$G$ -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})

G-Set