Let $\sigma \in A_n$ show that $C_{S_{n}}(\sigma) \subset A_n$ if and only if the descomposition in disjoint cycles of $\sigma$ contains all most $1$ cycles of length $1,3,5,7, \ldots$ and $0$ cycles of length $2,4,6,8, \ldots$.
I tried the following:
$\Rightarrow $) If $C_{S_{n}}(\sigma) \subset A_n$ then $\sigma$ commutes only with even permutations but $\sigma$ commutes with its cycles of descomposition, then these cycles are even and therefore our length are odd.
How can I conclude that at most there is one?
$\Leftarrow )$ Assume the cyclic decomposition of $\sigma$ as in the statement, and now let $\tau \in C_{S_{n}}(\sigma )$ then $\tau$ commutes with every cycle $\alpha$ in the descomposition of $\sigma$
I tried to show that $\tau$ acts as a power of each $\alpha$ but I'm not sure about that.
Any hint or help, I will be very grateful for