**Contemporary Mathematics**

Volume: 694;
2017;
229 pp;
Softcover

MSC: Primary 01; 05; 17; 20;

Print ISBN: 978-1-4704-3678-0

Product Code: CONM/694

List Price: $111.00

AMS Member Price: $88.80

MAA Member Price: $99.90

**Electronic ISBN: 978-1-4704-4168-5
Product Code: CONM/694.E**

List Price: $111.00

AMS Member Price: $88.80

MAA Member Price: $99.90

# Finite Simple Groups: Thirty Years of the Atlas and Beyond

Share this page *Edited by *
*Manjul Bhargava; Robert Guralnick; Gerhard Hiss; Klaus Lux; Pham Huu Tiep*

This volume contains the proceedings of the
international conference Finite Simple Groups: Thirty Years of the
Atlas and Beyond Celebrating the Atlases and Honoring John Conway,
which was held from November 2–5, 2015, at Princeton University,
Princeton, New Jersey.

Classification of Finite Simple Groups, one of the most monumental
accomplishments of modern mathematics, was announced in 1983 with the
proof completed in 2004. Since then, it has opened up a new and
powerful strategy to approach and resolve many previously inaccessible
problems in group theory, number theory, combinatorics, coding theory,
algebraic geometry, and other areas of mathematics. This strategy
crucially utilizes various information about finite simple groups,
part of which is catalogued in the Atlas of Finite Groups (John
H. Conway et al.), and in An Atlas of Brauer Characters (Christoph
Jansen et al.). It is impossible to overestimate the roles of the
Atlases and the related computer algebra systems in the everyday life
of researchers in many areas of contemporary mathematics.

The main objective of the conference was to discuss numerous
applications of the Atlases and to explore recent developments and
future directions of research, with focus on the interaction between
computation and theory and applications to number theory and algebraic
geometry. The papers in this volume are based on talks given at the
conference. They present a comprehensive survey on current research
in all of these fields.

#### Readership

Graduate students and research mathematicians interested in algebra, group theory, representation theory, combinatorics, coding theory, number theory, and algebraic geometry.

# Table of Contents

## Finite Simple Groups: Thirty Years of the Atlas and Beyond

- Cover Cover11
- Title page iii2
- Contents v6
- Preface vii8
- Acknowledgments ix10
- Moonshine and the meaning of life 112
- The Monster is fabulous 314
- Majorana representation of the Monster group 1122
- Letter to Donna Testerman 1930
- Reliability and reproducibility of Atlas information 2132
- Characters and Brauer trees of the covering group of Β²πΈβ(2) 3344
- Maximal subgroups of sporadic groups 5768
- Construction of the Thompson Chain of subgroups of the Conway group β π and complete graphs on π letters 7384
- Conwayβs groupoid and its relatives 91102
- The subgroup structure of finite groups 111122
- 1. Reductions for permutation groups 111122
- 2. The finite simple groups 113124
- 3. The alternating and symmetric groups 113124
- 4. Groups of Lie type 115126
- 5. Exceptional groups of Lie type 116127
- 6. The Palfy-Pudlak Question 116127
- 7. Lower signalizer lattices. 118129
- 8. A question and a theorem 119130
- References 120131

- Some remarks on maximal subgroups of finite classical groups 123134
- Toward a classification of endotrivial modules 139150
- 1. Introduction 139150
- 2. Definitions and preliminaries 140151
- 3. A little history 141152
- 4. The kernel of restriction and Green correspondents 143154
- 5. The kernel of restriction to the Sylow subgroup 144155
- 6. Another viewpoint 145156
- 7. Simple endotrivial modules 147158
- 8. Acknowledgments 148159
- References 148159

- Some remarks on global/local conjectures 151162
- Minuscule weights and Chevalley groups 159170
- A method for building permutation representations of finitely presented groups 177188
- Character ratios for finite groups of Lie type, and applications 193204
- Conjugacy classes, growth and complexity 209220
- Permutation groups where non-trivial elements have few fixed points 223234
- Back Cover Back Cover1242