Permutation Representation defines Group Action

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

Let $X$ be a set.

Let $\Gamma \left({X}\right)$ be the symmetric group 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:
 * $\phi \left({g, x}\right) = \rho \left({g}\right) \left({x}\right)$

is a group action.

Proof
Let $g, h \in G$ and $x \in X$.

We verify that $g * \left({h * x}\right) = \left({g h}\right) * x$:

We verify that $e * x = x$.

Let $\operatorname{id}$ be the identity mapping on $X$.