Group Action of Symmetric Group

Theorem
Let $\N_n$ denote the set $\set {1, 2, \ldots, n}$.

Let $\struct {S_n, \circ}$ denote the symmetric group on $\N_n$.

The mapping $*: S_n \times \N_n \to \N_n$ defined as:
 * $\forall \pi \in S_n, \forall n \in \N_n: \pi * n = \map \pi n$

is a group action.

Proof
The group action axioms are investigated in turn.

Let $\pi, \rho \in S_n$ and $n \in \N_n$.

Thus:

demonstrating that holds.

Then:

demonstrating that holds.