# Finitely presented periodic group

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

## Definition

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

1. It is a finitely presented group -- it has a presentation that uses a finite number of generators and a finite number of relations.
2. 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