# Burnside group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This term is related to: combinatorial group theory

View other terms related to combinatorial group theory | View facts related to combinatorial group theory

## Contents

## Definition

### Definition with symbols

The **Burnside group** (sometimes called the **free Burnside group**) is defined as the quotient of the free group on generators by the normal subgroup generated by all powers. A Burnside group is a group that occurs as for some choice of and .

## Relation with other properties

### Stronger properties

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

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

### Weaker properties

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

Finitely generated group | ||||

Reduced free group | |FULL LIST, MORE INFO |