Parity of Conjugate of Permutation

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Then:
 * $\forall \pi, \rho \in S_n: \map \sgn {\pi \rho \pi^{-1} } = \map \sgn \rho$

where $\map \sgn \pi$ is the sign of $\pi$.

Proof
As $\map \sgn \pi = \pm 1$ for any $\pi \in S_n$, we can apply the laws of commutativity and associativity: