## Linear Representation of Finite Groups |

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### Contents

2 Preliminarxes on Finitely Generated Modules | 12 |

3 Torsion Modules | 24 |

4 Structure of Finitely Generated Modules | 35 |

Copyright | |

4 other sections not shown

### Common terms and phrases

abelian group algebraic integer algebraically closed field Ann(x artinian ring assume character characteristic roots Clearly coefficients commutative completely reducible module complex numbers conjugate classes Corollary cyclic group decomposition deduce defined by setting denote descending chain diagonalizable direct sum distinct elementary divisors equivalent finite group finite number finitely generated modules follows form f g 6 G Galois group G hence hermitian form indecomposable invariant factors invertible involution irreducible A-module irreducible polynomials irreducible representations isomorphic K-basis K-space K[X]-module KG-module left-ideal lemma Let G linear representation linear transformation mapping matrix representation minimal left ideal minimal polynomial Moreover nilpotent non-trivial number of elements one-to-one orthogonal pairwise permutation polynomial q prime element prime number principal ideal domain Proof proposition prove R/Rp representation of G representation space roots of unity semi-simple ring sentation sesquilinear form skewfield subgroup of G subspace theorem torsion-free trivial two-sided ideal uniquely vector space zero