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?

Let $|G|=p^nm,~~(p,m)=1$ and let $H=\langle g\rangle$. So we see that $|H|=p^k$ and morover $|N_G(P)|=p^n.t$ such that $t\mid m$. Try to use this point that $H\le N_G(P)$ and that $HP$ is a subgroup of the normalizer of $P$. what will you get when you write $|HP|\big||N_G(P)|$...
