# Finitely presented periodic group

From Groupprops

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

View other group property conjunctions OR view all group properties

## Contents

## Definition

A group is termed a **finitely presented periodic group** if it satisfies both the following conditions:

- It is a finitely presented group -- it has a presentation that uses a finite number of generators and a finite number of relations.
- It is a periodic group -- every element has finite order.

## Conjecture

The conjecture that every finitely presented periodic group is finite is currently open. Note that there do exist finitely *generated* periodic groups that are not finite, such as the Grigorchuk group and Tarski monsters and other negative solutions to the Burnside problem (see periodic not implies locally finite).

## Relation with other properties

### Stronger properties

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

finite group | underlying set is finite | obvious | implication conjectured to not be strict, i.e., the properties are conjectured to be equal |
-- |

### Weaker properties

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

finitely presented group | has a finite presentation | |FULL LIST, MORE INFO | ||

periodic group | every element has finite order | Finitely generated periodic group|FULL LIST, MORE INFO | ||

finitely generated periodic group | finitely generated and periodic | |FULL LIST, MORE INFO |