Permutations, cycles and conjugacy

Question

Let $u \in S_n$ be a cycle, where $S_n$ is the group of permutations of the set with $n$ elements. Let $\sigma \in S_n$ such that the support of $\sigma \circ u \circ \sigma^{-1}$ is the same as the support of $u$. Does this imply that $u$ commutes with $\sigma$ ?

Answer 1

I'm guessing that the support of $\sigma$ is $\{x \mid \sigma(x) \neq x\}$.

With that in mind, let $S_4$ act on $\{1,2,3,4\}$ and take $\sigma = (1\; 2),$ $u = (1\; 2\; 3)$. Then the support of $u$ is $\{1,2,3\}$.

$\sigma u \sigma^{-1} = (1\; 3\; 2) \neq u$, so $u$ and $\sigma$ don't commute, and the support of $\sigma u \sigma^{-1}$ is $\{1,2,3\}$ also.