Normalizer of a subgroup

#group_theory

Let $H$ be a subgroup of a group $G$ . The normalizer of $H$ in $G$ is the set $N_{G} (H) = {g \in G ∣ g H = H g}$ .

Prove $N_{G} (H)$ is a subgroup of $G$ containing $H$ .
Prove $N_{G} (H)$ is the largest subgroup of $G$ in which $H$ is normal.

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Let $H$ be a subgroup of a group $G$. The {\bfseries normalizer} of $H$ in $G$ is the set ${\bf N}_G(H)=\{g\in G\,\mid\, gH=Hg\}$.
\begin{enumerate}[label=\alph*)]
	\item Prove ${\bf N}_G(H)$ is a subgroup of $G$ containing $H$.
	\item Prove ${\bf N}_G(H)$ is the largest subgroup of $G$ in which $H$ is normal.
\end{enumerate}