## An introduction to group representations |

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### Contents

PRELIMINARIES | 3 |

REPRESENTATIONS AND CHARACTER RELATIONS | 20 |

INDUCED REPRESENTATIONS AND THE FROBENIUS | 52 |

2 other sections not shown

### Common terms and phrases

A-module A-submodules abelian group acter algebraic integer Burnside char character table characters of G classes of G clearly commutative completely reducible conjugate classes Corollary 2.19 decomposition define Definition Let denote direct sum direct summand distinct irreducible ducible elements of G exists field finite dimensional finite group follows form a basis gkKH GL(M group algebra group G group of order group representations Hence HomCG homomorphism HomR Homs ideal of CG idempotent induced representations inequivalent irreducible CG-module irreducible characters irreducible matrix representations irreducible representation isomorphic KG-module left S-module Let G linear transformation Maschke's Theorem minimal ideal module non-trivial non-zero order of G orthogonality relations Proof repre representation of G representation space roots of unity scalar product Schur's Lemma sentation solvable groups subgroup of G sum of irreducible Suppose tation of G TCg^gg tion trivial vector space Z(CG