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).

A left

-module consists of[1] an abelian group

together with a left group action

denotes

ρ ( g , a )

-module is defined similarly.

-module by defining

A function

is called a morphism of

-linear map, or a

is both a group homomorphism and

The collection of left (respectively right)

-modules and their morphisms form an abelian category

(resp.

Mod-

The category

) can be identified with the category of left (resp.

-modules, i.e. with the modules over the group ring

is a subgroup

that is stable under the action of

, the quotient module

is the quotient group with action

is a topological group and

is an abelian topological group, then a topological G-module is a G-module where the action map

is continuous (where the product topology is taken on

[3] In other words, a topological G-module is an abelian topological group

together with a continuous map

satisfying the usual relations

The torus can be made an abelian group isomorphic to the product of the circle group . This abelian group is a Klein four-group -module, where the group acts by reflection in each of the coordinate directions (here depicted by red and blue arrows intersecting at the identity element).