**Graduate Studies in Mathematics**

Volume: 59;
2003;
212 pp;
Hardcover

MSC: Primary 20;

Print ISBN: 978-0-8218-3222-6

Product Code: GSM/59

List Price: $52.00

Individual Member Price: $41.60

**Electronic ISBN: 978-1-4704-2104-5
Product Code: GSM/59.E**

List Price: $52.00

Individual Member Price: $41.60

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#### Supplemental Materials

# Representation Theory of Finite Groups: Algebra and Arithmetic

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*Steven H. Weintraub*

“We explore widely in the valley of ordinary representations, and we take the reader over the mountain pass leading to the valley of modular representations, to a point from which (s)he can survey this valley, but we do not attempt to widely explore it. We hope the reader will be sufficiently fascinated by the scenery to further explore both valleys on his/her own.”

—

Representation theory plays important roles in geometry, algebra, analysis, and mathematical physics. In particular, it has been one of the great tools in the study and classification of finite groups. The theory contains some particularly beautiful results: Frobenius's theorem, Burnside's theorem, Artin's theorem, Brauer's theorem—all of which are covered in this textbook. Some seem uninspiring at first but prove to be quite useful. Others are clearly deep from the outset. And when a group (finite or otherwise) acts on something else (as a set of symmetries, for example), one ends up with a natural representation of the group.

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.

#### Table of Contents

# Table of Contents

## Representation Theory of Finite Groups: Algebra and Arithmetic

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents v6 free
- Preface vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. Semisimple Rings and Modules 718 free
- Chapter 3. Semisimple Group Representations 4152
- §3.1. Examples and General Results 4152
- §3.2. Representations of Abelian Groups 5364
- §3.3. Decomposition of the Regular Representation 5667
- §3.4. Applications of Frobenius's Theorem 5970
- §3.5. Characters 6475
- §3.6. Idempotents and their Uses 8192
- §3.7. Subfields of the Complex Numbers 8697
- §3.8. Fields of Positive Characteristic 96107

- Chapter 4. Induced Representations and Applications 99110
- Chapter 5. Introduction to Modular Representations 155166
- Chapter 6. General Rings and Modules 161172
- Chapter 7. Modular Group Representations 187198
- Appendix. Some Useful Results 203214
- Bibliography 209220
- Index 211222
- Back Cover Back Cover1226

#### Readership

Graduate students and research mathematicians interested in algebra, representation theory, number theory, and algebraic geometry.