normalizerNormalizer2002-04-25 17:22:362007-08-22 08:10:09Definitionself-normalizing\usepackage{amssymb}
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\section*{Definitions}
Let $G$ be a group, and let $H \subseteq G$.
The {\em 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 {\em self-normalizing} if $N_G(H) = H$.
\section*{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 $N_G(H)=G$.