1. ## Maximal normal subgroup

Prove that $H$ is a normal subgroup of $N_{G}(H)$ and that $N_{G}(H)$ is the largest subgroup of $G$containing $H$ as a normal subgroup.

2. Originally Posted by didact273

Prove that $H$ is a normal subgroup of $N_{G}(H)$ and that $N_{G}(H)$ is the largest subgroup of $G$ containing $H$ as a normal subgroup.
very straightforward! just follow the definition of the normalizer of a subgroup.

3. Originally Posted by didact273
Prove that $H$ is a normal subgroup of $N_{G}(H)$ and that $N_{G}(H)$ is the largest subgroup of $G$containing $H$ as a normal subgroup.
I want to add more to what NonCommAlg said. You need to understand the meaning of "largest". This problem is saying that if $K$ is a subgroup of $G$ so that $H$ is a normal subgroup of $K$ then $K\subseteq N(G)$ i.e. " $N(G)$ is the largest". Show that any element in $K$ must be automatically be an element of $N(G)$ to complete the proof. That is all you need to show, now follow the definition of normalizer.