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