## Representations and Characters of GroupsThis is the second edition of the popular textbook on representation theory of finite groups. The authors have revised the text greatly and included new chapters on Characters of GL(2,q) and Permutations and Characters. The theory is developed in terms of modules, since this is appropriate for more advanced work, but considerable emphasis is placed upon constructing characters. The character tables of many groups are given, including all groups of order less than 32, and all but one of the simple groups of order less than 1000. Each chapter is accompanied by a variety of exercises, and full solutions to all the exercises are provided at the end of the book. |

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This is the best book to know what "Representation" actually means....

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Simple and Excellent book on representations of groups

### Contents

Groups and homomorphisms | 1 |

Vector spaces and linear transformations | 14 |

Group representations | 30 |

FGmodules | 38 |

FGsubmodules and reducibility | 49 |

Group algebras | 53 |

FGhomomorphisms | 61 |

Maschkes Theorem | 70 |

Tensor products | 188 |

Restriction to a subgroup | 210 |

Induced modules and characters | 224 |

Algebraic integers | 244 |

Real representations | 263 |

Summary of properties of character tables | 283 |

Characters of groups of order pq | 288 |

Characters of some pgroups | 298 |

Schurs Lemma | 78 |

Irreducible modules and the group algebra | 89 |

More on the group algebra | 95 |

Conjugacy classes | 104 |

Characters | 117 |

Inner products of characters | 133 |

The number of irreducible characters | 152 |

Character tables and orthogonality relations | 159 |

Normal subgroups and lifted characters | 168 |

Some elementary character tables | 179 |

### Common terms and phrases

abelian group algebraic integer b~lab basis bilinear form calculate CG-homomorphism CG(g CG(x character of G character table classes of G column orthogonality relations complex numbers conjugacy classes conjugate Corollary deduce define Definition direct sum eigenvalue eigenvectors element g elements of G endomorphism entries equivalent Example Let Exercises for Chapter finite group follows g e G given group algebra group G group of order HomcG homomorphism inner product invertible involution IRG-module irreducible CG-module irreducible CG-submodules irreducible characters irreducible representations isomorphic Ker9 Let G linear characters linear transformation Maschke's Theorem matrix non-zero normal subgroup prime number Proof Let Proposition Let Prove representation of G result root of unity Schur's Lemma Show simple group subgroup of G submodule subspace sum of irreducible Summary of Chapter Suppose that G symmetric symmetry group table of G values vector space Z(CG