Normal Subgroup iff Normalizer is Group

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Then $H$ is normal in $G$ iff the normalizer of $H$ is equal to $G$:


 * $H \lhd G \iff N_G \left({H}\right) = G$

Proof
Follows directly from Normalizer of Subgroup is Largest Subgroup containing that Subgroup.