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'normalizer'
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| Title of object: |
normalizer |
| Canonical Name: |
Normalizer |
| Type: |
Definition |
| Created on: |
2002-04-25 17:22:36 |
| Modified on: |
2007-08-22 06:05:22 |
| Classification: |
msc:20A05 |
| Defines: |
self-normalizing |
| Synonyms: |
normalizer=normaliser |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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Content:
\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 $(g,H)\mapsto gHg^{-1}$.
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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