Equivalence Relation on Symmetric Group by Image of n is Congruence Modulo Subgroup

Theoerm
Let $S_n$ denote the symmetric group on $n$ letters

Let $\sim$ be the relation on $S_n$ defined as:
 * $\forall \pi, \tau \in S_n: \pi \sim \tau \iff \map \pi n = \map \tau n$

Then $\sim$ is an equivalence relation which is congruence modulo a subgroup.