Permutation Representation defines Group Action

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

Let $X$ be a set.

Let $\map \Gamma X$ be the symmetric group of $X$.

Let $\rho: G \to \map \Gamma X$ be a permutation representation, that is, a homomorphism.

The mapping $\phi: G \times X \to X$ associated to $\rho$, defined by:
 * $\map \phi {g, x} = \map {\map \rho g} x$

is a group action.

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

We verify that $g * \paren {h * x} = \paren {g h} * x$:

We verify that $e * x = x$.

Let $I_X$ denote the identity mapping on $X$.