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In mathematics, given a group ''G'', a ''G''-module is an abelian group ''M'' on which ''G'' acts compatibly with the abelian group structure on ''M''. This widely applicable notion generalizes that of a representation of ''G''. Group (co)homology provides an important set of tools for studying general ''G''-modules. The term ''G''-module is also used for the more general notion of an ''R''-module on which ''G'' acts linearly (i.e. as a group of ''R''-module automorphisms). ==Definition and basics== Let ''G'' be a group. A left ''G''-module consists of〔.〕 an abelian group ''M'' together with a left group action ρ : ''G''×''M'' → ''M'' such that :''g''·(''a'' + ''b'') = ''g''·''a'' + ''g''·''b'' (where ''g''·''a'' denotes ρ(''g'',''a'')). A right ''G''-module is defined similarly. Given a left ''G''-module ''M'', it can be turned into a right ''G''-module by defining ''a''·''g'' = ''g''−1·''a''. A function ''f'' : ''M'' → ''N'' is called a morphism of ''G''-modules (or a ''G''-linear map, or a ''G''-homomorphism) if ''f'' is both a group homomorphism and ''G''-equivariant. The collection of left (respectively right) ''G''-modules and their morphisms form an abelian category ''G''-Mod (resp. Mod-''G''). The category ''G''-Mod (resp. Mod-''G'') can be identified with the category of left (resp. right) modules over the group ring Z(). A submodule of a ''G''-module ''M'' is a subgroup ''A'' ⊆ ''M'' that is stable under the action of ''G'', i.e. ''g''·''a'' ∈ ''A'' for all ''g'' ∈ ''G'' and ''a'' ∈ ''A''. Given a submodule ''A'' of ''M'', the quotient module ''M''/''A'' is the quotient group with action ''g''·(''m'' + ''A'') = ''g''·''m'' + ''A''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「G-module」の詳細全文を読む スポンサード リンク
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