I am currently taking Abstract Algebra II and am stuck on the last question of my assignment. I asked the Professor about it and he said that he didn't remember how to do it and that there might be a typo. Would anyone be able to help answer the question (or provide a counterexample if there is a typo)?
Let $P$ be a Sylow p-subgroup of a group $G$. If $H$ is a subgroup of $G$ and $N(P) \subseteq H$, show that $N(H) = H$.
This is what I have so far - I'm not entirely sure if it's useful:
Since $H\subseteq N(H)$, all we need to do is prove that $N(H) \subseteq H$. Let $a\in N(H)$, then $aHa^{-1}=H$. But $N(P)\subseteq H$ so $N(P)\subseteq aHa^{-1}$, or $aN(P)a^{-1} \subseteq H$. Also, $P\subseteq N(P) \subseteq H$ so $aPa^{-1}\subseteq H$. But everything of the form $aPa^{-1}$ is also a Sylow p-subgroup by Sylow's second theorem.
Thanks!