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The cohomology and homology groups
Presentations and resolutions
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abelian groups additive group Algebra assume augmentation ideal automorphism called cd G chapter coefficient ring cohomological dimension cohomological functors cohomology sequence commutative ring Corollary Dedekind domain def G define direct summand element epimorphism equivalent exact sequence Exercise exists finite groups finitely presentable free abelian free extension free group free object G is finite G-free G-module Gaschutz given group G group theory H G,A Hence homology hypothesis implies indecomposable induces infinite cyclic injective KG-projective Lemma Let G mapping Math maximal minimal free minimal projective minimum number module monomorphism morphism natural homomorphism natural isomorphism nilpotent nilpotent group non-trivial normal subgroup Note p-group p-subgroup prime projective cover projective resolution Proof of Theorem Proposition 9 prove Remark restriction result right ideal soluble split extension stem cover subgroup of G Suppose surjective Swan Sylow p-subgroup torsion-free trivial write yields zero