Group action

An action of a group $G$ on a set $X$ is a map
$$\begin{aligned} G \times X &\longrightarrow X \\ (g, x) &\longmapsto g \cdot x \end{aligned}$$verifying the following properties for all $x\in X$:

1. Action of the identity: $e\cdot x=x$
2. Successive actions : $(gh)\cdot x= g\cdot (h\cdot x)$ for all $g,h\in G$.