1. ## Maximal normal subgroup

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

2. Originally Posted by didact273

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

3. Originally Posted by didact273
Prove that $\displaystyle H$ is a normal subgroup of $\displaystyle N_{G}(H)$ and that $\displaystyle N_{G}(H)$ is the largest subgroup of $\displaystyle G$containing $\displaystyle H$ 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 $\displaystyle K$ is a subgroup of $\displaystyle G$ so that $\displaystyle H$ is a normal subgroup of $\displaystyle K$ then $\displaystyle K\subseteq N(G)$ i.e. "$\displaystyle N(G)$ is the largest". Show that any element in $\displaystyle K$ must be automatically be an element of $\displaystyle N(G)$ to complete the proof. That is all you need to show, now follow the definition of normalizer.