Conjugates of a cycle $\sigma$ in $S_n$

Question

I'm trying to study the different cycle types and their conjugates in a symmetric group $S_n$.

For more of a concrete example, suppose we are given a cycle $\sigma \in S_5$. My questions are:

How to determine what elements are conjugate to $\sigma$,

how many elements are of the same cycle type as $\sigma$,

lastly how to determine what the elements of the centralizer $C_G(\sigma)$ are.

Any help would be great!

Answer 1

Let $\sigma$ be a cycle of order $m$ in $S_n$

1. The elements that are conjugate to $\sigma$ are also cycle of order $m$.

2. The number of elements that are of same cycle type of $\sigma$ is $(1/m)(n)(n-1)\dots(n-m+1)$

3. $|C_G(\sigma)|=(n-m)!m$