Local Analysis for the Odd Order Theorem

By: Helmut Bender, George Glauberman, J. W. S. Cassels (Editor)

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Paperback | 24 July 1995

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188 Pages

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22.86 x 15.24 x 1.09

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In 1963 Walter Feit and John G. Thompson proved the Odd Order Theorem, which states that every finite group of odd order is solvable. The influence of both the theorem and its proof on the further development of finite group theory can hardly be overestimated. The proof consists of a set of preliminary results followed by three parts: local analysis, characters, and generators and relations (Chapters IV, V, and VI of the paper). Local analysis is the study of the centralizers and normalizers of non-identity p-subgroups, with Sylow's Theorem as the first main tool. The main purpose of the book is to present a new version of the local analysis of the Feit-Thompson Theorem (Chapter IV of the original paper and its preliminaries). It includes a recent (1991) significant improvement by Feit and Thompson and a short revision by T. Peterfalvi of the separate final section of the second half of the proof. The book should interest finite group theorists as well as other mathematicians who wish to get a glimpse of one of the most famous and most forbidding theorems in mathematics. Current research may eventually lead to a revised proof of the entire theorem, but this goal is several years away. For the present, the authors are publishing this work as a set of lecture notes to contribute to the general understanding of the theorem and to further improvements.

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'This book is written well ... the authors have succeeded both in simplifying the proof of the Odd Order Theorem and in making it accessible to a wider audience.' Paul Flavell, Bulletin of the London Mathematical Society

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Non-Fiction Mathematics Calculus & Mathematical Analysis Groups & Group Theory Algebra

Preface
Preliminary Results	p. 1
Elementary Properties of Solvable Groups	p. 1
General Results on Representations	p. 9
Actions of Frobenius Groups and Related Results	p. 17
p-Groups of Small Rank	p. 33
Narrow p-Groups	p. 44
Additional Results	p. 49
The Uniqueness Theorem	p. 55
The Transitivity Theorem	p. 55
The Fitting Subgroup of a Maximal Subgroup	p. 61
The Uniqueness Theorem	p. 64
Maximal Subgroups	p. 69
The Subgroups M[subscript [alpha]] and A[subscript [sigma]]	p. 69
Exceptional Maximal Subgroups	p. 80
The Subgroup E	p. 83
Prime Action	p. 97
The Family of All Maximal Subgroups of G	p. 105
Maximal Subgroups of Type [actual symbol not reproducible] and Counting Arguments	p. 105
The Subgroup M[subscript F]	p. 117
The Main Results	p. 123
App. A: Prerequisites and p-Stability	p. 135
App. B: The Puig Subgroup	p. 139
App. C: The Final Contradiction	p. 145
App. D: CN-Groups of Odd Order	p. 153
App. E: Further Results of Feit and Thompson	p. 157
Bibliography	p. 167
List of Symbols	p. 169
Index	p. 172
Table of Contents provided by Blackwell. All Rights Reserved.

Local Analysis for the Odd Order Theorem

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