**Mathematical Surveys and Monographs**

Volume: 40;
2018;
344 pp;
Hardcover

MSC: Primary 20;

**Print ISBN: 978-0-8218-4069-6
Product Code: SURV/40.7**

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**Electronic ISBN: 978-1-4704-4387-0
Product Code: SURV/40.7.E**

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AMS Member Price: $97.60

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#### Supplemental Materials

# The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a

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*Daniel Gorenstein; Richard Lyons; Ronald Solomon*

The classification of finite simple groups is
a landmark result of modern mathematics. The multipart series of
monographs which is being published by the AMS (Volume 40.1–40.7 and
future volumes) represents the culmination of a century-long project
involving the efforts of scores of mathematicians published in
hundreds of journal articles, books, and doctoral theses, totaling an
estimated 15,000 pages. This part 7 of the series is the middle of a
trilogy (Volume 40.5, Volume 40.7, and forthcoming Volume 40.8)
treating the Generic Case, i.e., the identification of the alternating
groups of degree at least 13 and most of the finite simple groups of
Lie type and Lie rank at least 4. Moreover, Volumes 40.4–40.8 of
this series will provide a complete treatment of the simple groups of
odd type, i.e., the alternating groups (with two exceptions) and the
groups of Lie type defined over a finite field of odd order, as well
as some of the sporadic simple groups. In particular, this volume
completes the construction, begun in Volume 40.5, of a collection of
neighboring centralizers of a particularly nice form. All of this is
then applied to complete the identification of the alternating groups
of degree at least 13.

The book is suitable for graduate students and researchers
interested in the theory of finite groups.

#### Readership

Graduate students and researchers interested in the theory of finite groups.

#### Reviews & Endorsements

The presentation is just perfect. It is a tough business to present a proof which runs over three volumes. The authors take the reader by the hand, tell her or him what they are doing, and why and how it fits into the general picture. Thus, even if you are not willing to follow all the details, it gives you a good picture what the authors are doing and why this eventually yields a proof of the so-called generic case.

-- Gernot Stroth, Mathematical Reviews

#### Table of Contents

# Table of Contents

## The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7-11: The Generic Case, Stages 3b and 4a

- Cover 11
- Title page 44
- Contents 88
- Preface 1010
- Chapter 7. The Stages of Theorem \Cβ* 1212
- Chapter 8. General Group-Theoretic Lemmas 3030
- Chapter 9. Theorem \Cβ*: Stage 3b 5050
- 1. Introduction 5050
- 2. Theorem 2: The Setup and Special Cases 5252
- 3. The Cases πΎβ π΄_{π} or πΎβ π»π, with π=2 5656
- 4. The πΉβ Case 6565
- 5. π-Thin Configurations, π Odd 6767
- 6. The General Lie Type Case, Begun 7474
- 7. The General Lie Type Case, Concluded 8686
- 8. The π=2, ππΏβ(π) case 9999
- 9. Theorem 3 106106
- 10. Theorem 4: The Prime π 113113
- 11. The π -Uniqueness Case 122122
- 12. Theorem 5: The General Case 126126
- 13. Theorem 5: Residual Cases, π=2 132132
- 14. Theorem 5: Residual Cases, π>2 137137
- 15. Theorem 6 154154

- Chapter 10. Theorem \Cβ*, Stage 4a: Constructing a Large Alternating Subgroup πΊβ 156156
- 1. Introduction 156156
- 2. Theorems 1 and 2: Most Sporadic Components 158158
- 3. Theorems 1 and 2: π½β and Ree Components 160160
- 4. The Alternating Case: Theorem 3βCentralizers of Rank 2 163163
- 5. Theorem 4βCentralizers of Rank 1: Preliminaries 168168
- 6. Theorem 4βAll Alternating Group Neighbors 171171
- 7. Theorem 4βNon-Alternating Group Neighbors 178178
- 8. Theorem 4βThe Final Case: 2π»π Neighbors 186186

- Chapter 11. Properties of \K-Groups 192192
- 1. Everyday Tools 192192
- 2. Automorphisms and Coverings 200200
- 3. π-Rank and π-Sylow Structure of Quasisimple Groups 202202
- 4. Computations in Alternating Groups 207207
- 5. Computations in Sporadic Groups 209209
- 6. Computations in Groups of Lie Type 214214
- 7. Small Groups 228228
- 8. Embeddings 233233
- 9. Recognition Theorems 234234
- 10. Balance 234234
- 11. Subcomponents 235235
- 12. Pumpups and the Ranking Function \FF 237237
- 13. Pumpups 242242
- 14. Acceptable Subterminal Pairs, I 273273
- 15. Acceptable Subterminal Pairs, II 282282
- 16. Neighborhoods 288288
- 17. Generation 299299
- 18. Splitting Primes 303303
- 19. Double Pumpups 309309
- 20. The Double Pumpup Propositions 314314
- 21. Monotonicity of \FF, and \J_{π}Β²(πΊ) 331331
- 22. Miscellaneous 348348

- Bibliography 352352
- Index 354354
- Back Cover 362362