G-module

In mathematics, given a group ${\displaystyle G}$, a G-module is an abelian group ${\displaystyle M}$ on which ${\displaystyle G}$ acts compatibly with the abelian group structure on ${\displaystyle M}$. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general ${\displaystyle G}$-modules.

The term G-module is also used for the more general notion of an R-module on which ${\displaystyle G}$ acts linearly (i.e. as a group of ${\displaystyle R}$-module automorphisms).