G-module


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 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.

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
where g·a denotes ρ. 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 if f is both a group homomorphism and G-equivariant.
The collection of left G-modules and their morphisms form an abelian category G-Mod. The category G-Mod can be identified with the category of left ZG-modules, i.e. with the 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· = g·m + A.

Examples

If G is a topological group and M is an abelian topological group, then a topological G-module is a G-module where the action map G×M → M is continuous.
In other words, a topological G-module is an abelian topological group M together with a continuous map G×M → M satisfying the usual relations g = ga + ga′, a = g, and 1a = a.