# Periodic nilpotent group

From Groupprops

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

View other group property conjunctions OR view all group properties

## Contents

## Definition

A **periodic nilpotent group** or **locally finite nilpotent group** is defined in the following equivalent ways:

- It is both locally finite (every finitely generated subgroup is finite) and nilpotent.
- It is both periodic (every element has finite order) and nilpotent.
- It is a nilpotent group and also a group generated by periodic elements: it has a generating set where all the elements have finite orders.
- Its abelianization is a periodic abelian group.
- It is a nilpotent group and all the quotient groups between successive members of its lower central series are periodic abelian groups.
- It is a restricted direct product of nilpotent p-groups, with a common bound on their nilpotency class.

### Equivalence of definitions

`For full proof, refer: Equivalence of definitions of periodic nilpotent group`

## Relation with other properties

### Stronger properties

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

finite nilpotent group | nilpotent and finite | |FULL LIST, MORE INFO | ||

nilpotent p-group | nilpotent and every element has order a power of a fixed prime | |FULL LIST, MORE INFO | ||

periodic abelian group | abelian and every element has finite order | |FULL LIST, MORE INFO |

### Weaker properties

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

periodic group | Locally finite group, Periodic solvable group|FULL LIST, MORE INFO | |||

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

locally finite group | Periodic solvable group|FULL LIST, MORE INFO |