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The Classification of Quasithin Groups: I. Structure of Strongly Quasithin \(\mathcal{K}\)-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; 477 pp; hardcover
Volume: 111
ISBN-10: 0-8218-3410-X
ISBN-13: 978-0-8218-3410-7
List Price: US$109
Member Price: US$87.20
Order Code: SURV/111
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See also:

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

Volume I: Structure of strongly quasithin \(\mathcal{K}\)-groups
  • Introduction to volume I
  • Elementary group theory and the known quasithin groups
  • Basic results related to failure of factorization
  • Pushing-up in SQTK-groups
  • The \(qrc\)-lemma and modules with \(\hat{q}\leq 2\)
  • Generation and weak closure
  • Weak BN-pairs and amalgams
  • Various representation-theoretic lemmas
  • Parameters for some modules
  • Statements of some quoted results
  • A characterization of the Rudvalis group
  • Modules for SQTK-groups with \(\hat{q}(G, V) \leq 2\)
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
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