Finite Groups -Coming of Age Proceedings of the Canadian Mathematical Society Conference held on June 15-28, 1982

AMERICAn MATHEMATICAL SOCIETY UOLUME 45

Titles in this Series

Volume 1 Markov random fields and their

applications. Ross Kindermann and J. Laurie Snell

2 Proceedings of the conference on integration, topology, and geometry In linear spaces. William H. Graves. Editor

3 The closed graph and P-closed graph properties in general topology, T. R. Hamlett and L. L. Herrington

4 Problems of elastic stability and vibrations. Vadim Komkov. Editor

5 Rational constructions of modules for simple Lie algebras, George B. Seligman

6 Umbral calculus and Hopf algebras, Robert Morris. Editor

7 Complex contour Integral representation of cardinal spline functions. Walter Schempp

8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio. Editors

9 Papers in algebra, analysis and statistics. R. lidl. Editor

. 0 Operator algebras and K-theory, Ronald G. Douglas and Claude Schochet. Editors

1 Plane ellipticity and related problems. Robert P. Gilbert. Editor

2 Symposium on algebraic topology In honor of Jose Adem, Samuel Gitler. Editor

3 Algebraists' homage: Papers In ring theory and related topics. S. A. Amitsur. D. J. Saltman and G. B. Seligman. Editors

4 Lectures on Nielsen fixed point theory, Boju Jiang

COnTEMPORARY MATHEMATICS

15 Advanced analytic number theory. Part 1: Ramification theoretic methods. Carlos J. Moreno

16 Complex representations of GL(2. K) for finite fields K, llya Piatetski-Shapiro

17 Nonlinear partial differential equations, Joel A. Smoller. Editor

18 Fixed points and nonexpansive mappings, Robert C. Sine. Editor

19 Proceedings of the Northwestern homotopy theory conference, Haynes R. Miller and Stewart B. Priddy, Editors

20 Low dimensional topology, Samuel J. Lomonaco. Jr .. Editor

21 Topological methods in nonlinear functional analysis. S. P. Singh, S. Thomeier. and B. Watson. Editors

22 Factorizations of b" ± 1. b = 2. 3. 5. 6. 7.10,11,12 up to high powers. John Brillhart. D. H. Lehmer. J. L. Selfridge, Bryant Tuckerman. and S. S. Wagstaff. Jr.

23 Chapter 9 of Ramanujan's second notebook-Infinite series identities, transformations. and evaluations . Bruce C. Berndt and Padmini T. Joshi

24 Central extensions, Galois groups, and Ideal class groups of number fields, A. Frohlich

25 Value distribution theory and its applications. Chung-Chun Yang, Editor

26 Conference in modern analysis and probability, Richard Beals. Anatole Beck. Alexandra Bellow and Arshag Hajian. Editors

27 Microlocal analysis, M. Salah Baouendi. Richard Beals and Linda Preiss Rothschild. Editors

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Titles in this Series

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28 Fluids and plasmas: geometry and dynamics, Jerrold E. Marsden. Editor

29 Automated theorem proving, W. W. Bledsoe and Donald Loveland. Editors

30 Mathematical applications of category theory, J. W. Gray. Editor

31 Axiomatic set theory, James E. Baumgartner. Donald A. Martin and Saharon Shelah. Editors

32 Proceedings of the conference on Banach algebras and several complex variables. F. Greenleaf and D. Gulick. Editors

33 Contributions to group theory, Kenneth I. Appel. John G. Ratcliffe and Paul E. Schupp. Editors

34 Combinatorics and algebra. Curtis Greene. Editor

35 Four-manifold theory, Cameron Gordon and Robion Kirby. Editors

36 Group actions on manifolds, Reinhard Schultz. Editor

37 Conference on algebraic topology in honor of Peter Hilton, Renzo Piccinini and Denis Sjerve. Editors

38 Topics in complex analysis. Dorothy Brown Shaffer. Editor

39 Errett Bishop: Reflections on him and his research, Murray Rosenblatt. Editor

40 Integral bases for affine Lie algebras and their universal enveloping algebras. David Mitzman

41 Particle systems, random media and large deviations. Richard Durrett. Editor

42 Classical real analysis, Daniel Waterman. Editor

43 Group actions on rings, Susan Montgomery. Editor

44 Combinatorial methods in topology and algebraic geometry, John R. Harper and Richard Mandelbaum. Editors

45 Finite groups-coming of age, John McKay. Editor

Finite Grou.ps - Coming of Age

COnTEMPORARY MATHEMATICS

Finite Groups -Coming of Age Proceedings of the Canadian Mathematical Society Conference held on June 15-281 18821 with partial support from the FCAC (Quebec) and the NSERC (Canada)

John McKay. Editor

Volume45

AMERICAn MATHEMATICAL SOCIETY Providence • Roode Island

EDITORIAL BOARD

Jan Mycielski R. 0. Wells, Jr., managing editor Johannes C. C. Nitsche

Adriano M. Garsia Carl M. Pearcy James I. Lepowsky Irving Reiner

Alan D. Weinstein

The Canadian Mathematical Society Conference on Finite Groups-Coming of Age was held at Concordia University, Montreal, Quebec on June 15-28, 1982, with partial support from FCAC (Quebec) and NSERC (Canada).

1980 Mathematics Subject Classification. Primary 10012, 10020, 14E15, 20C15, 20C20, 20005, 20008,20C15, 20F05,22E65,51E25.

Library of Congress Cataloging-in-Publication Data Conference on Finite Groups-Coming of Age (1982: Concordia University)

Finite groups-coming of age.

(Contemporary mathematics, ISSN 0271-4132: v. 45) "Proceedings of the Conference on Finite Groups-Coming of Age, held at Concordia

UniversitY, Montreal, Quebec, June 15-28, 1982"-T.p. verso. Bibliography: p. 1. Finite groups-Congresses. I. McKay, John. II. Title. 111. Series: Contemporary

mathematics (American Mathematical Society); v. 45. QA171.C6788 1982 512'.22 85-15049 ISBN 0-8218-5047-4 (alk. paper)

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CONTENTS

Preface

Francis Buekenhout Diagram geometries for sporadic groups

Colin M. Campbell Symmetric representations and linear groups

Leonard Chastkofsky Rationality of certain zeta functions associated with modular representation theory

John H. Conway and Larissa Queen Computing the character table of a Lie group

D. Dummit, H. Kisilevsky and J. McKay Multiplicative products of 11-funetions

I. B. Frenkel, J. Lepowsky and A. Meurman An E8 approach to F 1

Robert L. Griess, Jr. The Monster and its nonassociative algebra

William M. Kantor Some consequences of the classification of finite simple groups

Horst Knorrcr Group representations and the resolution of rational double points

Geoffrey Mason M24 and certain automorphic forms

Robert V. Moody Generalized root systems and characters

vii

ix

1

33

41

51

89

99

121

159

175

223

245

viii CONTENTS

Simon P. Norton The uniqueness of the Fischer-Griess Monster

Edmund F. Robertson Efficiency of finite simple groups and their covering groups

Mark A. Ronan Buildings and sporadic groups

Stephen D. Smith On the Head characters of the Monster simple group

Stephen D. Smith Residual geometries for sporadic and classical groups - A survey

Robert Steinberg Some consequences of the elementary relations in SL

n

271

287

295

303

315

335

Preface

There has been widely expressed an undue pessimism concerning the

future of the theory of finite simple groups following their classification;

however, there are grounds for believing that the excitement following the

discoveries of the sporadic simple groups will be more than matched when

they can be placed in a broader context. It is curious that the theory

developed for the classification touches so few other domains of mathematics.

Can Chevalley's classification of the finite-dimensional simple Lie groups

which, over finite fields, yields all but the 26 sporadic simple groups, be

generalized to include these by an extension to infinite dimensions!

What understanding can be made of the empirically observed

phenomenon of 'moonshine' in which the classical 19th century Hauptmoduln

are reincarnated as character-generating functions of the Fischer-Griess

Monster? How general is moonshine? It is disquieting that a proof does not

necessarily bring understanding.

There are tantalizing suggestions that the existence of the Monster, the

Baby Monster, & Fischer's F 24 may be related to the exceptional groups of

type E6, E7, & E8 respectively. Are there connections with singularities?

How well can sporadic geometries describe the sporadic groups! What can

be said of the finite subgroups of complex Lie groups? How much of the

Brauer theory of characters for finite groups can be extended to Lie groups?

The purpose of this conference has been to stimulate interested persons

and focus on activity related to finite groups which has not been central to

the classification.

ix

X PREFACE

The conference was sponsored by the Canadian Mathematical Society

and financed by a grant from the Nat ural Sciences & Engineering Research

Council of Canada.

These proceedings have been prepared on equipment of the Computer

Science Department of Concordia University & special mention must be made

to many people including Donna & Larry Thiel and Lily Lam for seeing this

task through with inadequate facilities.

John McKay

Montreal, March, 1985