## Character Theory of Finite GroupsExcellent text approaches characters via rings (or algebras). In addition to techniques for applying characters to "pure" group theory, much of the book focuses on properties of the characters themselves and how these properties reflect and are reflected in the structure of the group. "A pleasure to read." — American Mathematical Society. 1976 edition. |

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### Contents

Algebras modules and representations | 1 |

Group representations and characters | 13 |

Characters and irttegrality | 33 |

Products of characters | 47 |

Induced characters | 62 |

Normal subgroups | 78 |

TI sets and exceptional characters | 99 |

Brauers theorem | 126 |

Changing the ﬁeld | 144 |

The Schur index | 160 |

Projective representations | 174 |

Character degrees | 198 |

Character correspondence | 219 |

Linear groups | 240 |

Changing the characteristic | 262 |

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### Common terms and phrases

1e lrr(G absolutely irreducible afforded algebraic integer Brauer character CG(x character of G character table character theory character triple choose class function classes of G compute conclude conjugacy classes conjugate contradiction CoRoLLARY Let coset cyclic deﬁned deﬁnition e lrr(G elements exists extendible to G ﬁnd ﬁrst ﬁxed Frobenius group G and let g e G G is solvable Galois group G hence IBr(G invariant in G irreducible characters irreducible constituent irreducible F representation isomorphism LEMMA Let Let 1elrr(G Let F Let G Let H Q G linear character linear group lrr(C lrr(H lrr(N lt follows matrix module nilpotent nonabelian normal subgroup Note p-block p-group p-rational permutation Problem projective representations proof is complete Proof Let prove Q ker representation of G result follows root of unity S-invariant satisﬁes Show that G splitting ﬁeld subgroup of G sufﬁces to show Sylow p-subgroup THEoREM Let theory unique write yields