Representation Theory of Finite Groups: Algebra and Arithmetic (Graduate Studies in Mathematics, 59)

A graduate textbook introducing group representation theory to students who have no prior knowledge of it, and may not even know what a representation is. They are expected to have a sound knowledge of linear algebra and a good familiarity with basic module theory. Annotation (c) Book News, Inc., Portland, OR (booknews.com) The Book Assumes Only The Material Of A Standard Graduate Course In Algebra. It Is Suitable As A Text For A Year-long Graduate Course. The Subject Is Of Interest To Students Of Algebra, Number Theory And Algebraic Geometry. The Systematic Treatment Presented Here Makes The Book Also Valuable As A Reference.--book Jacket. Ch. 1. Introduction -- Ch. 2. Semisimple Rings And Modules -- 2.1. Basic Notions -- 2.2. Structure Theorems -- 2.3. Idempotents And Blocks -- 2.4. Behavior Under Field Extensions -- 2.5. Theorems Of Burnside And Frobenius-schur -- Ch. 3. Semisimple Group Representations -- 3.1. Examples And General Results -- 3.2. Representations Of Abelian Groups -- 3.3. Decomposition Of The Regular Representation -- 3.4. Applications Of Frobenius's Theorem -- 3.5. Characters -- 3.6. Idempotents And Their Uses -- 3.7. Subfields Of The Complex Numbers -- 3.8. Fields Of Positive Characteristic -- Ch. 4. Induced Representations And Applications -- 4.1. Induced Representations -- 4.2. Mackey's Theorem -- 4.3. Permutation Representations -- 4.4. M-groups -- 4.5. Theorems Of Artin And Brauer -- 4.6. Degrees Of Irreducible Representations -- Ch. 5. Introduction To Modular Representations -- Ch. 6. General Rings And Modules -- 6.1. Jordan-holder And Krull-schmidt Theorems -- 6.2. The Jacobson Radical -- 6.3. Rings Of Finite Length -- 6.4. Finite-dimensional Algebras -- Ch. 7. Modular Group Representations -- 7.1. General Results -- 7.2. Characters And Brauer Characters -- 7.3. Examples -- App. Some Useful Results. Steven H. Weintraub. Includes Bibliographical References (p. 209) And Index. This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. The approach is to develop the requisite algebra in reasonable generality and then to specialize it to the case of group representations. Methods and results particular to group representations, such as characters and induced representations, are developed in depth. Arithmetic comes into play when considering the field of definition of a representation, especially for subfields of the complex numbers. The book has an extensive development of the semisimple case, where the characteristic of the field is zero or is prime to the order of the group, and builds the foundations of the modular case, where the characteristic of the field divides the order of the group. The book assumes only the material of a standard graduate course in algebra. It is suitable as a text for a year-long graduate course. The subject is of interest to students of algebra, number theory and algebraic geometry. The systematic treatment presented here makes the book also valuable as a reference. We shall begin by giving some basic examples of the objects and questions we will be studying.