G-module

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.

Definition and basics

Let be a group. A left -module consists of an abelian group together with a [Group action (mathematics)|left group (mathematics)|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 if is both a group homomorphism and -equivariant.
The collection of left -modules and their morphisms form an abelian category . The category can be identified with the category of left -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.

Examples

Given a group, the abelian group is a -module with the trivial action.
Let be the set of binary quadratic forms with integers, and let . Define
If is a representation of over a field, then is a -module.

Topological groups

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.
In other words, a topological -module is an abelian topological group together with a continuous map satisfying the usual relations,, and.