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# $G$-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. $gv\in V$.

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 $GL(V)$.

Keywords:

group,representation

Related:

GroupRepresentation,Group

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

20C99*no label found*

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