Normalizers and centralizers

#group_theory

Let $G$ be a group and suppose $H \leq G$ . The normalizer of $H$ in $G$ is defined to be $N (H) = {g \in G | g H = H g}$ and the centralizer of $H$ in $G$ is defined to be $C (H) = {g \in G | g h = h g for all h \in H}$ .

Prove that $N (H)$ is a subgroup of $G$ .
Prove that $C (H)$ is a normal subgroup of $N (H)$ and that $N (H) / C (H)$ is isomorphic to a subgroup of $Aut (H)$ .

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Let $G$ be a group and suppose $H\leq G$. The {\bfseries normalizer} of $H$ in $G$ is defined to be $N(H)=\{g\in G\,|\, gH=Hg\}$ and the {\bfseries centralizer} of $H$ in $G$ is defined to be $C(H)=\{g\in G\,|\,  gh=hg\text{ for all }h\in H\}$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $N(H)$ is a subgroup of $G$.
	\item Prove that $C(H)$ is a normal subgroup of $N(H)$ and that $N(H)/C(H)$ is isomorphic to a subgroup of $\operatorname{Aut}(H)$.
\end{enumerate}