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Local Analysis for the Odd Order Theorem : London Mathematical Society Lecture Note Series - Helmut Bender

Local Analysis for the Odd Order Theorem

London Mathematical Society Lecture Note Series

Paperback

Published: 24th July 1995
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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.

'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

Preface
Preliminary Resultsp. 1
Elementary Properties of Solvable Groupsp. 1
General Results on Representationsp. 9
Actions of Frobenius Groups and Related Resultsp. 17
p-Groups of Small Rankp. 33
Narrow p-Groupsp. 44
Additional Resultsp. 49
The Uniqueness Theoremp. 55
The Transitivity Theoremp. 55
The Fitting Subgroup of a Maximal Subgroupp. 61
The Uniqueness Theoremp. 64
Maximal Subgroupsp. 69
The Subgroups M[subscript [alpha]] and A[subscript [sigma]]p. 69
Exceptional Maximal Subgroupsp. 80
The Subgroup Ep. 83
Prime Actionp. 97
The Family of All Maximal Subgroups of Gp. 105
Maximal Subgroups of Type [actual symbol not reproducible] and Counting Argumentsp. 105
The Subgroup M[subscript F]p. 117
The Main Resultsp. 123
App. A: Prerequisites and p-Stabilityp. 135
App. B: The Puig Subgroupp. 139
App. C: The Final Contradictionp. 145
App. D: CN-Groups of Odd Orderp. 153
App. E: Further Results of Feit and Thompsonp. 157
Bibliographyp. 167
List of Symbolsp. 169
Indexp. 172
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780521457163
ISBN-10: 0521457165
Series: London Mathematical Society Lecture Note Series
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 188
Published: 24th July 1995
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 23.5 x 15.88  x 1.91
Weight (kg): 0.27