# Representation Theory of Finite Groups

1st Edition - January 1, 1965

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• Author: Martin Burrow
• eBook ISBN: 9781483258218

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## Description

Representation Theory of Finite Groups is a five chapter text that covers the standard material of representation theory. This book starts with an overview of the basic concepts of the subject, including group characters, representation modules, and the rectangular representation. The succeeding chapters describe the features of representation theory of rings with identity and finite groups. These topics are followed by a discussion of some of the application of the theory of characters, along with some classical theorems. The last chapter deals with the construction of irreducible representations of groups. This book will be of great value to graduate students who wish to acquire some knowledge of representation theory.

• ﻿Preface

Chapter I. Foundations

1. Introduction

2. Group Characters

3. Representation Modules

4. Application of Ideas and Results from Group Theory

5. The Regular Representation

Exercises

Chapter II. Representation Theory of Rings with Identity

6. Some Fundamental Lemmas

Exercise

7. The Principal Indecomposable Representations

8. The Radical of a Ring

9. Semisimple Rings

10. The Wedderburn Structure Theorems for Semisimple Rings

11. Intertwining Numbers

12. Multiplicities of the Indecomposable Representation

13. The Generalized Burnside Theorem

Exercises

Chapter III. The Representation Theory of Finite Groups

14· The Group Algebra

15. The Regular Representation of a Group

16. Semisimplicity of the Group Algebra

17. The Center of the Group Algebra

18. The Number of Inequivalent Irreducible Representations

19. Relations on the Irreducible Characters

20. The Module of Characters over the Integers

21. The Kronecker Product of Two Representations

Exercises

22. Linear Characters

Exercises

23. Induced Representations and Induced Characters

Exercises

Chapter IV. Applications of the Theory of Characters

24. Algebraic Numbers

25. Some Results from the Theory of Characters

26. Normal Subgroups and the Character Table

A. The Existence of Normal Subgroups

B. The Determination of All Normal Subgroups

27. Some Classical Theorems

Exercises

Chapter V. The Construction of Irreducible Representations

28. Primitive Idempotents

29. Some examples of Group Representations

1. Cyclic Groups

2. Abelian Groups

3. The Symmetric Groups Sn

Exercises

Chapter VI. Modular Representations

30. General Remarks

31. p-Regular Elements of a Finite Group

32. Conditions for Two Representations to Have the Same Composition Factors

33. The Brauer Characters

34. Integral Representations

Exercise

35. Ordinary and Modular Representations of Algebras

1. Arithmetic in an Algebra

Exercise

2. Connection with Integral Representations

1. General Definition and Properties

2. Ordinary Valuation Metrical Properties

3. Completion of a p-Adic Field

4. p-Adic Valuation of the Rational Field

5. Extension of the p-Adic Valuation to Algebraic Number Fields

37. Algebras over a p-Adic Field

1. Notation

2. Preliminary Results

38. A Connection between the Intertwining Numbers

39. Modular Representations of Groups

40. Cartan Invariants and Decomposition Numbers

41. Character Relations

42. Modular Orthogonality Relations

Appendix

1. Groups

2. Rings, Ideals, and Fields

Bibliograpy

Subject Index

## Product details

• No. of pages: 196
• Language: English