Conjugate Permutations have Same Cycle Type

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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 $\rho = \sigma \pi \sigma^{-1}$.