Maximal normal subgroup

**didact273** · March 29th 2009, 02:05 PM

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.

**NonCommAlg** · March 29th 2009, 06:15 PM

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.

**ThePerfectHacker** · March 29th 2009, 07:11 PM

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.

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