**Mathematical Surveys and Monographs**

Volume: 111;
2004;
477 pp;
Hardcover

MSC: Primary 20;

**Print ISBN: 978-0-8218-3410-7
Product Code: SURV/111**

List Price: $116.00

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MAA Member Price: $104.40

**Electronic ISBN: 978-1-4704-1338-5
Product Code: SURV/111.E**

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# The Classification of Quasithin Groups: I. Structure of Strongly Quasithin \(\mathcal{K}\)-groups

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*Michael Aschbacher; Stephen D. Smith*

Around 1980, G. Mason announced the classification of a certain
subclass of an important class of finite simple groups known as "quasithin
groups". The classification of the finite simple groups depends upon a proof
that there are no unexpected groups in this subclass. Unfortunately Mason
neither completed nor published his work. In the Main Theorem of this two-part
book (Volumes 111 and 112 in the AMS series, Mathematical Surveys and
Monographs) the authors provide a proof of a stronger theorem classifying a
larger class of groups, which is independent of Mason's arguments. In
particular, this allows the authors to close this last remaining gap in the
proof of the classification of all finite simple groups.

An important corollary of the Main Theorem provides a bridge to the program
of Gorenstein, Lyons, and Solomon (Volume 40 in the AMS series,
Mathematical Surveys and Monographs) which seeks to give a new, simplified
proof of the classification of the finite simple groups.

Part I (the current volume) contains results which are used in the proof of
the Main Theorem. Some of the results are known and fairly general, but their
proofs are scattered throughout the literature; others are more specialized and
are proved here for the first time.

Part II of the work (Volume
112) contains the proof of the Main Theorem, and the proof of the corollary
classifying quasithin groups of even type.

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

#### Readership

Graduate students and research mathematicians interested in the theory of finite groups.

#### Table of Contents

# Table of Contents

## The Classification of Quasithin Groups: I. Structure of Strongly Quasithin $\mathcal{K}$-groups

- Contents vii8 free
- Preface xiii14 free
- Volume I: Structure of strongly quasithin K-groups 116 free
- Introduction to Volume I 318
- 0.1. Statement of Main Results 318
- 0.2. An overview of Volume I 520
- 0.3. Basic results on finite groups 722
- 0.4. Semisimple quasithin and strongly quasithin K-groups 722
- 0.5. The structure of SQTK-groups 722
- 0.6. Thompson factorization and related notions 823
- 0.7. Minimal parabolics 1025
- 0.8. Pushing up 1025
- 0.9. Weak closure 1126
- 0.10. The amalgam method 1126
- 0.11. Properties of K-groups 1227
- 0.12. Recognition theorems 1328
- 0.13. Background References 1530

- Chapter A. Elementary group theory and the known quasithin groups 1934
- Chapter B. Basic results related to Failure of Factorization 6782
- B.1. Representations and FF-modules 6782
- B.2. Basic Failure of Factorization 7489
- B.3. The permutation module for A[sub(n)] and its FF*-offenders 8398
- B.4. F[sub(2)]-representations with small values of q or q 85100
- B.5. FF-modules for SQTK-groups 98113
- B.6. Minimal parabolics 112127
- B.7. Chapter appendix: Some details from the literature 118133

- Chapter C. Pushing-up in SQTK-groups 121136
- Chapter D. The qrc-lemma and modules with q ≤ 2 171186
- Chapter E. Generation and weak closure 209224
- Chapter F. Weak BN-pairs and amalgams 259274
- F.1. Weak BN-pairs of rank 2 259274
- F.2. Amalgams, equivalences, and automorphisms 264279
- F.3. Paths in rank-2 amalgams 269284
- F.4. Controlling completions of Lie amalgams 273288
- F.5. Identifying L[sub(4)](3) via its U[sub(4)](2)-amalgam 299314
- F.6. Goldschmidt triples 304319
- F.7. Coset geometries and amalgam methodology 310325
- F.8. Coset geometries with b > 2 315330
- F.9. Coset geometries with b > 2 and m(V[sub(1)]) = 1 317332

- Chapter G. Various representation-theoretic lemmas 327342
- G.1. Characterizing direct sums of natural SL[sub(n)](F[sub(2[sup(e)])])-modules 327342
- G.2. Almost-special groups 332347
- G.3. Some groups generated by transvections 337352
- G.4. Some subgroups of Sp[sub(4)](2[sup(n)]) 338353
- G.5. F[sub(2)]-modules for A[sub(6)] 342357
- G.6. Modules with m(G,V) ≤ 2 345360
- G.7. Small-degree representations for some SQTK-groups 346361
- G.8. An extension of Thompson's dihedral lemma 349364
- G.9. Small-degree representations for more general SQTK-groups 351366
- G.10. Small-degree representations on extraspecial groups 357372
- G.11. Representations on extraspecial groups for SQTK-groups 364379
- G.12. Subgroups of Sp(V) containing transvections on hyperplanes 370385

- Chapter H. Parameters for some modules 377392
- H.1.Ω[sup(ε)][sub(4)](2[sup(n)]) on an orthogonal module of dimension 4n (n > 1) 378393
- H.2. SU[sub(3)](2[sup(n)]) on a natural 6n-dimensional module 378393
- H.3. Sz(2[sup(n)]) on a natural 4n-dimensional module 379394
- H.4. (S)L[sub(3)](2[sup(n)]) on modules of dimension 6 and 9 379394
- H.5. 7-dimensional permutation modules for L[sub(3)](2) 385400
- H.6. The 21-dimensional permutation module for L[sub(3)](2) 386401
- H.7. Sp[sub(4)](2[sup(n)]) on natural 4n plus the conjugate 4n[sup(t)] 388403
- H.8. A[sub(7)] on 4 ⊕ 4 389404
- H.9. Aut(L[sub(n)](2)) on the natural n plus the dual n* 389404
- H.10. A foreword on Mathieu groups 392407
- H.11. M[sub(12)] on its 10-dimensional module 392407
- H.12. 3M[sub(22)] on its 12-dimensional modules 393408
- H.13. Preliminaries on the binary code and cocode modules 395410
- H.14. Some stabilizers in Mathieu groups 396411
- H.15. The cocode modules for the Mathieu groups 398413
- H.16. The code modules for the Mathieu groups 402417

- Chapter I. Statements of some quoted results 407422
- I.1. Elementary results on cohomology 407422
- I.2. Results on structure of nonsplit extensions 409424
- I.3. Balance and 2-components 414429
- I.4. Recognition Theorems 415430
- I.5. Characterizations of L[sub(4)](2) and Sp[sub(6)](2) 418433
- I.6. Some results on Tl-sets 424439
- I.7. Tightly embedded subgroups 425440
- I.8. Discussion of certain results from the Bibliography 428443

- Chapter J. A characterization of the Rudvalis group 431446
- Chapter K. Modules for SQTK-groups with q(G,V) ≤ 2 451466
- Bibliography and Index 461476
- Index 471486 free