normalizer

Definitions

Let $G$ be a group, and let $H\subseteq G$.The normalizer of $H$ in $G$, written $N_G(H)$, is the set

$$\{g\in G\mid gH{g}^{-1}=H\}.$$

A subgroup $H$ of $G$ is said to be self-normalizing if $N_G(H)=H$.

Properties

$N_G(H)$ is always a subgroup of $G$,as it is the stabilizer of $H$ under the action $(g,H)\mapsto gH{g}^{-1}$of $G$ on the set of all subsets of $G$(or on the set of all subgroups of $G$, if $H$ is a subgroup).

If $H$ is a subgroup of $G$, then $H\le N_G(H)$.

If $H$ is a subgroup of $G$, then $H$ is a normal subgroup of $N_G(H)$;in fact, $N_G(H)$ is the largest subgroup of $G$of which $H$ is a normal subgroup.In particular, if $H$ is a subgroup of $G$,then $H$ is normal in $G$ if and only if $N_G(H)=G$.