In mathematics, given a groupG, a G-module is an abelian groupM on which Gacts 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).
Let be a group. A left -module consists of[1] an abelian group together with a left group action such that
g·(a1 + a2) = g·a1 + g·a2
for all a1 and a2 in M and all g in G, 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.
The collection of left (respectively right) G-modules and their morphisms form an abelian categoryG-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 ringZ[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 moduleM/A is the quotient group with action g·(m + A) = g·m + A.
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).[3]
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 1a = a.
^Kim, Myung-Hwan (1999), Integral Quadratic Forms and Lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea, American Mathematical Soc.