**Mathematical Surveys and Monographs**

Volume: 40;
2018;
488 pp;
Hardcover

MSC: Primary 20;

**Print ISBN: 978-1-4704-4189-0
Product Code: SURV/40.8**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-5059-5
Product Code: SURV/40.8.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

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#### Supplemental Materials

# The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed

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*Daniel Gorenstein; Richard Lyons; Ronald Solomon*

This book completes a trilogy (Numbers 5, 7, and 8) of the series The
Classification of the Finite Simple Groups treating the generic case
of the classification of the finite simple groups. In conjunction
with Numbers 4 and 6, it allows us to reach a major milestone in our
series—the completion of the proof of the following theorem:

Theorem O: Let G be a finite simple group of odd
type, all of whose proper simple sections are known simple groups.
Then either G is an alternating group or G is a finite group of Lie
type defined over a field of odd order or G is one of six sporadic
simple groups.

Put another way, Theorem O asserts that any minimal counterexample
to the classification of the finite simple groups must be of even
type. The work of Aschbacher and Smith shows that a minimal
counterexample is not of quasithin even type, while this volume shows
that a minimal counterexample cannot be of generic even type, modulo
the treatment of certain intermediate configurations of even type
which will be ruled out in the next volume of our series.

#### Readership

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

#### Table of Contents

# Table of Contents

## The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12-17: The Generic Case, Completed

- Cover 11
- Title page 44
- Preface 1010
- Chapter 12. Introduction 1414
- Chapter 13. Recognition Theory 2222
- Chapter 14. Theorem \C₇*: Stage 4b+. A Large Lie-Type Subgroup 𝐺₀ for 𝑝=2 4242
- 1. Introduction 4242
- 2. A 2-Local Characterization of 𝐿₄^{±}(𝑞), 𝑞 Odd 4646
- 3. The Case 𝐾≅𝐿₃^{±}(𝑞) 5252
- 4. The Case 𝐾≅𝐺₂(𝑞) or \td4𝑞 5454
- 5. The Non-Level Case 6060
- 6. The Other Exceptional Cases 7171
- 7. The Case 𝐾≅𝑃𝑆𝑝_{2𝑛}(𝑞)=𝐶_{𝑛}(𝑞)^{𝑎} 7575
- 8. The 𝑆𝑝𝑖𝑛_{𝑛}(𝑞) Cases, 𝑛≥7 8787
- 9. The 𝑆𝑝_{2𝑛} Cases 9393
- 10. The Linear and Unitary Cases 100100
- 11. The Orthogonal Case, Preliminaries 125125
- 12. The Orthogonal Case, Completed 135135
- 13. The Cases in Which 𝐺₀ is Exceptional 154154
- 14. Summary: 𝑝=2 167167

- Chapter 15. Theorem \C₇*: Stage 4b+. A Large Lie-Type Subgroup 𝐺₀ for 𝑝>2 168168
- 1. Introduction 168168
- 2. A Choice of 𝑝 169169
- 3. The Weyl Group 170170
- 4. The Field Automorphism Case 173173
- 5. Some General Lemmas 180180
- 6. The Case 𝑚_{𝑝}(𝐵)=4 191191
- 7. The Case \Aut_{𝐾}(𝐵)≅𝑊(𝐵𝐶_{𝑛}) or 𝑊(𝐹₄) 200200
- 8. The Case \Aut_{𝐾}(𝐵)≅𝑊(𝐷_{𝑛}), 𝑛≥4 206206
- 9. Some Exceptional Cases 219219
- 10. The Case 𝐾/𝑍(𝐾)≅𝑃𝑆𝐿_{𝑚}^{±}(𝑞) 220220
- 11. The Final Case: 𝐾/𝑍(𝐾)≅𝐸₆^{𝜀}(𝑞) 232232
- 12. Identification of 𝐺₀: Setup 237237
- 13. 𝐺₀≅𝑆𝑝_{2𝑛+2}(𝑞) or 𝐴_{𝑛+1}^{-\epsq}(𝑞) 238238
- 14. 𝐺₀≅𝐷_{𝑛+1}^{±}(𝑞) 245245
- 15. 𝐺₀≅𝐿_{𝑘}^{\epsq}(𝑞) 261261
- 16. 𝐺₀≅𝐸₈(𝑞) 265265
- 17. 𝐺₀≅𝐸₆^{\epsq}(𝑞) and 𝐸₇(𝑞) 270270
- 18. The Remaining Cases for 𝐺₀ 277277
- 19. Γ_{𝐷,1}(𝐺) Normalizes 𝐺₀ 290290

- Chapter 16. Theorem \C₇*: Stage 5+. 𝐺=𝐺₀ 304304
- Chapter 17. Preliminary Properties of \K-Groups 334334
- 1. Weyl Groups and Their Representations 334334
- 2. Toral Subgroups 348348
- 3. Neighborhoods 368368
- 4. CTP-Systems 389389
- 5. Representations 396396
- 6. Computations in Groups of Lie Type 400400
- 7. Outer Automorphisms, Covering Groups, and Envelopes 419419
- 8. 𝑝-Structure of Quasisimple \K-groups 421421
- 9. Generation 428428
- 10. Pumpups 436436
- 11. Small Groups 468468
- 12. Subcomponents 477477
- 13. Acceptable Subterminal Pairs 483483
- 14. Fusion 487487
- 15. Balance and Signalizers 493493
- 16. Miscellaneous 494494

- Bibliography 498498
- Index 500500
- Back Cover 506506