The Jacobson Radical of Group Algebras (North-Holland Mathematics Studies)

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. 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.

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