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Homenormalizer

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# 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 $N_{G}(H)=G$.

Defines:

self-normalizing

Related:

Centralizer,NormalSubgroup, NormalClosure2

Synonym:

normaliser

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

20A05*no label found*

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