Concept

# G-module

Summary
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 \rho:G\times M\to M such that :g·(a1 + a2) = g·a1 + g·a2 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
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