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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 $N_G(H) = H$
$N_G(H)$ 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 $N_G(H)=G$
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