Finite Groups - Coming of Age Proceedings of the Canadian Mathematical Society Conference held on June 15-28, 1982 AMERICAn MATHEMATICAL SOCIETY UOLUME 45
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
http://dx.doi.org/10.1090/conm/045
Titles in this Series
Volume
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