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 (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
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) ZG-modules, i.e. with the modules over the group ring Z["G"].
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".
Topological groups.
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 (where the product topology is taken on "G"×"M").
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"("a" + "a′") = "ga" + "ga′", ("gg′")"a" = "g"("g′a"), and 1"a" = "a".