[Math] Show that $N(H)$ is the largest subgroup of $G$ in which $ H$ is normal.

Question

Let $G$ be a group and $H \leq G$. The normalizer of $H$ in $G$ is
$$N(H) = \{g \in G|gHg^{−1} = H\}$$
If $H$ is a normal subgroup of $K \leq G$ then $K \leq N(H)$. Show that $N(H)$ is the largest subgroup of $G$ in which $ H$ is normal.

Can anyone give me a hit to do this question?

Thanks

Answer 1

Let $x\in K$. Since $H\lhd K$, $xHx^{-1}=H$ which implies $x\in N(H)$. Therefore $K\subseteq N(H)$.