# Divisible nilpotent group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: divisible group and nilpotent group

View other group property conjunctions OR view all group properties

## Contents

## Definition

A group is termed a **divisible nilpotent group** if it satisfies the following equivalent conditions:

- is a divisible group.
- The abelianization of is a divisible abelian group.
- For every positive integer , the quotient group of successive members of the lower central series is a divisible abelian group.
- For any two positive integers , if denote respectively the and members of the lower central series of , then the quotient group is a divisible group.

## Relation with other properties

### Stronger properties

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

rationally powered nilpotent group | |FULL LIST, MORE INFO | |||

divisible abelian group | can also be characterized as an injective object in the category of abelian groups | |FULL LIST, MORE INFO |

### Weaker properties

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

nilpotent group | ||||

divisible group |

### Prime set-parametrized version

- Nilpotent group that is divisible for a set of primes: Given a set of primes , we can talk of the notion of a -divisible nilpotent group.