Normal Subgroup iff Normalizer is Group

Theorem
Let $G$ be a group.

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

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


 * $H \lhd G \iff \map {N_G} H = G$

Sufficient Condition
Let $H$ be normal in $G$.

Then $G$ is trivially the largest subgroup of $G$ in which $H$ is normal.

Thus from Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup:
 * $\map {N_G} H = G$

Necessary Condition
Let $\map {N_G} H = G$.

From Subgroup is Normal Subgroup of Normalizer, $H$ is normal in $\map {N_G} H$.

Hence $H$ is normal in $G$.