Parity of Conjugate of Permutation

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


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

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

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

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