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).

Let 👁 {\displaystyle G}
be a group. A left 👁 {\displaystyle G}
-module consists of^[1] an abelian group 👁 {\displaystyle M}
together with a left group action 👁 {\displaystyle \rho :G\times M\to M}
such that

👁 {\displaystyle g\cdot (a_{1}+a_{2})=g\cdot a_{1}+g\cdot a_{2}}

for all 👁 {\displaystyle a_{1}}
and 👁 {\displaystyle a_{2}}
in 👁 {\displaystyle M}
and all 👁 {\displaystyle g}
in 👁 {\displaystyle G}
, where 👁 {\displaystyle g\cdot a}
denotes 👁 {\displaystyle \rho (g,a)}
. A right 👁 {\displaystyle G}
-module is defined similarly. Given a left 👁 {\displaystyle G}
-module 👁 {\displaystyle M}
, it can be turned into a right 👁 {\displaystyle G}
-module by defining 👁 {\displaystyle a\cdot g=g^{-1}\cdot a}
.

A function 👁 {\displaystyle f:M\rightarrow N}
is called a morphism of 👁 {\displaystyle G}
-modules (or a 👁 {\displaystyle G}
-linear map, or a 👁 {\displaystyle G}
-homomorphism) if 👁 {\displaystyle f}
is both a group homomorphism and 👁 {\displaystyle G}
-equivariant.

The collection of left (respectively right) 👁 {\displaystyle G}
-modules and their morphisms form an abelian category 👁 {\displaystyle G{\textbf {-Mod}}}
(resp. 👁 {\displaystyle {\textbf {Mod-}}G}
). The category 👁 {\displaystyle G{\text{-Mod}}}
(resp. 👁 {\displaystyle {\text{Mod-}}G}
) can be identified with the category of left (resp. right) 👁 {\displaystyle \mathbb {Z} G}
-modules, i.e. with the modules over the group ring 👁 {\displaystyle \mathbb {Z} [G]}
.

A submodule of a 👁 {\displaystyle G}
-module 👁 {\displaystyle M}
is a subgroup 👁 {\displaystyle A\subseteq M}
that is stable under the action of 👁 {\displaystyle G}
, i.e. 👁 {\displaystyle g\cdot a\in A}
for all 👁 {\displaystyle g\in G}
and 👁 {\displaystyle a\in A}
. Given a submodule 👁 {\displaystyle A}
of 👁 {\displaystyle M}
, the quotient module 👁 {\displaystyle M/A}
is the quotient group with action 👁 {\displaystyle g\cdot (m+A)=g\cdot m+A}
.

• Given a group 👁 {\displaystyle G}
  , the abelian group 👁 {\displaystyle \mathbb {Z} }
  is a 👁 {\displaystyle G}
  -module with the trivial action 👁 {\displaystyle g\cdot a=a}
  .
• Let 👁 {\displaystyle M}
  be the set of binary quadratic forms 👁 {\displaystyle f(x,y)=ax^{2}+2bxy+cy^{2}}
  with 👁 {\displaystyle a,b,c}
  integers, and let 👁 {\displaystyle G={\text{SL}}(2,\mathbb {Z} )}
  (the 2×2 special linear group over 👁 {\displaystyle \mathbb {Z} }
  ). Define

👁 {\displaystyle (g\cdot f)(x,y)=f((x,y)g^{t})=f\left((x,y)\cdot {\begin{bmatrix}\alpha &\gamma \\\beta &\delta \end{bmatrix}}\right)=f(\alpha x+\beta y,\gamma x+\delta y),}

where

👁 {\displaystyle g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}}

and 👁 {\displaystyle (x,y)g}
is matrix multiplication. Then 👁 {\displaystyle M}
is a 👁 {\displaystyle 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 👁 {\displaystyle V}
  is a representation of 👁 {\displaystyle G}
  over a field 👁 {\displaystyle K}
  , then 👁 {\displaystyle V}
  is a 👁 {\displaystyle G}
  -module (it is an abelian group under addition).

If 👁 {\displaystyle G}
is a topological group and 👁 {\displaystyle M}
is an abelian topological group, then a topological G-module is a 👁 {\displaystyle G}
-module where the action map 👁 {\displaystyle G\times M\rightarrow M}
is continuous (where the product topology is taken on 👁 {\displaystyle G\times M}
).^[3]

In other words, a topological 👁 {\displaystyle G}
-module is an abelian topological group 👁 {\displaystyle M}
together with a continuous map 👁 {\displaystyle G\times M\rightarrow M}
satisfying the usual relations 👁 {\displaystyle g(a+a')=ga+ga'}
, 👁 {\displaystyle (gg')a=g(g'a)}
, and 👁 {\displaystyle 1a=a}
.

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