**Mathematical Surveys and Monographs**

Volume: 112;
2004;
743 pp;
Hardcover

MSC: Primary 20;

Print ISBN: 978-0-8218-3411-4

Product Code: SURV/112

List Price: $135.00

Individual Member Price: $108.00

**Electronic ISBN: 978-1-4704-1339-2
Product Code: SURV/112.E**

List Price: $135.00

Individual Member Price: $108.00

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# The Classification of Quasithin Groups: II. Main Theorems: The Classification of Simple QTKE-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 (Volume 111) 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 (the current volume) 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.

#### Table of Contents

# Table of Contents

## The Classification of Quasithin Groups: II. Main Theorems: The Classification of Simple QTKE-groups

- Contents of Volumes I and II vii8 free
- Volume II: Main Theorems; the classification of simple QTKE-groups 47914 free
- Introduction to Volume II 48116
- Part 1. Structure of QTKE-Groups and the Main Case Division 49732
- Part 2. The treatment of the Generic Case 627162
- Part 3. Modules which are not FF-modules 693228
- Chapter 7. Eliminating cases corresponding to no shadow 695230
- 7.1. The cases which must be treated in this part 696231
- 7.2. Parameters for the representations 697232
- 7.3. Bounds on w 698233
- 7.4. Improved lower bounds for r 699234
- 7.5. Eliminating most cases other than shadows 700235
- 7.6. Final elimination of L[sub(3)](2) on 3 ⊕ 3 701236
- 7.7. mini- Appendix: r > 2 for L[sub(3)](2).2 on 3 ⊕ 3 703238

- Chapter 8. Eliminating shadows and characterizing the J[sub(4)] example 711246
- Chapter 9. Eliminating Ω[sup(+)][sub(4)](2[sup(n)]) on its orthogonal module 729264

- Part 4. Pairs in the FSU over F[sub(2[sup(n)])] for n > 1 739274
- Chapter 10. The case L ε L*[sub(f)](G,T) not normal in M 741276
- Chapter 11. Elimination of L[sub(3)](2[sup(n)]), Sp[sub(4)](2[sup(n)]), and G[sub(2)](2[sup(n)]) for n > 1 759294
- 11.1. The subgroups N[sub(G)](V[sub(i)]) for T-invariant subspaces V[sub(i)] of V 760295
- 11.2. Weak-closure parameter values, and 〈V[sup(N[sub(G)](V[sub(1)]))]〉 766301
- 11.3. Eliminating the shadow of L[sub(4)](q) 770305
- 11.4. Eliminating the remaining shadows 775310
- 11.5. The final contradiction 778313

- Part 5. Groups over F[sub(2)] 785320
- Chapter 12. Larger groups over F[sub(2)] in L*[sub(f)](G,T) 787322
- 12.1. A preliminary case: Eliminating L[sub(n)](2) on n ⊕ n* 787322
- 12.2. Groups over F[sub(2)], and the case V a Tl-set in G 794329
- 12.3. Eliminating A[sub(7)] 807342
- 12.4. Some further reductions 812347
- 12.5. Eliminating L[sub(5)](2) on the 10-dimensional module 816351
- 12.6. Eliminating A[sub(8)] on the permutation module 822357
- 12.7. The treatment of A[sub(6)] on a 6-dimensional module 838373
- 12.8. General techniques for L[sub(n)](2) on the natural module 849384
- 12.9. The final treatment of L[sub(n)](2), n = 4, 5, on the natural module 857392

- Chapter 13. Mid-size groups over F[sub(2)] 865400
- 13.1. Eliminating L ε L*[sub(f)](G,T) with L/O[sub(2)](L) not quasisimple 865400
- 13.2. Some preliminary results on A[sub(5)] and A[sub(6)] 876411
- 13.3. Starting mid-sized groups over F[sub(2)], and eliminating U[sub(3)](3) 884419
- 13.4. The treatment of the 5-dimensional module for A[sub(6)] 896431
- 13.5. The treatment of A[sub(5)] and A[sub(6)] when 〈V[sup(G[sub(1)])][sub(3)]〉 is nonabelian 915450
- 13.6. Finishing the treatment of A[sub(5)] 926461
- 13.7. Finishing the treatment of A[sub(6)] when 〈V[sup(G[sub(1)])]〉 is nonabelian 935470
- 13.8. Finishing the treatment of A[sub(6)] 946481
- 13.9. Chapter appendix: Eliminating the A[sub(10)]-configuration 969504

- Chapter 14. L[sub(3)](2) in the FSU, and L[sub(2)](2) when L[sub(f)](G,T) is empty 975510
- 14.1. Preliminary results for the case L[sub(f)](G,T) empty 975510
- 14.2. Starting the L[sub(2)](2) case of L[sub(f)] empty 981516
- 14.3. First steps; reducing 〈V[sup(G[sub(1)])]〉 nonabelian to extraspecial 989524
- 14.4. Finishing the treatment of 〈V[sup(G[sub(1)])]〉 nonabelian 1005540
- 14.5. Starting the case 〈V[sup(G[sub(1)])]〉 abelian for L[sub(3)](2) and L[sub(2)](2) 1013548
- 14.6. Eliminating L[sub(2)](2) when case 〈V[sup(G[sub(1)])]〉 is abelian 1020555
- 14.7. Finishing L[sub(3)](2) with case 〈V[sup(G[sub(1)])]〉 abelian 1042577
- 14.8. The QTKE-groups with L[sub(f)](G,T) ≠ ∅ 1078613

- Part 6. The case L[sub(f)](G,T) empty 1081616
- Chapter 15. The case L[sub(f)](G,T) ≠ ∅ 1083618
- 15.1. Initial reductions when L[sub(f)](G,T) is empty 1083618
- 15.2. Finishing the reduction to M[sub(f)]/C[sub(M[sub(f)])](V(M[sub(f)])) ≈ O[sup(+)][sub(4)](2) 1104639
- 15.3. The elimination of M[sub(f)]/C[sub(M[sub(f)])](V(M[sub(f)])) = S[sub(3)] wr Z(sub(2)] 1120655
- 15.4. Completing the proof of the Main Theorem 1155690

- Part 7. The Even Type Theorem 1167702
- Chapter 16. Quasithin groups of even type but not even characteristic 1169704
- 16.1. Even type groups, and components in centralizers 1169704
- 16.2. Normality and other properties of components 1173708
- 16.3. Showing L is standard in G 1177712
- 16.4. Intersections of N[sub(G)](L) with conjugates of C[sub(G)](L) 1182717
- 16.5. Identifying J[sub(1)], and obtaining the final contradiction 1194729

- Bibliography and Index 1205740
- Index 1215750 free

#### Readership

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