Finite Group Algebras and their Modules

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Overview

Originally published in 1983, the principal object of this book is to discuss in detail the structure of finite group rings over fields of characteristic, p, P-adic rings and, in some cases, just principal ideal domains, as well as modules of such group rings. The approach does not emphasize any particular point of view, but aims to present a smooth proof in each case to provide the reader with maximum insight. However, the trace map and all its properties have been used extensively. This generalizes a number of classical results at no extra cost and also has the advantage that no assumption on the field is required. Finally, it should be mentioned that much attention is paid to the methods of homological algebra and cohomology of groups as well as connections between characteristic 0 and characteristic p.

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Table of Contents

Preface; Part I. The Structure of Group Algebras: 1. Idempotents in rings. Liftings; 2. Projective and injective modules; 3. The radical and artinian rings; 4. Cartan invariants and blocks; 5. Finite dimensional algebras; 6. Duality; 7. Symmetry; 8. Loewy series and socle series; 9. The p. i. m.'s; 10. Ext; 11. Orders; 12. Modular systems and blocks; 13. Centers; 14. R-forms and liftable modules; 15. Decomposition numbers and Brauer characters; 16. Basic algebras and small blocks; 17. Pure submodules; 18. Examples; Part II. Indecomposable Modules and Relative Projectivity: 1. The trace map and the Nakayama relations; 2. Relative projectivity; 3. Vertices and sources; 4. Green Correspondence; 5. Relative projective homomorphisms; 6. Tensor products; 7. The Green ring; 8. Endomorphism rings; 9. Almost split sequences; 10. Inner products on the Green ring; 11. Induction from normal subgroups; 12. Permutation models; 13. Examples; Part III. Block Theory: 1. Blocks, defect groups and the Brauer map; 2. Brauer's First Main Theorem; 3. Blocks of groups with a normal subgroup; 4. The Extended First main Theorem; 5. Defect groups and vertices; 6. Generalized decomposition numbers; 7. Subpairs; 8. Characters in blocks; 9. Vertices of simple modules; 10. Defect groups; Appendices; References; Index.

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