# $G$-module

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

1. 1.

$gv\in V$.

2. 2.

$g(hv)=(gh)v$

3. 3.

$ev=v$

where $gv$ stands for $\psi(g,v)$ and $e$ is the identity element of $G$.

 $g(cv+dw)=c(gv)+d(gw)$
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 Gmodule 2013-03-22 14:57:53 2013-03-22 14:57:53 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Definition msc 20C99 GroupRepresentation Group