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?	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	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