Permutation Representation defines Group Action

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

Let $X$ be a set, and let $\Gamma \left({X}\right)$ be the group of permutations of $X$.

Let $\rho : G \to \Gamma \left({X}\right)$ be a permutation representation, that is, a homomorphism.

The mapping $\phi : G \times X \to X$ associated to $\rho$, defined by:
 * $\phi(g, x) = \rho(g)(x)$

is a group action.

Proof
Let $g, h \in G$.

For $g \in G$, define the mapping $\phi_g: X \to X$ as:
 * $\phi_g \left({x}\right) = \phi \left({g, x}\right)$