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*×

*M*→

*M*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* : *M* → *N* is called a **morphism of G-modules** (or a

**, or a**

*G*-linear map**) if**

*G*-homomorphism*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-**). The category

*G**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 *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 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*) =*ax*^{2}+ 2*bxy*+*cy*^{2}with*a*,*b*,*c*integers, and let*G*= SL(2,**Z**) (the 2×2 special linear group over**Z**). Define

- where
- and (
*x*,*y*)*g*is matrix multiplication. Then*M*is a*G*-module studied by Gauss.^{[2]}Indeed, we have

- 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*×

*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 1*a* = *a*.

## Notes

**^**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.**^**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.**^**D. Wigner (1973). "Algebraic cohomology of topological groups".*Trans. Amer. Math. Soc*.**178**: 83–93. doi:10.1090/s0002-9947-1973-0338132-7.

## References

- Chapter 6 of Weibel, Charles A. (1994).
*An introduction to homological algebra*. Cambridge Studies in Advanced Mathematics.**38**. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.