Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup

Theorem
Let $G$ be a group.

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

Then $\map {N_G} H$, the normalizer of $H$ in $G$, is the largest subgroup of $G$ containing $H$ as a normal subgroup.

Proof
From Subgroup is Subgroup of Normalizer, we have that $H \le \map {N_G} H$.

Now we need to show that $H \lhd \map {N_G} H$.

For $a \in \map {N_G} H$, the conjugate of $H$ by $a$ in $\map {N_G} H$ is:

so:
 * $\forall a \in \map {N_G} H: H^a = H$

and so by definition of normal subgroup:
 * $H \lhd \map {N_G} H$

Now we need to show that $\map {N_G} H$ is the largest subgroup of $G$ containing $H$ such that $H \lhd \map {N_G} H$.

That is, to show that any subgroup of $G$ in which $H$ is normal is also a subset of $\map {N_G} H$.

Take any $N$ such that $H \lhd N \le G$.

In $N$, the conjugate of $H$ by $a \in N$ is $N \cap H^a = H$.

Therefore:
 * $H \subseteq H^a$

Similarly, $H \subseteq H^{a^{-1} }$, so:
 * $H^a \subseteq \paren {H^a}^{a^{-1} } = H$

Thus:
 * $\forall a \in N: H^a = H, a \in \map {N_G} H$

That is:
 * $N \subseteq \map {N_G} H$

So what we have shown is that any subgroup of $G$ in which $H$ is normal is a subset of $\map {N_G} H$, which is another way of saying that $\map {N_G} H$ is the largest such subgroup.