TITLE: Why are two permutations conjugate iff they have the same cycle structure?
QUESTION [59 upvotes]: I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?

REPLY [5 votes]: Suppose $\rho=\pi\sigma\pi^{-1}$, for any $m\in Z$,
we have $\rho^m=\pi\sigma^m\pi^{-1}$, i.e. $\rho^m\pi=\pi\sigma^m$.
For a cycle $(i,\sigma(i),\ldots,\sigma^{r-1}(i))$, we have
$$(\pi(i),\pi\sigma(i),\ldots,\pi\sigma^{r-1}(i))
=(j,\rho(j),\ldots,\rho^{r-1}(j))$$
where $j=\pi(i)$.  This is intuitive, isn't it?