**Mathematical Surveys and Monographs**

Volume: 40;
1994;
165 pp;
Softcover

MSC: Primary 20;

Print ISBN: 978-0-8218-0960-0

Product Code: SURV/40.1.S

List Price: $45.00

Individual Member Price: $27.00

**Electronic ISBN: 978-1-4704-1267-8
Product Code: SURV/40.1.S.E**

List Price: $45.00

Individual Member Price: $27.00

# The Classification of the Finite Simple Groups

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*Daniel Gorenstein; Richard N Lyons; Ronald M Solomon*

The classification of the finite simple groups is one of the major feats of contemporary mathematical research, but its proof has never been completely extricated from the journal literature in which it first appeared. This book serves as an introduction to a series devoted to organizing and simplifying the proof. The purpose of the series is to present as direct and coherent a proof as is possible with existing techniques. This first volume, which sets up the structure for the entire series, begins with largely informal discussions of the relationship between the Classification Theorem and the general structure of finite groups, as well as the general strategy to be followed in the series and a comparison with the original proof. Also listed are background results from the literature that will be used in subsequent volumes. Next, the authors formally present the structure of the proof and the plan for the series of volumes in the form of two grids, giving the main case division of the proof as well as the principal milestones in the analysis of each case. Thumbnail sketches are given of the ten or so principal methods underlying the proof. Much of the book is written in an expository style accessible to nonspecialists.

#### Table of Contents

# Table of Contents

## The Classification of the Finite Simple Groups

- Contents ix10 free
- Preface xiii14 free
- Preface to the Second Printing xv16 free
- Part I, Chapter 1: Overview 322 free
- Introduction to the Series 322
- A. The Finite Simple Groups 625
- B. The Structure of Finite Groups 1231
- C. Classifying Simple Groups 2746
- D. The Background Results 4463
- E. Sketch of the Simplified Proof 5170
- 19. Centralizers of semisimple elements 5170
- 20. Uniqueness subgroups 5271
- 21. The sets L[sub(p)](G) and groups of even type 5372
- 22. Generic simple groups and neighborhoods 5574
- 23. The main case division 5877
- 24. Special simple groups 5978
- 25. Stages of the proof 6180
- 26. Generic simple groups 6382
- 27. The identification of G 6887

- F. Additional Comments 7291

- Part I, Chapter 2: Outline of Proof 7998
- Introduction 7998
- A. The Grids 8099
- B. The Uniqueness Grid 87106
- C. The Classification Grid: Generic and Special Simple Groups 99118
- D. The Classification Grid: The Stages of the Proof 106125
- 17. Theorem C[sub(1)] 106125
- 18. p-terminal p-components 108127
- 19. Centralizer of involution patterns 109128
- 20. Theorem C[sub(2)] 110129
- 21. Theorem C[sub(3)] 114133
- 22. Theorem C[sub(4)] 114133
- 23. Theorem C[sub(5)] 116135
- 24. Theorem C[sub(6)] 118137
- 25. Vertical neighborhoods 118137
- 26. Theorem C[sub(7)] 120139

- E. Principal Techniques of the Proof 122141
- 27. Fusion 122141
- 28. The Bender method 123142
- 29. Signalizer functors and k-balanced groups 124143
- 30. L[sub(p')]-balance, the B[sub(p)]-property, and pumpups 127146
- 31. The signalizer functor method 128147
- 32. Near components, pushing up, and failure of factorization 129148
- 33. The amalgam method 131150
- 34. Local analysis at two primes 134153
- 35. Character theory and group order formulas 135154
- 36. Identification of the groups of Lie type 137156
- 37. Properties of K-groups 138157

- F. Notational Conventions 139158

- Background References 140159
- Expository References 141160
- Glossary 148167
- Index 155174

#### Readership

The material here ranges from exposition suitable to a first- or second-year graduate student to more technical portions suitable for specialists.

#### Reviews

This new, second-generation proof is a great improvement while still being mostly based on the methods developed for the first proof… The entire organization of the proof is given in great detail. The wait for this project to reach this stage has been worth it.

-- Mathematical Reviews

Written in extremely careful and clear style … readable, and will repay those who work through it by giving them a detailed picture of the strategy of the whole proof.

-- Bulletin of the London Mathematical Society