I found it In Exercise in abstract algebra by Dummit and Foote.
Let $P$ be a Sylow $p$-group of $S_p$.
What is the order of $N_{S_p}(P)$?
2026-03-26 20:41:12.1774557672
The order of the normalizer of a $p$-subgroup of $S_{p}$
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If $\;n_p\;$ denotes the number of Syloy $\;p\,-$ subgroups, then we know that
$$n_p=[S_p:N_G(P)]=\frac{p!}{|N_G(P)|}$$
and we know that $\;n_p=(p-2)!\;$