If $g$ is in $N_G(P)$ then $g\in P$ where $P$ is a $p$-Sylow subgroup Please help me to solve this problem. Let $P$ is a $p$-Sylow subgroup of the finite group $G$ and $g$ is an element such that $\lvert g \rvert=p^k$ then if $g$ is in $N_G(P)$ then $g\in P$. Where to start?

Hint: $P$ is a normal $p$ Sylow subgroup of $N_G(P)$, and because of its order, $g$ must be in a Sylow p-subgroup of this group...
