# Nilpotent group that is finitely generated as a powered group for a set of primes

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: nilpotent group and group that is finitely generated as a powered group for a set of primes

View other group property conjunctions OR view all group properties

## Contents

## Definition

A group is termed a **nilpotent group that is finitely generated as a powered group for a set of primes** if it satisfies the following equivalent conditions:

- is a nilpotent group as well as a group that is finitely generated as a powered group for a set of primes. The latter means that there exists a subset (possibly empty, possibly finite, possibly infinite, possibly including all primes) such that is a -powered group and there is a finite subset of such that there is no proper -powered subgroup of containing .
- is a nilpotent group and there is a set of primes such that is a pi-powered group, and further, the abelianization of is finitely generated as a -module.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian group that is finitely generated as a module over the ring of integers localized at a set of primes | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group that is finitely generated as a powered group for a set of primes | |FULL LIST, MORE INFO | |||

nilpotent group | |FULL LIST, MORE INFO |