# normalizer

## Definitions

Let $G$ be a group, and let $H\subseteq G$. The 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 $N_{G}(H)=G$.

Title normalizer Normalizer 2013-03-22 12:36:53 2013-03-22 12:36:53 yark (2760) yark (2760) 15 yark (2760) Definition msc 20A05 normaliser Centralizer NormalSubgroup NormalClosure2 self-normalizing