# 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 |