In mathematics, given a group , a G-module is an abelian group on which acts compatibly with the abelian group structure on . This widely applicable notion generalizes that of a representation of. Group (co)homology provides an important set of tools for studying general -modules.
The term G-module is also used for the more general notion of an -module on which acts linearly (i.e. as a group of -module automorphisms).
Let be a group. A left -module consists of an abelian group together with a left group action such that
for all and in and all in , where denotes . A right -module is defined similarly. Given a left -module , it can be turned into a right -module by defining .
A function is called a morphism of -modules (or a -linear map, or a -homomorphism) if is both a group homomorphism and -equivariant.
The collection of left (respectively right) -modules and their morphisms form an abelian category (resp. ). The category (resp. ) can be identified with the category of left (resp. right) -modules, i.e. with the modules over the group ring .
A submodule of a -module is a subgroup that is stable under the action of , i.e. for all and . Given a submodule of , the quotient module is the quotient group with action .
If is a topological group and is an abelian topological group, then a topological G-module is a -module where the action map is continuous (where the product topology is taken on ).
In other words, a topological -module is an abelian topological group together with a continuous map satisfying the usual relations , , and .