Conjugate Permutations have Same Cycle Type

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

Let $$\pi, \rho \in S_n$$ be two permutations in $$S_n$$.

Then $$\pi$$ and $$\rho$$ are conjugate iff they have the same cycle type.

Proof

 * Let $$\pi, \rho \in S_n$$ be conjugate.

Then from Cycle Decomposition of Conjugate, the cycle decomposition of $$\pi \rho \pi^{-1}$$ can be obtained from that of $$\rho$$ by substituting all instances of $$i$$ in $$\rho$$ with $$\pi \left({i}\right)$$.

Thus the cycle type of $$\rho$$ does not change when $$\rho$$ is conjugated with $$\pi$$.

Thus, if two permutations are conjugate, they have the same cycle type.


 * Now suppose $$\pi$$ and $$\rho$$ are of the same cycle type. Then there is an element $$\sigma \in S_n$$ such that $$\rho = \sigma \pi \sigma^{-1}$$.