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additive group assumption Aut G automorphism central extension Chapter composition series conjugate consider Corollary corresponding cosets of H Coxeter group Coxeter system cyclic group defined definition denote direct product direct sum element g element of G elements of order endomorphism Example Exercise extension G factor group factor set field F finite group formula free abelian group free group function G-set GL(V group G group of order Hence Hint homomorphism homomorphism from G identity implies induces integer invariant irreducible lemma Let F Let G Let H mapping matrix maximal subgroup natural number normal subgroup notation p-group permutation prime number Proof prove Q-invariant relations relatively prime representation group semidirect product sequence Show Similarly simple groups solvable Sp-subgroup of G subgroup H subgroup of G subnormal series subset subspace Suppose Sylow's Theorem T-module transvection trivial uniquely determined vector space Weyl group wreath product