$G$-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 $GL(V)$.

Title	$G$-module
Canonical name	Gmodule
Date of creation	2013-03-22 14:57:53
Last modified on	2013-03-22 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