TRENDS IN MATHEMATICS

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Groups and Geometries Siena Conference, September 1996

Lino di Martino William M . Kantor Guglielmo Lunardon Antonio Pasini Maria Clara Tamburini Editors

Springer Basel AG

Editors' address:

Antonio Pasini Department of Mathematics University of Siena Via del Capitano 15 53100 Siena

1991 Mathematical Subject Classification 20-06, 51-06, 05-06

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data

Groups and geometries : Siena conference, September 1996/Antonio Pasini ed. - Basel; Boston ; Berlin : Birkhäuser, 1998 (Trends in Mathematics)

ISBN 978-3-0348-9785-3

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con­cerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.

© 1998 Springer Basel AG Originally published by Birkhäuser Verlag in 1998 Softcover reprint of the hardcover 1 st edition 1998

Printed on acid-free paper produced of chlorine-free pulp. TCF °=

ISBN 978-3-0348-9785-3 ISBN 978-3-0348-8819-6 (eBook)

9 8 7 6 5 4 3 2 1

DOI 10.1007/978-3-0348-8819-6

Contents

Preface 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 vii

Talks 0000000000000000000000000000000000000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 ix

Participants 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 xi

Mo Go Aschbacher and So Do Smith, Quasithin groups 0000000000000000000000000 1

Bo Baumeister and Go Stroth, The non-canonical gluings of two affine spaces 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9

R. Jo Blok and Ao E. Brouwer, The geometry far from a residue 0000000000000 29

Fo Buekenhout, Mo Dehon and Do Leemans, On flag-transitive incidence geometries of rank 6 for the Mathieu group M12 0000000000000000 39

Bo No Cooperstein, On a connection between ovoids on the hyperbolic quadric Q+(lO,q) and the Lie incidence geometry E6 ,1(q) 0000000000000000 55

Ho Gottschalk and Do Leemans, The residually weakly primitive geometries of the Janko group J1 0000000000000000000000000000000000000000 65

R. Mo Guralnick and Co Hoffman, The first cohomology group and generation of simple groups 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 81

Co Huybrechts and Ao Pasini, A characterization of the Hall-Janko group J2 by a coL * -geometry 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 91

Ao Ao Ivanov, Affine extended dual polar spaces 0000000000000000000000000000 107

No Lo Johnson, Derivable nets may be embedded in nonderivable planes 00000 123

Mo Wo Liebeck, Regular orbits and the k(GV)-problem 000000000000000000000 145

Ao Lucchini, Generating minimally transitive groups 000000000000000000000000 149

Go Mo Seitz, Maximal subgroups of finite exceptional groups 0000000000000000 155

Ao Shalev, Subgroup structure, fractal dimension, and Kac-Moody algebras 000000000000000000000000000000000000000000000000 163

Eo Shult, Aspects of buildings 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 177

Ao 1. Steinbach, Generalized quadrangles arising from groups generated by abstract transvection groups 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 189

Jo Ao Thas, Embeddings of geometries in finite projective spaces: a survey 0000000000000000000000000000000000000000000000000000 0 00000000000 201

Jo Van Bon and Ho Cuypers, Affine extensions of near hexagons related to the spin module of type B3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 215

Ho Van Maldeghem, Ovoids and spreads arising from involutions 0 0 0 0 0 0 0 0 0 0 0 231

So Yoshiara, The Borel-Tits property for finite groups 00000000000000000000000 237

Fo Zara, Generalized reflection groups 1. Constructing the algebra 00000000000 251

v

Preface

On September 1-7, 1996 a conference on Groups and Geometries took place in lovely Siena, Italy. It brought together experts and interested mathematicians from numerous countries. The scientific program centered around invited exposi­tory lectures; there also were shorter research announcements, including talks by younger researchers.

The conference concerned a broad range of topics in group theory and geometry, with emphasis on recent results and open problems. Special attention was drawn to the interplay between group-theoretic methods and geometric and combinatorial ones.

Expanded versions of many of the talks appear in these Proceedings. This volume is intended to provide a stimulating collection of themes for a broad range of algebraists and geometers. Among those themes, represented within the conference or these Proceedings, are aspects of the following:

1. the classification of finite simple groups,

2. the structure and properties of groups of Lie type over finite and algebraically closed fields of finite characteristic,

3. buildings, and the geometry of projective and polar spaces, and

4. geometries of sporadic simple groups.

We are grateful to the authors for their efforts in providing us with manuscripts in LaTeX. Barbara Priwitzer and Thomas Hintermann, Mathematics Editors of Birkhauser, have been very helpful and supportive throughout the preparation of this volume.

The organizing committee consisted of L. Di Martino (Milan), W. M. Kan­tor (Eugene), G. Lunardon (Naples), A. Pasini (Siena), and M. C. Tamburini (Brescia). The Conference Service of the University of Siena provided valuable as­sistance for the organization of the Conference. We thank the University of Siena, the Italian National Research Council (C.N.R.), the team ofthe National Research Project 'Non-commutative Algebra' (supported by the Italian Ministry for Uni­versity and Scientific Research) and the Bank Monte dei Paschi di Siena for their financial support to the conference. The Universities of Brescia and Naples also gave a precious support to the Conference. We warmly thank them for this.

vii

Talks

M. Aschbacher, Quasithin groups.

L. Bader, On infinite flocks of quadratic cones.

B. Baumeister, A new computer free existence proof of the third group of Janko.

A. Bonisoli, On two transitive ovals in finite projective planes.

A. Brouwer, The geometry far away from a point or a chamber.

F. Buekenhout, The flag-transitive incidence geometries of small almost simple groups and an application to the Mathieu group M 12 .

M. Buratti, Constructions for block designs by difference sets.

A. Camina, Can we classify line-transitive linear spaces'?

C. Casolo, Character degrees of finite groups.

A. Cohen, Integral representations of finite groups in algebraic groups.

B. Cooperstein, On a connection between the hyperbolic quadric Q+(9, q) and the E6,1 -geometry.

H. Cuypers, Local recognition of graphs related to polarities.

H. Gottschalk, Geometries for the group J1 .

R. Guralnick, The first cohomology group and generation of simple groups.

S. Heiss, On the P3 sequenceability of finite groups.

J. Hirschfeld, The geometry of the ternary unitary group.

C. Huybrechts, A characterization of the Hall-Janko group J2 by a c.L*-geometry.

A. Ivanov, Flag-transitive extended classical dual polar spaces.

N. Johnson, Derivable nets in non derivable planes.

M. Liebeck, Regular orbits of linear groups.

R. Liebler, Towards a chamber systems representation theory.

A. Lucchini, Generating minimally transitive groups.

G. Malle, Finite irreducible linear groups with polynomial ring of invariants.

J. Saxl, Generating classical groups.

G. Seitz, Maximal subgroups of finite exceptional groups.

E. Shult, Some aspects of buildings.

ix

x Talks

S. Smith, Application of geometry to group cohomology.

L. Soicher, Computing fundamental groups and covers of simplicial complexes.

A. Shalev, Hausdorff dimension, pro-p groups and Kac-Moody algebras.

S. Shpektorov, A characterization of rank 2 Chevalley groups in even characteristic.

A. Steinbach, Generalized quadrangles weakly embeddable in projective spaces.

B. Stellmacher, Locally s-transitive groups.

G. Stroth, Diagram geometries and sporadic simple groups.

M.C. Tamburini, Carter subgroups in classical groups.

J. Thas, Embedding of geometries in finite projective spaces: a survey.

F. Timmesfeld, Moufang polygons and abstract root-subgroups.

H. Van Maldeghem, Spreads and ovoids in finite Moufang hexagons.

N. Vavilov, The geometry of tori.

S. Yoshiara, The Borel-Tits property for finite groups.

F. Zara, Generalized reflection groups.

Participants

P. ARROYO JORDA (Valencia, Spain) E. DETOMI (Padova, Italy)

M. ARROYO JORDA (Alcoy, Spain) L. DI MARTINO (Milano, Italy)

M. ASCHBACHER (Pasadena, M. ENEA (Palermo, Italy) California, USA) G. FALCONE (Palermo, Italy)

1. BADER (Roma, Italy) P. FRANCINI (Siena, Italy) C. BARDINI (Genova, Italy)

C. BARTOLONE (Palermo, Italy)

A. BASILE (Verona, Italy)

B. BAUMEISTER (Halle, Germany)

M. BIANCHI (Milano, Italy)

G. BINI (Roma, Italy)

A. BLUNCK (Darmstadt, Germany)

A. BONISOLI (Potenza, Italy)

L. BROUNS (Gent, Belgium)

A. BROUWER (Eindhoven, The Netherlands)

F. BUEKENHOUT (Bruxelles, Belgium)

M. BURATTI (L Aquila, Italy)

C. CASOLO (Firenze, Italy)

M. CAZZOLA (Milano, Italy)

P. CECCHERINI (Roma, Italy)

D. CHILLAG (Haifa, Israel)

A. COHEN (Eindhoven, The Netherlands)

B. COOPERSTEIN (Santa Cruz, California, USA)

A. COSSIDENTE (Potenza, Italy)

H. CUYPERS (Eindhoven, The Netherlands)

F. DALLA VOLTA (Milano, Italy)

U. DARDANO (Napoli, Italy)

A. DEL FRA (L'Aquila, Italy)

D. FROHARDT (Detroit, Michigan, USA)

A. FUKSCHANSKI (Halle, Germany)

N. GAVIOLI (L'Aquila, Italy)

A. GILLIO (Milano, Italy)

H. GOTTSCHALK (Giessen, Germany)

T. GRUNDHOFER (Wiirzburg, Germany)

R. GURALNICK (Los Angeles, California, USA)

S. HEISS (Halle, Germany)

M. HERZOG (Tel Aviv, Israel)

xi

J. HIRSCHFELD (Brighton, UK)

D. HUGHES (Torrita di Siena, Italy)

A. HULPKE (Aachen, Germany)

C. HUYBRECHTS (Bruxelles, Belgium)

A. IVANOV (London, UK)

E. JABARA (Mestre, Italy)

V. JHA (Glasgow, UK)

N. JOHNSON (Iowa City, Iowa, USA)

A. JUHASZ (Haifa, Israel)

W. KANTOR (Eugene, Oregon, USA)

O. KING (Newcastle upon Tyne, UK)

G. KORCHMAROS (Potenza, Italy)

R. LAWTHER (Lancaster, UK)

D. LEEMANS (Bruxelles, Belgium)

xii

M. LIEBECK (London, UK)

R. LIEBLER (Fort Collins, Colorado, USA)

P. LONGOBARDI (Napoli, Italy)

H. LOWE (Braunschweig, Germany)

R. LOWEN (Braunschweig, Germany)

A. LUCCHINI (Brescia, Italy)

G. LUNARDON (Napoli, Italy)

K. LUX (Aachen, Germany)

M. MAINARDIS (Udine, Italy)

M. MAJ (Napoli, Italy)

A. MAKHNEV (Ekaterinburg, Russia)

G. MALLE (Heidelberg, Germany)

A. MARTINEZ PASTOR (Valencia, Spain)

S. MATTAREI (Padova, Italy)

F. MENEGAZZO (Padova, Italy)

M. MORIGI (Padova, Italy)

C. O'KEEFE (Adelaide, Australia)

G. PARMEGGIANI (Padova, Italy)

D. PASECHNICK (Eindhoven, The Netherlands)

A. PASINI (Siena, Italy)

G. PICA (Napoli, Italy)

P. QUATTROCCHI (Modena, Italy)

G. RINALDI (Modena, Italy)

S. RINAURO (Potenza, Italy)

G. ROHRLE (Bielefeld, Germany)

L. ROSATI (Firenze, Italy)

D. SAELI (Potenza, Italy)

Participants

P. SANCHINI (Brescia, Italy)

L. SERENA (Firenze, Italy)

J. SAXL (Cambridge, UK)

G. SEITZ (Eugene, Oregon, USA)

A. SHALEV (Jerusalem, Israel)

S. SHPEKTOROV (Columbus, Ohio, USA)

E. SHULT (Manhattan, Kansas, USA)

S. SMITH (Chicago, Illinois, USA)

L. SOICHER (London, UK)

A. STEINBACH (Giessen, Germany)

B. STELLMACHER (Kiel, Germany)

G. STROTH (Halle, Germany)

M. TAMBURINI (Brescia, Italy)

J. THAS (Gent, Belgium)

C. TIBILETTI (Milano, Italy)

F. TIMMESFELD (Giessen, Germany)

M. VACCARO (Napoli, Italy)

J. VAN BON (London, UK)

H. VAN MALDEGHEM (Gent, Belgium)

N. VAVILOV (St. Petersburg, Russia)

T. WEIGEL (Freiburg im Br., Germany)

C. WIEDORN (Halle, Germany)

J. WILSON (Birmingham, UK)

S. YOSHIARA (Osaka, Japan)

G. ZACHER (Padova, Italy)

F. ZARA (Amiens, France)

P. ZUCCA (Palermo, Italy)