Question: Suppose $p$ is prime, $P \subset S_p$ is a Sylow $p$-subgroup of $S_p$ (the symmetric group on $p$ elements) and $N = N(P)$ is the normalizer of $P.$ Show that $$ N \cong (\mathbb{Z}/p\mathbb{Z}) \rtimes_{\varphi} (\mathbb{Z}/p\mathbb{Z})^{\times} $$ and describe $\varphi$.
I understand the arguments given in some similar questions on this site for $|N|=p(p-1).$ However, I am not quite sure how to show what the question is actually asking. I wanted to use the semidirect recognition theorem, but that doesn't quite apply here, because one of the hypotheses of the recognition theorem is $H \cap K = 1$ if we want to conclude $G \cong H \rtimes_{\varphi} K$ where $H$ and $K$ are subgroups of $G.$ However, $\mathbb{Z}/p\mathbb{Z} \cap (\mathbb{Z}/p\mathbb{Z})^{\times} = (\mathbb{Z}/p\mathbb{Z})^{\times}.$ How can I proceed?