Stabilizer of a coset

Let $G$ be a group and $H \leq G$ a subgroup. For each coset $a H$ of $H$ in $G$ , define the set

G_{a H} = {b \in G | b a H = a H} .

Prove that $G_{a H}$ is a subgroup of $G$ .
Suppose that $H$ is normal in $G$ . Prove that $G_{a H} = H$ .

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Let $G$ be a group and $H\leq G$ a subgroup. For each coset $aH$ of $H$ in $G$, define the set
\[
	G_{aH}=\{b\in G\,|\,baH=aH\}.
\]
\begin{enumerate}[label=\alph*)]
	\item Prove that $G_{aH}$ is a subgroup of $G$.
	\item Suppose that $H$ is normal in $G$. Prove that $G_{aH}=H$.
\end{enumerate}