Finitely generated nilpotent group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and nilpotent group
View other group property conjunctions OR view all group properties

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: slender group and nilpotent group
View other group property conjunctions OR view all group properties

Definition

A finitely generated nilpotent group is a group satisfying the following equivalent conditions:

1. It is finitely generated and nilpotent.
2. It is finitely presented and nilpotent.
3. It is Noetherian (i.e., every subgroup is finitely generated -- this is also described using the adjective "slender") and nilpotent.
4. It is nilpotent and its abelianization is finitely generated.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of finitely generated nilpotent group

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finitely generated abelian group	both a finitely generated group and an abelian group			|FULL LIST, MORE INFO
Finite nilpotent group	both a finite group and a nilpotent group			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group in which every subgroup is finitely presented				|FULL LIST, MORE INFO
Finitely presented group				Finitely presented solvable group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
Noetherian group (also called slender group)	every subgroup is finitely generated			Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
Finitely generated group				Finitely generated group for which all homomorphisms to any finite group can be listed in finite time, Finitely generated solvable group, Finitely presented solvable group, Group in which every subgroup is finitely presented, Group with a finite complete rewriting system, Group with polynomial-time solvable word problem, Group with solvable word problem, Noetherian group, Polycyclic group|FULL LIST, MORE INFO
Nilpotent group				|FULL LIST, MORE INFO
Supersolvable group	has a normal series where all successive quotients are cyclic			|FULL LIST, MORE INFO
Polycyclic group	has a subnormal series where all successive quotients are cyclic			Supersolvable group|FULL LIST, MORE INFO
Finitely generated solvable group				Finitely presented solvable group, Polycyclic group, Supersolvable group|FULL LIST, MORE INFO