLL Hans-Joachim LANCELLOTTI P a o l a LARATO Bambina LAURI J . LESTER J u n e L I Z Z I O Angelo LO RE P i a Maria LORIMER J . W . Michael LOWE S t e f a n LUWEN R a i n e r LUNARDON Guglielmo L ~ N E B U R GH .

Gent - Belgium Roma - I t a l y Catania - I t a l y Tubingen - Germany Gent - Belgium Miinchen - Germany Munchen - Germany Mainz - Germany Brigbton - Great B r i t a i n Bergen - Norway L'Aquila - T t a l y Glasgow - Great B r i t a i n D u s s e l d o r f - Germany Dortmund - Germany Munchen - Germany E s k i s e h i r - Turkey Guildford - Great B r i t a i n Gent - Belgium Hamburg - Germany Birmingham - G r e a t B r i t a i n K i e l - Germany K i e l - Germany Potenza - I t a l y Munchen - Germany Modena - I t a l y Bari - I t a l y Msida - Malta Los Angeles - C a l i f o r n i a - U.S.A. Catania - I t a l y Napoli - I t a l y Toronto - Canada Braunschweig - Germany Tubingen - Germany Napoli - I t a l y K a i s e r s l a u t e r n - Germany Sofia - Bulgaria Brescia - Ttaly

MANEV b l i c k o l a i Lazarov MARCH1 Mario

-

- Italy

-

Participants MARINO Maria C o r i n n a M'AURER Helmut MAYER E r i k a MELONE Nicola MENICHETTI Giampaolo MEYER Rita M I G L I O R A T O Renato MIGLIORI Grazia M I SFELD J u r g e n NAGAMUNY Reddy NOLTE Wolfgang OLANDA Domenico OTT Udo PASINI Antonio PAYNE S t a n l e y E . PELLEGRINI S i l v i a PELLEGRINO C o n s o l a t o PERELLI CIPPO C l a u d i o PERTICHINO Michele PETIT Jean-Claude PFALZGRAF Jochern PFLUGFELDER Hala PIANTA S i l v i a PIEPER-SEIER I r e n e PIRILLO Giuseppe PLAUMANN P e t e r POTT Alexander PRIESS-CRAMPE S i b y l l a PROCESI CIAMPI R i t a PUCCIO L u i g i a QUATTROCCHI Gae t a n o QUATTROCCHI P a s q u a l e

RAGUSO G r a z i a RAJOLA Sandro RAO S a l v a t o r e RELLA L u i g i a REMUS Dieter ROSA Alexander ROSATI L u i g i Antonio ROTA R o s a r i a ROZERA Guglielmo RUOFF D i e t e r SAELI r o n a t o SASSO-SANT Maic SCAFATI T A L L I N I Maria SCAPELLATO R a f f a e l e SCHULZ Ralph H . SEIDEL H a n s - P e t e r SENATO Domenico SIEMON Helmut SIEMONS Johannes SIMON1S J u r i a a n

Messina - I t a l y Darmstadt - Germany L'Aquila - I t a l y Napoli - I t a l y Bologna - I t a l y Hannover - Germany Messina - I t a l y Roma - I t a l y Hannover - Germany Tirupati - India Darmstadt - Germany Napoli Italy Braunschweig - Germany Siena - I t a l y Denver - Colorado - U.S.A. Brescia - I t a l y Modena Italy Brescia - I t a l y Bari - I t a l y Limoges - F r a n c e Saarbriicken - Germany P h i l a d e l p h i a - U.S.A. Brescia - I t a l y Oldenburg - Germany Firenze - I t a l y E r l a n g e n - Germany G i e s s e n - Germany Munchen - Germany Roma - I t a l y Messina - I t a l y Catania - I t a l y Modena - I t a l y Bari - Italy Roma - I t a l y Napoli - I t a l y Bari - Italy Hagen - Germany Hamilton - O n t a r i o - Canada Firenze - I t a l y Roma - I t a l y Roma - I t a l y Regina - Canada Potenza - I t a l y S a a r l o u i s - Germany Roma - I t a l y Parrna - I t a l y B e r l i n - Germany Tubingen - Germany Napoli - I t a l y Reichenberg - Germany Norwich - Great B r i t a i n D e l f t - Holland

-

-

xv

xvi SPANICCIATI R e n a t a S P A R T A L I S Stefanos STANGARONE R o s a STEINKE G u n t e r STRAMBACH K a r l SYCHOWICZ A n d r z e j SZAMBIEN H o r s t SZCZERBA L e s l a w TALLINI G i u s e p p e THOMSEN Momme Johs TONCHEV V l a d i m i r D . VAN MALDEGHEM H e n d r i k J VEDDER K l a u s VENEZIA A n t o n i e t t a VOUGIOUKLIS T h o m a s WAGNER A s c h e r WOLFF K a r l E r i c h ZAGAGLIA N o r m a Z A K S Joseph ZANELLA C o r r a d o ZEITLER H e r b e r t ZIESCHANG Paul H e r m a n n ZIZIOLI E l e n a

Participants Roma - Italy Xanthi - Greece B a r i - Italy K i e l - Germany E r l a n g e n - Germany Roma - Italy Hannover - Germany W a r s a w - Poland Roma - Italy Hamburg - Germany Sofia - B u l g a r i a Gent - Belgium G i e s s e n - Germany Roma - I t a l y Xanthi - Greece Birmingham - G r e a t B r i t a i n Darmstadt - Germany M i l a n o - Italy H a i f a - Israel Rorna - I t a l y Bayreuth - Germany K i e l - Germany B r e s c i a - Italy

Annals of Discrete Mathematics 37 (1988) 1-6

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

NET OF RATIONALITY IN A MINKOWSKI PLANE

Rafael ARRY Departmentof Mathematics University of Haifa, 3 1999Haifa, Israel ABmcT

In one of the derived affine planes of a Minlcowskiplane II over an infinite KT-nearfield, an infinite net is construM by means of joining points intersecting lines and drawing allels to lines through points, and so on ad hinitum,&s out from two given nonparallel points and the enerators through them. It is shown that the result is a plane W6bius net-) over a pr@e field. The Construetion is based on pmving that the validlty of certain special Desargues mdltions in 4-websis implied b the Rectangle Axiom in a.Al ebfaically this confirms &e theorem of Kerby and Wefebeid which states that every KT-nearfield has a prime subfield.

%

1. THE MINKOWSKI PLANE

DEFINITION.A A&r&oWp&?e COZlSiStS of a set ll of points and of Wee disjoint subsets of the p o w set of 11: the set of cycles, the set of (+)generators,and the set of (-)generators. Two points are called (+)parallel olotation $1 if some (+)generatorcontainsboth, (-)parallel (notation It-) if some (->generatorcontains both. Parallel (11) will mean 1' 1, or II-'. Nonparallel means -neither ll+nor 11-'. Points lying on the same cycle are called concyclic. Points will be denoted by capital letters, cycles by lower case letters. We postulate I. For every point P there is a unique (+lgenefatofand a unique (-lgenerator containingP. 11. Every generator intersects every cycle in exactly one point. Every (+)generatorintersects every (-)generator in exactly one point. I I I. For every three nonparallel points A, B, C there is a unique cycle containing them, denoted ABC. IV. For each cycle c, each point P on c, and each Q in ll\c, nonparallel to P, there is a unique cycle k mtaining P and Q such that c n k = {P}. V. There is a cycle c with kl 1 3, and Tl\c f 0. DEFINITION. The ukr-vdplme with respect to the point P is the affine plane obtained from a Minkowski plane as follows: its points are ll\{P}, and its lines are all generators of the Minkowski plane which do not contain P and all those cycles that contain P,with the provision that P be removed

R.Artzy

i

from all these cycles.

DEFINITION. A ET+w?YeM is a planar nearfield in whose multiplicative structure there exists an involutory automorphism o identically satisfying thefunctionalequation~o+l)o+(x+l)o= 1. 1.1. RESULT (for instance, 141). A Minkowski plane ~t can be coordinatized by a KT-nearfield if and only if r holds in IC. 1.2. PROPOSITION (Fig. 1). In a Minkowski plane with r, let p:= P 1P2P3 q:= Q 1QzQ3 r:=R 1R2R3 s:= S 1S2S3 A E p n q, C E r, A 11- C, and for i=1.2,3.

Pi ILQi 1 I-t

II-Pi-

C E S.

Proof. Suppose C 4 s. Let S be a point on ssuch that S t C, S IGC, P E p,

Q E q, S IIP II+Q. In view of r, we have Q 11-C. But A 11-C and A E q, h a = A = Q- since A E p, this implies A =P,a d thus also S = C a d therefore C E s.

1.3. PROPOSITION (Converseof 1.2). In Fig.1, let, for 1 = 1, 2, 3, Pill+411- Ri ll+Si Il+Pi, A E p n % C E fn S, P = P 1PzP3, q Q 1Q2Q3 f =R1RzR3 s=S1S2C. TheftS3liesOnS.

-

Net of Rationality in a Minkowski Plane

c.

3

Fig. 1

proof. Suppose S3 4 s. Let s’: = S S2S3. Then, by 1.1, C E s’, and s = s’

1.4. PROPOSITION (Fi.2). In a m o w s k i plane with

r,

-

let p P1P2A,

r = R1R2C, s = S1S2C, B E q n f, B ILD, D E p, Pi lLQi 1I-qlL$ II-Pi, for 1.: 1,2. Then D lies on s. Proof. Suppose D 6 s. Then,by 1.3, there are S o n s with S z D, R E r, Q E q, such that D 11-S I[+R 11-Q ll+D and B E q. Hence B = Q. Since B E r, this implies B R a d thus also S = D, D E S. 3 :

-

1.5. PROPOSITION Wi.3). In a Minkowski plane with r, let p DPA, q = AQB, r = BRC, s = CSD, All-C. Then s n p = {D} if and only if q n r = {B}, and B ILD.

Proof. Lets n p = {D}. Suppose r n q = {B,B’}, B nonparallel to B’. Then there is a point S’ on s such that S’ ILB’, a point P’ on p such that P’ 11-S’, a point

Q’on q such that Q’ ll+P’, and by 1.4, Q’ 11-B’. But both Q’ and B’ lie on q thus B’ = Q’. This implies that P’ and S’must coincide and lie on both p and S, thatis, P’ = S’ = p n S = D, and B’ $D. Hence B = B’.The Converse is proved analogously.

R.Artzy

4

1.6. PROPOSITION (Fig.4). With the same hypotheses as in 1.5,let q n f = {B}and B E p. Then p n s = {B}.

Proof. Suppose B 4 p n S. Tha, by 1.4, B E pzh a m B = D.

n SI=

1, and B ILD, p n s = {D}. But

1.7. COROLLARY (Fig.4). In the derived affine plane with respect to B, the followingspeaal Desargues theorem A2 holds true: Let P Il+Q 11-R ll+S 11-P, A II-C, and the straight lines CR 11 AQ (usual line parallelism !). Then PA II CS.

Fig.4

Fig5

Proof. In the derived plane, all cycles through B become straight lines. StraIghtlines are pallel if, as cycles, their single point of intersection is B.

1.8. COROLLARY (Fig.5). The special Desargues theorem A 1, obtained from A2 bp intefchangmg ll+and II-,holds true: Let P II+Q11-R ll+S kP, D ll+B, and the lines PD II QB. Then SD II RB.

We summarize: 1.9. THEOREM. The speual Desargues theorems A 1 (Fig.4) and A2 (Fig.5) are valid in every derived affine plane of a Minkowski plane with axlorn r.

In an affine plane, let three noncoflinear points be given. Staftmg from them, we join existing points, intersect erdsfing lines, and draw parallels through exisllng points to existing lines. We continue these operations ad infmtum, without assuming the validity of any incidence theorems. The resultlng plane is called an affine m&fBafffa?6stkmkm p&m The plane is called Wnimal' because no such plane can be obtained if the numberofstartingpointsislessthan 3.

Net of Rationality in a Minkowski Plane

If, however, we assume the validity of the Desargues theorem (and no other incidence condition) from the start, the resulting plane is called an affine netdra&ima&@of an affine MWusmt.I t is well known that a M6bius net is coordinatiaed by a prime field, that is, in the case of the absence of any finiteness conditions,the field Q of rational numbers.In other words, if the three starhng points have, for instance,the CooTdinateS (O,O), ( l,O), (0,1), then all the points of the M6bius net have coordinates of the form 0;s)with r, s E Q .On the other hand, every point with such rational coordinates indeed errlsts in the Mdbius net.

However, the affine Mdbius net can already be obtained from the affine minimalfree extension plane by imposing on it incidence theorems which are weaker than the full Desargues theorem. An analogous situation exists also in projecUve planes, where the projective Mdbius net can be based on the threefold degenerationof the Desarguesmeorem ((51 p.275). In the following, we will describe such weaker conditions on which to base the construction of the affine Mdbius net. A ,nwWo-mb is a parUal affine plane in which each line is parallel to exactly one of n given mutually nonparallel lines. Thus, in a parallel n-web, there is not always a line corm- twogiven points, however Euclid's parallel axtom holds in every parallel n-web.The n line familie will be numbered:we will speak of the 1-lines, 2- lines*...,n-lines of the n-web.

In one derived affine plane of a Minkowskiplane K, consider the minimal affinefreeextmsion produced by three distinct points L, M, and N such that L II+M11-N.Let the (+)generatorsbe the 1-lines of a pafael web and the (-)generatorsits 2-lines. All the parallels to the line LN will be called 3-lines. We complete this parallel 3-web to a parallel 4-web W 1 by adding one further family of parallels to it. We repeat this procedure wit3 every other eSsthg fourth family of parallels and obtain thus 4-webs W2.W5.. The union of the 4-weh W 1, W2,. . . is a pautial affine plane I. 2.1. RESULT 111. A minimalaffine free extension is constructed, Starting out from the points L, M, and N.A 3-wb is chosen so that LM is a 1-line, MN a a-line, and LN a 3-line. The 3-web is completed to a 4-vmb by adding one further family of parallels to it. This is r e p W for every existing family of parallels distinct from the families 1,2,3. If the speual Desargues conditions A 1 and A2 are made to hold in each of these 4 - w , then the result is a M6bius net. 2.2. THEOREM. If the Axiom r holds in the infinite Minkowskiplane rr, then tbe partial plane y is an affine M6bius net, and hence an affine plane over a prime field.

5

R. Artzy

6

For the points of y we introduce coordinates (q), with rational x and y. Through every nonallinear triple of points which are mutually nonparallel in A, we draw the hyperbola with asymptotes x = const and y = const. These asymptotes then represent generators in A.The result is the classical hyperbola model of a Minkowski plane [21. We auld be tempted to call this plane a Minkowski M6bius net. However, it has a flaw: we cannot guarantee the existence of points of intersection for hyperbolas with lines of hyperbolas with hyperbolas. Intersectionsof hyperbolas with generators have rational coordinates,though, and we do have here a Minkowski plane over the rational field Q. W e have thus proved the following theorem, a special case of a well known result I3l.

2.3. THEOREM. Every mfinite KT-nearfield has a prime subfield F, such thatxa = x-l for every x f 0 in P.

REFERENCES Ill Artzy, R., 4 - w b s and the Mdbius net. Riveon Lematematika 7 ( 1954). 1-7. 121 Benz, W.,Vorlesungen uber Geometric der Algebren (Springer Verlag, Berlin-Heidelberg-New York, 1973). 131 Kerby, W. and Wefelscheid, H., Uber eine schaff 3-fach transitiven Gruppen zugeordnete algebraische Stfuktuf. Abh. Math. Seminar Hambug 37 (19721,225-235. 141 Percsy, N.. A remark on the introduction of coordinates in Minkowski planes. J. of Geometry 12 (1979). 175-163.

I51 Pickeft, G., Projektive m e n (Springer Verlag, Berlin- HeidelbergNew Yo*, 1975).

Annals of Discrete Mathematics 37 (1988) 7-20 0 Elsevier Science Publishers B.V. (North-Holland)

A NEW CLASS OF TRANSLATION PLANES

R. D. Baker and G. L. Ebert Department of Mathematical Sciences University of Delaware Newark, Delaware 19716

We define a nest of reguli to be a collection P of reguli in a regular spread S of PG(3,q) such t h a t every line of S is contained in exactly 0 or 2 reguli of P . Let U denote the lines of S contained in the reguli of some nest. If V is a partial spread of PC(3,q) covering the same points as U but having no lines in common with U , then V will be called a replacement set for U . Clearly, ( S - U ) u V is a spread of PC(3,9), yielding a (potentially new) translation plane of order q 2 which is 2-dimensional over its kernel. Nests of size (q+3)/2 were first studied (under another name) by Bruen and later by many others. Whether such (q+3)/2-nests exist for q > 13 and whether such nests are necessarily reversible are still open questions. In this paper we consider nests of size q . We exhibit an infinite family of q-nests, one for each odd prime q , and show that each nest is reversible. The translation planes so obtained appear to be new, a t least for q 2 11.

1. INTRODUCTION

Andre' [2] and Bruck and Bose [4] have shown that there is a one-to-one correspondence between spreads of odd-dimensional projective spaces and translation planes. Spreads of PG (3,q) and the corresponding "two-dimensional" translation planes are those that

have been most heavily studied. Since every spread of PC(3,q) may be obtained by replacing a subset of lines in a regular spread, we construct new spreads by finding appropriate replaceable subsets of a regular spread. In section 2 we define a nest of reguli in a regular spread, generalizing the notion of chains that was earlier introduced by Bruen [ 5 ] . In section 3 we show that nests of

size q exist for any odd prime q , and in section 4 show that this leads to a replaceable subset of lines in a regular spread of PG(3,q). The corresponding translation planes of order q 2 appear to be new, a t least for q

2

11. In section 5 we look a t

R.D. Baker and G.L. Ebert

8

some geometric properties of the spreads obtained, discuss derivation of these spreads, and consider the orbit structure induced by their collineation groups.

2. PRELIMINARY RESULTS

Let

C

C

= P G ( 3 , q ) denote projective 3-space over the finite field G F ( q ) . A spread of

is any collection of q 2

+1

skew lines, necessarily partitioning the points of

+1

and a regulus of

C

t o 3 lines of R

is transversal t o all lines of R . Any three skew lines of

is any set R of q

determine a regulus, and a spread

R

C,

skew lines such t h a t any line transversal

C

uniquely

is called reguiar if and only if the regulus deter-

mined by any three of its lines is contained in R. T h e translation plane obtained from a regular spread is desarguesian.

We define a nest of reguli t o be a collection P of reguli in a regular spread of

C

such t h a t every line of

R is contained

R

in exactly 0 or 2 reguli of P . Letting

U denote the lines of R contained in the reguli of P , a set replacement s e t for U if V is a partial spread of

C

V will be called a

covering the same points as ci

b u t having no lines in common with U . Clearly, (R-V) U V then becomes a spread

of

C

yielding a (potentially new) translation plane of order 9'.

Nests of size (q+3)/2 were first studied (under the name "chains") by Bruen [5] and Bruen and T h a s [6], and later by many others (see [l],[ 7 ] ,[ll], [14], for instance). Whether such (q+3)/2-nests exist for q

> 13 a n d

whether such nests are necessarily

reversible are still open questions. In this paper we study nests of size q , show that they always exist for odd primes

q,

and show that each such nest constructed is

reversible. The translation planes so obtained appear t o be new, a t least for q

2

11.

In [3] Bruck has shown there is a one-to-one correspondence between the lines and reguli of a regular spread

R

and the points a n d circles of a miquelian inversive

plane M ( q ) of order q . T h e reader is referred to [9] for the definition and relevant properties of a miquelian plane. Among the several models for M ( q ) , we will use the one given by W. F. Orr in [13]. T h a t is, the points will be represented by elements of G F ( q 2 ) U {m}, a n d the circles by 1-dimensional subspaces over

G F ( q ) of certain

9

A New Class of Translation Planes

2 X 2 matrices. More explicitly, a circle will be represented by a matrix of the form

[;

-Lq],

where

(Y

t

GF(qZ), a , b

6

G F ( q ) , and

(~‘7+’

+ ab # 0.

Two such

matrices represent the same circle if and only if one is a GF(q)-scalar multiple of the other.

It should be noted that the unique inversion associated with the circle

C = z

E

[;-zq]

is the semilinear fractional collineation

azq

z +

G F ( q 2 )u {m}, where the usual conventions concerning

+a

bzq -

for all

( ~ q

are invoked.

00

By

definition this inversion fixes the points of the circle C and moves all other points of M ( q ) in orbits of size two. Thus the q satisfying the equation z =

(YZ*

+a

bzq -

+1

points of C are simply those points

. We will alternately represent a circle in M ( q )

(Y‘I

by a matrix of the above type or by the set of points incident with it.

One advantage of this model is the ease with which images of circles under various maps can be computed. For instance, let R = with a6 - /37

#

0, and let Qg : z

to see that the linear fractional map

+

~

dR

’

*’ 72 + 6 +

(; f ) where

for all z

t

a, P, 7, 6

GJ’(q2)

c F ( q 2 ) u {m). It is easy

is a collineation of M ( q ) . Moreover, if C is

any matrix representing a circle of M ( q ) as described above, it is shown in [13] that

QR(C)= RCR”, where R ’ denotes the matrix obtained by raising each entry of R to the q‘* power and R

’

denotes the classical adjoint of R .

A second advantage of this model is the associated collection of computational

tools for easily determining the intersection pattern of a given set of circles. For this purpose we restrict to odd q , and let C and D be matrices representing distinct circles of M ( q ) . Define

h ( C , D ) = IIC Then define

+ Dll - IlCll - 11D11,

C

x

1 D = (-h(C,D))’ 2

where 11 I1 denotes the determinant function.

- llCll IIDII.

The following result is found in

[13], and will be stated here without proof.

Lemma (1): Using the above notation, C and D represent circles that are dis-

R.D.Baker and G . L . Ebert

10

joint, tangent, or secant accordingly as

C

x D

is a nonzero square, zero, or a

nonsquare in G F ( q ) .

We now exploit these computational tools to construct q-nests of circles in M ( q ) for each odd prime q .

3. THE EXISTENCE QUESTION

For the remainder of this paper we restrict to odd primes (rather than prime powers) q . Using the above-mentioned correspondence between the lines and reguli of a regular

spread in P G ( 3 , q ) and the points and circles of a miquelian plane M ( q ) , we construct a nest of size q

by working in M ( q ) . To accomplish this we distinguish the

3 (mod 4).

1 (mod 4) and q

cases q

Suppose first t h a t

G F ( q 2 ) , a n d let c =

3 (mod 4).

q

p(qt1)/2.

Then

cq-'

/3 denote a primitive element of

Let

= - 1 and

tive element of the subfield G F ( q ) . Let R =

(h i )

w =

= ,W+'

6'

is a primi-

and let 0 = q5R be the linear

(;

fractional collineation of M ( q ) associated with R . Let D =

')

represent

a circle in M ( q ) . Using the techniques described above, it is easy t o compute that (5

0'(D) =

+ 1)c

(s+l)2w-l

for s = 0, 1, 2,

...,

q - 1. Clearly the order of the

(s+l)c

map 0 is q , as q compute t h a t D,

is a prime. Letting D, = 6 ' - ' ( D ) for s = 1, 2, 3,

1 x D, = -(s-t)'w

[(s--t)'w

4

-

4) for s

#

- t

(see theorem (67) of

(q+1)/2 D , ' s

[lo], for

instance), we see that

!

q , we

t.

Since ( s - t ) ' ~ - 4 is a nonzero square for precisely (q+1)/2 s

...

D,

nonzero choices of

is secant to precisely

from lemma (1) and the fact t h a t w is a nonsquare in G F ( q ) . But

each circle has precisely q

+1

points. Thus, if we can show that no point lies on 3

Di's, we will have shown t h a t {D = D , , D,,..., Dp} is a q-nest of circles. To t h a t end suppose that z c D, is easy to see t h a t 0,

00

6 D, for

n Dt n D,

for distinct subscripts s , t , u . It

i = 1, 2, ..., q3 a n d hence

t 6

G F ( q ) # . Using the

A New Class of Translation Planes

inversions associated with each circle, we obtain t =

SEZq

+ s2w - 1 + SE

tEzq

zq

+ t2W zq

+ tt

- 1 - uczq

Letting z = zq and solving simultaneously, we obtain z 2 and

z2

+ ( t + u ) c z + tuw + 1 = 0.

+ uzw - 1 + UE

29

+ ( s + t ) c z + stw + 1 = 0 s = u,

This implies that

Hence the existence of a q-nest has been established for all primes q

a contradiction. f3

(mod 4).

It should also be noted that 9 fixes the point co and leaves invariant each circle of the pencil

with

carrier

C,

x D

a2

1 +4

We

00.

= a2

1

+ -.4

again

easily

compute

Since there are (q-3)/2

C, X D = 1

that

choices for 0

is a nonzero square (see theorem (67) of

[lo], for

#

a

t

and

G F ( q ) such t h a t

instance), the corresponding

circles C , together with C, form a partial pencil of (q-1)/2

circles which are dis-

joint from D by lemma (1). Hence these circles are disjoint from each circle of the q nest constructed above, since the q-nest is nothing other than the orbit of D under 0.

A trivial counting argument now shows that the q 2

+ 1 - q(q+1)/2

= (q2-q+2)/2

points of M(q) not incident with the circles in our q-nest are covered by the (q-1)/2 circles in the above partial pencil with carrier

00.

We now turn our attention to the case q

= 1 (mod 4).

Let 6 = d R be the

same linear fractional collineation of M(q) as above, but this time let D be the circle represented by

[i y ] .

Then

lemma (1) implies that D

C, x D = w

and

C,

x

D = apw

is once again disjoint from (q-1)/2

+ -,41

and

circles in the pencil

Lo (none of which is C,). Computing s = 0, 1, 2,

compute D,

as

..., q - 1.

above

we

obtain

V ( D ) = [('+')'1

Again letting D, = e'-'(D)

x D, = -1( ~ - - t ) ~ w[(s--t)2 ~ 4

- 41 for s

for s = 1, 2, 3,

#

s(sf2)w) (s+l)t

..., q ,

we easily

t . Since theorem (67) of

implies that ( ~ - t ) ~4 is a nonzero square in GF(q) for (q-5)/2

for

[lo]

nonzero choices

R.D. Baker and G.L. Eberr

12

of s

t

-

is a nonzero square in G F ( q ) , we see tha t ( s - t ) ,

and since -4

(nonzero) choices of s - t . Hence by lemma (1) each D, is

nonsquare for (9-1)/2

D,'s,secant to (9-1)/2

disjoint from (q-5)/4

- 4 is a

D t ' s , and tangent to 2 Df ' 5 . Just

as in the previous case it can easily be shown t ha t no point is incident with 3 distinct

Df 's, and hence {Dl,D,,..., D q } is a q-nest of circles by elementary counting. We thus have the following theorem.

Theorem @ Let

q

be any odd prime, and let R

be a regular spread of

PG(3,q). Then R contains a q-nest of reguli as defined above. Moreover, this q-nest can be chosen so that if U denotes t h e q(q+1)/2 distinct lines contained in the reguli of this nest, then the

lines of

(q2-q+2)/2

R - U constitute a partial pencil of

reguli.

(q-1)/2

4 . THE REPLACEMENT QUESTION

In this section we show t h at the partial spreads replaceable.

More explicitly, if

u R,, q

U =

U C R of theorem

(2) are always

{ R l , R2,..., R q } is a q-nest of reguli such t h a t

we will exhibit a partial spread of

C = PC(3,q)

covering the same

i=l

points as U a n d consisting of (q+1)/2 lines from the opposite regulus

i = 1, 2,

RPPP

for each

.., q .

To this end we again use the correspondence established in [3]between the points and circles of M ( q ) and the lines and reguli of R. Letting

6

= P(q+')i2 as defined in

the previous section, every element z of GF(q2) can be uniquely expressed as a

+ be

for some a , b e G F ( q ) . Representing the points, lines, and planes of P G ( 3 , q ) by the 1-, 2-,

and 3-dimensional subspaces of a 4-dimensional vector space over G F ( q ) , the

point z = a

+ be

of M ( q ) corresponds to the line l ( a , b ) =

and the point co of M ( q ) corresponds to the line I , = . before

R = {I,}

zu = e 2 = P(q+')

U {l(a,b)

:

Q,

is

a

primitive

element

b c G F ( q ) } is a regular spread of

again we distinguish the cases q

= 1 (mod 4) and q

F

of

GF(q).

As

Then

P G (3, q) (see 131). Once

3 (mod 4).

A New Class of Translation Planes

Suppose first that M ( q ) , D , = Oq-'(D) =

fying

zP+l =

- 1.

3 (mod 4).

q

13

Using the notation established above for

(y il)and hence consists of the points Dq = {a

That is,

P G ( 3 , q ) , we obtain the regulus R , = {Z(a,b) reguli of our q-nest, recall that 8 : a

+ bc

: b2w = a2

+ l}.

+ l}.

: b2w = a'

+ bc + a + ( b + l ) t

z

G F ( q 2 ) satis-

E

Pulling back to

To describe the other

and 8 : 00

as a col-

+ 00

lineation of M ( q ) . Defining T , to be the collineation of P G ( 3 , q ) defined homogeneously by

T I : (21,22,%3,24) Tl : (a,wb,l,O)

( z ~ + z ~ , z ~ + U ) Z ~ , Z ~ , Z ~we ),

-+

-+

T, : (b,a,O,l) -+ (b+l,a,O,l).

and

(a,w(b+l),l,O)

and similarly T , : f ,

T1 : f ( a , b ) + I ( a , b + l ) ,

see that

+

I,.

Thus

Hence T , is a collineation of the

regular spread R which is a pre-image of B under the canonical map

T

taking the

collineation group of R onto the collineation group of M ( q ) . It is easy to see that the projective order of Tl is q and

T i : (%1,z2,Z3,zq)

-+

(z1+~z,,z2+~zoz3,z3,z4) for

t'

= 1, 2, 3 ,

..., q

- 1. Hence, reading

the subscripts modulo q , we obtain

Ri = T [ ( R o )= T f ( R , ) = { I ( , , * ) : w(b-i)' i = 0, 1, 2,

..., q - 1

we know there is a unique i

0 (mod q ) such that

E$

Ri. A simple computation shows that this means w(b-i)* = a'

and hence i since q

= 26

and a

#

0 since w

#

0

is a nonsquare of G F ( q ) . One similarly

R2b = {L(a,b),

Temporarily fix a , b c G F ( q ) with

R o and P = R,OPP

+ 1 = wb2

(mod q ) . It should be noted that Ita,b) c R , implies that b

= 3 (mod 4)

shows that R , fl

of

+ 1 ) for

as the reguli of our q-nest.

If l ( = , b ) is any line of R,, t

= a'

containing

L = .

wb2 = a2 + 1. Thus l(a,b)

is a line of

is some point on this line. If L denotes the unique line

P,

an

In fact,

coordinate argument now shows that L

easy

computation

=

n Ti(L)= 4

=L

shows

n

for i = 1, 2,

that

A simple

..., q

- 1. Thus,

given a line of R,OPP and letting TI act on that line, we obtain a set of q mutually skew lines of

C,

one from each RPPP for i = 0, 1, 2,

..., q - 1.

We now more carefully look at ker(n), where z is the canonical map taking Aut(f2) onto Aut(M(q)). Thus ker

(T)

consists of the collineations of

C

leaving

R.D. Baker and G.L. Ebert

14

invariant every line of R. As before let Choose s , t

= /3q+'.

UJ

t

f(z)= (z-s)'

looks like

p

G F ( q ) such that the minimal polynomial of /3 over G F ( q ) - w-'t2.

T h e irreducibility of

choice of s and t , and also forces t h a t t vector space over

be a primitive element of G F ( q Z ) and let

#

z -+ /3z. Relative to the basis (1, -(/3-s)},

t

[i

G F ( q 2 ) defined by

the matrix representation for this opera-

The projective order of this operator, treated as a collineation

of P G ( l , q ) , is q

+ 1. T,

Next let

L!-@

0. Treating G F ( q 2 ) as a 2-dimensional

G F ( q ) , consider the linear operator on 1

tor is A =

f(z) allows for such a

denote the collineation of

P G ( 3 , q ) induced by the matrix

relative to our original basis. Since

I0 l A 1

d] 1

and

T, n.

is a collineation of order q

+1

leaving invariant each line of the regular spread

T, = (T,)' is a collineation of order (q+1/2 in ker(7r). Since T, commutes with T, and gcd (q,(q+1)/2) = I, we obtain a collineation T = T , T, of R Thus

whose order is q(q+1)/2. We now claim that the orbit under T of any line if R,OPP is a replacement set for

U . Since the lines of such an orbit consist of (q+1)/2 lines

..., q -

from RPPP for i = 0, 1, 2,

1, it suffices t o show that these lines are pairwise

skew.

To t h a t end let R , and R , be any two distinct reguli of our q-nest that share a

common

R,

n

line.

Ri = {Z(a,b),

wb2 = a'

+ 1.

Previous

work

shows that

R,

and

Ri

where i e 2b (mod q ) . As before

are secant a

#

Let L = . As indicated above,

0, b

#

with

0, and

L is a line of

15

A New Class of Translation Planes

T,, say T i ,

Suppose by way of contradiction that some nonzero power of

P'

mapped

P.

onto

As

the

matrix

- [A),

[&,)

A2"10

representation

for

-

2';

looks like

where denotes that the two , this would force , A 4 vectors are GF(q)-scalar multiples of one another. A simple induction argument shows I

0

that A Z k has the form have y,a

+ y,bw

:1 1,

= 0, and therefore

where y 2 # 0 as k wb2 = u2

# 0. Hence

+ 1 = (y:w2b2+yi) /

we must

yi. This

implies that

w~(Y?-w-~Y;)b 2 = - y l

Y:

is a nonsquare in

G F ( q ) , since q

- w -l y; = det(AZk)= (det A)2k is necessarily a square in

3 (mod 4).

But

G F ( q ) , yielding a

T, maps Q' onto Q , and hence no line in the orbit of T : b ( L ) under T , meets L. The transitivity inherent in contradiction. One similarly shows that no power of

the problem and the commutativity of T, with T , now implies that the orbit of L under T = T , T , is a replacement set for U. Finally, when q

+w

Ri = { l ( a , b ): a' line of R,, containing

n Rzb

l(=,b).

=

of

#0

iW = {) a + b e : az + w

[

for i = 0, 1, 2,

= w(b-i)'}

then necessarily b

#

0 and

Moreover, if a = O

l(5+5)

l(a,b)7

a

1 (mod 4), a completely analogous situation occurs. This

D, = D , =

time

Ro

3

+a,~)

if a

Z0

1.

If

= wb2},

..., q

and

- 1 . If l ( a , b )

hence

denotes any

R2b is the other regulus of the q-nest

l(a,b)

is a line of

R o chosen so that

and L is the unique line of R,OpP passing through the point

l(a,b),

then

= < ( a , w b , l , O ) , ( O , w , b , a ) > where

= L

Similar arguments to those given above show that the orbit of L

n

under T = T , T ,

is a replacement partial spread for U in this case as well. Hence we have the following theorem.'

R.D. Baker and G.L. Ebert

16

Theorem

a Let

q

be any odd prime, and let

tinct lines of a regular spread

R

that are contained in a q-nest

tial spread V can be obtained as the orbit of any line in whose order is q ( q + 1 ) / 2 .

i=O,1,2

q ( q + 1 ) / 2 dis-

{ R o , R,, ..., Rq-,}

U is replaceable. Moreover, a replacement par-

constructed as in theorem (2). Then

R

U denote the

R,OPP

under a collineation of

V consists of ( q + 1 ) / 2

The set

lines from

for

R:pp

,...) 4 - 1 .

Corollary. For any odd prime q a nondesarguesian translation plane of order q2 can be constructed from the spread (R-U)

u

0

V of theorem (3).

For q = 5 the spread obtained via theorem (3) is equivalent to the spread S, found in a computer search by Oakden [la]. It was a new plane a t the time it was found. For q = 7 the spread obtained is most likely equivalent to the one recently found by Cohen and Ganley [8] using Chebyshev polynomials. However, for q

2

11

the translation planes obtained here appear to be new.

5. DERIVATION AND COLLINEATION GROUPS

In this section we discuss the reguli contained in the spreads constructed above, some derivations of these spreads, and the orbit structure of the spreads under their collineation groups.

(J Theorem 4

If S = ( 0 - U )

u

V is a spread obtained via theorem (3), then S

contains at least q reguli.

Proof. From theorem (2) we know that the lines of pencil of (9-1)/2

R-U

comprise a partial

reguli with carrier loo. Our work above shows that

T of any line in

taken to be the orbit under

L = < ( a , w b , l , O ) , (-b,-a,O,l)> (mod 4) and wb2 = u2

+w

is a line of (a

# 0)

if q

points < ( l , O , - ~ , - w b ) > and points < ( l , O , - a w - l , - - b ) >

V may be

R,OPP.

An easy computation shows that

R,OPP,

where wb2 = a'

+1

1 (mod 4). In fact, L if q

= 3 (mod 4),

and < ( O , w , b , a ) > if q

G

if q

G

3

contains the

while L contains the

1 (mod 4).

A New Class of Translation Planes

As shown above L , T , ( L ) , T : ( L ) , ..., 2'-' We claim that this set, together with I,,

17

( L ) is a set of

q skew lines in V.

forms a regulus. To see this simply let

M = , N = ,

{

and if q

= 3 (mod 4)

< ( O , w , b , a ) , ( a , w ( b + l ) , b , a ) > if q

= 1 (mod 4)



More easy calculations show that M , N , and J are three skew lines that meet each

L , T , ( L ) , ..., Tf-'(L),and therefore the latter set of

of I,

a regulus R of S . Since T,(I,) = I, a

regulus

(q-1)/2

of

S

for

+ (q+1)/2 = q

the line I,

+1

q

skew lines forms

and T 2 ( V )= V , we obtain that T i ( R ) is

..., ( q - 1 ) / 2 .

i = 0, 1, 2,

Hence

S

contains at

least

reguli. It should be remarked that all of these reguli contain

and any two of them share only that line.

0

When q = 5 or q = 7, it has been verified that the above reguli are the only ones contained in S, and it is conjectured that this is always the case. Reversing any one of these reguli in S when q = 5 yields a spread S' that contains exactly one regulus.

The spread 5"

is not subregular and, in fact, is isomorphic to Oakden's

spread S9 (see [ 1 2 ] ) . When q = 7 one of the three reguli contained in Ci-U cial. It corresponds to the circle

c,, as defined previously.

Whenever q

= 3 (mod 4),

S will always contain such a "special" regulus. Reversing this regulus when again yields a non-subregular spread one of the other six reguli of

3

S yields

is spe-

q =7

containing exactly one regulus. Reversing any

a spread inequivalent to

s, although still non-

subregular and containing exactly one regulus. It appears as if the six spreads so obtained are all equivalent, but this has not been completely verified. Finally we discuss the orbit structure of S under its collineation group. This is equivalent to looking a t the orbit structure of the line at infinity under the translation complement of the corresponding translation plane. As shown before, the collineation

TI maps each of the (9-1112

reguli of R-CI

upon itself by fixing the line I ,

and

moving the remaining q lines in a single orbit. We also know that V itself is a single orbit under

T = T, T , . Thus, using only the collineations T, and T, of S , we

obtain one orbit of size 1, (q-1)/2

orbits of size q , and one orbit of size q ( q + 1 ) / 2 .

R.D. Baker and G.L. Ebert

18

When q = 7 these orbits can be further combined by using two other collinea-

[fi]

tions of S , one being an involution induced by the matrix

and the other

being an element of order 3 found by ad hoc methods. With the addition of these col-

S

lineations

is reduced to one orbit of size 1, one orbit of size 7, and one orbit of size

42. It is because of this orbit structure that we feel this particular spread is probably

equivalent t o the one found by Cohen and Ganley [8]. The one orbit of size 7 corresponds t o the "special" regulus of Rsize 7 were combined by the mapping

U

mentioned above. The other two orbits of

[1 ;,:*] ,

and then this orbit was combined

with the orbit of size 28 by the ad hoc collineation of order 3.

It should be mentioned that this collineation of order 3 for q= 7 is not inherited from the collineation group of the regular spread R. behavior cannot happen for large

q.

The mapping corresponding to

3 (mod 4 ) .

however, is a collineation of S whenever q

Proposition (5J- Let

C = PG(3,q) the matrix regulus of R-

Proof. T 3 : l(a,b)

R-U =

+

q

G

3 (mod 4) be a prime, and let S

be a spread of

obtained via theorem (3). Then the collineation T , of

[*]

is a collineation of

U

C

induced by

S . Moreover, T , leaves invariant one

and pairs off the remaining ( q - 3 ) / 2 reguli of R- U.

Straightforward '(-a,-b).

: a2

Thus

computations

show

T,

involution

+ 1 is a square in

L =

is

T,(L) = 2", by deciding t h a t i t i s true i f f Q/Z i s centrally nilpotent of class ( k - 1 ) . Despise several analogies, the theory of C M L s display f a r more features t h a n the theory of groups. 1

- The f i r s t sign of t h i s i s the identity n(x,y,z) = ( n

t

:

3m)(x,y,z)

which holds for any two integers n and m. As a consequence, b o t h the central factor and the derived subloop have exponent 3, since (x,y,3z) = 0 = 3(x,y,z). 2

- B u t above a l l there i s the deep Bruck-Slaby Theorem

: any CML

on n genera-

tors, n > 2 i s centrally nilpotent of class l e s s than n . As a consequence, the C M L s enjoy very strong properties which are t o be compared

t o those of locally nilpotent groups having a locally f i n i t e derived subgroup. 2.12. Now each terentropic quasigroup (Q,.) i s isotropic t o an e s s e n t i a l l y unique CML, t h a t may be constructed by taking an a r b i t r a r y element e from Q and by endowing the s e t Q w i t h the binary multiplication : X,Y

-

X+Y =

(x/e).((e/e)\y)

where x/u and u \ x are defined as usual by :

(x/u).u = x = U.(U\X). Conversely i f (Q,t) i s a CML, one may recover a l l the terentropic quasigroups isotopic t o ( Q , i ) by considering binary operations of the form : x*y = c t f ( x )

t

g(y)

where c is a central element of ( 0 , t ) and f,g are commuting automorphisms of

L. Beneteau

26

(a,+)

such t h a t , f o r any x i n Q, f ( x ) + x and g ( x ) + x a r e c e n t r a l elements. When

(Q,+) i s an a b e l i a n group, t h e n (Q,

0 )

i s e n t r o p i c , and c o n v e r s e l y . F u r t h e r i f

( Q , * ) i s g e n e r a t e d b y n elements w i t h n > 1 t h e n t h e i s o t r o p i c CML can be gener a t e d b y ( n - 1 ) elements "up t o c e n t r a l e l e m e n t s " . As an i m p o r t a n t consequence one may g e n e r a l i z e S o u b l i n ' s r e s u l t about t h e n i l p o t e n c e o f d i s t r i b u t i v e q u a s i groups : Theorem 2.12 : Any n-generated t e r e n t r o p i c q u a s i g r o u p w i t h n > 2 i s c e n t r a l l y n i p o t e n t o f c l a s s l e s s t h a n (n-1) w i t h r e s p e c t t o t h e e n t r o p i c l a w . T h e r e f o r e s e v e r a l p r o p e r t i e s o f t h e CMLs c a r r y o v e r t o t h e t e r e n t r o p i c q u a s i groups. As an i n t e r m e d i a r y r e s u l t i n t h e p r o o f , one o b t a i n s t h e f o l l o w i n g byproduct : P r o p o s i t i o n 2.13 : I n any CML (G,+)

t h e minimal subsets t h a t g e n e r a t e G up t o

c e n t r a l elements f o r m t h e f a m i l y o f t h e bases o f a m a t r o i d i n t h e u n d e r l y i n g s e t G. L e t u s t u r n Oow t o t h e s t u d y o f t h e m u l t i p l i c a t i o n group PI = Mlt(G,.)

2.14.

of

any c u b i c q u a s i g r o u p (G, ) . Though s e v e r a l p a i r w i s e non-isomorphic c u b i c q u a s i groups can be i s o t o p i c t o t h e same CML, t h e y a l l have t h e same m u l t i p l i c a t i o n group. L e t us say t h a t a " r e f l e x i o n " o f a CML(G,+) the form : x

+

i s a permutation u o f G o f

d-x = x ' , where d i s any f i x e d element o f G ; d i s then unique

and x ' = a ( x ) i s c h a r a c t e r i z e d b y x

+ x'

t h e s e t o f a l l t h e r e f l e x i o n s and s e t

6?

= d ; n o t a t i o n : u = ud. Designate by

M" =

< R > . The subgroup generated by t h e

N

squares o f elements o f M i s denoted by i ( 2 ) . P r o p o s i t i o n 2.15 : f? i s t h e m u l t i p l i c a t i o n group o f e v e r y c u b i c q u a s i g r o u p

61

i s o t o p i c t o (G,+).

I? i s

i s a u n i o n o f complete conjugacy c l a s s e s . Any element o f

a p r o d u c t o f p a i r w i s e d i s t i n c t elements f r o m

equals

6 . The

d e r i v e d subgroup

M"'

and i s c o n t a i n e d i n M=Mlt(G,+). t a r y a b e l i a n 2-group, [ M : MI = 2. 0

Whenever (G,+)

P r o o f : A c u b i c q u a s i g r o u p i s o t o p i c t o (G,+)

has a m u l t i p l i c a t i o n o f t h e f o r m

i s n o t an elemen-

N

x * y = c + f ( x ) + g ( x ) where f = - I d = g , x

+ x'

= d and L a ( x )

+ La(

XI

) = (a

thus M l t ( G , * ) = < - I d > M . I f x ' = o d ( x ) ,

+ x) + (a + x ' )

2a + d = 6, hence LaudLa1 = ag.

T h i s shows t h a t @ i s s t a b l e under c o n j u g a t i o n by any element . The r e mainder of t h e s t a t e m e n t i s a s t a n d a r d consequence o f t h e f a c t t h a t fi i s generated b y a normal s u b s e t o f i n v o l u t i o n s . I n case (G,+) a b e l i a n 2-group, Inn(G,+).

i s n o t a elementary

- I d i s an o r d e r 2 element, and t h u s i t cannot b e l o n g t o

0

3 . GEOMETRICAL MOTIVATIONS

3.1.

I n what f o l l o w s k i s a f i e l d whose a l g e b r a i c c l o s u r e i s

k

andIPn(k) i s

t h e n-dimensional p r o j e c t i v e space o v e r k . L e t V be an a b s o l u t e l y i r r e d u c t i b l e

Quasigroups and Groups Arising from Cubic Surfaces

27

cubic hypersurface d e f i n e d over k : the p o i n t s o f V are those whose homogeneous coordinates s a t i s f y an equation o f the type

. .Xn)

F(Xo, X1,.

= 0 where F i s a

homogeneous form o f the t h i r d degree over k which i s assumed t o be i r r e d u c t i b l e i n ECXo,

Denote by E = Vr(k) the s e t o f non-singular k - p o i n t s o f V.

X1,...Xn].

By d e f i n i t i o n three p o i n t s x, y, z o f E are s a i d t o be co&ea/r

exists a l i n e

i f f there

such t h a t :

-

e i t h e r x, y, z

-

o r k * V = x + y + z ( i n t e r s e c t i o n c y c l e ; each p o i n t t u r n i n g up e q u a l l y o f t e n

k c V

E

as i t s i n t e r s e c t i o n mu1 t i p 1 i c i t y ) . The so-defined three-place r e l a t i o n o f c o l l i n e a r i t y L(x, y, z) i s c l e a r l y i n v a r i a n t under any permutation o f x, y, z, and f o r any x, y i n E t h e r e e x i s t s a t l e a s t one p o i n t z i n E such t h a t L(x, y, z ) . We must now d i s t i n g u i s h between the case o f a curve and t h e case dim V 3.2. The case n = 2 ( i .e. dim V

2

2.

1)

P r o p o s i t i o n : I f V i s a b s o l u t e l y i r r e d u c t i b l e cubic plane curve o f P 2 ( k ) then :

( i ) V admits a t most one s i n g u l a r p o i n t ( e i t h e r E V o r E = V \ {s} where s i s the unique s i n g u l a r p o i n t ) ; ( i i ) no l i n e i s contained i n V ; ( i i i ) f o r any two ( n o t n e c e s s a r i l y d i s t i n c t ) p o i n t s x and y from E t h e r e i s o n l y one p o i n t z i n E such t h a t L(x,y,z), operation x,y

-+

and o n l y one l i n e s a t i s f y i n g k . V = x + y + z

; ( i v ) the

x o y = z t u r n s E i n t o an e n t r o p i c cubic quasigroup : ( v ) each

o p e r a t i o n o f t h e form x, y

-+

x;;y=uo ( x o y ) . where u i s an a r b i t r a r y f i x e d

element o f E, t u r n s E i n t o an a b e l i a n group (Lame's theorem).

17

The f o r e g o i n g f a c t s are mostly c l a s s i c a l , though p a r t ( i v ) i s v e r y seldom exp l i c i t y s t a t e d i n the l i t e r a t u r e .

3.3.

From now on u~nurnen > 2, so t h a t dim V = n - l r 2 ( t h e conclusions are n o t

v a l i d f o r a curve). We a l s o assume t h a t k i s i n f i n i t e ( t h i s ensures the e x i s and (iii) of tence o f some p o i n t s ) . One observes a t once t h a t p a r t s ( i ) (ii) p r o p o s i t i o n 3.2 f a i l t o h o l d here. I n o r d e r t o recover a uniquely d e f i n e d b i n a r y o p e r a t i o n one must replace E = Vr(k) by a s u i t a b l e f a c t o r s e t T = E/R. I n the sequel R designates a b i n a r y r e l a t i o n on E, say R f o r (x,y)

c

E

x

E ; we w r i t e xRy

R.

D e f i n i t i o n : R i s s a i d t o be an ad&nible

trd&tion i f

R i s an equivalence r e -

l a t i o n on E such t h a t f o r any f i v e p o i n t s x, y, z, y ' , z ' from E such t h a t L(x, y, z), L(x, y ' , z ' ) and yRy' one has zRz'. I n t h e n e x t statements R i s any given admissible r e l a t i o n on E, and a t y p i c a l element o f the f a c t o r s e t E = E/R i s where x E E.

x=x

Lemma 3.4 : Given any two elements X and Y from Z = {Z

E

El

3x

E

X,

3y

E

Y, L ( x , y,

f the Z)

set :

holds}

L. Benetem

28

i s an equivalence class modulo R . The operation X , Y + Z = XoY makes T i n t o a symmetric quasigroup. 0 Proof : Let xo, yo be representatives of X and Y respectively. There e x i s t s zo i n E such t h a t L(xo, yo, zo) holds. Consider any t r i p l e of collinear p o i n t s x, y, z with x E X and y E Y . Let z ' be a p o i n t from E such t h a t L ( x o y y, z ' ) holds. Since yoRy the admissibility implies t h a t zoRz'. Likewise xoRx implies z'Rz, hence z ' E fo, namely Z e Z o . Conversely i f t E io, then any p o i n t y' such t h a t L ( x , y ' , t ) holds belongs t o Y , so t h a t Z=.?,. The i n i t i a l inclusion was sufficient t o ensure the existence of an operation o on E such t h a t = f. This operation i s of course symmetric. L ( x , y, z ) , d X o 3.5. Let us come back t o the hypothesis of 3.1 and 3.3. For any x

~ denote E by

Tx the tangent hyperplane a t x w i t h V . Let Cx = Tx n V . Of course Cx i s a (possibly degenerate) cubic hypersurface in Tx. Following Manin we shall say t h a t a p a M x from E i s 06 g e n u d a p e i f the following two conditions are satisfied : (1) C, i s geometrically irreductible and reduced ; ( 2 ) there exists a l i n e L containing x and another point y from C x \ {x} which i s n o t completely contained i n Cx ( t h i s l a t e r condition means t h a t x i s not "conical" i n C x ) . Denote by the (possibly empty) open s e t of a l l the points of general type of E. For any x i n E there e x i s t s a unique birational map tx : V + V such t h a t t x ( y ) i s defined for every y E V \ C x and such t h a t L ( x , y , t x ( y ) ) holds (Manin, c51, 11, 12-13).

q(V)

Theorem 3.6. (Manin'n theatrem) : Let V be an absolutely irreductible cubic hypersurface of dimension 2 2 over an i n f i n i t e f i e l d k . Let E = Vr(k) be the set of i t s n o n - s i n g u l a t k - p o i n t s . Assume t h a t Q ( V ) 0. Consider some admiss i b l e relation R on E and l e t ( r , o ) be the corresponding symmetric quasigroup. Then every class modulo R i s dense in the Zariski topology and admits a representative i n g(V).Further (r,o) i s a cubic quasigroup in which ( X2 )2 = x2 . 0

.[

Remark 3.7. F can be non-&ivhL o d y i f k ( i f k is algebraically closed there i s just one class [Manin 74, 11.1.3.1. ( i ) ] ) . B u t an open quenfion whe&h e m be non-e.nOtuph .the gene&L e a e (see [Manin, [5], 11, Problem 11.111).

(r,o)

2 3.8. Let (9, ) be a cubic quasigroup. Since x2-y2 = (xy) xhe n q w e mapping x + p(x) = x 2 42 m endomo~pkinmad (Q,.). Hence i t s image Q2 = {x2 / x Q} i s a subquasigroup and the s e t Idemp(Q,-) of idempotent elements of (Q,.) i s e i t h e r empty o r a subquasigroup. Proposition 3.9. The three following conditions are equivalent : ( i ) a l l the squares are idempotent ( ( x ' ) ~ = x2) ; ( i i ) (Q,.) admits an idempotent and i t s

Quasigroups and Groups Arising from Cubic Surfaces

29

(Q, -) i s a d i r e c t product A D o f an elemenr e l a t e d CML has exponent 6. (iii) t a r y abelian 2-group A by a d i s t r i b u t i v e cubic quasigroup D. When these condit i o n s are s a t i s f i e d the f a c t o r s o f the decomposition i n ( i i i ) are e s s e n t i a l l y unique, and m r e p r e c i s e l y : - D - Q2 = Idemp(Q, *) f.

-

a l l the f i b r e s o f p are groups isomorphic t o A.

0

Proof : Each o f the conditions (i)(ii) ( i i i ) ensures the existence o f a t l e a s t one idempotent e . L e t us f i x e f o r a l l t h e p r o o f and s e t x t y = e . x y . Since e 2 = e , x y = - x - y and x 2 = -2x. 2 2 2 S t a r t i n g from (i) x2-x2 = x2, one deduces : 2x = x t x = e.x2x2 = e.xx = 2x, hence -4x= 2x and 6x = e . I f one assumes t h a t (Q, s a t i s f i e s ( i i ) , then 2 p(x) = x = -2x i s an idempotent endomorphism o f (a,+) and any x can be w r i t t e n Im(p) and ax Ker(p) ( i n uniquely as a sum o f the form x = a x + d , where d 2 2 f a c t : d = x and aX = x - x ) . Since a CPlL w i t h o u t order 3 elements must be an a b e l i a n group, (A,+) i s an elementary abelian 2-group and V a, b A, 2 a t b = - ( a t b ) = a b ( t h e operations ( t ) and (.) coincide i n A ) . Since y = y f o r 2 2 2 any y i n Q , ( Q , a ) i s a d i s t r i b u t i v e cubic quasigroup. Further f o r x = a t x 2 2 x2 and y = b t y we have x y = - x - y = a - x 2 - y = a b +x2y2, whence x -t (ax,x ) Y X x x i s an isomorphism from (Q,.) onto (A,.) x (Q',.). L a s t l y i f one assumes (iii), 2 2 i f x = (a,d), a E A, d E D, then x (OA,d) = (x2)', so t h a t x = x i f f a = O A . 0 0)

The preceding p r o p o s i t i o n has two immediate consequences, b o t h o f geometrical interest

.

C o r o l l a r y 3.10 : I f

(a,.)

i s a cubic quasigroup i n which ( X ~ ) ~ = then X ~ ,using

the notations o f XI.3.9 i t s m u l t i p l i c a t i o n group i s isomorphic t o the d i r e c t 2 product o f A by the centerless Fischer group Mt(Q ,.). 0 2 C o r o l l a r y 3.11 : I f (Q,.) i s a cubic quasigroup i n which (x2)' = x i d e n t i c a l l y , 2 2 then (a,.) i s non-entropic i f f Q i s non-entropic, o r e q u i v a l e n t l y i f f Q cont a i n s an 81-order subquasigroup isomorphic t o (L3, .)). 0 We j u s t mention here t h a t CMLs arose i n combinatorial contexts. There i s a number of papers devotes t o the H a l l t r i p l e systems ; now i t i s well-known t h a t these special S t e i n e r t r i p l e systems are merely another way t o describe the exponent 3 CMLs ( o r e q u i v a l e n t l y , the d i s t r i b u t i v e t o t a l l y symmetric quasigroups). REFERENCES

C 1 1 Beneteau L ., Free Commutative Moufang Loops and Anticommutative Graded

Rings, Journ. Alg. 67 (1980) pp. 1-35 ; MR 82c 20 118. C21 Beneteau L., Contribution Z l ' g t u d e des Boucles de Pbufang c o m u t a t i v e s e t des espaces apparent& (AlgCbre, Combinatoire, G e o k t r i e ) , These d ' E t a t , Univ. de Provence, M a r s e i l l e (1981).

L. Beneteau

30

131 Bruck R.H., A s u r v e y o f b i n a r y systems, S p r i n g e r , B e r l i n , New-York, (1958) ; M.R. 20 + 7 6 . C41 K e p t a T . S t r u c t u r e o f t r i a b e l i a n quasigroups, Comment. Math. U n i v . Carol inae 17 (1976). C51 Manin Yu.I., Cubic forms ; a l g e b r a , geometry, a r i t h m e t i c s , N o r t h - H o l l a n d P u b l i s h i n g Company, Amsterdam, London, New York, 1974. 1 6 1 S o u b l i n J.P., Etude a l g e b r i q u e de l a n o t i o n de moyenne, J o u r n . Math. pures e t appl S @ r . 9, 50, (1971) pp. 53-264 ; M.R. 45 Jt436,# 437 e t 9 438.

.,

Annals of Discrete Mathematics 37 (1988) 31-42 0 Elsevier Science Publishers B.V. (North-Holland)

31

BLOCKING SETS I N THE LARGE MATHIEU DESIGNS,I : THE CASE S ( 3 , 6 , 2 2 ) L u i g i a BERARDI Dipartimento d i Ingegneria E1ettrica.Universita'

de L ' A q u i l a , I t a l i a

A c l a s s i f i c a t i o n o f t h e b l o c k i n g sets i n S ( 3 , 6 , 2 2 ) i s g i v e n . Each of them i s c h a r a c t e r i z e d by i n c i d e n c e p r o p e r t i e s . 1. INTRODUCTION Denote by ( S , a )a p a i r where S is a v - s e t of e l e m e n t s c a l l e d p o i n t s and 93 i s a f a m i l y of s u b s e t s of S c a l l e d b l o c k s . The p a i r ( S , g ) is c a l l e d a S t e i n e r s y s t e m S ( t , k , v ) , where t , k , v are i n t e g e r s w i t h 2 l t < k < v , i f : i) e v e r y b l o c k c o n t a i n s k p o i n t s , i i ) e v e r y t - s u b s e t o f S i s c o n t a i n e d i n e x a c t l y o n e b l o c k of g . Denote by r s ( s = O , l , t ) t h e c o n s t a n t number o f b l o c k s t h a t c o n t a i n a f i x e d s - s u b s e t o f S. Then:

...,

s=o,1 , .

.., t .

C o n s i d e r a f i x e d X E S and t h e f a m i l y B x =+-(XI, x E B E ~ ) ; t h e p a i r (S- { x } , B x ) i s a S t e i n e r s y s t e m S(t-1,k-1,v-1) c a l l e d t h e c o n t r a c t i o n of (S,B) a t p o i n t x. We d o n ' t know e x a m p l e s o f S t e i n e r systems w i t h t > 5 . T h e r e a r e two v e r y and S ( 5 , 8 , 2 4 ) s p e c i a l t y p e s o f S t e i n e r s y s t e m s w i t h t = 5 , namely S ( 5 , 6 , 1 2 ) which t o g h e t h e r w i t h t h e i r c o n t r a c t i o n s are r e s p e c t i v e l y c a l l e d t h e l i t t l e and t h e l a r g e M a t h i e u d e s i g n s . T h e s e s y s t e m s a r e u n i q u e l y d e t e r m i n e d by t h e i r p a r a m e t e r s . They h a v e been t h e s u b j e c t o f s e v e r a l s t u d i e s f o r t h e f o l l o w i n g r e a s o n s : t h e authomorphism g r o u p s o f S ( 5 , 6 , 1 2 ) , S ( 4 , 5 . 1 1 ) , S ( 5 , 8 , 2 4 ) , S ( 4 , 7 , 2 3 ) S ( 3 , 6 , 2 2 ) are r e s p e c t i v e l y M a t h i e u ' s f i v e c l a s s i c a l g r o u p s M , , , M,, , M,, M,, M,, i . e . t h e f i r s t f i v e s p o r a d i c g r o u p s d i s c o v e r e d i n 1850, c f . [6],[lO]. My p u r p o s e i s t o c o n t r i b u t e t o t h e s t u d y of t h e s e d e s i g n s w i t h r e g a r d t o t h e i r blocking sets. A blocking set i n a S t e i n e r system i s a set C such t h a t every block i n t e r s e c t s C b u t n o b l o c k i s c o n t a i n e d i n C. Note t h a t t h e complement of a b l o c k i n g set i s a b l o c k i n g s e t t o o . A b l o c k i n g set C i s s a i d t o b e r e d u c i b l e i f t h e r e e x i s t s a p o i n t X E C s u c h t h a t C-{x} i s a b l o c k i n g s e t . O t h e r w i s e , C is said to be irreducible. I n [l] w e o b t a i n e d a v e r y s i m p l e c h a r a c t e r i z a t i o n f o r b l o c k i n g s e t s i n S ( 5 , 6 , 1 2 ) and t h e i r c o n t r a c t i o n s . The b l o c k i n g sets i n S ( 5 , 6 , 1 2 ) are t h e 6 - s e t s which are n o t b l o c k s . The s y s t e m S(4.5.11) d o e s n o t c o n t a i n b l o c k i n g s e t s , w h i l e i n t h e i n v e r s i v e p l a n e S ( 3 , 4 , 1 0 ) , o f o r d e r 3 , t h e b l o c k i n g sets are t h e u n i o n o f two 2 - s e c a n t b l o c k s i n which we d e l e t e a common p o i n t , and w i t h s y m m e t r i c d i f f e r e n c e d i s t i n c t from a b l o c k (cf.[5] ). F i n a l l y , a s proved i n [8], t h e s y s t e m S ( 2 , 4 , 9 ) , t h a t i s AG(2,3),does n o t c o n t a i n b l o c k i n g sets. I n C2J w e o b t a i n e d t h e c h a r a c t e r i z a t i o n of b l o c k i n g sets i n S ( 2 , 5 , 2 1 ) PG(2,4), t h a t i s t h e c o n t r a c t i o n of S ( 3 , 6 , 2 2 ) . The a i m of t h i s p a p e r i s t o g i v e a c h a r a c t e r i z a t i o n of b l o c k i n g s e t s i n S(3,6,22) t h a t i s t h e second c o n t r a c t i o n of t h e big system S(5,8,24). In S ( 3 , 6 , 2 2 ) w e h a v e t h e f o l l o w i n g e x a m p l e s o f b l o c k i n g sets: EXAMPLE (a).(F-Fano s e t s ) . I n S ( 3 , 6 , 2 2 ) i . e . e v e r y b l o c k i s 1 - s e c a n t or 3 - s e c a n t . regard to t h e p a t t e r n of S(3,6,22) given them. I n S e c t i o n 3 w e s h a l l c h a r a c t e r i z e

t h e r e e x i s t s a 7 - s e t of t y p e ( 1 , 3 ) , The s e t { 1 , 2 , 3 , 8 , 1 5 , 1 6 , 1 9 } , w i t h i n S e c t i o n 2 ( C o n s t r . I ) is one o f t h e s e s e t s t h a t we c a l l Fano sets.

L. Berardi

32

EXAMPLE ( b ) . (D-sets). Suppose B,B' B n B ' = @ . F i x u E B , V E B ' . The s e t D :=(B-{u})

u (B'

are

-

two

blocks i n

S(3,6,22)

with

{v} )

i s an i r r e d u c i b l e blocking set. (See next 4.1). Example ( c ) . ( E - s e t s ) .

I n S ( 3 , 6 , 2 2 ) w e c o n s i d e r t h e 11-sets

E := F U B -{x} where B i s a 1 - s e c a n t b l o c k of t h e Fano s e t F a t t h e p o i n t x. I n s e c t i o n 7 w e s h a l l p r o v e t h a t s u c h s e t s are s e t s o f t y p e ( 1 . 3 . 5 ) . They are i r r e d u c i b l e b l o c k i n g 2ets. I n t h i s paper w e prove t h a t t h e above examplesare t h e i r r e d u c i b l e set7 of 5 ( 3 , 6 , 2 2 ) . Namely I a m g o i n g t o p r o v e .

blocking

1.1 THEOREM. Denote by C a b l o c k i n g set i n S ( 3 , 6 , 2 2 ) . Then IC.1 2 7 , and

( a ) 4 C I = 7 i m p l i e s t h a t C i s a Fano s e t , ( s e c t i o n 3). (b)

1 Cl= 8

o r 9 i m p l i e s C=FUX l y ( s e c t i o n 4 and 5).

with

X n F = # and 1 X ( = l o r I X ( = 2 , r e s p e c t i v e -

( c ) l C I = l O . Then e i t h e r C = D ( i r r e d u c i b l e case) o r C=FUX, where X i s a 3-set w i t h F O X =@,such t h a t t h e b l o c k c o n t a i n i n g X i s 1 - s e c a n t t o F . ( S e c t i o n 6 ) (d)

(e)

1 C /=

11. Then C i s o n e of t h e f o l l o w i n g sets ( i ) C i s a n E - s e t i . e . a set o f t y p e (1,3,5), ( i r r e d u c i b l e c a s e ) . ( i i ) C=DU{w} , w # D . ( i i i ) C=F U X where X i s a 4 - s e t w i t h F n X = # s u c h t h a t e v e r y c o n t a i n i n g a 3-set o f X i s 1 - s e c a n t t o F. ( S e c t i o n 7 ) . IC1212 i m p l i e s t h a t C i s r e d u c i b l e and C is t h e s e t c o n s i d e r e d above. ( S e c t i o n 8 ) .

complement of a

block

blocking

The c l a s s i f i c a t i o n o f b l o c k i n g s e t s i n S ( 4 , 7 , 2 3 ) and i n S ( 5 , 8 , 2 4 ) a p p e a r i n two n e x t p a p e r s .

will

be

2. RESULTS AND LEMMAS we Denote by C a c-set i n S ( t , k , v ) . F o l l o w i n g G.TALLIN1 ( c f . [ B J ) , k ) t h e number o f b l o c k s t h a t a r e i - s e c a n t C . W e have: by t , ( i = O , l ,

...,

(s=O,l, I n t h e case 'of S ( 3 , 6 , 2 2 ) w e h a v e r3 = 1, becomes : t o +t , + t , + t 3 + t,+ t , t

(2.2)

r2 = 5,

t, =

...,t ) . r, =21,

r,,=77

77

t , + 2 t 2 + 3 t , + 4 t , + 5 t , + 6t, = 2 1 ~ 2t2+6t,+12t,+20t,+30t,

=

denote

5~(~-1)

6 t , +24t, + 6 0 t 5 + 1 2 0 t , = c( c-1 ) ( c - 2 )

.

and

(2.1)

Blocking Sets in the Large Mathieu Designs

33

l a t e r on.

.

If CONSTRUCTION I. Let u s c o n s i d e r TODD'S t a b l e o f S ( 5 ,8 ,2 4 ) , g i v e n i n [lo] we c o n t r a c t a t p o i n t s 0,- w e o b t a i n t h e f o l l o w i n g l i s t o f b l o c k s of S ( 3 , 6 , 2 2 ) for the set {1,2,...,22}. 5

6

7 13 16 17

2

5

6

8 12 14 21

2

5

7 1 1 14 18 19

5

8 9 I0 17 I 8

5 10 13 1 4 15 22 5 I 1 12 15 17 20

6 7 8 91120 6 7 10 1 2 15 I 8

3 4 8 921 3 6 12 16 26

3 7111315 2 3 10 18 19 22 2 4 5 6 10 I 1 2 4 7 17 18 20 2 4 12 14 15 19 2 5 7 91222 2

6 9 10 16 21 22 6 10 14 17 19 20

2

5

2

5 15 16 I 8

6 I 1 I 2 13 19 22 7 8 13 I 8 21 22

2

7 9 14 15 17 21 7 15 10 19 20 22 8 9 12 13 15 16

?

8 LO I 1 15 I9 21

21

6 8151722 2 6 9 13 14 I 8

2

3

51417

3

4

5121318

1

2

4131622

3

4

6

3 4 I0 15 16 17 3 5 6 91519

4 10 14 18 21

19 21

I

7 I2 13 14 20

1 13

6 18 20 22 9 1 1 13 21 10 12 I 6 19 10 11 17 12 9 14 19 22 I5 17 I 8 19

I 8 16 17 20 21 4 8 I 1 13 I 4 17

4 9

I1

7

lo

2 9 11 16 I? 19 2 I 1 14 20 21 22

1

5

2 LO I2 13 17 21 I 6 11 14 15 I6

1

7

1

8

9 I 2 18 19 20 21

8 10 I 2 20 22

1

8

1 4 16

7 1 4 2 2

1 2 6 71921 I 2 8 I 1 I2 18 1 2 9 LO 15 20 1 3 4 11 19 20 1 3 6 8 10 13 1 3 7 91618 1 3 12 15 21 22 1 4 5 7 8 15 1 4 6 91217 5 1 5

LO I 1 13 16 I8 20 12 1 4 16 17 I 8 22 4

8 13 19 20

1

1

15

7 10 20 2 1

3

5

3 3

5 8 I 1 I6

3

7

3

8 14 15 I 8 20

8 12 17

9 10 1 1 9 13 17 3 13 1 4 I6 4 5 9 14

3 3

4

22

6 11 17 I 8 2 1

19

I 2 14 20 2 2

16 20

5 17 19 21 22

4 6 8 I 6 I 8 19 4 6 13 15 20 21 9 10 13 19

L

7

4

7 11 12 16 21

18 2 2

CONSTRUCTION I1 (Liineburg [7]). L e t u s c o n s i d e r t h e p o i n t - s e t of PG(2,4) and a new p o i n t m . I n S , w i t h I S I = 22, w e g i v e t h e f o l l o w i n g f a m i l i e s o f blocks: a ) S e t { L U {m}] , where L i s a l i n e of PG(2,4). b) A c l a s s o f 56 h y p e r o v a l s c o n s t r u c t e d i n t h e f o l l o w i n g way. Let % . be t h e set of t h e 168 h y p e r o v a l s of PG(2,4). I f H 1 , H 2 € & , w e say that

-

H,*H,

iff

lHln H , [

or 6

= 0,2

.

The r e l a t i o n is an equivalence r e l a t i o n . W e have e x a c t l y 3 eq u i v al en ce classes, each of w h i c b c o n t a i n s 56 h y p e r o v a l s. We can assume each of t h e s e 3 classes a s t h e c l a s s of 56 blocks.

We r e c a l l t h e f o l l c w i n g 2.1 RESULT (W.Jdnsson [6], 3.2, 3.4, 3.5).Let BIB' be two b l o c k s of S ( 3 ,6 ,2 2 ) . Then e i t h e r I B n B ' I = 0 o r IBnB11=2.For any f i x e d B t h e r e are 1 6 , o r 6 0 b l o ck s B' such t h a t B n B ' = @ o r IBflB'I-2, r e s p e c t i v e l y Moreover, if x i s a f i x e d p o i n t , x # B , t h e r e are 6 b lo c k s B' w i t h X E B ' and B n B ' = @ and 10 b l o c k s w i t h x # B ' and B n B ' = @ .

.

The f o l l o wi n g lemmas w i l l be u s e f u l throughout t h e p ap er . 2.2 LEMMA. Let B,B' b e two b lo c k s of S ( 3 , 6 , 2 2) w i t h IBflB'I =2. L e t p o i n t wi t h x g B U B ' . The set B U B ' U { x } h a s two e x t e r n a l b l o ck s.

x

be

a

PROOF. S t e p 1. F i r s t , w e prove t h a t B U B ' h a s e x a c t l y 4 e x t e r n a l blocks. Denot e by ti t h e c h a r a c t e r s o f B U B ' . We have t,=2. Moreover, t 5 = t l = 0 , s i n c e each The block i n t e r s e c t i n g B U B ' i s a t l e a s t 2-secant and a t most 4-secant B U B ' . s y s t e m of c h a r a c t e r s h a s t h e s o l u t i o n : t s = 2 , t , = O , t,=12, t 3 = 3 2 , t ,= 2 7 , t , =0, t,=4. These S t e p 2. Denote by E , , E , , E 3 , E, t h e f o u r e x t e r n a l b l o c k s t o B U B ' . b l o ck s are 2-secant two by two. I n f a c t , i f Ei n E i = @ , ev er y o t h e r block which

L. Berurdi

34

i s d i f f e r e n t from them, s h o u l d i n t e r s e c t a t l e a s t B o r Then E i n E j # @ , t h a t i s / E i n E j / = 2 .

B',

a

contradiction.

S t e p 3. B l o c k s E h and E k a r e 4 - s e c a n t E i U E j . I n f a c t , i n S ( 3 , 6 , 2 2 ) t h e r e a r e o n l y two p o i n t s x , y o u t s i d e t o B U B ' U E i U E . . Moreover E h and Ek a r e e x t e r n a l t o B U B ' . C o n s e q u e n t l y , t h e y must h a v e 4 p o i n t s i n Ei U E , and E h n E,= { x , y }

.

S t e p 4. implies through Then

F i n a l l y , w e p r o v e t h a t s e t B U B ' U (x} h a s two e x t e r n a l b l o c k s . S t e p 3 (Ei U E . ) f l E , , n E k = F . S o , t w o of f o u r e x t e r n a l b l o c k s t o B U B' p a s s e a c h p L i n t x o u t s i d e B U B' t h e a s s e r t i o n i s proved.

.

2 . 3 LEMMA. L e t B , B ' be two b l o c k s w i t h / B n B ' / =2. Denote by W t h e complement p o i n t - s e t of B U B ' , and by 6 = { E , , E , , E , , E,} t h e s e t o f external blocks t o B U B ' . Then t h e p a i r ( W , b ) i s a n 1 - ( 1 2 , 6 , 2 ) d e s i g n .

.

PROOF. L e t E l , E 2 b e two e x t e r n a l b l o c k s t o B U B ' Then / E l n E, 1 = 2 , n e c e s s a r i l y . P u t {y,z} :=W - E,U E,. The b l o c k E, ( a n d t h e n E , ) must h a v e 2 p o i n t s of W o u t s i d e E,U E,, and so i t c o n t a i n s x and y. Thus, e a c h p o i n t o f W i s i n c i d e n t t o two b l o c k s o f 8 , a n d ( W , b )i s a 1 - ( 1 2 , 6 . 2 ) d e s i g n . 2.4 LEMMA. L e t x , y b e two p o i n t s of S ( 3 , 6 , 2 2 ) . Denote by B,,B,,B, t h r e e b l o c k s t h r o u g h x , y . Then t h e s e t C=B, U B , U B, h a s t h e s e c h a r a c t e r s : t , = 7 , t , = 5 6 , t 2 = 1 4 , t , = O , t l = t 3 = t 5 = 0 ,i n o t h e r words C i s of t y p e ( 2 , 4 , 6 ) . PROOF. I t i s c l e a r t h a t t , = t , = t , = O .

Then ( 2 . 2 ) i m p l i e s t h e a s s e r t i o n .

2 . 5 REMARK. The complement o f t h e s e t C d e f i n e d (0,2,4).

in

2.4

2.6. LEMMA. The s e t of f o u r b l o c k s which p a s s t h r o u g h two t h e s e c h a r a c t e r s : t , = 2 0 , t , = 2 4 , t , = l , t,=t,=t3=0,t 5 = 3 2 .

is

set

a

fixed

of

points

type has

PROOF. S i m i l a r t o 2.4. 2.7 LEMMA. The 1 2 - s e t o f t w o d i s j o i n t b l o c k s B,B' (2,4,6).

of

S(3,6,22)

is

of

type

PROOF. S e t B U B ' h a s t , = t 3 = 0 , t 6 = 2 , s i n c e e a c h o t h e r bloclc i n t e r s e c t s B ( o r B ' ) i n 0 o r 2 p o i n t s . From ( 2 . 2 ) w e o b t a i n t o =t , = O and t2=30, t , = 4 5 .

.

Fix uEB\B' 2 . 8 LEMMA. L e t B,B' b e two b l o c k s o f S ( 3 , 6 , 2 2 ) w i t h B n B ' = { p , q } and a E B' B. Denote by B", B"' t h e b l o c k s t h r o u g h p , u , a and q , u , a r e s p e c t i v e l y . The set := B U B ' U B"U B"' i s t h e complement of a b l o c k .

PROOF. P u t I= { p , q , u , a ) a n d X'={B,B' ,B'',B"') .We i n v e s t i g a t e t h e c h a r a c t e r s o f H . O b v i o u s l y t , =O. W e p r o v e t h a t t , = O . I f t h e r e e x i s t s a b l o c k B, 3 - s e c a n t t o H, t h e n B, c o n t a i n s a t l e a s t one e l e m e n t o f I , o t h e r w i s e B, s h o u l d be 1 - s e c a n t t o one t o a b l o c k o f & . I f B, c o n t a i n s o n l y one e l e m e n t o f I , B , i s 1 - s e c a n t b l o c k of 2V , s i n c e 3 b l o c k s p a s s t h r o u g h e a c h p o i n t of I and B, should c o n t a i n two o t h e r p o i n t s o n t w o o f t h e s e 3 b l o c k s . So B, c o n t a i n s a t l e a s t 2 p o i n t s o f I. I f B, c o n t a i n s 2 p o i n t s o f I and one o u t s i d e I , B, is 2 - s e c a n t t o t h r e e b l o c k s of & ( t h o s e c o n t a i n i n g t h e two e l e m e n t s o f I and t h e o n e which t h e t h i r d p o i n t i s o n ) and 1 - s e c a n t t o one b l o c k o f % ( t h a t c o n t a i n s o n l y one of t h e 2 p o i n t s of I s a i d a b o v e ) . So B, s h o u l d h a v e 3 p o i n t s of I. B u t i t i s i m p o s s i b l e , s i n c e t h r o u g h any 3 e l e m e n t s of I t h e r e i s one b l o c k of&, which i s 6 - s e c a n t t o E. C o n s e q u e n t l y t , = O . set R , t h e n Do c o n t a i n s W e p r o v e t h a t t,=O. I f B, i s 5 - s e c a n t t o t h e a t l e a s t a p o i n t o f I , o t h e r w i s e B, s h o u l d b e 1 - s e c a n t t o o n e b l o c k of &. B, c a n n o t c o n t a i n o n l y one p o i n t o f I , s i n c e B, s h o u l d c o n t a i n a n o t h e r p o i n t on

35

Blocking Sets in the Large Mathieu Designs

e a c h of t h e 3 b l o c k s t h r o u g h t h i s p o i n t and o n l y o n e p o i n t on t h e r e m a i n i n g b l o c k of X . T h e b l o c k B, c a n n o t c o n t a i n t w o p o i n t s of I , s i n c e , i n t h i s case, Bo c o u l d b e 4 - s e c a n t t o H. B, c a n n o t c o n t a i n 3 p o i n t s of-I, s i n c e B, s h o u l d b e a b l o c k of X , which i s 6 - s e c a n t t o G . Hence t , = O . Then H i s a 1 6 - s e t w i t h t , = =t,=t,=O. By ( 2 . 2 ) w e o b t a i n t o = l ,t , = 2 , t,=60, t,=16. So B h a s o n e e x t e r n a l block, exactly.

3. BLOCKING SETS OF MINIMAL CARDINALITY I n [ l ] w e proved t h a t i f C i s a b l o c k i n g s e t i n S ( 3 , 6 , 2 2 ) ,

(3.1)

then

7 1 ICI 115.

I n t h i s s e c t i o n w e d e a l w i t h b l o c k i n g sets o f s e v e n p o i n t s . W e s h a l l prove Theorem 1.1 (a). I n view of example ( a ) , t h e r e e x i s t s a b l o c k i n g set of minimal c a r d i n a l i t y 7. By example ( a ) t h e f o l l o w i n g i s a non-empty d e f i n i t i o n :

3.1 DEFINITION. A s e t F of p o i n t s i n S ( 3 , 6 , 2 2 ) of t y p e (1,3) i s c a l l e d a set. W e prove t h a t : 3 . 2 THEOREM. The b l o c k i n g s e t s i n S ( 3 , 6 , 2 2 ) of minimal s i z e Fano sets.

are

exactly

Fano the

PROOF. Suppose F i s a Fano s e t , i . e . a set o f t y p e (1,3) i n S ( 3 , 6 , 2 2 ) . By ( 2 . 2 ) w e o b t a i n t , = 4 2 , t,=35, c=7. So a Fano s e t i s a b l o c k i n g set o f minimal s i z e i n view of (3.1). Suppose now t h a t C is a b l o c k i n g set w i t h ICI =7. By ( 2 . 2 ) w i t h t o =t, =0, c=7 we obtain: t,= t,= t 2 = 0 , t,= 35, t , = 42. So, t h e a s s e r t i o n f o l l o w s .

We r e c a l l t h a t i n PG(2,4) t h e r e a r e two classes o f s e t s o f t y p e (1,3): t h e Fano ( o r B a e r ) s u b p l a n e s and t h e H e r m i t i a n arcs. I n S ( 3 , 6 , 2 2 ) o n l y t h e class o f Fano s e t s i s . N e x t , l e t u s l o o k a t t h e c o n n e c t i o n between a Fano s u b p l a n e i n PG(2,4) and a Fano s e t o f S ( 3 , 6 , 2 2 ) . I n PG(2,4) w e c o n s i d e r a c o n i c V, a t a n g e n t T a t a p o i n t t of V and t h e k n o t k o f %? which a l s o l i e s on T. The s e t V U T- { t } i s a b l o c k i n g s e t o f PG(2,4) which i s r e d u c i b l e s i n c e t h e s e t V U T - { t , k ) i s a b l o c k i n g s e t ( o f 7 p o i n t s ) t o o . O b v i o u s l y , e a c h Fano s u b p l a n e c a n b e c o n s t r u c t e d i n t h i s way. We prove: 3 . 3 THEOREM. L e t F ( 2 ) b e a Fano s u b p l a n e o f P G ( 2 , 4 ) . C e n o t e by T a 3 - s e c a n t l i n e o f F ( 2 ) . The s y m m e t r i c d i f f e r e n c e s e t X : = T A F ( 2 ) i s a h y p e r o v a l of PG(2,4). Vice v e r s a , i f a 6-arc of PG(2,4) and L i s a 2-secant line of X , t h e n s e t X A L i s a Fano s u b p l a n e .

xis

PROOF.We p r o v e t h a t a n y l i n e M h a s e i t h e r zero or two p o i n t s i n X . S u p p o s e t h a t M ?'.s1 - s e c a n t t o F ( 2 ) . I f M i n t e r s e c t s F ( 2 ) a t a p o i n t o f F ( 2 ) ll T , t h e n M i s 0 - s e c a n t t o z I f M i n t e r s e c t s F ( 2 ) a t a p o i n t o u t s i d e T , t h e n M is 2 - s e c a n t t o X.1f M=T, t h e n M i s 2 - s e c a n t t o X b y c o n s t r u c t i o n . I f M + T i s 3 - s e c a n t t o F ( 2 ) , t h e n M i s 1 - s e c a n t t o T a t a p o i n t of T \ F ( 2 ) , s i n c e two 3 - s e c a n t s 0: F ( 2 ) i n t e r s e c t a t a p o i n t of F ( 2 ) . C o n s e q u e n t l y M i s 2 - s e c a n t t o X. C o n v e r s e l y , l e t & b e a 6-arc o f PG(2,4) and L a 2 - s e c a n t o f X. P u t Ii : = = X A L . L i n e L i s 3 - s e c a n t t o H. I f M i s a 0 - s e c a n t l i n e of x , t h e n M i s I - s e c a n t t o L a t a p o i n t of L \A?. SuFpose t h a t M i s a 2 - s e c a n t l i n e o f z I f M i n t e r s e c t s .f a t a p o i n t of L flX, t h e n M i s 1 - s e c a n t of H; i f M i n t e r s e c t s T a t two p o i n t s of y \ L , t h e n M i s 3 - s e c a n t t o H. So X A L i s a Fano s u b p l a n e . L e t F(2.) b e a Fano s u b p l a n e and d t h e

family

of

hyperovals

of

PG(2,4)

L. Berardi

36

c o n s t r u c t e d by s t a r t i n g from a n y 3 - s e c a n t l i n e t o F ( 2 ) . We c a l l d t h e f a m i l y o f t h e 6-arcs a s s o c i a t e d w i t h F ( 2 ) . C l e a r l y , e a c h 6-arc d e t e r m i n e s o n e s u b p l a n e F ( 2 ) , b u t o n e s u b p l a n e F ( 2 ) f i x e s a f a m i l y d of 6 - a r c s . We p r o v e . 3 . 4 THEOREM. L e t F ( 2 ) b e a Fano s u b p l a n e o f PG(2.4) and d t h e f a m i l y o f 6 - a r c s a s s o c i a t e d w i t h F ( 2 ) . Any two 6-arcs o f d h a v e t w o common p o i n t s .

the

PROOF.Let T, and T, b e t w o l i n e s o f PG(2.4) which a r e 3 - s e c a n t t o F ( 2 ) . Denote by &= T, A F ( 2 ) and X,= T, A F ( 2 ) t h e two arcs of d c o n s t r u c t e d by s t a r t i n g from T, or T2 , r e s p e c t i v e l y . It i s v e r y e a s y t o p r o v e t h a t -%l;nGF(2)-U, U T2). so 2.

I-%x;nx:(=

We n o t e t h a t e a c h Fano s e t o f S = ( 3 , 6 , 2 2 ) i s a Fano s u b p l a n e i n t h e c o n t r a c t i o n a t a p o i n t x $ F . We a s k a b o u t t h e v i c e v e r s a . F o r t h e v i c e v e r s a w e work i n PG(2,4). We c o n s i d e r a Fano s u b p l a n e F ( 2 ) and t h e f a m i l y d of t h e 6-arcs a s s o c i a t e d w i t h F ( 2 ) . Note t h a t o n e o f t h e s e 6-arcs i s a b l o c k i f and o n l y i f e a c h 6-arc o f d i s a b l o c k , s i n c e by 3 . 4 t w o 6-arcs o f d are i n t h e same Liineburg class. Now w e a r e r e a d y t o c h a r a c t e r i z e t h e Fano s u b p l a n e which are Fano s e t s t o o . We p r o v e :

3.5 THEOREM. A Fano s u b p l a n e F ( 2 ) i n t h e c o n t r a c t i o n o f s = S ( 3 , 6 , 2 2 ) a t a p o i n t x $ F ( 2 ) i s a Fano set o f S i f and o n l y i f t h e f a m i l y d o f 6 - a r c s a s s o c i a t e d w i t h F ( 2 ) d o e s n o t c o n t a i n b l o c k s of S. PROOF. L e t F 2 Fano s e t . W e c o n s i d e r t h e c o n t r a c t i o n o f S a t x 4 F. S i n c e F i s of t y p e ( 1 , 3 ) , i t f o l l o w s t h a t F i s a Fano s u b p l a n e i n PG(2,4). Denote by F ( 2 ) a Fano s u b p l a n e o f PG(2.4). So t h e r e i s a c o n i c V , a p o i n t t €%?,a 1 - s e c a n t l i n e T t o % ' a t p o i n t t , s u c h t h a t F ( 2 ) = V U T - { t , k } , where k i s t h e k n o t of V. Denote by f i = V U { k } o n e o f t h e 6-arcs a s s o c i a t e d w i t h F ( 2 ) . Note t h a t if % i s a b l o c k o f S ( 3 , 6 , 2 2 ) , s e t F ( 2 ) c a n n o t b e a Fano s e t , s i n c e n o t a b l o c k o f S ( 3 , 6 , 2 2 ) . So, % s h o u l d b e a 4 - s e c a n t b l o c k . Assume t h a t by 3 . 4 , n o 6-arc of d c a n b e a b l o c k . Each b l o c k B of S ( 3 , 6 , 2 2 ) which d o e s n o t p a s s t h r o u g h p o l e a , h a s 1 or 3 common p o i n t s w i t h %, s i n c e % and B are two 6-arcs which are i n t w o d i f f e r e n t L k e b u r g ' s classes. a ) Suppose t h a t B h a s 1 common p o i n t w i t h %-{t,k}. Block B i n t e r s e c t s b l o c k TU {a}i n 0 o r 2 p o i n t s , s i n c e t h e y a r e b l o c k s of S ( 3 , 6 , 2 2 ) . So B i n t e r s e c t s F(2) i n 1 o r 3 points. b ) Suppose t h a t t E B and k 4 B . T h e n B c o n t a i n s a n o t h e r p o i n t x o f T-{t,k} ( s i n ce TU{oo}is a b l o c k ) and 0 o r 2 p o i n t s o f %- { t , k } . Then B i s 1 or 3 - s e c a n t t o F(2). V'= % ' -{t} U {k} i s a c o n i c and T i s C ) Now s u p p o s e t h a t k € B and t $ B. S i n c e a t a n g e n t l i n e of W ' a t k , w e c m c h a n g e t w i t h k . So a l l i s as i n p o i n t b ) . d ) Suppose t h a t t , k E B . T h e n B i s 0 - s e c a n t t o T - { t , k , a } and 1 - s e c a n t X - { t , k } . e ) F i n a l l y , s u p p o s e t h a t B h a s 3 common p o i n t s w i t h %-{t,k}. We s h a l l p r o v e t h a t B n ( T U { m ) ) =#.On t h e c o n t r a r y , s u p p o s e t h a t B fl (T - { t , k , a f ) ={x,y} C o n s i d e r t h e l i n e s x z and y z , where z i s a p o i n t of VflB. T h e s e two l i n e s d o n o t c o n t a i n k , s o t h e y are s e c a n t t o V. Then o n e o f them, s a y xz, c o n t a i n s two p o i n t s of B n ( W - { t , k } ) . So t h e b l o c k s B and x z U { a ) s h o u l d h a v e 3 common points, a contradiction.

xis

.

4 . BLOCKING SETS WITH EIGHT POINTS I n t h i s s e c t i o n w e s h a l l p r o v e p o i n t ( b ) of 1.1 f o r t h e b l o c k i n g s e t s C w i t h I C I = 8 . Moreover, we s h a l l p r o v e t h a t t h e 10-set D d e f i n e d i n Example ( b ) of I n t r o d u c t i o n i s a n i r r e d u c i b l e b l o c k i n g s e t . 4.1 LEMMA. L e t B,B' b e two b l o c k s o f S ( 3 , 6 , 2 2 ) w i t h B n B ' = @ . L e t u , v p o i n t s w i t h u € B , V E B ' . Then t h e 10-set

be

two

37

Blocking Sets in the Large Mathieu Designs

i s a n i r r e d u c i b l e blocking set. PROOF. S e t D i s a b lo c k in g set by 2.7. Now w e prove t h a t D is i r r e d u c i b l e . F i x a p o i n t X E B . Four b l o c k s d i f f e r e n t from B p a s s t h r o u g h u,x. Only 3 o f t h e s e 4 blo ck s i n t e r s e c t B'. So, e a c h p o i n t x is i n c i d e n t t o a b l o ck which is 1-secant t o D. I f X E B ' , t h e r e a s o n i n g i s t h e same. 4.2 THEORM. I n S ( 3 , 6 , 2 2 ) t h e b lo c k in g sets w i t h 8 p o i n t s are a l l r e d u c i b l e . They are e x a c t l y t h e sets F U { x }, b e i n g F a Fano set and x a p o i n t w i t h x @ F . PROOF. Let: C be a b l o c k i n g set o f S ( 3 , 6 , 2 2 ) w i t h ( C I = 8 . S i n c e C i s a s e t , w e have t , = t , = O . Then w e o b t a i n by (2.2):

,

t:,=7-4t5

tJ = 2 8 +6 t5

,

t 2= 1 4 - 4 t ,

,

t1=28+t,

b l o ck i n g

.

From t , z 0 , i t f o l l o w s O s t , 51. W e prove t h a t t , = O . Suppose t,= 1. L e t B b e t h e 5-secant b lo c k t o C. Denote by B' t h e block t h r o u g h t h e p o i n t of B \ C and two p o i n t s of C-B. C i s co n t ai n ed i n t h e union of B U B ' wi t h a point.So C is n o t a b lo c k in g set by 2.2.Then t 5 # l , i . e . t , = O . S i n c e t,=O,set C i s a l s o a b l o c k i n g set i n t h e c o n t r a c t i o n o f S( 3 ,6 ,2 2 ) wi t h r e s p e c t t o a p o i n t b o u t s i d e C.Then C i s e i t h e r t h e s e t FU{x},where F i s a Fano s u b p l an e and x $ F , o r t h e t r i a n g l e 0,[2].We r e c a l l t h a t i n PG(2,4), c o n t r a c t i o n o f S ( 3 , 6 , 2 2 ) , t h e t r i a n g l e A is d e f i n e d as f o l l o w s . L e t L.M be two l i n e s wi t h LnM = ( p } F i x u E L , vEM. F i x a p o i n t W E U V w i t h w+ U , V . Then A : = ( L { u } ) U(M-{v})U{w} Suppose C=A i n PG(2,4). It f o l l o w s t h a t B, := LU( b } and B,:=MU{b} are two b lo c k s of S ( 3 ,6 ,2 2 ) w i t h B,n B,= { b ,p ) .Si n ce C i s c o n t a i n e d i n B , U B , u ( w } , C has e x t e r n a l b l o c k s i n view of 2.2. So, s i n c e C i s r e d u c i b l e , t h e r e e x i s t s a p o i n t x s u c h t h a t C-{x) i s a Fano su b p l an e of PG(2,4). F i n a l l y , w e prove t h a t such Fano s u b p l a n e i s a Fano s e t of S(3.6.22). n e c e s s a r i l y . I n PG(2,4) w e have F = V U T - { t , k } , where V is a c o n i c w i t h k n o t k and T is a 1 - s e c a n t l i n e t o 4& a t T E ~ .I f F is n o t a Fano s e t of S ( 3 , 6 , 2 2 ) , t h en g U { k } , by 3.3, i s a block o f S(3,6,22). I n t h i s case t h e b l o ck i n g s e t C i s co n t ai n ed i n t h e union o f two b lo c k s g U { k } and TU{ b} , a c o n t r a d i c t i o n by 2.2.

.

.

5. BLOCKING SETS WITH NINE POINTS I n t h i s s e c t i o n w e s h a l l prove p o i n t ( b ) o f 1.1 f o r t h e b l o ck i n g sets C w i t h ICI=9. T h i s i s a consequence of some lemmas. W e a l s o e x p l a i n t h e b l o ck i n g set T d ef i n ed i n 5.3 which w i l l a p p e a r i n t h e n e x t 6.2 The b l o ck i n g s e t s N u , N, d ef in e d i n t h i s S e c t i o n ( c f . 5.2, 5.4) are o n l y a u s i l i a r y s e t s and t h ey appear n e i t h e r i n 1.1 nor i n o t h e r S e c t i o n s .

.

5.1 LFMMA. L e t B,B' be two b l o c k s o f S ( 3 , 6 , 2 2 ) w i t h I B n B ' I = 2. Denote by a , b two p o i n t s of B ' \ B. L e t 8 = {El ,E, ,E3,E, } b e t h e s e t of t h e f o u r b l o c k s e x t e r n a l t o B UB' (a) There e x i s t two b l o c k s S,,S, which are 2-secant t o B U B ' a t a and b. ( b ) For ev er y x E S , ( o r S , ) , w i th x +a, b t h e two b l o ck s E, , E, of 8 t h r o u g h x have t h e o t h e r common p o i n t y on S, (or S,). (c) One of t h e t w o p o i n t s o u t s i d e BUB'U E,U E, i s i n S, and t h e o t h e r i n

.

s,

-

PROOF. ( a ) Four b lo c k s d i f f e r e n t from B' pass t h r o u g h a and b. Ex act l y two of t h e s e are 2-secant t o B \ B ' , s o t h e o t h e r two i n t e r s e c t B U B ' o n l y a t a , b . ( b ) Suppose x ~ S , - { a , b } . L e t E l , E, be t h e two b l o c k s o f 8 through x ( c f . 2.3) and d en o t e by y t h e o t h e r p o i n t of E , n E ,

.

L. Berardi

38

Assume o n t h e c o n t r a r y t h a t x , y E S 1 through x,y. Each o f t h e s e contradiction, since i n the c o n t r a c t i o n w i t h r e s p e c t t o x ( o r t o y ) t h e r e would b e t h r e e l i n e s of PG(2,4) t h r o u g h y (or x ) e x t e r n a l t o 6 - a r c B. S t e p 1. F i r s t w e p r o v e t h a t y $ S , - i a , b i .

- { a , b t . Then t h e t h r e e b l o c k s E l , E , , S, p a s s b l o c k s h a s n o common p o i n t w i t h B. T h i s i s a

.

S t e p 2. Now w e p r o v e t h a t y e s z - [ a , b ) Assume t h e c o n t r a r y . I n t h e c o n t r a c t i o n w i t h r e s p e c t t o x , t h e s e t s E , - ( x ) and E , - { x ) are two l i n e s t h r o u g h y , w h i l e S , - i x ) i s a l i n e which i n t e r s e c t s 6 - a r c S, a t a and b. The l i n e s E , - { x ) and E , - i x i i n t e r s e c t S, a t p o i n t s d i f f e r e n t from a and b. S i n c e t h e r e c a n n o t b e 4 l i n e s c o n t a i n i n g y and s e c a n t t h e 6-arc S, , n e c e s s a r i l y Y E S , - j x , a , b i T h i s i s a c o n t r a d i c t i o n i n view of S t e p 1. ( c ) The p r o o f f o l l o w s by c o u n t i n g a r g u m e n t s .

.

5 . 2 LEMMA. Use t h e same n o t a t i o n s o f 5.1. F i x a p o i n t u € B \ B ' t h e o n l y p o i n t of S, o u t s i d e B U B ' U E , U E , . Then

(5.1)

:= B U B ' -

N,

and d e n o t e by z

{ a,b,u}U (x,z)

i s a blocking set with n i n e p o i n t s .

, since PROOF. The b l o c k s t h a t a r e n o t e l e m e n t s o f 8 U ( S l , S z } i n t e r s e c t N , t h e y i n t e r s e c t B U B ' . Moreover, El ,E,,S, c o n t a i n x , and E,,E,,S, c o n t a i n z . So each block i n t e r s e c t s N, Suppose t h a t t h e r e e x i s t s a b l o c k 6 - s e c a n t t o N , . This block n e c e s s a r i l y c o n t a i n s x and z ; m o r e o v e r , i t i s 2 - s e c a n t t o B \ B ' and B ' - B a t p o i n t s of N , . C o n s e q u e n t l y , i t s h o u l d b e 1 - s e c a n t t o S, and S , , a c o n t r a d i c t i o n .

.

We s h a l l u s e t h e f o l l o w i n g Lemma i n t h e s e q u e l .

5 . 3 LEMlYA. L e t B , B ' b e two b l o c k s o f S ( 3 , 6 , 2 2 ) w i t h I B n B ' l =2. F i x and a E B ' \ B. T h e r e a r e t w o p o i n t s x , y o u t s i d e BUB' s u c h t h a t : (5.2)

T :=[BUB'

u E B

1

B'

- (u,a))J{x,y}

i s a r e d u c i b l e b l o c k i n g set w i t h t e n p o i n t s .

b l o c k which i n t e r s e c t s B U B ' , i n t e r s e c t s T. T h e r e a r e 4 b l o c k s E , , E z , E 3 ,E., e x t e r n a l t o B U B ' . I n view of 2.2 ( c f . S t e p 2 ) , i f x E E , n E, , y E E , n E , , then T has no e x t e r n a l block. Suppose t h a t E, i s a 6 - s e c a n t b l o c k t o T , n e c e s s a r i l y u n i q u e ; B, i n t e r s e c t s B h B a t two p o i n t s , B \ B ' a t two p o i n t s and m u s t c o n t a i n x a n d y.So, I E , n B,I =1, a contradiction. F i n a l l y , w e p r o v e t h a t T i s r e d u c i b l e . D e n o t e by p and q t h e p o i n t s o f B n B ' . L e t B",B"' b e t h e b l o c k s t h r o u g h p , u , a and q , u , a r e s p e c t i v e l y . By Lemma 2 . 8 t h e complement of set B U B ' U B ' ' U B " ' i s a b l o c k H. S i n c e b o t h x , y c a n n o t b e i n H ( o t h e r w i s e t h e r e e x i s t s a n e x t e r n a l b l o c k t o T by ( 2 . 2 ) , t h e n o n e of them l i e s i n B" o r i n B"'. I f , f o r e x a m p l e , x ~ B " , t h e n T - i p ) i s a b l o c k i n g set. S o , T is reducible. PROOF. Each

5.5 LEMMA. Let B,B',B" be t h r e e b l o c k s w i t h I B n B ' n B " / = l . F i x a p o i n t z o f B"\(BUB'). Denote by E , , E , t h e two e x t e r n a l b l o c k s t o B U B ' U 1 2 1 . Let y be a p o i n t o f E , n E , . Then (5.3)

N,:=

[(BuB~)\B"

]

UiY9Z

1

i s a b l o c k i n g set. w i t h n i n e p o i n t s .

.

PROOF. P u t B n B ' n B " = { u t and ( B U B ' ) n B " = ( u , v , w ) The b l o c k s i n t e r s e c t i n g B U B ' \ B " , i n t e r s e c t N, The b l o c k t h r o u g h u , v , w , i . e . B", c o n t a i n s z . Each

.

Blocking Sets in the Large Mathieu Designs

39

b l o c k t h r o u g h u and o n e o f p o i n t s v,w i n t e r s e c t s N o . F i n a l l y , b l o c k s E, and E, c o n t a i n y. Suppose t h a t B, i s a 6 - s e c a n t b l o c k t o No.Then B, c o n t a i n s y , z and i s 4-sea t p o i n t s o f N o . So IB B"I = 1, a c o n t r a d i c t i o n . c a n t t o (B-B')U(B'.B)

n

Now w e g i v e a c h a r a c t e r i z a t i o n o f b l o c k i n g s e t s w i t h 9 p o i n t s , i n terms N o and N ,

of

5 . 6 LEMNA. Let C b e a b l o c k i n g s e t o f S ( 3 , 6 , 2 2 ) w i t h I C I = 9. C=N, where N, a n d N , are d e f i n e d i n (5.3) and (5.1).

or

.

Then

C=N,

~

PROOF. F i r s t , w e p r o v e t h a t t h e r e i s a t most o n e 5 - s e c a n t b l o c k t o C. On t h e It f o l o t h e r h a n d , s u p p o s e t h a t B, ,B, are two o f them. O b v i o u s l y B, B,j; $5 lows t h a t C i s c o n t a i n e d e i t h e r i n B , U B , o r i n t h e u n i o n o f B, ,B2 and o n e p o i n t . T h i s i s a c o n t r a d i c t i o n by 2 . 2 . Suppose t h a t B i s a 5 - s e c a n t b l o c k t o C. I f w e p u t t o = t 6 = 0 ,t5=1, ICI =9 i n ( 2 . 1 ) , w e o b t a i n t 4 = 1 2 . L e t B ' b e a 4 - s e c a n t b l o c k t o C. W e prove t h a t B B' B' c o n t a i n s two p o i n t s o f C. Assume t h e c o n t r a r y . Then e i t h e r B n B ' = @ o r B c o n t a i n s one p o i n t o f C and o n e p o i n t o u t s i d e C. I n t h e f i r s t case C i s s t r i c t l y c o n t a i n e d i n t h e b l o c k i n g set D ( c f . 4 . 1 ) , which i s i r r e d u c i b l e , a c o n t r a d i c t i o n . The o t h e r case i s i m p o s s i b l e i n view o f 2.2. P u t l u ) = B \ C , { a , b ) = B ' - - C and I X , ~ ) = C \ ( B U B ' ) . Denote by S , , a n d S, t h e t w o b l o c k s t h r o u g h a and b , which a r e 2 - s e c a n t t o B U B'. So x , z are n e c e s s a r i l y a s i n (5.1). C o n s e q u e n t l y , C=N,. Suppose now t h a t t h e r e i s no 5 - s e c a n t b l o c k t o C. L e t B b e a 4 - s e c a n t b l o c k to C I n t h e c o n t r a c t i o n PG(2,4) of S ( 3 , 6 , 2 2 ) a t a p o i n t u o f B - C , s e t C i s a l s o a b l o c k i n g s e t . The l i n e L=B- { u ) i s a 4 - s e c a n t o f C i n PG(2,4). So i n PG(2,4) s e t C c a n n o t b e a h e r m i t i a n a r c a n d it i s n e c e s s a r i l y formed by (L- Iv1)U (M-iw ) ) U { z , y I where L,M a r e l i n e s , v E L , W E M , z,v,w a r e c o l l i n e a r , y @ L U M ( c f . [ 2 ] , Theor. 2 . 3 ) . I n S ( 3 , 6 , 2 2 ) w e c o n s i d e r t h e b l o c k s B = L U i u ) , B ' = M U ( u ) and p o i n t z l y i n g on b l o c k B" t h r o u g h u,v,w. So C v e r i f i e s t h e C=N, and o u r h y p o t h e s i s o f 5 . 5 and p o i n t y must b e f i x e d as i n (5.3). Then lemma i s c o m p l e t e l y proved.

.

n

n n

.

5.7 THEOREM. Denote by F a Fano s e t i n S ( 3 , 6 , 2 2 ) . Suppose t h a t x , y @ F. Then t h e set N : = F U { x , y } i s a r e d u c i b l e b l o c k i n g s e t w i t h 9 p o i n t s . Moreover, if B i s a b l o c k s u c h t h a t I B n F I =3 a n d x , y € B , t h e n N=N,,otherwise N=N,

.

PROOF. S e t F i s of t y p e ( 1 , 3 ) , c o n s e q u e n t l y N i s a r e d u c i b l e b l o c k i n g s e t . Suppose t h a t x , y are o n a 3 - s e c a n t b l o c k B o f F. Then B i s 5 - s e c a n t t o N , and i n view o f 5.6 N=N,. I f x , y are n o t i n t h e same 3 - s e c a n t b l o c k o f F, t h e n t h e r e i s n o 5 - s e c a n t b l o c k t o N . Then N=N, by 5.6. So, t h e case l . l ( b ) , I C I = 9 i s a c o n s e q u e n c e of t h e a b o v e t h e o r e m s .

6. BLOCKING SETS WITH TEN POINTS I n t h i s s e c t i o n w e s h a l l p r o v e p o i n t ( c ) of 1.1. I f C w i t h ICI = 10, w e h a v e t , = t , = O a n d w e o b t a i n by (2.2):

t,=8+t5

,

T h e s e imply t , 1 7 .

t,=8+6t5

,

t,=33-4t5

,

t4=28-4t,

a

is

blocking

set

.

So w e h a v e many d i s t i n c t p o s s i b i l i t i e s f o r t ,

.

We b e g i n w i t h t h e f o l l o w i n g 6.1 THEOREM. L e t C b e a b l o c k i n g set i n S ( 3 , 6 , 2 2 ) w i t h I C I =lo. Then t 5 = 2 i f and o n l y i f C is t h e i r r e d u c i b l e b l o c k i n g set D d e f i n e d i n 4.1.

L. Berardi

40

PROOF. I f t 5 = 2 , w e h a v e t,=10, & = 2 0 , t , = 2 5 , t , = 2 0 . I f t h e two b l o c k s B,B' t h a t are 5 - s e c a n t t o C h a v e n o common p o i n t , t n e n E D ( d e f i n e d by 4 . 1 ) . On t h e o t h e r h a n d , i f ( B n B ' \ = 2 , w e h a v e a c o n t r a d i c t i o n . I n f a c t b o t h 2 common p o i n t s must b e I n C, o t h e r w i s e C would b e c o n t a i n e d i n t h e u n i o n o f two b l o c k s and a p o i n t , a c o n t r a d i c t i o n by 2.2. Then C i s c o n t a i n e d i n B B ' U I x,y ) , where x , y $ B U B ' . T h e r e a r e a t most 4 d i s t i n c t b l o c k s t h r o u g h x , y t h a t a r e 4 - s e c a n t t o C.(Denote by u i t h e c h a r a c t e r s o f b l o c k s t h r o u g h x and y. We h a v e u,+u,+u4=5, u 3 + 2 u 4 = 8 , s o u 4 < 4).The o t h e r 4 - s e c a n t b l o c k s t o C are e i t h e r t h o s e 4 - s e c a n t t o C a t p o i n t s o f B U B ' or t h o s e 3 - s e c a n t t o C a t p o i n t s of B U B ' and c o n t a i n i n g x o r y.Count t h e 4 - s e c a n t b l o c k s t o C a t p o i n t s o f B U B ' ways. Denote by u i t h e c h a r a c t e r s of F i x two p o i n t s p , q o f C i n B L B i n b l o c k s t h r o u g h p and q. W e have: u 2 + u 3 + u 4 + u , = 5 , u3+2u4+3u5 = 6 , u , = l . SO, u4 5 1 . Then t h e r e q u i r e d b l o c k s are a t most , ( : ) . Count t h e 3 - s e c a n t b l o c k s t o C a t p o i n t s o f B U B ' and c o n t a i n i n g one p o i n t of {X,Y). T h e s e b l o c k s c a n b e e i t h e r t h o s e c o n t a i n i n g o n e p o i n t o f B n B ' , which are a t most 6 , o r t h o s e c o n t a i n i n g t h e p o i n t v of B \ C ( o r u o f B'\ C ) o n e p o i n t z o f B n C (or B ' n C ) and two p o i n t s of B ' n C (or B n C ) . Denote by u , t h e c h a r a c t e r s o f b l o c k s t h r o u g h v , z w i t h r e s p e c t t o s e t ( B U B ' ) n C . We h a v e u,+u,+u,+u,=5, v2+2u,+4u,=7, u , = l . It f o l l o w s u , i l . C o n s e q u e n t l y t h e s e l a s t 4 - s e c a n t b l o c k s t o C are a t most 6. Then t 4 1 4 + 3 + 6 + 6 = 19, a c o n t r a d i c t i o n . Vice v e r s a i s t r i v i a l .

u

(:I

6.2 THEOREM. L e t C b e a b l o c k i n g set w i t h I C I = l O i n S ( 3 , 6 , 2 2 ) . I f t , + 2, C = FUX, where =3, X n F = @ and t h e b l o c k t h r o u g h X i s I-secant F.

1x1

then

PROOF. If t , 2 3 , l e t B,B' b e two 5 - s e c a n t b l o c k s t o C. It i s l B n B ' I = 2 , s i n c e i f B n B ' = @, t h e n i t would b e t , = 2 . Moreover B n B ' C C , o t h e r w i s e C would h a v e e x t e r n a l b l o c k s . S o , C i s c o n t a i n e d i n BUB' (x,yl with x,y $ B U B'. The p o i n t s x , y are n e c e s s a r i l y as i n ( 5 . 2 ) and C=T. S i n c e T i s r e d u c i b l e t h e a s s e r t i o n i s proved. I f t , < l , t h e n t , 1 9 , so C i s r e d u c i b l e , a n d t h e assert i o n i s a l s o t r u e i n t h i s case.

u

7. BLOCKING SETS WITH ELEVEN POINTS I n t h i s s e c t i o n w e s h a l l p r o v e p o i n t ( d ) o f 1.1. We know t h a t i n S ( 3 , 6 , 2 2 ) t h e r e are 11-sets of t y p e (1,3,5) ( c f . Example ( c ) , I n t r o d u c t i o n ) . If I C I =11 and t , = t t , = O , w e h a v e by ( 2 , 2 ) : (7.1)

t, = t , = 4 ( 1 1 - t , )

,

t3=6t,-11,

t, = t ,

,

n e c e s s a r i l y w i t h 2 < t , _ < 11. We b e g i n by p r o v i n g : 7.1 LEMMA. A b l o c k i n g set C i n S ( 3 , 6 , 2 2 ) w i t h I C l o n l y i f C i s a n 11-set o f t y p e (1,3,5).

=

11 i s i r r e d u c i b l e

if

and

PROOF. Suppose C i s i r r e d u c i b l e . Then t , = t1 ->. I C I = l l , and so t , =11 and t , = t , = O . T h i s means C i s of t y p e (1,3,5). N o w w e p r o v e t h a t a n 11-set C of t y p e (1,3,5) i s a n i r r e d u c i b l e b l o c k i n g s e t . Assume t h e c o n t r a r y and s u p p o s e t h a t no t a n g e n t p a s s e s t h r o u g h a p o i n t x E C . Then t h e b l o c k s t h r o u g h x are e i t h e r 3 - s e c a n t or 5 - s e c a n t C. In t h e c o n t r a c t i o n w i t h r e s p e c t t o x t h e s e t C - ( x ) i s o f t y p e ( 2 , 4 ) . Denote t h e characters o f t h e 10-set C - t x ) w i t h r e s p e c t t o l i n e s o f PG(2,4) by y i . bJe s h o u l d have: y, +y4 =21, 2y, +4y4 =50, 2y2 +12y4 =90, a c o n t r a d i c t i o n . 7.2 REMARK. I n S ( 3 , 6 , 2 2 ) t h e complement of a n 11-set of t y p e (1,3,5) a n 11-set of t y p e (1.3.5).

is

also

7 . 3 REMARK. Each 11-set of t y p e (1,3,5) i s a n example of b l o c k i n g set which i s i r r e d u c i b l e w i t h i t s complement.

Blocking Sets in the Large Mathieu Designs

41

7.4 THEOREM. Let F b e a Fano set o f S(3,6,22). Denote by B a block such t h a t F n B = { x } I n S ( 3 , 6 , 2 2 ) t h e sets of t y p e ( 1 , 3 , 5 ) a r e e x a c t l y t h e s e t s E def i n ed by E :=FUB - { x }

.

.

PROOF.First, w e prove t h a t set E is o f t y p e (1,3,5).The block B i s 5-secant t o E. Each block B' w i th x $ B ' i s m-secant F and n-secant B w i t h m=1,3 and n=0,2. Hence B' is (n+n)-secant t o E w i t h mtn=1,3,5. Each b l o ck 8'1 B w i t h x E B' i s 1-secant t o B n E and m-secant t o F n E w it h m=0,2. So B ' i s (m+n)-secant to E w i t h mtn= =1,3. Let C be a s e t o f t y p e (1,3,5). We have t,=11. Denote by B a 5-secant block t o C and by x t h e p o i n t of B--C.We prove t h a t t h e set H : = ( C \ B ) U ( x , i s a Fano s e t . Let B' be a b l o c k w i t h x $ B ' . I f B' is 5 - secan t t o C, t h e n I B n B' I =2, n e c e s s a r i l y . Assume t h e c o n t r a r y , i . e . B n B ' = @ . Then C sh o u l d be t h e union of two d i s j o i n t b l o c k s B,B',each without one p o i n t , w i t h a p o i n t i outside t o BUB'.Then C s h o u l d b e > a r e d u c ib le . blocking set, s i n c e , c - { w } , ;hould b e b l o ck i n g s e t D d ef i n ed i n 4.1. Hence I B n B ' I =2. Consequently 3 ' i s 3-secant t o C \ B. I f B' is 3-secant t o C, th e n B' i s e i t h e r 1-secant o r 3-secant t o C \ B . If B' i s I - s ecan t t o C , t h e n B' is I - s e c a n t t o C \ B s i n c e B' can n o t be I - secan t t o B. L e t Li' be a b l o c k d i f f e r e n t from B w i t h x E B ' . Then B' i n t e r s e c t s B a t e claim t h a t B' c a n n o t b e 5-secant t o C.Assume t h e c o n t r a r y . a n o t h er p o i n t w. W Then C should be t h e union o f B U B ' - ( x i w i t h 2 p o i n t s a , b Q B U B ' . The block through a , b , x n e c e s s a r i l y c o n t a i n s w, o t h e r w i s e i t sh o u l d be 4-secant t o c. Then, t h e r e i s a t l e a s t a b lo c k c o n t a i n i n g a , b , which h a s a n even number of Consep o i n t s i n C, a c o n t r a d i c t i o n . Hence B ' ( f B) i s m-secant C w i t h m = 1 , 3 . q u e n t l y B' is n-secant t o C \ B w i t h n=0,2, and t h e n s - s e c a n t t o ( C \ B ) U ( x ) wi t h s=1,3. F i n a l l y w e prove 7.5 THEOREM, Suppose C i s r e d u c i b l e w i t h ICl =11. Then 1.1 ( d ) h o l d s. PROOF. I f C i s r e d u c i b l e , t h e n t h e r e e x i s t s a p o i n t w € C su ch t h a t C- ( w , i s a bl o ck i n g set o f t e n p o i n t s . Then w e n e c e s s a r i l y have t h e f o l l o w i n g two cases. (i) C - I W , = D , w @ D ( i i ) C - I W I = F U ( x , y , z ] , where t h e block t h r o u g h x , y , z i s 1 - secan t t o F. I n t h i s case w e r e q u i r e t h a t whenever w e t a k e 3 p o i n t s i n ( x , y , z , w } , a block through them i s I - s e c a n t t o F.

.

8. BLOCKING SETS WITH MORE THAN ELEVEN POINTS

All the b l o ck i n g s e t s w i t h 7 , 8 , 9 , 1 0 , 1 1 S ( 3 , 6 , 2 2 ) . Now, w e prove t h e f o l l o w i n g

p o i n t s have

been

characterized

8.1 THEOREM. Denote by C an i r r e d u c i b l e b lo c k i n g set i n S(3,6,22).Then The e q u a l i t y h o l d s i f and o n ly i f C i s an 11-set o f t y p e ( 1 , 3 , 5 ) .

in

IC/Cll.

PROOF. By 7.1 i t i s s u f f i c i e n t t o prove t h a t i f 1CI 2 1 2 , t h e n C i s r e d u c i b l e . T h i s is t r i v i a l i f l C l = 15, s i n c e C i s a set o f t y p e (3,s). I n t h e o t h e r cases: ICI = 12 i m p l i e s t,= 60-4t3 , t,= ts-8, t h e n t , S l 5 and t , S 7 p

where p(n) = minimal m with an,m = 0. Proof: This is exactly what is being done when playing backwards. Proposition 2 : A direct description of the n-th row can be given by induction on m : a = 0 and a = remainder when dividing n,m n,1 m- 1 n - 1 a by m(m22). p=lntp Proof:

By induction on n using proposition 1 .

Definition 2 : Let {a,}, be the sequence of those k = 1,2,3,..., n for which the length of the row is increasing. By figure 1 the . first members of this sequence are 1,2,4,6,10,12,18,22,30,34

,...

Looking in Sloane's book 1 3 1 , one finds that this sequence might be the sequence No. 377 given there. This sequence is constructed

Kalahari and the Sequence ‘Sloane No. 377’

53

in [ 1 , 2 ] by a sieving process in the following way: Write in the first column all natural numbers. To get the second column cross out 1 and every second number. From the remaining numbers cross out the first and every third to get the third column. Then cross out the first and every fourth number and so on. The resulting sequence is the sequence No. 3 7 7 (figure 2 ) . S o in figure 2 a bracket around a number means that this number has to be deleted in the following column. Observation 2: The sequence of definition 2 coincides with the sequence No. 377. The places of brackets in figure 2 coincide with the places of brackets in figure 1. m = 2

n= 1 2 3 4 5

6

7 8

9 10 11 12 13 14 15 16 17 18 19 20

3

2

1 1 0 0 (2) 2

41 42 43

44 45 46 47 48

2 2 2 2

1 1 0 0 0 0 (2) 0 2 0 1 (;) 1 0 2 2 0 (2) 2 2 2 1

1 0 0 1

40

6

7

8

9

10

11

12

(;I

1

(2) 2

34 35 36 37 38 39

5

(2)

1

24

4

1

1 0 0 0 0

(3) 1 3 0 2 0 2 (2) 2 2 2 1 1 1

0 0 (2) 2 1 1 0 0 (2) 2 1 1 0

1

0 0

0

2

(4)

4

4 4

3 3 2 2 2 2 2 2 1 1 1 1

0

4

4 4 4 3

3

4 4

6

4

6

4 4

6

4 3 3 3 3 2 2 2 2 2 2 2 2

6

5

5

5 5 4 4 4 4 4 4 4 4 3 3

1

3

3 2 2

1

1 1 1 1 1 0 0 0 0

1 0

(5) 5

3

‘5’ 0 5 5 5

5

3 2 2

6 6 5

5 5 5

7

6 6

2

6 6

2 2 2 2

6 6 6

2 1

7 7

3 3 3 3

2 Figure 1

6 5 5 5 5 5 5 4

9 9 8

8 8 0 8

6

0 8

7 7 7 7 7 7 6

(10) 10 10 10 10 10 10

10

9 9 9 9 9 9 8

(;;I 11 11 11 11

10

13

D.Betten

54

m = 2 n=

1 2 3 4 5 6 7 .8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

3

( 1) 2 ( 2) ( 3) 4 4 ( 5) 6 6 ( 7) 8 ( 8) ( 9) 10 10 (11) 12 12 (13) 14 (14) (15) 16 16 (17) 18 18 (19)

20

(21 1 22 (23) 24 (25) 26 (27) 28 (29) 30 (31) 32 (. 3 3 .) 34 (351 36 (37) 38 (39) 40 (41 1 42 (43) 44 (45) 46 (47) 48

5

6

10

10

(10)

12

12

12

(12)

22

22

22

(22)

30

30

30

30

(30)

34

34

34

34

(34)

4

7

8

9

10

11

12

13

( 4 )

6

(16) 18

(20) 22

22

22

24

24

(24)

(26) 28

(28)

30

30

30

34

34

34

34

36

36

36

(36)

42

42

42

42

42

42

(42)

48

48

48

48

48

48

48

(32)

(38) 40

(40)

42

42

42

46

46

(46)

48

48

48

(44)

148)

Figure 2

The numbers a occur also in [l], so the two procedures (Kalahan,m ri and Sieving) are in fact related to each other. Erdos and Jabotinski [2]study the growth of the sequence ak and prove k2 ak = 7 + 0(k4’3). Another proof for this result is given by David [ l ] . Looking at the rows of the scheme 1 for great n it can be seen that the row ends with an arithmetic progression of difference 2; this progression is preceeded by an arithmetic progression of difference 4 , which in turn is preceeded by an arithmetic progression of difference 6 and so on. Those progressions are studied by David [ l ] . As an illustration we give here the sum

Kalahari and the Sequence Sloane No. 377'

55

representation for n = 100000 (figure 3 ) . The numbers in brackets give the differences in the arithmetic progressions. 1oO000

=

O + 1 + 3 t t

t 1 3 t 1 7 t +43 +35 t

1 t

5 + 2 + 4 + 3 + 1 + 1 + 7 + 2 t 1 0 + O + 8 +

9 + 7+15+ 1 + 2+16+13t 1 + 1 3 t 9+20+26+ 2 + 8+21+

41

t

29

3 + 6 t 3 0 t

5+ 3+28+ 4 + 9 + 5+35+15+31+41+45t

1 + 2 7 + 5 0 + 16 + 3 1 + 4 3 + 5 2 + 0 +

t

21 +

t

1 4 t 6 2 + 43 + 21 t 6 5 (48) (44)

t

1 t 5 9 +56 +50 (58)

t

39 + 10 + 5 0 + 17 + 55 + (40) (38)

+18+54+13+47+ 2+.. +66+16+.. +46+ (36) (34) (32) (30) +21+..+77+17+..+69+ 4+..+76+ 5+..+93+ (28) (26) (24) (22)

..

..

..

..

.. ..

.. ..

..

..

+15 + +95 +10 + +loo+ 7 + +103+ 1 + +113 (14) (20) (18) (16) + 0 + +132+ 5 + +145+ 1 + +169+ 1 + +205 (12) (8) (6) (10) t 0 t +280+ 2 + +560 (4) (2)

t

Figure 3

Wheras David has studied the rows of scheme 1, we now want to look at the columns and show Theorem: The m'th column of the scheme {a 3 is periodic of n,m length 1.c.m. {2,3,4,...,m}. In order to prove this theorem we define another scheme {AN,m}, N z 0 , m z 2 , which gives the numbers of vertical repetitions of the digits an the quotient scheme We call the scheme {AN ,m' ,m of Ian 1 , see figure 4 . ,m Proposition 3 : duction on m:

The columns of the scheme Ak (m-1 ) ,m+l

=

Ak (m-l)+i,m+l = (k = 0,1,2,... , i =

%m,m

{AN ,m}

are given by in-

+Akm+l ,m

*km+l +i,m 1,2,..., m-2)

.

Proof:

We look at the original scheme {a 1 and compare two n,m a a numbers with the corresponding numbers an,m+l n+l ,m n+l ,m+l of the next column. Using proposition 1 we note: an+l,m = an,m then also an+l,m+l= an,m+l; if

=

an,,,-1

then a l s o

a = a -1; but if a = 0 and a = m-1 n+l ,m+l n,m+l n,m n+l ,m

then

D. Betten

56

= a So, to get column m+l from column m in the a n+l ,m+l n,m+l. 1 the number of 0 ' s and the number of the following scheme {AN ,m (m-1)'s must be added, all other lengths of repetition are carried over without change. m = 2

N

=

3

6

5

4

O

0

7

0

0 12

1

5

21

6

61

2

4

6

5

4

3

3

4

4

0

4

2

8

3

4

5

1 -

4

2

8

6

0

2

1 -

6

7

5

6'

0

6

8

4

6

6

4'

3

6

5 2

10

2

4

4 12

11

-1 -2

3

6

12

2

4

13

-

8

1

14 15

L

=

1

1

2

3

12

10

1

=

2

6

12

60

60

42 0

Figure 4

Note: In the first row (N =0)we get the sequence {a,], so Proposition 3 gives another method to construct this sequence. Definition 3 : We define l ( m ) = length of period of column m of the scheme {a 3 , L(m) = length of period of column m of the quotient n,m scheme

{ANlm).

Proposition 4 : (a) (b)

Induction rules for the numbers L and 1 : m- 1 L(m+l) = [m,L(m)1 m 1 (m-1) l(m) = [m,L(m)l -~(m7

57

Kalahari and the Sequence ‘Sloane No. 377’

Proof: (a) The procedure of combining Akm,m and Akm+l ,m in column m is periodic of length [m,L(m)1. The period L(m+l) is somewhat smaller, because the two numbers Akm,m and %m+l,m in column m give only one number in column m+l. Therefore we must subtract [mtk(m)7 and get L(m+l) = [m,~(m)]- [m,L(m’il = [m,~(m)l m

m1 m e

(b) The number l(m) is the sum of the A’s in column rn from N = 0 to N = [m,L(m)] - 1 . This period consists of r ~ ~ ~ ) ( m ) l sequences of length L(m) and the sum in each of those sequences is l(m-I). Therefore we get l(m) = [m,L(m)1 1 (m-1) . L (m)

,..., m+ll m

Lemma:

ml [m+l, [ 2 ,..., m

Proof:

Compare factors on both sides.

= [2

for every

m 2 2

.

Proof of the theorem: (1) L(m)

=

[2,-..,*-11 m-l

for all

m 2 3 .

Proof: This is true for m = 3 . Suppose that (1) is true for some m 2 3 , then by proposition 4(a) and Lemma it follows m-Ill m-l = m- 1 = [m,[ 2 L(m+l) = [m,L(m)] m- 1 m m - [ Z ,...,ml = 12 ,...,ml m- 1 m m

,...,

(2)

i(m)

=

L(m+l) m for every

m 2 2

,

Proof: This is true for m = 2. If ( 2 ) has already been proved for some m 2 2 , then using proposition 4 (b), the lemma and (1 it follows l(m) - tm+l,L(m+l)l m = l(m+l) = [m+I,L(m+l)l -(m+l)-

-

= [m+l, [2,.

- [2,

.. ,ml .

m

..., m+ll m+ 1

(m+l)

= [2,

...,m+ll m

= L(m+2)

*

.m =

(m+l) ,

so, by induction, ( 2 ) is proved.

Using ( 2 ) and (1) the theorem follows: l(m) = L(m+l) m = [ 2 , 3 , . ..,ml

-

m = [2,3,

...,m1 .

D.Betten

58

References:

Y. David: On a sequence generated by a sieving process, Riveon Lematematika 1 1 , 26-31 ( 1 9 5 7 ) . P. Erdos and E. Jabotinsky: On a sequence of integers

generated by a sieving process, Indagationes Math. 2 0 , 1 1 5 - 1 2 8 ( 1 9 5 8 ) . N.J.A.

Sloane: A Handbook of integer sequences, Academic Press, New York 1 9 7 3 .

C.D. Grupp: Brettspiele-Denkspiele, Humboldt Taschenbuchverlag, Miinchen 1 9 7 6 .

Mathematisches Seminar der Universitat OlshausenstraBe 40, 2 3 0 0 Kiel 1 Bundesrepublik Deutschland

Annals of Discrete Mathematics 37 (1988) 59-68 0 Elsevier Science Publishers B.V. (North-Holland)

59

ENCIPHERED GEOMETRY. SOME APPLICATIONS OF GEOMETRY TO CRYPTOGRAPHY Albrecht BEUTELSPACHER Siemens AG, ZT ZT! SYS 42, Otto-Hahn-Ring 6, 0-8000Munchen, West Germany We present three applications of finite geometry t o cryptography. These applications comprise threshold schemes, binary sequences and aut henti cation systems. The main geometric objects are classical projective spaces.

1. MUCH ADO ABOUT NOTHING: INTRODUCTION

There are two major branches in todays cryptography. On the one hand one wants t o conceal the content of a message. An al orithm which achieves this aim is called an encr ption algorithm. On the other h a n z encryption does not guarantee the integrity o f t h e message; a bad guy can easily change the message without being able t o understand what the message says. This is a serious problem in many applications. (Think for instance of changing the address (i.e. the account number) of an electronic cheque.) The part of cryptography in which one studies techniques in order t o controll the integrity of a message is called authentication. For both, encryption and authentication, one needs keys, usually secret keys. These keys have t o be generated, administrated, changed, etc. All these procedures are controlled by a "master key". It is clear that the access t o this master key is a critical point in the system. In other words, before installing a crypto system, one has t o have an efficient access control system. Consequently, we shall deal with three topics: Access control, encryption and authentication. Our aim is neither t o search for a complete survey, nor to present the most frequently used algorithms; we focus on applications of finite geometry! One could think that new types of problems need new methods for solutions. This is true for quite a few parts of cr ptography (such as the idea of public keys). On the other hand, surprisingly enougi, many classical mathematical structures and results can be applied in cryptography. In particular, number theor was injected with new life by the applications in the RSA-algorithm and signature sc\emes. Here, I would like t o show that also classical projective geometry has ver clever applications in crypto systems. Clearly, geometry is not the center of appliedlmathematics; but I want to confirm that some parts of geometry are very close t o the most modern and exciting parts o f computer science.

2. TWO HEADS ARE BETTER THAN ONE: AN ACCESS CONTROL SYSTEM

Threshold schemes are very clever access control systems which were invented t o control access t o top secret operations such as changing the master key of a computer security system. A widely used method t o solve this problem is the following: The critical operation should only be performed if t or more authorized usersagree on it.

A . Beutefspacher

60

in order to handle such a situation, threshold schemes have been introduced in the literature (see for instance 111, (51, [61, [71,[101, [131). A t-threshold scheme consists of a certain number of pieces of information such that the following properties are fulfilled:

(i) a secret datum x can be retrieved from any t shadows, (ii) determining x with knowledge of only t-1 of the shadows is impossible. In geometric /an uage, a t-threshold scheme can be described as follows. In a geometry (consisting ofpoints and blocks) one choses a block B and n points PI, ..., Pn on B in such a way that (i)any t ofthe n points determine B uniquely, (ii) through any t-1 or fewer of the n points there are 'many' blocks. One choses also a set 5 which intersects B in one point X (or a non-empty set X of points). The system has to know only the geometry (i.e. the points and blocks) and the point X. If a certain number of shadows (i.e. points) enter the system, it tries t o determine the block through these points. If there is no unique block through the given shadows, the process stops. If, on the other hand, there is a unique such block C, the systems computes Pc : = C n 5. Only if Pc = X, the users get access t o the system and may perform the operation in question.

Clearly, such a scheme works equally well, no matter where or who the users are. Also, the system has a certain guarantee that the users are authorized, since the datum X has t o be computed. (If an unauthorized user takes part in the process, the system is virtually certain to determine a block C which is different from B; then the s stem will com Ute a point Pc different from X.) As a consequence we have that tKe existence o only one unauthorized user U guarantees that no group of users which U belongs to can perform the critical operation.

P

Example 1. Curves One may take as 6 a randomly chosen conic in a Desarguesian projective plane P = PG(2,K) over the field K. As 5 we take a (randomly chosen) tangent t o B a t a point X. Since the number of conics through four points in general position equals IKI -2 (which can be chosen as big as one wants to), we have a 5-threshold scheme.

Enciphered Geometry

61

If we restrict ourselves t o circles, we get 3-threshold schemes. These examples generalize in the following way. A rational normal curve in the d-dimensional projective space Ps has the property that it is determined by any d + 3 of its points. Moreover, through any d + 3 points of Ps, no d + 1 o f which are contained in a common hyperplane, there is precisely one rational normal curve (cf. B. Segre [12] and van der Waerden 1141). If d = 2, the rational normal curves are precisely the conics. Example 2. Flats Consider now an affine or a projective geometry G of dimension d. 1. As B we take a (t-1)-dimensional (flat) subspace; PI,...,^, are n points on B which are in general position and for 5 we take a (d-t + 1)-dimensional subspace which intersects B in just one point X. We get a t-threshold scheme for every positive integer t.

2. Now we consider a variation of the above example. Suppose that we have t w o types of users which want to get access, say programs and human users. We would like t o design our system in such a way that

(i) Any t o f the total number of n users determine the block B uniquely. (ii) The programs alone, even if their number is bigger than t, do not determine B. In order t o achieve this, we take a (t-1)-dimensional flat B and a (t-2)-dimensional subflat B*. The points which correspond t o programs will be chosen as points in general position of B*, whereas the points which correspond t o human users are points of B outside B*. The following picture describes the case t = 3 .

Users of type 1

/

users

*..................... Now we consider the problem where there is a hierarchy o f users with the property that the levels o f the hierarchy correspond t o the amount of privileges a certain user has. A trivial solution is t o design a threshold scheme for every level seperately and "join" the different systems disjointly. For example, suppose we have users of two types.

62

A . Beutelspacher

The users in group 1 are representated by a 2-threshold scheme (in our example, by a line t), whereas users in group 2 are representated by a 3-threshold scheme (in our situation by a circle C). The line P and the circle C intersect in a point P,which is also the intersectuion of C (or C) by the line 5.

C

This solution clearly has the disadvantage that no set of users of one group can replace one or more users of another group. This is not the case in the following more sophisticatedsystem. In this system, a user i is usually not represented by a point, but by a subspace Ui - in such a way that the dimension of U i corresponds t o the hierarchical level of i. In a military world, w e might design our system as follows:

Generals are represented by planes, colonels by lines, ordinary soldiers by points,

all within a (say) 5-dimensional space W. The critical operation will be executed if the subspaces which correspond t o the users in question span the subspace W. In contrast to the above system, here a (big) number of soldiers might replace a general. Example 3. Designs A t-(v,k,h)-design is a geometry D consisting of points and blocks (which we always think of sets of points) satisfying the following properties: (a) D has v points. (b) Every block of D consists of exactly k points. (c) through any t points there are exactly A blocks.

For definitions, background and the theory of block designs see Beth, Jungnickel, Lenz [3], Beutelspacher 141, Hughes and Piper [9]. If A = 1, D is also called a Steinersystem. So, by definition, given a block B and n ( < k) points on 6, the unique block through any t of these points is B. Hence, every Steiner system yields many (usually different) t-threshold schemes. In a design with A > 1, t y ically, throulh a certain number s of points there are many blocks. (If s = t, this Pollows from t e definition; but in many cases, this is also true for certain values of s > t.) But there are certain situations, where for a given number s, any set of s points determine a block uniquely. In other words, any set of s points lies on at most one block.

Enciphered Geometry

63

The non-negative integer p is said t o be an intersection number o f the design D, if there are t w o different blocks o f D which intersect in precisely p points. The following statement is obvious: Denote by D a t-(v,k,h)-design ,a n d fix an arbitrary number s wich is bigger than the maximal intersection number o f D.Then through any set o f s points o f D there Is at most one block o f D. As a consequence w e have: If s is a number bigger than any intersection number o f a given design D such that through some s-1 points there are many blocks, then there exists an s-threshold scheme. This applies in particular t o biplanes, i.e. symmetric 2-(v,k,h) designs.

3. I KNOW SOMETHING YOU DON'T KNOW: ENCRYPTION Before we can send a message d o w n an "electronic chanel" we have t o "encode" i t into a binary sequence, th a t is a se uence o f 0's and 1's. The most obvious way (due t o Vernam [15]) t o encipher such a t i n a r y message M consisting of the bits m , m2, is adding t o it, b i t by bit, another binary sequence K (consisting o f the bils k,, m,. m ek , c =mjekr.. (where @ denotes k, k, ,.._).Thesequence c, = m,@k c binary addition) is called the crphe&xf a7d 4his:s the text w ich IS being sent d o w n the chanel.

Message (cleartext) M = m,, m2, m3,... Ciphertext c,, ,c, c3,...

Key sequence K = k,, k.,

*

k, ,...

Since under modulo 2 addition (mieki)eki = mi fo r all mi, ki in {O,l}, the recipient can recover th e message just by adding the sequence K (called the key) t o the ciphertext. If the sequence K is a truly random 0,l-sequence, then also the ciphertext is a random sequence, and so it is ' unbreakable"! But the price one has t o pay for this super cipher is t hat t he key K is extreme1 long (its length equals the len th o f the message) -and, o f course, the key has t o e forwarded t o the receiver be ore the message is transmitted (otherwise the receiver would be in the same unpleasant situation as an eavesdropper). Moreover, th e ke has t o be transmitted in a secure way (otherwise, an eavesdropper would be in tKe same pleasant situation as the receiver). Usually, this problem is solved by taking as the key a "pseudo-random" sequence instead o f a truly random sequence. A sequence is pseudo-random if it behaves a t a first lance as a random sequence, but is determined by ver few data. W hat do w e mean %y "behaves at a first glance as a random sequence"? TKe three famous postulates o f Golomb (cf. for instance [2]) provide one possible answer t o this question. First o f all,

i!l

9

A . Beutelspacher

64

these postulates deal only with periodic sequences. Let p be the length of the period, and denote by C a generating cycle. The first postulate is very easy t o state P1. The number ofzeros in C equals the numberofones.

If p is an odd number, then this postuilate can never be true in a strict sense. In this case "equals" should be interpreted as "differs by at most 1". A similar interpretation should take place in the following postulates. By a gap we mean a maximal sequence of O's, whereas a string is a maximal sequence of 1's of C. So, the following sequence has two gaps of length 2 and one string of length 1: Example:

c = 01 1101100101000 Now, we can formulate the second postulate: P2. For any i, the number o f gaps o f length i equals (that is, differ by at most one) the number ofstrings of length i.

Clearly, if C is a enerating cycle of a periodic sequence, then also the cycle C(a) which is obtained %om C by a cyclic shift of a positions is a generating cycle. If C denotes the cycle in the above example, then C(2) = 11011001010001 For a fixed a f 0, we denote by A the number of positions in which C and C(a) agree and by D the number o f positions in which C and C(a) disagree. (So D = p A.) The number (A-D)/p = (2A-p)/p is called the out-of-phase autocorrelation. Again considering our above example (with a = 2) we get A = 7, D = 8 and asthe out-of-phase autocorrelation

(A-D)/p = -1/15. The third postulate of Golornb readsas follows: P3. The out-of-phase autocorrelation is a constant (for a Z 0)

At this point the natural question arises, whether there are sequences satisfying Golombs postulates. Surprisingly enough, most of the known sequences with the above discussed properties arise from projective spaces! Consider the projective spaces P = PG(d.2) of dimension d over the field with 2 elements. It is well known that P admits a Sin er cycle, that is a cyclic group which acts transitively on the points (as well as on t f e hy erplanes) of P. We label the points of P by 1, ...,v = 2d '-1 in such a way that tRe map u which ma s i onto i + 1 (mod v) is a generating element o f the Singer cycle. Then we have the Ellowing +

Theorem. Let C = (a ,...,av) be the incidence vector of a hyperplane H o f P with respect to the above defined labeling. (That is aj = 1 if H is incident with the point i, and ai = 0 otherwise.) Then the cycle C satisfies Golomb'spostulates. Proof. By definition, the number a of 1's in C equals the number of points in the hyperplane H. So,

Enciphered Geometry

65

On the other hand, the number b of 0's in C is nothing else than the number of points outside H, hence

b = 2d In other words, P1 is true. In order t o check P2 we have t o take into consideration that our labeling corresponds t o the Singer c cle Let f be an irreducible polynomial of degree d + 1 over GF(2). Then the set o/po;nts of the projective space P can be seen as the set of all non-vanishing polynomials of degree 5 d. Moreover, the generating element u of the Singer cycle is just multiplication by x (mod f). We take as our hyperplane the hyperplane H which is spanned by the points 1, x, x2, ..., xd-l. In other words, H consists of all polynomials of degree 5 d-1. We claim that the corresponding incidence,vector C has one strin o f length d. one gap o f length d + 1, and 2' stringsand 2' gapsoflengths d-1-i (? = O,l, ...,d-2). Clearly, this will show that P2 is true. H and l/x = (f-l)/x P H, there.is one String of length d (namely Since xd 1,x,x2 ,...,xtl). Now take a polynomial h = XI + six' + ... + a,x + 1 of degree I + 1 which has a non-vanishing absolute term. Obviously, there are exactly 2' such polynomials. I t follows that +

h, x-h, x2*h,... xd-2-i*h are points of H. Now, xd-lei-h and hlx are polynomials o f degree d (note that hlx = h(f-l)/x, which has degree d, since h has 1 as its absolute term). Hence h ives rise t o a string o f len t h d-1-i. In such a way we get at least 2' strings o f length 3-1-i. Since we have consi ered every polynomial of de ree 5 d-1 just once, we have covered every point of H excactly once. So, the consi ered strings are all possible strings. This proves the first part of our assertion.

3

3

The number of gaps can be computed in a similar way by observing that x" h = 0 or h € H.

+h P H

if

For P3 we a ain make use of the fact that our labeling is induced by a collineation. that not only C is the incidence vector o f a hyper lane (namely From this it H), but also C(a). More precisely, C(a) is the incidence vector o f H' = u (H), which is a hyperplane as well, since u (and ua) isa collineation.

?allows

.f

The rest is easy: The number A o f positions in which C and C(a) aggree is the number o f common 1's plus the number of common 0's;so it is the number of points in H and H' plus the number of points off H and off H'. Consequently, A = 2d-l-1 + 2d-l = Zd-l.

From this it follows D=

zd

+

'-1 - (Zd-l) = 2d.

Therefore, the out-of-phase autocorrelation is a constant.

Remark. There is only one se uence known which satisfies Golomb's postulates, but which cannot be constructed?rorn a projective space in the avove described way. The interested reader is referred t o [ l 11, where further details can be found.

A . Beutelspacher

66

4. MAKING LIFE AS DIFFICULTAS POSSIBLE FOR THE BAD GUY: A PERFECT AUTHENTICATION SYSTEM

In this section we study the following question: How can one guarantee the integrity of a message? Or a little bit more modest: Are there systems which guarantee that the receiver notices whether the message has been changed or not?We illustrate the problem by an example. (The example and the theory can be found in [ 8 ] . ) The local manager (a bad guy) of a certain gambling casino has t o transfer the dollars from the slot machines every night to the owner. The bad guy cheats the owner by reporting the daily takings from the slot machines t o be less than they actually are and keeping the rest for himself. But the owner is suspicious and therefore he installes at every slot machine a device which takes as inputs the day's takings M and a (secret) key K and has as i t s output an authenticator (or, a message authentication code) A = f(M,K) The bad guy's dut is t o transfer not only the dollars, but also the authenticator A. The owner takes tKe transfered number M' of dollars (which might differ from M) and computes, using the secret key K, the authenticator A' = f(M',K).

Casino

Bad Guy

Owner of the casino

If M' iM, he is virtually certain that A' # A. So, if the outcome A' of his computationsdoes not aggree with A, he "knows" that the bad guy ischeating him. Such a system is called an authenticatlon system. Although the bad guy does not know the correct key, it is strongly advisable for the owner that he asks himself the question what chances the bad guy has of guessing the correct authenticator. Of course, the bad guy's chance depends on many things, for instance on the chosen authentication function f. But one principal constraint is the number k of keys. The following theorem shows that there are no authentication systems which are arbitrarily good.

Enciphered Geometry

67

Theorem. (Gilbert, MacWilliams, Sloane [ 8 ] )Let k be the total number o f keys. Then there is no authentication system in which the bad guy's chance of success is worse than l l d k . We shall not prove this theorem here. We shall, however, discuss the natural question, whether there are systems which are as ideal as possible, that is systems in which the bad guy is confronted with the unpleasant situation that he has on1 a chance of l / d k t o guess a correct authenticator. This question will be answeredYin the affirmative by the following Example. Let P be a finite projective plane of order n. Fix one line t',. The messagesare the points on to; the keys are all points outside to; the authenticators are alle lines different from t',.

So, the number of keys is n2. By the above theorem we know that the bad guy has at least a chance of l / n t o cheat without being noticed. It is t o show that he can do no better.

Assume that the bad guy has replaced the correct number M of dollars by a (smaller) number M' and that he wants t o cheat the owner's authentication scheme. He may use additional information, such as the original correct authenticator A. But he does not know the correct key K. The only thing the bad guy knows is that K is one if the n remaining points on A. On the other hand, any hypothetical authenticator through M' intersects A in a point, which is a hypothetical key. Therefore, the bad guy has only a chance of l / n to guess the correct key. In other words, the projectiveplane-authentication-scheme is as good as one can expect from the theory.

ACKNOWLEDGEMENT The author would like t o thank very much Dr. Jean Georgiades for many helpful discussions and for his strong coffee.

A . Beutelspacher

68

REFERENCES G.R. Blakley, Safeguarding cryptographic keys. Proc. NCC 48, AFIPS Press, Montvale, N.J., 317-319(1979). H. Beker and F.C. Piper, Cipher Systems. Northwood Books, London 1982. T. Beth, D. Jungnickel and H.Lenz, Design Theory. 6.1.-Wissenschaftsverlag, Mannheim - Wien -Zurich 1985. A. Beutelspacher, Einfuhrung in die endliche Geometrie I. Blockplane. B.1.Wissenschaftsverlag, Mannheim - Wien - Zurich 1982. A. Beutelspacher and K. Vedder, Geometric structures as threshold schemes. To appear. D. Chaum, Computer systems established, maintained, and trusted by mutually suspicious groups. Memorandum No. UCB/ERL M79110,University of California, Berkeley, CA, February 22,1979. D.E.R. Denning, Cryptography and Data Security. Addison-Wesley 1983. E.N. Gilbert, F.J. MacWilliams, N.J.A. Sloane, Codes which detect deception. Bell. Syst. Tech. J. 53 (1974).405-424. D.R. Hu hes and F.C. Piper, Design Theory. Cambridge University Press, 1985. S.C. Kotlari, Generalized Linear Threshold Scheme. Adavances in Cryptology (Proceedings of CRYPT0 84),Lecture Notes in Computer Science 196, Springer

1985,231-241.

F.C. Piper and M. Walker, Binary sequences and Hadamard designs. B. Segre, Lectures on modern geometry. Cremonese, Roma 1961. A. Shamir, How t o share a secret. Comm. ACM Vol. 22(1),612-613(1979). B. van der Waerden, Einfuhrung in die algebraische Geometrie G.S. Vernam, Cipher printing telegraph systems for secret wire and radio telegraphiccommunications. J.AlEE 45(1926),109-115.

Annals of Discrete Mathematics 37 (1988) 69-74 0 Elsevier Science Publishers B.V. (North-Holland)

69

ON FINITE GRASSMANN SPACES

Paola B I O N D I 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 V i a Mezzocannone 8 80134 N a p o l i , I t a l i a *

I n t h i s paper a c h a r a c t e r i z a t i o n o f t h e Grassmann spaces r h ( l P ) a s s o c i a t e d w i t h a f i n i t e p r o j e c t i v e space IP i s o b t a i n e d by means o f i n c i d e n c e p r o p e r t i e s o f p o i n t s and l i n e s o n l y .

1.

INTRODUCTION

L e t IP be a p r o j e c t i v e space o f dimension a t l e a s t t h r e e . For each i n t e g e r h, 1< h < dim IP-1, Gh(lP) and Fh(lP) denote t h e f a m i l i e s o f h-dimensional subspaces o f IP and o f p e n c i l s o f h-dimensional subspaces o f IP, r e s p e c t i v e l y . Such a p e n c i l c o n s i s t s o f a l l t h e elements i n Gh(IP) t h r o u g h a f i x e d ( h - 1 ) - d i m e n s i o n a l subspace o f IP and which a r e c o n t a i n e d i n a g i v e n subspace o f IP o f dimension h + l . The i n c i d e n c e s t r u c t u r e Fh(lP) = (Gh(lP),Fh(lP)) i s a p a r t i a l l i n e a r space, t h e h - t h Grassmann space a s s o c i a t e d w i t h IP. I f dimIP< 00 and IP i s c o o r d i n a t i z e d by a f i e l d , t h e n rh(lP) i s i s o m o r p h i c t o t h e Grassmann v a r i e t y r e p r e s e n t i n g t h e h-dimensional subspaces o f IP. I n r e l a t i o n t o s p e c i a l v a l u e s o f h, more c h a r a c t e r i z a t i o n o f t h e spaces rh(lP) a r e known ( C l l , [21, C31, C41, C51, C61). The spaces rh(lP), f o r each h, a r e c h a r a c t e r i z e d i n C71, w i t h t h e h e l p o f i n c i d e n c e p r o p e r t i e s o f t h e maximal subspaces i n rh(IP). rh(lP), w h i c h o n l y uses i n c i d e n c e p r o p e r t i e s A c h a r a c t e r i z a t i o n o f t h e spaces o f p o i n t s and l i n e s , i s o b t a i n e d , i n t h e s p e c i a l case IP f i n i t e and i r r e d u c i b l e i n C31 as a consequence o f a more g e n e r a l theorem. T h i s paper p r o v i d e s a c h a r a c t e r i z a t i o n , i n t h e f i n i t e case, o f t h e p a r t i a l l i n e a r spaces (P,L) i s o m o r p h i c t o r h ( l P ) , by means o f i n c i d e n c e p r o p e r t i e s o f p o i n t s and l i n e s o n l y . T h i s c h a r a c t e r i z a t i o n a l s o h o l d s i f IP i s r e d u c i b l e . Moreover, t h i s r e s u l t i s e x t e n d a b l e t o t h e i n f i n i t e case i f t h e f o l l o w i n g c o n d i t i o n h o l d s : " t h e r e e x i s t s i n (P,L) a maximal subspace o f f i n i t e r a n k " .

2.

INCIDENCE

PROPERTIES

OF POINTS

AND LINES

IN

rh(lp)

L e t (P,L) be a p a r t i a l l i n e a r space (C81). F i r s t o f a l l , we r e c a l l some w e l l known b a s i c d e f i n i t i o n s . Two d i s t i n c t p o i n t s a , b r P a r e collinear, a - b , i f t h e l i n e (a,b) t h r o u g h them e x i s t s . I f t h i s i s n o t t h e case, a and b a r e noncoZZinear, a c b . Moreover, any p o i n t i s assumed t o be c o l l i n e a r w i t h i t s e l f . *Work supported by N a t i o n a l Research P r o j e c t on " S t r u t t u r e geometriche, Combinatoria, l o r 0 a p p l i c a z i o n i " o f I t a l i a n M.P.1

.

P. Biondi

70

A p a r t i a l l i n e a r space i s connected i f f o r any two d i s t i n c t p o i n t s i n P t h e r e e x i s t s a p o l y g o n a l p a t h i n (P,L) j o i n i n g them. A subspace o f (P,L) i s a subset o f P c o n s i s t i n g o f p a i r w i s e c o l l i n e a r p o i n t s and c o n t a i n i n g t h e l i n e t h r o u g h any two o f i t s ( d i s t i n c t ) p o i n t s . A subspace i s maximal i f no subspace e x i s t s i n (P,L) which p r o p e r l y c o n t a i n s i t . Two p o i n t s b and c a r e of t h e same type w i t h r e s p e c t t o a p o i n t a, b $ c, if b o t h o f them a r e e i t h e r c o l l i n e a r o r n o n c o l l i n e a r w i t h a. L e t aEP and 1eL . The p o i n t a i s coZZinear w i t h 1, a - 1 , i f a i s c o l l i n e a r i f i t i s collinear with w i t h any p o i n t on 1, one-collinear w i t h 1, a-1, e x a c t l y one p o i n t on 1 and n o y o l l i n e a r w i t h 1, a + l , i f i t i s c o l l i n e a r w i t h no p o i n t on 1 . I f a + 1 , t h e n 1, = { X E P : x - a , x - 1 } . L e t 1,mEL . The l i n e s 1 and m a r e collinear, 1-m, i f x - y f o r any x s l and y m . I f t h i s i s n o t t h e case, 1 and m a r e noncollinear, 1 + m . F i n a l l y , l e t 1,mcL We d e f i n e a chain o f l e n g t h k and w i t h e n d - l i n e s 1 and m as an o r d e r e d k - t u p l e (11, 1 k ) o f elements i n L such t h a t : i ) 11'1 and 1k'm ; i i ) li - 1 i + l , i=l, ..., k-1, and 1 i + l i + 2 , i=l, ..., k-2. I f i i ) h o l d s f o r any i (mod. k ) and lln ... n 1 k = 13, t h e n t h e c h a i n i s c a l l e d a c y c l e . 1k) i s minimal i f Denote by V t h e s e t o f c y c l e s i n (P,L). The c h a i n (11, t h e r e e x i s t s no c h a i n o f l e n g t h h, h2.

I t i s easy t o v e r i f y t h a t , g i v e n a p r o j e c t i v e space IP o f dimension a t l e a s t three, r h ( l P ) i s connected and s a t i s f i e s ( 2 . 1 ) - ( 2 . 5 ) , f o r each h, 1< h3. Then, a c h a i n w i t h e n d - l i n e s 1 and ( a k - l , a k ) e x i s t s , e i t h e r 1 n ( a k - l , a k ) = g or l n ( a k - 1 , a k ) # n . Moreover, a c h a i n w i t h t h e two i n c i d e n t e n d - l i n e s ( a k - l , a k ) and m exists. W i t h t h e h e l p o f t h i s two chains, a c h a i n w i t h e n d - l i n e s 1 and m can be e a s i l y constructed.

...,

...,

PROPOSITION 3.5. dimension.

The maximal subspaces i n (P,L)

a r e p r o j e c t i v e spaces o f f i n i t e

P r o o f . Since (P,L) i s f i n i t e , t h e s t a t e m e n t f o l l o w s f r o m prop.3.10

i n C21.

12

4.

P. Biondi

DISTANCE BETWEEN TWO MAXIMAL SUBSPACES I N (P,L)

L e t 1,m~L. We d e f i n e t h e distance between 1 and m , d ( l . m ) , as t h e i n t e g e r k-2, where k i s t h e l e n g t h o f a m i n i m a l c h a i n w i t h e n d - l i n e s 1 and m ( s e e prop.3.4). Now, l e t L and M be two maximal subspaces. S e t d(L,M) The i n t e g e r d(L,M)

= Cd(1,m)l

1cL ,mcM

w i l l be s a i d t o be t h e distance between L and M.

PROPOSITION 4.1. L e t V and W be t h e two maximal subspaces t h r o u g h a l i n e 1. Moreover, l e t M be a maximal subspace such t h a t V n M = W n M = 0. Then, d(V,M)= d(W,M)+l

.

P r o o f . Set d(V,M)=k and d(W,M)=k'. Moreover, denote by (v1, ...,vk+2) and (wl,.. ..,wkl+z) two c h a i n s o f l e n g t h k+2 and k'+2, r e s p e c t i v e l y , w i t h v l c V , wlcW and Vk+2% Wk'+2 c M. F i r s t o f a l l , r e c a l l t h a t two l i n e s a r e c o l l i n e a r i f f a maximal subspace t h r o u g h them e x i s t s . I f v1=1, t h e n v 2 c W . T h e r e f o r e , k'(k-1. I f v l f l , t h e n v 2 + 1 by pr0p.V i n C41,sect.2. Thus, t h e c h a i n ( l , v l , ...,vk+2) o f l e n g t h k+3 e x i s t s , so t h a t k'(k+l. I n any case, k ' ( k + l . By t h e same argument, k(k'+l. T h e r e f o r e , k-l 2 , set q = p and let u be (*) work done within the activity of G.N.S.A.G.A. Italian Ministry of Public Education

.

and supported by the

A . Bonisoli

16

an automorphism of the finite field the following subset of X:=PG(l,q)=GF(q)u{m}

-

G:= { x u

x

PTL(2.q)

such that

o2 # 1 ; define

G

to be

in its natural action on the projective line

:

(ax+b)/(cx+d)

c+

GF(q)

ad-bc

:

(ao(x)+b)/(co(x)+d)

} u

.

1

ad-bc a non-square in GF(q)*

:

A family of regular subsets of

a square in GF(q)*

G,under certain conditions on

q , is given in

.

C2l

The classification of the sharply 3-transitive finite permutation groups goes back to H. Zassenhaus and J. Tits stated as follows : the finite set PG(1,q)

=

if

G

for some prime power

G

(1.2)

q = ~ *for ~ an odd prime

PGL(2,q)

and can be

can be identified with the projective line

(1.1)

is

C6, XI 2.61 )

is a sharply 3-transitive group of permutations on

X , then X

GF(q)u{=}

[4l or

(cfr.

q

and either

in its natural action on

PG(1,q)

or

p

denotes the unique involution in Aut(GF(q)) lowing subgroup of the group

-

PG(1,q) M(p2f) U{

x

R

:= {

x

(ax+b)/(cx+d)

(ao(x)+b)/(co(x)+d)

is the fol-

in its natural action on

: :

ad-bc

p-l

is taken from

R

a square in GF(q)*

C6l )

(1.1)

or

G , we may always assume

for a fixed element

(1.2)

.

}

. p

aim of this paper to determine all the regular subgroups one of the groups in

}u

ad-bc a non-square in GF(q)*

is a regular subset of a permutation group (otherwise consider

1 ER

PTL(2,q)

then G

0

:

(the notation M(p2f)

If

f , and if

and a positive integer

of of

R ) ; it is the

G when G

Of course the condition that

is

R be

a subgroup and not merely a subset is a strong one; in order to find all the regular of

G

subsets

of

G

one should investigate all the

transitive

subgroups

(since a regular subset necessarily generates a transitive subgroup).

but this seems to be a harder task

.

Our results are summarized in the following

The Regular Subgroups PROPOSITION

.

The regular subgroups of

PGL(2,pm)

can be classified into the

(T)

following families : a family of

cyclic subgroups forming a single conjugacy class

in PGL(2,pm) for p

77

;

odd, p m # 3 , a family of (p2m)

dihedral subgroups forming a

single conjugacy class in PGL(2,pm) ; for p m = 3 , a unique dihedral normal subgroup of

PGL(2,3)

even lie in PSL(2,pm)

if

;

for p = l l , m = l , a family of of which lie in PSL(2,ll) for p = 2 3 , m = l

,

pm

-1 (mod

4) all such subgroups

and form there a single conjugacy class ; 55

subgroups isomorphic to A4

all

and form there a single conjugacy class ;

a family of

of which lie in PSL(2,23)

506 subgroups isomorphic to

S4

all

and split there into two conjugacy classes

of equal size, while they form a single conjugacy class in PGL(2,23) ; for p = 5 9 , m = l , a family of 3422 subgroups isomorphic to A5 all of which lie in PSL(2,59)

and split there into two conjugacy classes

of equal size, while they form a single conjugacy class in PGL(2,59)

No subgroup of M(P*~)

.

operates regularly on PG(l,p2f)

The basic observation for the proof is that the groups in contain PSL(2,q) group

R

as a subgroup of index at most

is either contained in PSL(2,q)

has index

2

on X

reduces to the identity

, i.e.

ClOl

El01

(q+1)/2

, but F.

groups ! )

or to

of

PGL(2,q)

of

C71

.

PGL(2,9)

the stabilizer of each point

.

PSL(2,q) x

C5,II 8 . 2 7 1

of X

PSL(2,q)

: we refer to

for such a list and to Theorem 3

of

PGL(2,q)

.

V. Franceschini showed that every regular subset

is a subgroup, while no regular subset exists in

Zironi just announced the existence of regular subsets

in M(25)

PSL(2,q)

of

for a list of all the abstract groups which are subgroups of

with identity of

(1.2)

(cfr. [12,page83) ; we use then Dickson's classifi-

Exhaustive computer search by

M(9)

and

or its intersection with

cation of the abstract groups appearing as subgroups of Theorem 4 of

(1.1)

2 , therefore a regular sub-

in R , thus yielding a subgroup of order

operating semireguZarly

.

(no sub-

The conjecture that every regular subset with identity

must be a subgroup has a further piece of evidence in the result

A . Bonisoli

78

We remark that the problem we approached is also motivated by the general search for sharply k-transitive subsets inside n-transitive permutation groups

111 and [lll

(see for instance

.

)

Of course the main question in this di-

rection is the determination of the finite 2-transitive permutation groups containing a sharply 2-transitive subset : such subsets do not exist in a finite sharply 3-transitive permutation group

El11

also in

with a few exceptions

ce of regular subsets in M(9)

.

(even a set)

11

in fact M(9)

elements :

C91 and

yields an easy proof of the non-existence of

sharply 2-transitive subsets in the Mathieu group sitively on

as proved in

We also point out that the non-existen-

We finally metion the article

C8l

M l l acting sharply 4-tran-

is the one-point-stabilizer in M l l .

, which contains a method based on character

theory for the determination of regular subsets in a transitive permutation

If

2.

contains a regular subgroup R , then this will also be a

PSL(2,pm)

regular subgroup of

, and in case m = 2f , p odd, also of M(pm)

PGL(2,pm)

of order

(pm+l)

and obtain the following possibilities :

(2.1) cyclic subgroups for

p=2 ;

(2.2) dihedral subgroups for p (2.3) A4

for

p = 11

,m= 1

;

(2.4) S4

for

p = 23

, m= 1

;

(2.5) A5

for p = 5 9

,m=

.

(2.3)

(2.1)

and

, PSL(2,ll)

(2.2)

1

12- IA41

PG(1,ll)

.

, we

resp.

resp.

PGL(2,23)

resp.

A5

PSL(2,59)

PG(1,59)

resp.

PSL(2,ll)

have order

S4

.

11.5

and sin-

which is rela-

resp.

resp. PSL(2,59)

A5

Furthermore, every subgroup of

PGL(2,59)

this is clear for A4

which is isomorphic to A4

and

A5

possesses sub-

and they all operate regularly

is actually contained in PSL(2,ll) ;

A s to case

see that each of these subgroups operates re-

Similarly PSL(2,23)

groups which are isomorphic to PG(1,23)

.

does possess subgroups which are isomorphic to A4

tively prime with

on

odd ;

will be dealt with in the next section

ce the one-point-stabilizers in

gularly on

.

, looking for possible subgroups

PSL(2,pm)

We check the list of subgroups of

Cases

.

we have not tried to use this method in our situation

group :

resp.

PSL(2,23)

PGL(2,ll) resp.

S4

resp.

since they have no subgroup of

The Regular Subgroups index

2 , while for

S4

Assume

RSPGL(2,23)

, R

79

it can be seen by the following easy argument R$PSL(2,23)

.

Then RnPSL(2,23)

morphic to A4 , the unique subgroup of index

2

in

JI

4 in R

of order

has the form Ry

,

and since PGL(2,23)

...

(m.0,l.c)

in PGL(2,23) ; then

this mapping is

S4

E

c = 2 and

JI

x

:

-

[3, pages 282-2851

PSL(2,pm)

, p odd, and end

(2.6)

cyclic subgroups ;

(2.7)

dihedral subgroups ;

(2.8)

A4

for p = 2 3

(2.9)

S4

for p = 4 7 , m = l . and

(2.7)

For the number of gi-

. is when

is a semiregular subgroup of order

Again we look for possible subgroups of order

.

and their partition into con-

The other possibility for a regular subgroup R

groups o f

2 / (2-x) ; the determinant of

, 2 = 5 2 , and so JIERnPSL(2,23)aA4,

(2.3) , (2.4) , (2.5)

jugacy classes we refer to

RnPSL(2,pm)

There is an element

3-transitive we may assume JI

has no element of order 4

a contradiction since A4

p

is odd and

(pm+ 1) / 2

(em+ 1) /

2

,m=l;

(2.7)

we only remark here that a dihedral subgroup of order

will be dealt with in the next section; as to case

possesses a cyclic subgroup of order

(pm+ 1) / 4 , yielding

which is never fulfilled in case m = 2 f

.

In both cases

we only have to deal with the linear group PGL(2,pm)

in A4

-1 (mod 4 )

J

and

(2.9)

because here m = l

have order

23.11

. and

which is re-

possesses subgroups which are

and they all operate semiregularly on PG(1,47)

less, there exists no subgroup of order PSL(2.23)

pm

(2.8)

12- IA41 , we see that each such subgroup operates semi-

regularly on PG(1,23) ; similarly, PSL(2,47) S4

(pm+ 1) / 2

does possess subgroups which are isomorphic to A4

since the one-point-stabilizers in PSL(2,23)

isomorphic to

.

PSL(2,pm)

up with the following possibilities :

(2.6)

latively prime with

of

in the list of sub-

Cases

The group PSL(2.23)

is iso-

, otherwise we replace R by a suitable conjugate

2 , a square in GF(23)

ven subgroups in cases

is

.

S4

.

, nor

intersecting PSL(2.47)

24 in PGL(2,23)

.

Neverthe-

intersecting

does there exist a subgroup of order 48 in PGL(2.47) in S4 : we do have direct proofs of these facts, but

probably the easiest way to see them is to look at the list of subgroups of PGL(2,pm) , obtaining

S4 as the unique possibility in the first case and no

possibilities in the second one; the first case is then also ruled o u t once

A . Bonisoli

80

we know that each subgroup R

By

rL(2,q)

PGL(2,23)

=

Sh

.

we denote as usual the group of semilinear transformations

V

of a 2-dimensional vector space assume V

which is isomorphic to

, as we proved above

actually lies in PSL(2,23)

3.

of

.

GF(q2)

over

We denote by

-

, q=pm

GF(q)

@ : TL(2,q)

; we may of course

Aut( GF(q) )

the group-

-epimorphism taking every semilinear transformation into its associated field-automorphism: of course

.

ker@=GL(Z,q)

If

is a semilinear transforma-

$

tion we look at the determinant of its coefficient matrix with respect to a GF(q)-basis

x($)=

1

of

V : if this determinant is a square in GF(q)*

, while we set x ( $ ) = - 1

-

x:

we thus obtain another group-homomorphism - : I'L(2,q)

PTL(2,q)

-

@ ( $ ) : = @ ( $ ) and @ :

If

R

PTL(Z,q)

R

-

then we set

if the determinant is a non-square in GF(q)* rL(2,q)

{l,-ll

If

denotes the canonical epimorphism, then by setting

x($):=x($) Aut(GF(q)

-

we obtain the induced homomorphisms

x : PTL(2,q)

and

)

is a regular subgroup o f

PGL(2,q)

{l,-l]

.

then every non-identical element of

has no fixed points on the projective line X:=PG(l,q)=GF(q)U{m)

is a well-known-partition of the fixed-point-free elements of

introduce this partition as follows irreducible polynomial ( o r simply

$a

for all x E X

a E GF(q)

Let

To each element

when reference to

(ax+b)/(x+a+e)

that for

.

.

we have

h(z)

.

det

h(z)

define the mapping

.

that there exists precisely one element

For

al , a 2 f X

furthermore for a E X C : =C(h(z) ) : = We remark that if with

k(z)#h(z)

element of in the form

$, : a E X k(z) then

PGL(2,q)

= z2

+al$a2

we have

we have

$a,h(z)

)

be easily verified

.

be an

Observe that $, is the identity on X and b = h(-a) # 0 and so in any case the a+e PGL(2,q) If x , y are in X then it can

( a1

belongs to

=y

cor-

is clear) by setting $a,h(z) (x) : =

transformation 9,

$,(x)

PGL(2,q)

- ez - b E GF(q)[zl

= z2

aEX

; there

(see C5, I I 8 . 5 1 ) : we

responding to the so-called S i n g e r cyclic subgroups

=

.

;

$,$J-~-,=$~

=

.

Qa3

with

aEX

such that

a3=(ala2+b)/ (al+a2+e);

It follows that the subset

1 is a regular abelian subgroup of PGL(2,q)

- Ez - 6

E

GF(q)

C(h(z))nC(k(z))=

, then in particular

$(x)=(ax+b)/(x+a+e)

; from

.

[z]

is an irreducible polynomial

{$,I

.

$(m)#m

Jl(x)#x

If

$

is a fixed-piont-free

and thus

JI

for xEGF(q)

can be written it follows

81

The Regular Subgroups then that the polynomial

+ ez - b

z2

is also irreducible over GF(q)

\ {$-I

the sets C(h(z))

.

PGL(2,q)

GF(q)-basis

hc : V

4

where

V ,x

c---+

of

GL(2.q)

subgroup S

5

;

to the basis

{1,3 ,

then

.? has a non-trivial subgroup

in PSL(2,q)

E('):=

< F4 , ck? > and

regular on PG(1,q) case ( 2 . 7 ) , as

of index

is precisely the subgroup D(l)

exactly two dihedral subgroups of order mely

C

E(*):= ; both E(l)

and these are exactly the subgroups of

2 ;

; there are

containing and E(2)

PSL(2,q)

, na-

are semiyielding

runs over the semiregular cyclic subgroups of order (q+1)/4

A . Bonisoli

84

of

.

PSL(2,q)

For

v = 1,2

let

) ' ( F

we have that, for q > 7 , )'(F

PGL(2,q) ; by [ 3 , pages 267-2681 for q = 7 , F(")

denote the normalizer of

is a subgroup of

PSL(2,7)

in

E ( ' )

is D(l) ,while,

which is isomorphic to

S4 ; in

containing )'(E

either case we can check that every regular subgroup of )'(F

as a subgroup of index 2 is dihedral and falls therefore under those in (2.2).

In this last section we prove that M(pZf)

4.

has no regular subgroups

Indeed we know from sections 2 and 3 that no regular subgroups of

.

is odd

exist when p

all, it must intersect (pZf

+ 1)/2

Hence if a regular subgroup R

2 must necessarily be cyclic

which by section

regular cyclic subgroup of

A

PGL(Z.P~~) containing

which we have

c=inM(pZf)

normalizer o f

5

M(pZf) =

x(Pf)'

where

c

2 . of

From

2

IR:

?I

=2

in M(pZf)

2

is the normalizer o f

and

@

x ,

from

it

, for

(as well as the

t=0,1,

...

x(?)=

we obtain

?=?2f

1 and have therefore the following possibilities for

u { ~2t+l$ f f when x(ff) = 1

6

(with q=p2f):

, (q-1)/2 ; j = O , l ) u

...

: t=0,1.

, (q-1)/2

; j = 0,1

I

;

... , (q-1)/2; j = 0 , 1 ) u t=0,1, ... , (q-1)/2; j = 0 , 1 )

ij={~'~~j : t=0,1,

(4.2)

u{;2t?jpf when

:

x('if) = - 1 ;

(in both cases we have written the subgroup cnPSL(2,p2f)

In case

A

(4.2)

the subgroup U

A s a matter of fact

and

(2.6) ) ;

be the unique

S

.

(4.1) f?={BZtEj :

x

-

in P ~ L ( ~ , P * ~ ) Recalling the definition of the group

and of the homomorphisms =

(case

be such a subgroup and let

2

is contained in the normalizer

)

exists at

PSL(Z,P~~) in a semiregular subgroup of order

with the previous notation let

follows that R

PSL(Z,P~~)

M(pZf)

of

.

y

V=GF(q2)

PG(1,q)

Pzt Ej P f ( x )

deriving from the elements

respectively ; the point

c Z t , otherwise we would have Czt+'

,

. .

is not even transitive on PG(l,q)=PG(V)

p2t ~j(1) = pZt ~j~ ~ ( 1= cZt ) ; as in section

the points of

(q+l) I (Zt+l)

first)

y

1

3

denote by

and

of

5-l

has no representative in V of the form

EGF(q)

,

i.e.

(

c2t+1

)q-l

=

a contradiction because q+l is even ; therefore

# y , thus proving the assertion

.

1

, whence

Czt Pj(x)

# y ,

It follows that no regular sub-

The Regular Subgroups group

R

.

of

U

exists in case

(4.1) , assume R

In case

.

some $Ei?\PSL(2,q)

If

.

(4.2)

is a regular subgroup of

; then

$=i;2S+1?f

s~{O,l,

for some index

then, since ?= , we have that R

ii2t

3

12t+1 P ?f

as

t

85

varies in

R=