Representation Theory of Finite Groups

Representation Theory of Finite Groups

1st Edition - January 1, 1965

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  • Author: Martin Burrow
  • eBook ISBN: 9781483258218

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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.

Table of Contents

  • 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


    Chapter II. Representation Theory of Rings with Identity

    6. Some Fundamental Lemmas


    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


    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


    22. Linear Characters


    23. Induced Representations and Induced Characters


    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


    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


    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


    35. Ordinary and Modular Representations of Algebras

    1. Arithmetic in an Algebra


    2. Connection with Integral Representations

    36. p-Adic Fields

    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


    1. Groups

    2. Rings, Ideals, and Fields


    Subject Index

Product details

  • No. of pages: 196
  • Language: English
  • Copyright: © Academic Press 1965
  • Published: January 1, 1965
  • Imprint: Academic Press
  • eBook ISBN: 9781483258218

About the Author

Martin Burrow

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