$G$-Set

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

$$\lambda (\lambda (x,g_2),g_1)=\lambda (x,g_2{g}_1)$$