Locally nilpotent group

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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.

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 $G$ 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	|FULL LIST, MORE INFO
group in which every subgroup is subnormal	every subgroup is a subnormal subgroup			|FULL LIST, MORE INFO
Baer group	every cyclic subgroup is a subnormal subgroup			|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	|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			|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			|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