# normalizer condition

A group is said to satisfy the *normalizer condition* if every proper subgroup^{} is properly contained in its own normalizer^{}.
That is, a group $G$ satisfies the normalizer condition if and only if
$$ for all $$.
A group that satisfies the normalizer condition is sometimes called an *N-group*.

Every nilpotent group^{} is an N-group, and every N-group is locally nilpotent^{}.
In particular, a finitely generated group is an N-group
if and only if it is nilpotent.

A group satisfies the normalizer condition if and only if all its subgroups^{} are ascendant.

Title | normalizer condition |
---|---|

Canonical name | NormalizerCondition |

Date of creation | 2013-03-22 16:14:41 |

Last modified on | 2013-03-22 16:14:41 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 6 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20F19 |

Synonym | normaliser condition |

Related topic | LocallyNilpotentGroup |

Defines | N-group |