I need to find the centralizer of the permutations $\begin{pmatrix} 1 & 2 & \cdots & n \end{pmatrix}$ and $\begin{pmatrix} 1 & 2 & \cdots & n-1\end{pmatrix}$ in $S_{n}$. Now, perhaps I'm confusing it with the center, but I thought that individual elements didn't have centralizers. I thought that only sets of elements did. The definition I have in my class notes is the following:
Let $G$ be a group and $Y$ any set of elements of $G$. The centralizer $C_{G}(Y)$ is the set of elements of $G$ which commute with all $y \in Y$.
So, is this problem asking me to list which elements of $S_{n}$ commute with each other? I remember learning that any permutation can be written as a product of disjoint cycles and disjoint cycles commute, so is it true that two permutations commute iff they share no cycles in common?
If so, how would I prove this (conjugacy classes, perhaps?)? And if not, or if this statement is incorrect, how would I go about finding the centralizers of these two permutations in $S_{n}$?
Thank you.