Parity of Conjugate of Permutation

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


 * $\forall \pi, \rho \in S_n: \operatorname{sgn} \left({\pi \rho \pi^{-1}}\right) = \operatorname{sgn} \left({\rho}\right)$

where $\operatorname{sgn} \left({\pi}\right)$ is the sign of $\pi$.

Proof
As $\operatorname{sgn} \left({\pi}\right) = \pm 1$ for any $\pi \in S_n$, we can apply the laws of commutativity and associativity: