**Mathematical Surveys and Monographs**

Volume: 145;
2008;
556 pp;
Hardcover

MSC: Primary 03; 20; 51;
Secondary 17; 19

Print ISBN: 978-0-8218-4305-5

Product Code: SURV/145

List Price: $112.00

Individual Member Price: $89.60

**Electronic ISBN: 978-1-4704-1372-9
Product Code: SURV/145.E**

List Price: $112.00

Individual Member Price: $89.60

#### Supplemental Materials

# Simple Groups of Finite Morley Rank

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*Tuna Altınel; Alexandre V. Borovik; Gregory Cherlin*

The book gives a detailed presentation of the classification of the simple groups of finite Morley rank which contain a nontrivial unipotent 2-subgroup. They are linear algebraic groups over algebraically closed fields of characteristic two. Although the story told in the book is inspired by the classification of the finite simple groups, it goes well beyond this source of inspiration. Not only do the techniques adapted from finite group theory cover, in an unusual combination, various portions of the three generations of approaches to finite simple groups, but model theoretic methods also play an unexpected role. The book contains a complete account of all this material, part of which has not been published. In addition, almost every general result about groups of finite Morley rank is exposed in detail and the book ends with a chapter where the authors provide a list of open problems in the relevant fields of mathematics. As a result, the book provides food for thought to finite group theorists, model theorists, and algebraic geometers who are interested in group theoretic problems.

#### Table of Contents

# Table of Contents

## Simple Groups of Finite Morley Rank

- Contents v6 free
- Introduction ix10 free
- Part A. Methods 122 free
- Chapter I. Tools 324
- Overview 627
- 1. General group theory 1233
- 2. Rank 2344
- 3. Connected groups 3253
- 4. Fields 4465
- 5. Nilpotent groups 5778
- 6. Sylow theory 6586
- 7. Generalized Fitting subgroup 7293
- 8. Solvable groups 7394
- 9. Schur-Zassenhaus 82103
- 10. Automorphisms 90111
- 11. Modules 96117
- 12. Thompson A x B 99120
- 13. Complex reflection groups 100121
- 14. Notes 104125

- Chapter II. K-groups and L-groups 111132
- Chapter III. Specialized Topics 167188
- Overview 168189
- 1. Pseudoreflection groups 173194
- 2. Zassenhaus groups 177198
- 3. Suzuki groups 182203
- 4. Landrock-Solomon 191212
- 5. A theorem of Baumann 199220
- 6. Generalized n-gons 211232
- 7. Buildings and (B,N)-pairs 223244
- 8. A theorem of Niles 228249
- 9. Signalizer functors 231252
- 10. Generic identification 233254
- 11. Notes 240261

- Chapter IV. Generic Covering and Conjugacy Theorems 243264

- Part B. Mixed Type Groups 281302
- Part C. Even Type Groups 301322
- Chapter VI. Strong Embedding and Weak Embedding 303324
- 1. Weak solvability: Strong embedding 306327
- 2. Weak solvability: Weak embedding 311332
- 3. Recognition: Strong embedding, I 314335
- 4. Recognition: Strong embedding, II 325346
- 5. Recognition: Weak embedding, I 331352
- 6. Recognition: Weak embedding, II 338359
- 7. ¬(*), I: Toral blocks 345366
- 8. ¬(*), II: Rank 357378
- 9. ¬(*), III: Structure 364385
- 10.¬(*), IV: Contradiction 369390
- 11. Notes 374395

- Chapter VII. Standard components of type SL[sub(2)] 377398
- Chapter VIII. The C(G,T) Theorem and a Plan of Attack 403424
- Chapter IX. Quasithin groups 431452
- 1. The amalgam method 432453
- 2. Preparation 436457
- 3. Z[sub(δ)] 442463
- 4. Even s 448469
- 5. Odd s, S*[sub(γ,K)] 452473
- 6. Odd s, O°[sub(2)](G[sub(T)]) 460481
- 7. Odd s: Initial analysis 468489
- 8. Odd s: Detailed analysis 474495
- 9. A generalized polygon 487508
- 10. Identification 494515
- 11. Notes 497518

- Chapter X. Conclusion 499520

- Bibliography 539560
- Index of Notation 547568 free
- Index of Terminology 551572
- Index of Names 555576

#### Readership

Graduate students and research mathematicians interested in group theory and model theory related to logic.

#### Reviews

Not only is the lengthy and difficult proof presented in a very efficient way, readable both by model theorists and by finite and algebraic group theorists, but also the whole story is told in an informative and elegant style.

-- Mathematical Reviews