Group Action defines Permutation Representation

Theorem
Let $\Gamma \left({X}\right)$ be the set of permutations on a set $X$.

Let $G$ be a group.

A group action is a (group) homomorphism from $G$ to $\Gamma \left({X}\right)$.

Proof
Let $g, h \in G$.

From the definition of group action, $\forall \left({g, x}\right) \in G \times X: \phi \left({\left({g, x}\right)}\right) \in X = g \wedge x \in X$.

Let $\phi_g: X \to X$ be the mapping defined as $\phi_g \left({x}\right) = \phi \left({g, x}\right)$.

Let $\phi \left({g, x}\right) = \phi_g \left({x}\right)$.


 * First we show that $\phi_g \circ \phi_h \left({x}\right) = \phi_g \phi_h \left({x}\right)$.


 * We have:
 * $e \wedge x = x \implies \phi_e \left({x}\right) = x$

where $e$ is the identity of $G$.

Comment
Some treatments of this subject take this as the definition of a group action.