$G$-module

Let $V$ a vector space over some field $K$ (usually $K=\mathbb{Q}$ or $K=ℂ$). 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 $GL(V)$.