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Chapter IGroup Theory Section I Basic Notions
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A-module abelian group algebraic extension automorphism axioms called clearly coefficients column vectors commutative ring composition series congruent conjugates contains coset cyclic group defined degree denote diagonal direct sum division ring divisor of G divisors of zero element of G elementary divisors equation equivalence relation euclidean ring exists expressed fact finite number fixed field follows formally real Galois group group G group of order h-th roots Hence homomorphic mapping implies integrity domain inverse irreducible polynomial isomorphic Jordan-Holder K-automorphism law of composition left ideal left resp lemma linear basis linearly independent matrix of order module necessary and sufficient neutral element normal divisor notation one-to-one permutation primitive principal ideal ring rational integers real number roots of unity semi-group splitting field square matrix subfield subgroup of G submodule subring sufficient condition symbol symmetric Theory transformation uniquely determined unity element vector space vector subspace zero elements