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 $N_G(H)=G$.

Title	normalizer
Canonical name	Normalizer
Date of creation	2013-03-22 12:36:53
Last modified on	2013-03-22 12:36:53
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	15
Author	yark (2760)
Entry type	Definition
Classification	msc 20A05
Synonym	normaliser
Related topic	Centralizer
Related topic	NormalSubgroup
Related topic	NormalClosure2
Defines	self-normalizing