Definition:Group Action

Definition
Let $$X$$ 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 $$X$$.