Finitely generated nilpotent group
From Groupprops
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
Contents
Definition
A finitely generated nilpotent group is a group satisfying the following equivalent conditions:
- It is finitely generated and nilpotent.
- It is finitely presented and nilpotent.
- It is Noetherian (i.e., every subgroup is finitely generated -- this is also described using the adjective "slender") and nilpotent.
- 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 |