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!