# Finitely generated periodic group

From Groupprops

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

View other group property conjunctions OR view all group properties

## Contents

## Definition

### Symbol-free definition

A **finitely generated periodic group** is a group satisfying the following two conditions:

- It is finitely generated -- it has a generating set of finite cardinality.
- It is periodic -- every element in the group has finite order.

## Relation with other properties

### Stronger properties

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

Finite group | Follows from periodic not implies locally finite (see also examples: Grigorchuk group, infinite Burnside groups, and Tarski monsters) | Finitely presented periodic group|FULL LIST, MORE INFO | ||

Finitely presented periodic group | both periodic and finitely presented; conjectured to be equivalent to being a finite group | |FULL LIST, MORE INFO |

### Weaker properties

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

Periodic group | |FULL LIST, MORE INFO | |||

Finitely generated group | |FULL LIST, MORE INFO | |||

Group generated by finitely many periodic elements | |FULL LIST, MORE INFO |