Subgroup is Normal Subgroup of Normalizer

Theorem
Let $G$ be a group.

A subgroup $H \le G$ is a subgroup of its normalizer:


 * $H \le G \implies H \le N_G \left({H}\right)$

Subset of Normalizer
First we show that $H$ is a subset of $N_G \left({H}\right)$.

This follows directly from Coset Equals Subgroup iff Element in Subgroup:
 * $x \in H \implies x H = H$

As $x \in H \implies x^{-1} \in H$ it also follows that $x \in H \implies H x^{-1} = H$.

Thus $x \in H \implies x H x^{-1} = H^x = H$ and so $x \in N_G \left({H}\right)$.

So $H \subseteq N_G \left({H}\right)$ as we wanted to show.

Subgroup of Normalizer
By hypothesis, $H$ is a subgroup of $G$.

Thus $H$ is itself a group.

So by definition of subgroup:


 * 1) $H \subseteq N_G \left({H}\right)$;
 * 2) $H$ is a group,

it follows that $H$ is a subgroup of $N_G \left({H}\right)$.