G-module

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).

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 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 2×2 special linear group over 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 (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).

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.

• 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.