# Locally nilpotent group

From Groupprops

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 nilpotent group|Find other variations of nilpotent group |

## Contents

## Definition

A group is said to be **locally nilpotent** if every finitely generated subgroup of the group is nilpotent.

## Examples

- Any nilpotent group, including any abelian group or any finite nilpotent group, gives an example.
- Among non-nilpotent groups, an example is the generalized dihedral group for 2-quasicyclic group.

## Metaproperties

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

subgroup-closed group property | Yes | If is a locally nilpotent group and is a subgroup of , then is also locally nilpotent. | |

quotient-closed group property | Yes | If is a locally nilpotent group and is a normal subgroup of , then the quotient group is also locally nilpotent. | |

finite direct product-closed group property | Yes | If are all locally nilpotent groups, so is the external direct product . |

## Relation with other properties

### Stronger properties

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

nilpotent group | (direct) | locally nilpotent not implies nilpotent | Baer group, Group satisfying normalizer condition, Gruenberg group|FULL LIST, MORE INFO | |

group in which every subgroup is subnormal | every subgroup is a subnormal subgroup | Baer group, Group satisfying normalizer condition, Gruenberg group|FULL LIST, MORE INFO | ||

Baer group | every cyclic subgroup is a subnormal subgroup | Gruenberg group|FULL LIST, MORE INFO | ||

group satisfying normalizer condition | there is no proper self-normalizing subgroup; equivalently, every subgroup is ascendant | normalizer condition implies locally nilpotent | locally nilpotent not implies normalizer condition | Gruenberg group|FULL LIST, MORE INFO |

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

### Weaker properties

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

2-locally nilpotent group | subgroup generated by any two elements is nilpotent | 3-locally nilpotent group|FULL LIST, MORE INFO | ||

3-locally nilpotent group | subgroup generated by any three elements is nilpotent | |FULL LIST, MORE INFO | ||

Engel group | satisfies Engel conditions | 2-locally nilpotent group|FULL LIST, MORE INFO | ||

locally solvable group | every finitely generated subgroup is solvable | |FULL LIST, MORE INFO |

## Formalisms

### In terms of the locally operator

This property is obtained by applying the locally operator to the property: nilpotent group

View other properties obtained by applying the locally operator