# 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 |