G-module

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).

Definition and basics

Let ${\displaystyle G}$ be a group. A left ${\displaystyle G}$-module consists of 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 ${\displaystyle f:M\to N}$ 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 ${\displaystyle G\mathbf{\text{-Mod}}}$ (resp. ${\displaystyle \mathbf{\text{Mod-}}G}$). 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 ${\displaystyle g\cdot (m+A)=g\cdot m+A}$.