Group Action of Symmetric Group

Theorem
Let $S$ be a set.

Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$.

The mapping $*: \map \Gamma S \times S \to S$ defined as:
 * $\forall \pi \in \map \Gamma S, \forall s \in S: \pi * s = \map \pi s$

is a group action.

Proof
The group action axioms are investigated in turn.

Let $\pi, \rho \in \map \Gamma S$ and $s \in S$.

Thus:

demonstrating that holds.

Then:

demonstrating that holds.