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

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 $(g,H)\mapsto gHg^{-1}$ 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 $H\leq N_G(H)$ .

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 $N_G(H)=G$ .