# G-module

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).

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).

## Definition and basics

Let G be a group. A left G-module consists of[1] an abelian group M together with a left group action ρ : G×MM such that

g·(a + b) = g·a + g·b

(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.

A function f : MN is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant.

The collection of left (respectively right) G-modules and their morphisms form an abelian category G-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 ring Z[G].

A submodule of a G-module M is a subgroup AM that is stable under the action of G, i.e. g·aA for all gG and aA. Given a submodule A of M, the quotient module M/A is the quotient group with action g·(m + A) = g·m + A.

## Examples

• Given a group G, the abelian group Z is a G-module with the trivial action g·a = a.
• Let M be the set of binary quadratic forms f(x, y) = ax2 + 2bxy + cy2 with a, b, c integers and let G = SL(2, Z) (the two-by-two special linear group over Z). Define
${\displaystyle (g\cdot f)(x,y)=f((x,y)g^{t})=f((x,y)\cdot {\begin{bmatrix}\alpha &\gamma \\\beta &\delta \end{bmatrix}})=f(\alpha x+\beta y,\gamma x+\delta y)}$
where
${\displaystyle g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}}$
and (x, y)g is matrix multiplication. Then M is a G-module studied by Gauss.[2]
Indeed, we have
${\displaystyle g(h(f(x,y))=gf((x,y)h^{t})=f((x,y)h^{t}g^{t})=f((x,y)(gh)^{t})=(gh)f(x,y)}$
• If V is a representation of G over a field K, then V is a G-module (it is an abelian group under addition).

## Topological groups

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×MM 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×MM satisfying the usual relations g(a + a') = ga + ga' , (gg ')a = g(g 'a), 1a = a.

## Notes

1. ^ Curtis, Charles W.; Reiner, Irving (1962), Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons (Reedition 2006 by AMS Bookstore), ISBN 978-0-470-18975-7.
2. ^ 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.
3. ^ D. Wigner (1973). "Algebraic cohomology of topological groups,". Trans. Amer. Math. Soc. 178: 83–93. doi:10.1090/s0002-9947-1973-0338132-7.