$\newcommand{\Syl}{\operatorname{Syl}}$
This is an exercise (with hint about the second part) in my own language book in Group theory, however, maybe it is a lemma or theorem in an standard book which I don't have. It is:
Let $p$ is an odd prime number and $n$ is a natural number such that $p\leq n$. Therefore if $P\in \Syl_p(S_n)$ then $P\in \Syl_p(A_n)$ and $|N_{A_n}(P)|=\frac{1}{2}|N_{S_n}(P)|$.
What I have done: Suppose that $P\in \Syl_p(S_n)$ where in $S_n$ is the symmetric group on $n$ letters. For having $P\in \Syl_p(A_n)$, I clearly should prove $P\leq A_n$ first. I can see no clues here to prove that unless I assume $P\nleq A_n$. With this assumption, $P$ has an odd permutation. The rest is unclear to me. :( Maybe, I am losing some obvious facts for proving the first part?
For the second part that is $|N_{A_n}(P)|=\frac{1}{2}|N_{S_n}(P)|$; there is a hint noting use the Frattini Argument. I can handle this part. :) Indeed, I see $S_n=N_{S_n}(P)A_n$ and the rest is routine. Thanks for helping me about the first part.