# Group satisfying normalizer condition

From Groupprops

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

## Definition

A group is said to satisfy the **normalizer condition**, if it satisfies the following equivalent conditions:

- The normalizer of any proper subgroup properly contains it
- There is no proper self-normalizing subgroup of
- Every subgroup of is ascendant

Groups satisfying the normalizer condition have been termed **N-groups** but the term N-group is also used for groups with a particular condition on normalizers of solvable subgroups.

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | If is a group satisfying normalizer condition, and is a subgroup of , then also satisfies normalizer condition. | |

quotient-closed group property | Yes | If is a group satisfying normalizer condition, and is a normal subgroup of , then the quotient group also satisfies normalizer condition. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

nilpotent group | nilpotent implies normalizer condition | normalizer condition not implies nilpotent | Group in which every subgroup is subnormal|FULL LIST, MORE INFO | |

group in which every subgroup is subnormal | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Gruenberg group | every cyclic subgroup is ascendant | |FULL LIST, MORE INFO | ||

locally nilpotent group | every finitely generated subgroup is nilpotent | notmalizer condition implies locally nilpotent | locally nilpotent not implies normalizer condition | Gruenberg group|FULL LIST, MORE INFO |

group having no proper abnormal subgroup | there is no proper abnormal subgroup | |FULL LIST, MORE INFO | ||

group in which every maximal subgroup is normal | every maximal subgroup is a normal subgroup | |FULL LIST, MORE INFO |

## Metaproperties

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