Normal Subgroup iff Normalizer is Group

Theorem

Let $G$ be a group.

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

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

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

Proof

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:

$\map {N_G} H = G$

$\Box$

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 $G$.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$