Let $S_n,A_n$ denote the symmetric and alternating groups each on $n$ letters respectively. Let $\sigma \in S_n. $ I attempted to arrive at a handy equivalence condition for the statement
$$C_{S_n}(\sigma)\cap A_n=C_{S_n}(\sigma).\tag{$P$}$$
I found that this statement
$$\sigma\; \text{ fixes at most one letter and the cycle decomposition of } \sigma\; \text{does not contain any }\tau \notin A_n\tag{$Q$}$$
is a necessary condition for $(P)$.
I could not make progress in proving/disproving that the statement $(Q)$ is also sufficient for $(P)$.
Any help in this direction is appreciated.
Thank you.