The Jacobson Radical of Group Algebras

Gregory Karpilovsky (Eds.)0444701907, 9780444701909, 9780080872469

Let G be a finite group and let F be a field. It is well known that linear representations of G over F can be interpreted as modules over the group algebra FG. Thus the investigation of ring-theoretic structure of the Jacobson radical J(FG) of FG is of fundamental importance. During the last two decades the subject has been pursued by a number of researchers and many interesting results have been obtained. This volume examines these results. The main body of the theory is presented, giving the central ideas, the basic results and the fundamental methods. It is assumed that the reader has had the equivalent of a standard first-year graduate algebra course, thus familiarity with basic ring-theoretic and group-theoretic concepts and an understanding of elementary properties of modules, tensor products and fields. A chapter on algebraic preliminaries is included, providing a survey of topics needed later in the book. There is a fairly large bibliography of works which are either directly relevant to the text or offer supplementary material of interest.

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Table of contents :
Content:
Editor
Page ii

Edited by
Page iii

Copyright page
Page iv

Dedication
Page v

Preface
Pages vii-viii
Gregory Karpilovsky

1 Ring-theoretic background
Pages 1-66

2 Group algebras and their modules
Pages 67-103

3 The Jacobson radical of group algebras: Foundations of the theory
Pages 105-300

4 Group algebras of p-groups over fields of characteristic p
Pages 301-331

5 The Jacobson radical and induced modules
Pages 333-373

6 The Loewy length of projective modules
Pages 375-411

7 The nilpotency index
Pages 413-452

8 Radicals of blocks
Pages 453-500

Bibliography
Pages 501-519

Notation
Pages 520-526

Index
Pages 527-532