## Character Theory of Finite GroupsCharacter theory is a powerful tool for understanding finite groups. In particular, the theory has been a key ingredient in the classification of finite simple groups. Characters are also of interest in their own right, and their properties are closely related to properties of the structure of the underlying group. The book begins by developing the module theory of complex group algebras. After the module-theoretic foundations are laid in the first chapter, the focus is primarily on characters. This enhances the accessibility of the material for students, which was a major consideration in the writing. Also with students in mind, a large number of problems are included, many of them quite challenging. In addition to the development of the basic theory (using a cleaner notation than previously), a number of more specialized topics are covered with accessible presentations. These include projective representations, the basics of the Schur index, irreducible character degrees and group structure, complex linear groups, exceptional characters, and a fairly extensive introduction to blocks and Brauer characters. This is a corrected reprint of the original 1976 version, later reprinted by Dover. Since 1976 it has become the standard reference for character theory, appearing in the bibliography of almost every research paper in the subject. It is largely self-contained, requiring of the reader only the most basic facts of linear algebra, group theory, Galois theory and ring and module theory. |

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### Contents

Group representations and characters | 13 |

Characters and integrality | 33 |

Products of characters | 47 |

Induced characters | 62 |

Normal subgroups | 78 |

T I sets and exceptional characters | 99 |

The Schur index | 160 |

Projective representations | 183 |

Character degrees | 198 |

Character correspondence | 219 |

Linear groups | 240 |

Changing the characteristic | 262 |

Appendix Some character tables | 287 |

### Common terms and phrases

A-module absolutely irreducible afforded algebraic integer Brauer character C[G]-module CG(x character of G character table character theory character triple class function classes of G conclude conjugacy classes conjugate COROLLARY Let coset cyclic deﬁned deﬁnition exists ﬁnite ﬁrst Frobenius Frobenius group G and let g e G G is solvable Galois group G hence Hint Let homomorphism IBr(G invariant in G Irr(C Irr(H Irr(N irreducible characters irreducible constituent irreducible F-representation isomorphism LEMMA Let Let 9 Let F Let G Let H Let x e linear character linear group matrix mF(X module nilpotent nonabelian normal subgroup Note p-block p-group permutation Problem projective representations proof is complete Proof Let prove representation of G result follows root of unity S-invariant satisﬁes Show that G splitting field subgroup of G Sylow p-subgroup Sylp(G THEOREM Let unique write x e Irr(G XeIrr(G yields

### Popular passages

Page 4 - Suppose that ^ is a linear transformation of a finite- dimensional vector space V over a field K.