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FURTHER TOPICS IN GROUP THEORY
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additive group arbitrary assertion assume automorphism cent G coefficients commutative group commutative ring composition chain conjugate consider contains Corollary coset cyclic group defined Definition denoted direct product element of G equation establish Example exists field F finite group function Galois group gcd(a given group G group of order hence Hint homomorphism identity element implies integers modulo integral domain inverse invertible elements irreducible polynomial isomorphic Lemma mapping mathematics matrices maximal ideal multiplicative group nontrivial nonzero element normal extension normal subgroup notation nth root one-to-one ordered pairs p-group partially ordered permutation polynomial f(x positive integer prime ideal prime number Proof prove quotient group quotient ring real numbers result ring with identity root of unity semigroup simple groups solvable splitting field subfield subgroup H subgroup of G subring subspace Suppose Sylow p-subgroup symbol symmetric group Theorem theory vector space verify zero divisors