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The Classification of Quasithin Groups: II. Main Theorems: The Classification of Simple QTKE-groups
Michael Aschbacher, California Institute of Technology, Pasadena, CA, and Stephen D. Smith, University of Illinois at Chicago, IL
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Mathematical Surveys and Monographs
2004; 743 pp; hardcover
Volume: 112
ISBN-10: 0-8218-3411-8
ISBN-13: 978-0-8218-3411-4
List Price: US$135 Member Price: US$108
Order Code: SURV/112

The Classification of Quasithin Groups: I. Structure of Strongly Quasithin $$\mathcal{K}$$-groups - Michael Aschbacher and Stephen D Smith

The Classification of Finite Simple Groups: Groups of Characteristic 2 Type - Michael Aschbacher, Richard Lyons, Stephen D Smith and Ronald Solomon

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.

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

Volume II: Main theorems; the classification of simple QTKE-groups
• Introduction to volume II
Structure of QTKE-groups and the main case division
• Terminology
• Structure and intersection properties of 2-locals
• Classifying the groups with $$|\mathcal{M}(T)|=1$$
• Determining the cases for $$L \in \mathcal{L}^*_f(GT)$$
• Pushing up in QTKE-groups
The treatment of the generic case
• Introduction to part 2
• The generic case: $$L_2(2^n)$$ in $$\mathcal{L}_f$$ and $$n(H)>1$$
• Reducing $$L_2(2^n)$$ to $$n=2$$ and V orthogonal
Modules which are not FF-modules
• Introduction to part 3
• Eliminating cases corresponding to no shadow
• Eliminating shadows and characterizing the $$J_4$$ example
• Eliminating $$\Omega^+_4(2^n)$$ on its orthogonal module
Pairs in the FSU over $$F_{2^n}$$ for $$n>1$$
• Introduction to part 4
• The case $$L \in \mathcal{L}^*_f(G,T)$$ not normal in $$M$$
• Elimination of $$L_3(2^n), Sp_4(2^n)$$, and $$G_2(2^n)$$ for $$n>1$$
Groups over $$F_2$$
• Introduction to part 5
• Larger groups over $$F_2$$ in $$\mathcal{L}^*_f(G,T)$$
• Mid-size groups over $$F_2$$
• $$L_3(2)$$ in the FSU, and $$L_2(2)$$ when $$\mathcal{L}_f(G,T)$$ is empty
The case $$\mathcal{L}_f(G,T)$$ empty
• The case $$\mathcal{L}_f(G,T)=\emptyset$$
The even type theorem
• Quasithin groups of even type but not even characteristic
Bibliography and index
• Background references quoted (Part 1: also used by GLS)
• Background references quoted (Part 2: used by us but not by GLS)
• Expository references mentioned
• Index