## Linear Representations of Finite GroupsThis book consists of three parts, rather different in level and purpose: The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and charac ters. This is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics. I have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra. The examples (Chapter 5) have been chosen from those useful to chemists. The second part is a course given in 1966 to second-year students of I'Ecoie Normale. It completes the first on the following points: (a) degrees of representations and integrality properties of characters (Chapter 6); (b) induced representations, theorems of Artin and Brauer, and applications (Chapters 7-11); (c) rationality questions (Chapters 12 and 13). The methods used are those of linear algebra (in a wider sense than in the first part): group algebras, modules, noncommutative tensor products, semisimple algebras. The third part is an introduction to Brauer theory: passage from characteristic 0 to characteristic p (and conversely). I have freely used the language of abelian categories (projective modules, Grothendieck groups), which is well suited to this sort of question. The principal results are: (a) The fact that the decomposition homomorphism is surjective: all irreducible representations in characteristic p can be lifted "virtually" (i.e., in a suitable Grothendieck group) to characteristic O. |

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### Contents

3 | |

Character theory | 10 |

Subgroups products induced representations | 25 |

Compact groups | 32 |

Part I | 44 |

A theorem of Brauer | 74 |

Applications of Brauers theorem | 81 |

Rationality questions | 90 |

examples | 102 |

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### Common terms and phrases

abelian algebraic integer belongs character of degree character of G class function classes of G commutative conjugacy class conjugate Corollary corresponding cyclic subgroups decomposition defined denote direct sum elementary subgroup elements of G equal exists finite group follows formula function on G GL(V Grothendieck group group G hence homomorphism induced representation irreducible characters irreducible representations isomorphic lemma Let G Let H Let x e linear map linear representation matrix modular character module necessary and sufficient normal subgroup obtain order of G order prime p-group PA(G permutation prime number projective A[G]-module projective envelope proof prop Proposition prove R K G regular representation representation of G representations of degree resp ring RK G RK H roots of unity s e G scalar semisimple Show stable under G subgroup H subgroup of G subspace sufficiently large surjective Sylow p-subgroup tensor product theorem unique vector space zero