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
Symbol-free definition
A finitely generated nilpotent group is a group satisfying the following equivalent conditions:
- It is finitely generated and nilpotent.
- It is slender (i.e., every subgroup is finitely generated) 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 |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Noetherian group (also called slender group) | every subgroup is finitely generated | |FULL LIST, MORE INFO | ||
| Finitely generated 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 | |FULL LIST, MORE INFO | ||
| Finitely generated solvable group | |FULL LIST, MORE INFO |