Conjugate of Cycle

Theorem

Let $n \ge 1$ be a natural number.

Let $S_n$ be the symmetric group on $n$ letters.

Let $\pi, \sigma \in S_n$.

Let $\sigma$ be a cycle of length $k$.

Then the conjugate $\pi \sigma \pi^{-1}$ is a cycle of length $k$.

Proof

Follows directly from Conjugate Permutations have Same Cycle Type.

$\blacksquare$