Definition:Group Action

Definition
Let $\displaystyle X$ be a set.

Let $G$ be a group whose identity is $e$.

A group action is a mapping $\phi: G \times X \to X$ such that:


 * $\forall \left({g, x}\right) \in G \times X: \phi \left({\left({g, x}\right)}\right) \in X = g * x \in X$

in such a way that:
 * GA-1: $\forall g, h \in G, x \in X: g * \left({h * x}\right) = \left({g h}\right) * x$;
 * GA-2: $\forall x \in X: e * x = x$.

We say that the group $G$ acts on the set $X$.