# Periodic nilpotent group

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

## Definition

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

1. It is both locally finite (every finitely generated subgroup is finite) and nilpotent.
2. It is both periodic (every element has finite order) and nilpotent.
3. 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.
4. Its abelianization is a periodic abelian group.
5. It is a nilpotent group and all the quotient groups between successive members of its lower central series are periodic abelian groups.
6. 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
nilpotent p-group nilpotent and every element has order a power of a fixed prime $p$ |FULL LIST, MORE INFO