# $G$-Set

If $G$ is a group and $X$ a set, then $X$ is called a left $G$-Set if there exists a mapping $\lambda:G\times X\rightarrow X$ with

 $\lambda(g_{1},\lambda(g_{2},x))=\lambda(g_{1}g_{2},x)$

or shorter with $\lambda(g,x)=gx$

 $g_{1}(g_{2}(x))=(g_{1}g_{2})(x)$

for all $x\in X$ and $g_{1},g_{2}\in G$. And when $G$ acts on a set $X$, the set $X$ is always a $G$-set.

$X$ is called a right $G$-Set if there exists a mapping $\lambda:X\times G\rightarrow X$ with

 $\lambda(\lambda(x,g_{2}),g_{1})=\lambda(x,g_{2}g_{1})$
Title $G$-Set GSet 2013-03-22 17:55:44 2013-03-22 17:55:44 jwaixs (18148) jwaixs (18148) 5 jwaixs (18148) Definition msc 20-00 GroupAction