$G$module
Let $V$ a vector space over some field $K$ (usually $K=\mathbb{Q}$ or $K=\u2102$). Let $G$ be a group which acts on $V$. This means that there is an operation^{} $\psi :G\times V\to V$ such that

1.
$gv\in V$.

2.
$g(hv)=(gh)v$

3.
$ev=v$
where $gv$ stands for $\psi (g,v)$ and $e$ is the identity element^{} of $G$.
If in addition,
$$g(cv+dw)=c(gv)+d(gw)$$ 
for any $g\in G$, $v,w\in V$, $c,d\in K$, we say that $V$ is a $G$module. This is equivalent^{} with the existence of a group representation^{} from $G$ to $GL(V)$.
Title  $G$module 

Canonical name  Gmodule 
Date of creation  20130322 14:57:53 
Last modified on  20130322 14:57:53 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  6 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 20C99 
Related topic  GroupRepresentation 
Related topic  Group 