Finite Geometry and Combinatorics (London Mathematical Society Lecture Note Series)

L O N D O N MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England The books in the series listed below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 34 36 39 46 49 50 57 59 66 69 74 76 77 79 80 81 82 83 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 103 104 105 106 107 108 109 110 111 113 114 115 116 117 118 119 121 122 124 125 126 127

Representation theory of Lie groups, M.F. ATIYAH et al Homological group theory, C.T.C. WALL (ed) Affine sets and affine groups, D.G. NORTHCOTT p-adic analysis: a short course on recent work, N. KOBLITZ Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds) Commutator calculus and groups of homotopy classes, HJ. BAUES Techniques of geometric topology, R. A. FENN Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Several complex variables and complex manifolds II, M.J. FIELD Representation theory, I.M. GELFAND et al Symmetric designs: an algebraic approach, E.S. LANDER Spectral theory of linear differential operators and comparison algebras, H.O. CORDES Isolated singular points on complete intersections, E.J.N. LOOIJENGA Probability, statistics and analysis, J.F.C. KINGMAN & G.E.H. REUTER (eds) Introduction to the representation theory of compact and locally compact groups, A. ROBERT Skew fields, P.K. DRAXL Surveys in combinatorics, E.K. LLOYD (ed) Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE Topological topics, I.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, NJ. KALTON, NT. PECK & J.W. ROBERTS Polytopes and symmetry, S.A. ROBERTSON Classgroups of group rings, M J. TAYLOR Representation of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Representations of general linear groups, G.D. JAMES Low-dimensional topology 1982, R.A. FENN (ed) Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES k F. RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Groups and geometry, ROGER C. LYNDON Surveys in combinatorics 1985, I. ANDERSON (ed) Elliptic structures on 3-manifolds, C.B.THOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Syzygies, E.G. EVANS & P. GRIFFITH Compactification of Siegel moduli schemes, C-L. CHAI Some topics in graph theory, H.P. YAP Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, H. GONSHOR Analytical and geometric aspects of hyperbolic space, D.B .A. EPSTEIN (ed) Lectures on the asymptotic theory of ideals, D. REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG An introduction to independence for analysts, H.G. DALES & W.H. WOODIN Representations of algebras, PJ.WEBB(ed) Homotopy theory, E. REES & J.D.S. JONES (eds) Skew linear groups, M. SHIRVANI & B. WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Non-classical continuum mechanics, RJ. KNOPS & A.A. LACEY (eds) Lie groupoids and Lie algebroids in differential geometry, K. MACKENZIE Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE Van der Corpufs method of exponential sums, S.W. GRAHAM & G. KOLESNIK New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds)

128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 190 191

Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK Model theory and modules, M. PREST Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S. PUTCHA Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKES AKI (eds) Operator algebras and applications, 2, D. EVANS & M. TAKES AKI (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M. PEINADOR and A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) The geometry of jet bundles, D J. SAUNDERS The ergodic theory of discrete groups, PETER J. NICHOLLS Introduction to uniform spaces, I.M. JAMES Homological questions in local algebra, JAN R. STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO Continuous and discrete modules, S.H. MOHAMED & B J. MULLER Helices and vector bundles, A.N. RUDAKOV et al Solitons nonlinear evolution equations & inverse scattering, M. ABLOWITZ & P. CLARKSON Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C .B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & MJ. TAYLOR (eds) Number theory and cryptography, J. LOXTON(ed) Classification theories of polarized varieties, TAKAO FUJITA Twistors in mathematics and physics, T.N. BAILEY & RJ. BASTON (eds) Analytic pro-/? groups, J.D. DIXON, M.P.F. DU SAUTOY, A. MANN & D. SEGAL Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER (eds) Groups St Andrews 1989 volume 1, CM. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2, CM. CAMPBELL & E.F. ROBERTSON (eds) Lectures on block theory, BURKHARD KULSHAMMER Harmonic analysis and representation theory for groups acting on homogeneous trees, A. FIGA-TALAMANCA & C NEBBIA Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE Groups, combinatorics & geometry, M.W. LIEBECK & J. SAXL (eds) Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) Stochastic analysis, M.T. BARLOW & N.H. BINGHAM (eds) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B.BOTVINNIK Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY & C MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M.P. FOURMAN, P.T. JOHNSTONE, & A.M. PITTS (eds) Lower K-and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD, C PESKINE, G. SACCHIERO & S.A. STR0MME (eds) Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory II, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Polynomial invariants of finite groups, D J. BENSON Finite geometry and combinatorics, F. DE CLERCK et al

London Mathematical Society Lecture Note Series. 191

Finite Geometry and Combinatorics The Second International Conference at Deinze

Edited by A. Beutelspacher University ofGiessen R Buekenhout & J. Doyen University of Brussels R De Clerck & J.A. Thas University of Ghent and J.W.P. Hirschfeld University of Sussex

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521448505 © Cambridge University Press 1993 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 A catalogue recordfor this publication is available from the British Library ISBN-13 978-0-521-44850-5 paperback ISBN-10 0-521-44850-6 paperback Transferred to digital printing 2005

Contents L. Bader — G. Lunardon Generalized hexagons and BLT-sets

5

R. D. Baker — G. L. Ebert — G. Korchmaros — T. Szonyi Orthogonally divergent spreads of Hermitian curves 17 A. Blokhuis — F. Mazzocca Lifts of nuclei in finite projective spaces

31

A. Blokhuis — K. Metsch Large minimal blocking sets, strong representative systems, and partial unitals

37

A. E. Brouwer The complement of a geometric hyperplane in a generalized polygon is usually connected

53

A. E. Brouwer — D. G. Fon-der-Flaass — S. V. Shpectorov Locally co-Heawood graphs 59 F. Buekenhout A theorem of Parmentier characterizing projective spaces by polarities

69

F. Buekenhout — O.H. King Geometries with diagram o

73

L

o p* o

F. Buekenhout — H. Van Maldeghem Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group P. J. Cameron — C. E. Praeger Block-transitive t-designs, II: large t v

89 103

H. Cuypers Generalized Fischer spaces

121

V. De Smet — H. Van Maldeghem Ovoids and windows in finite generalized hexagons

131

D. Ghinelli Flag transitive L.C2 geometries

139

David Glynn On nonics, ovals and codes in Desarguesian planes of even order. 155 C. Gordon Orbits of arcs in projective spaces

161

W. Haemers There exists no (76,21,2,7) strongly regular graph

175

J. W. P. Hirschfeld J. F. Voloch Group-arcs of prime power order on cubic curves

177

C. Y. Ho Planar Singer groups with even order multiplier groups. . . . 187 C. Huybrechts On a footnote of Tits concerning Dn-geometries

199

V. Jha — G. P. Wene The structure of the central units of a commutative semifield plane 207 N. L. Johnson Partially sharp subsets of PPL(n, q)

217

G. Lunardon Partial ovoids and generalized hexagons

233

T. Meixner — A. Pasini A census of known flag-transitive extended grids

249

G. E. Moorhouse Root lattice constructions of ovoids

269

Bernhard Miihlherr Coxeter groups in Coxeter groups VI

277

A. Munemasa — S. V. Shpectorov A local characterization of the graphs of alternating forms. . . 289 A. Munemasa — D. V. Pasechnik — S. V. Shpectorov A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)

303

D. V. Pasechnik On some locally 3-transposition graphs

319

S. E. Payne Coherent configurations derived from quasiregular points in generalized quadrangles 327 E. E. Shult Veldkamp spaces

341

L. H. Soicher The Lyons group has no distance-transitive representation. . . 355 L. Storme — T. Szonyi Intersection of arcs and normal rational curves in spaces of odd characteristic 359 J. A. Thas — C. Herssens — F. De Clerck Flocks and partial flocks of the quadratic cone in PG(3, q). . . 379 John van Bon Some extended generalized hexagons

395

F. Wettl Nuclei in finite non-Desarguesian projective planes

405

vn

PREFACE This book contains articles based on talks at the Second International Conference on Finite Geometry and Combinatorics, which took place from 31 May to 6 June 1992, at the Conference center De Ceder in Astene-Deinze, Belgium. There were 76 participants and 52 talks. There are 35 articles in these proceedings. The editors, who were also the conference organisers, are grateful for the financial support of the National Fund for Scientific Research (NFWO) of Belgium and of the University of Ghent. Above all we are grateful to Zita Oost, the conference secretary, and to Ruth Lauwaert, who helped a great deal with the production of the copy.

Albrecht Beutelspacher Francis Buekenhout Frank De Clerck Jean Doyen James W. P. Hirschfeld Joseph A. Thas

April 1993

INTRODUCTION

Discrete mathematics has had many practical applications in recent years and this is only one of the reasons for its increasing dynamism. The study of finite structures is a broad area which has a unity not merely of description but also in practice, since many of the structures studied give results which can be applied to other, apparently dissimilar structures. Apart from the applications, which themselves generate problems, internally there are still many difficult and interesting problems in finite geometry and combinatorics, and we are happy to be able to demonstrate progress. It was a great pleasure to see several Russian colleagues participating both because they were able to do so, some for the first time, and because this is an area of Mathematics not as diffuse in Russia as elsewhere. It was also good to see the participation of a significant number of talented, younger colleagues, but at the same time sad to note the difficulty they are having in finding permanent positions. The conference papers are here divided into themes. The division is somewhat artificial as some papers could be placed in more than one group. The style of mathematics is very much resolving problems rather than the construction of grand theories. There are still many puzzling features about the sub-structures of finite projective spaces, as well as about finite strongly regular graphs, finite projective planes, and other particular finite diagram geometries. Finite groups are as ever a strong theme for several reasons. There is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in characterizing structures which either have a particular group acting on them or which have some degree of symmetry from a group action. Generalized polygons Bader and Lunardon together and Lunardon alone give new constructions of classical hexagons. Buekenhout and Van Maldeghem show that there are no surprises in the action of a classical group on a hexagon or octagon. De Smet and Van Maldeghem give a new geometrical characterization of the finite classical hexagons and characterise some finite Moufang hexagons, and van Bon 1

determines some extended generalized hexagons having certain geometric and group-theoretical properties. Payne constructs a coherent configuration inside a translation generalized quadrangle. Brouwer shows that the subgeometry of a polygon induced on the objects in general position with respect to a given flag is connected. Graphs and their groups Brouwer, Fon-der-Flaass, and Shpectorov find the three graphs which locally are the incidence graph of the unique biplane on 7 points. Haemers shows that a strongly regular graph on 76 points with consistent parametric conditions cannot exist. Munemasa, Pasechnik, and Shpectorov characterize the graphs of alternating and quadratic forms over GF(2)i whereas Munemasa and Shpectorov characterize alternating forms over all larger fields. In a similar vein, Pasechnik describes the graph of a certain GJP(3)-geometry, which leads to a characterization of Fischer's sporadic simple groups. Soicher shows that the Lyons simple group has no distance-transitive representation and hence determines all faithful multiplicity-free representations of this group. Finite Desarguesian planes A spread of a Hermitian curve in a Desarguesian plane is a set of non-tangent lines partitioning the points of the curve. BaJcer, Ebert, Korchmaros, and Szonyi study such spreads with the property that no line of the spread contains the pole of any other line in the spread and deduce a result on the linear code of the plane. A minimal blocking set in PG(2, q) has size b satisfying q + \Jq + 1 < b < qy/q + 1, with the bounds being achieved for square q by a Baer subplane and a Hermitian curve. Related to this, Blokhuis and Metsch study strong representative systems and show that, for q > 25, one cannot have b = q\Jq. A nucleus of a set of q + 1 points in PG(2,g) is a point not in the set such that every line through it meets the set; it is known that the number of nuclei is at most q — 1. Blokhuis and Mazzocca use a mapping to PG(3,g) to deduce more about the structure of sets in PG(2,^) with the maximum number of nuclei. Glynn relates the code of PG(2, q) with q even to the study of nonics, where a nonic is defined to be either a conic plus its nucleus or a line pair, real or imaginary, less the point of intersection. Gordon obtains a formula for the number of protectively distinct fc-arcs in PG(n, q) and also finds an efficient algorithm for determining this number; this is applied to finding the number of projectively distinct fc-arcs and projectively distinct complete fc-arcs in PG(2,11) and PG(2,13). Hirschfeld and Voloch obtain further results on the characterization of sets of points in PG(2, q) with at most three points on a line as cubic curves.

Higher-dimensional projective spaces Storme and Szonyi examine fc-arcs having many points in common with a normal rational curve. For odd q, it is elementary that a plane fc-arc not contained in a conic has at most (g + 3)/2 points on the conic. The problem is much more difficult in higher dimensions; it is shown that, for large q and bounded n, if an arc has more than (q + l)/2 points in common with a normal rational curve, it is contained in the curve. A partial flock of a quadric cone in PG(3, q) is a set of disjoint conies on the cone and a flock is a set of q conies forming a partition of the cone less its vertex. Thas, Herssens and De Clerck survey the knownflocksand construct a newflockfor q = 11 which generalizes to a partial flock of size 11 for any q = — 1 (mod 12). An ovoid of a quadric is a set of points meeting every generator (subspace of maximum dimension lying on the quadric) precisely once. Moorhouse describes a 9-dimensional lattice which defines simply an ovoid on a hyperbolic quadric in PG(7,p), p prime. These ovoids had previously been constructed by the author from the E8 root lattice. Non-Desarguesian planes Ho investigates Singer groups S acting on a projective plane of order n and proves, among other things, that, if the multiplier group M(S) has even order, each subgroup of S is invariant under the involution of M(5), except possibly if n = 16 and S is non-abelian. Jha and Wenefind the number of central units of a commutative semifield plane. Johnson characterises certain translation planes of order q2 as equivalent to particular subsets of cardinality q of the collineation group PFL(2, q). Wettl extends results on nuclei in a Desarguesian plane to the non-Desarguesian case. Block designs A t — (y, k, A) design is an incidence structure of v points and b blocks with k points on each block and the essential property that through t points there are precisely A blocks. Such structures exist for all t by Teirlinck's theorem, but the known examples mostly have very large A. Cameron and Praeger investigate block-transitive designs with 5 < t < 8. Among other results they determine the possible automorphism groups of block-transitive 6-designs and flag-transitive 5-designs. Polar spaces Buekenhout presents an old result of Parmentier which axiomatizes a pair (P, TT), where P is a projective space and IT a polarity, so that the axioms defining P alone are weaker than usual. Shult develops the axiomatization of Veldkamp spaces for point-line geometries.

Diagram geometries Buekenhout and King study flag-transitive diagrams of rank 3 such that the residues of O-elements are dual Petersen graphs, of 1-elements are generalized digons, and of 2-elements are finite linear spaces. The linear space turns out to be either a projective plane, in which case there are precisely two geometries, or the complete graph on four vertices. In the latter case there are no geometries for which the group acts primitively on the 2-elements, but examples are given of the imprimitive case. A grid is an incidence structure of points and lines such that any line has at least two points and any two points are incident with at most one line, with the additional property that the lines fall into two classes such that two lines intersect if and only if they belong to different classes; these intersections give all the points. Meixner and Pasini describe all known extensions of grids. Ghinelli classifies the flag-transitive rank 3 geometries in which the planes are linear spaces with constant line size and the point residues are classical generalized quadrangles other than grids. Huybrechts gives a new proof of the commutativity of the division ring for a thick, residually-connected Dngeometry. A generalized Fischer space is a partial linear space in which any two intersecting lines generate a subspace that is an affine plane or its dual. Subject only to some non-degeneracy conditions, Cuypers gives a complete classification of these spaces. Miihlherr describes a geometric method of constructing, from the diagrams, Coxeter groups as subgroups of other Coxeter groups.

Generalized hexagons and BLT-sets L. Bader

G. Lunardon* Abstract

An alternative construction for the dual (^(^-hexagon is given for q odd and different from 3 n .

1.

Introduction

In [4], W.M. Kantor has constructed the generalized quadrangle associated with the Fisher-Thas-Walker flock as a group coset geometry starting from the dual G2(^)-hexagon. Analyzing Kantor's construction, the following question arises in a natural way: is it possible to define new points and new lines in a generalized quadrangle Q associated with a flock of the quadratic cone, in such a way that the new point-line geometry H is a generalized hexagon? For q odd, we prove that the only possibility is that Q is the Kantor generalized quadrangle constructed in [4] and H is the dual C?2(