I encountered the following problem for the first time. I sketched a proof for it. I will be thankful if I know it is correct or not. Thanks.
$p$ is a prime and $H$ is a $p$-subgroup of a finite group $G$ such that $p\mid [G:H]$ . Prove that $p\mid [N_G(H):H]$.
I assume $|G|=p^\alpha m, (p,m)=1$ and $|H|=p^\beta, \beta\lneqq\alpha$. According to Sylow's theorem, there is a $p$-sylow subgroup of $G$ including $H$ as a subgroup, say $K$. I see that $H
(1): Once $H