**Mathematical Surveys and Monographs**

Volume: 40;
1998;
419 pp;
Hardcover

MSC: Primary 20;

Print ISBN: 978-0-8218-0391-2

Product Code: SURV/40.3

List Price: $101.00

Individual Member Price: $80.80

**Electronic ISBN: 978-1-4704-1268-5
Product Code: SURV/40.3.E**

List Price: $101.00

Individual Member Price: $80.80

# The Classification of the Finite Simple Groups, Number 3

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

This book offers a single source of basic facts about the structure of the finite simple groups with emphasis on a detailed description of their local subgroup structures, coverings and automorphisms. The method is by examination of the specific groups, rather than by the development of an abstract theory of simple groups. While the purpose of the book is to provide the background for the proof of the classification of the finite simple groups—dictating the choice of topics—the subject matter is covered in such depth and detail that the book should be of interest to anyone seeking information about the structure of the finite simple groups.

This volume offers a wealth of basic facts and computations. Much of the material is not readily available from any other source. In particular, the book contains the statements and proofs of the fundamental Borel-Tits Theorem and Curtis-Tits Theorem. It also contains complete information about the centralizers of semisimple involutions in groups of Lie type, as well as many other local subgroups.

#### Table of Contents

# Table of Contents

## The Classification of the Finite Simple Groups, Number 3

- Contents ix10 free
- List of Tables xi12 free
- Preface xiii14 free
- Part I, Chapter A: Almost Simple K-Groups 118 free
- Chapter 1. Some Theory of Linear Algebraic Groups 118
- 1.1. Fundamental Notions 219
- 1.2. Jordan Decomposition 421
- 1.3. Unipotent Groups; One-Parameter Groups 522
- 1.4. Tori 623
- 1.5. Solvable Algebraic Groups 623
- 1.6. Borel Subgroups 724
- 1.7. Radical; Reductive, Semisimple and Simple Groups 724
- 1.8. Abstract Root Systems 825
- 1.9. Weights, Roots and Root Subgroups in Reductive Groups 1330
- 1.10. Classification of Semisimple Groups; Isogenics and Versions 1532
- 1.11. BN-Structure 1633
- 1.12. Chevalley Generators and Relations 1734
- 1.13. Parabolic Subgroups 2239
- 1.14. Weights and Representations 2340
- 1.15. Isomorphisms, Automorphisms and Endomorphisms 2744
- 1.16. Notes 3047

- Chapter 2. The Finite Groups of Lie Type 3148
- 2.1. Steinberg Endomorphisms and Lang's Theorem 3148
- 2.2. The Finite Groups of Lie Type 3653
- 2.3. Bruhat Structure 4057
- 2.4. Chevalley Relations for Finite Groups of Lie Type 4562
- 2.5. Automorphisms 5370
- 2.6. Subsystem Subgroups and Parabolic Subgroups 6380
- 2.7. The Classical Groups 6784
- 2.8. Equicharacteristic Representations 7390
- 2.9. Presentations; Curtis-Tits Theorem 7996
- 2.10. Standard Notation 89106

- Chapter 3. Local Subgroups of Groups of Lie Type, I 91108
- Chapter 4. Local Subgroups of Groups of Lie Type, II 119136
- 4.1. Semisimple Automorphisms of Algebraic Groups 119136
- 4.2. Semisimple Automorphisms of Finite Groups of Lie Type 134151
- 4.3. Centralizers of Involutions in Aut[sub(o)](K): Statements 143160
- 4.4. Centralizers of Involutions in Aut[sub(o)](K): Proofs 153170
- 4.5. Centralizers of Involutions in Groups in Lie(r), r Odd: Statements 170187
- 4.6. Centralizers of Involutions in Groups in Lie(r), r Odd: Proofs 186203
- 4.7. Inner-Diagonal and Graph Automorphisms of Exceptional Odd Prime Orders 204221
- 4.8. Some Other p-Local Subgroups 219236
- 4.9. The Remaining Types of Non-Inner Automorphisms 227244
- 4.10. Cross-Characteristic Sylow Structure 236253

- Chapter 5. The Alternating Groups and the Twenty-Six Sporadic Groups 249266
- 5.1. Schur Multipliers and Coverings of Simple Groups 249266
- 5.2. The Alternating Groups 252269
- 5.3. Structure of the Twenty-Six Sporadic Groups 259276
- 5.4. Justification of the Tables: The Simple Case 288305
- 5.5. Justification of the Tables: The Non-Simple Case 294311
- 5.6. Ranks of Sporadic Groups 302319

- Chapter 6. Coverings and Embeddings of Quasisimple K-Groups 311328
- Chapter 7. General Properties of K-Groups 335352
- 7.1. Verification of the Fundamental K-Group Properties 335352
- 7.2. Generation of Semisimple Groups: Generalities 343360
- 7.3. Generation for Groups in Chev: Statements of Main Results 346363
- 7.4. Generation for Groups in Chev: Proofs 349366
- 7.5. Generation for Groups in Alt ∪ Spor 380397
- 7.6. Strongly p-Embedded Subgroups and Proper k-Generated Cores 383400
- 7.7. Balance and Signalizers 385402
- 7.8. Fusion 399416

- Background References 403420
- Expository References 404421
- Errata for Numbers 1 and 2 409426
- Glossary 410427
- Index 414431

#### Readership

Graduate students and research mathematicians interested in the subgroup structure of the finite simple groups of Lie type, the alternating groups and the sporadic simple groups.

#### Reviews

This is the third volume in a series in which the authors aim to write down a complete proof of the classification of simple finite groups. This third volume concentrates entirely on various basic properties of the known finite simple groups. The volume is written in the careful, clear and thorough style we have come to expect from the authors. Quite apart from its role in the series, it contains a wealth of information about the known simple groups which is essential for use in applications of finite group theory. For this reason, it will surely stand on its own as a standard text on simple groups.

-- Bulletin of the London Mathematical Society

The book is carefully written and much of the material presented has uses well beyond the task at hand. There is a wealth of information in this volume, including quite a number of useful tables … will be a valuable reference for future generations of mathematicians.

-- Mathematical Reviews