# Locally nilpotent group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This is a variation of nilpotent group|Find other variations of nilpotent group |

## Definition

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

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes If $G$ is a locally nilpotent group and $H$ is a subgroup of $G$, then $H$ is also locally nilpotent.
quotient-closed group property Yes If $G$ is a locally nilpotent group and $H$ is a normal subgroup of $G$, then the quotient group $G/H$ is also locally nilpotent.
finite direct product-closed group property Yes If $G_1, G_2, \dots, G_n$ are all locally nilpotent groups, so is the external direct product $G_1 \times G_2 \times \dots \times G_n$.

## 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

### 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