## Abstract and linear algebra |

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additive group algebraic arbitrary assertion assume basis Boolean ring cent G characteristic polynomial characteristic values characteristic vectors coefficients commutative group commutative ring consider contains Corollary coset cyclic group defined definition denoted diagonal direct product direct sum element a e equation equivalence relation establish Example exists field F finite group finite number finite-dimensional vector space function given group G group of order hence Hint homomorphism identity element implies integral domain inverse invertible elements irreducible isomorphic Lemma linear combination linear mapping linearly independent matrix maximal ideal multiplication nilpotent nontrivial nonzero element normal subgroup notation obtain one-to-one ordered pairs p-group partially ordered permutation positive integer prime ideal Proof prove quotient group quotient ring R-module result ring with identity scalar subgroup H subgroup of G subring subspace Suppose Sylow p-subgroup Theorem vector space verify