# Burnside problem

From Groupprops

*This article describes an open problem in the following area of/related to group theory*: presentation theory

## Statement

For what values of are the following equivalent conditions true?

- Every group of exponent is locally finite
- Every finitely generated group of exponent is finite
- For every positive integer , the Burnside group is a finite group.

Note that, at least prima facie, it is possible, for a given , that is finite for small but infinite for large .

## Facts

### Small exponent cases

Value of | Answer to Burnside's question | Nilpotency class of in terms of (assume ) | Order of in terms of | Explanation and comments |
---|---|---|---|---|

1 | Yes | 0 | 1 | The only possible group is the trivial group |

2 | Yes | 1 | exponent two implies abelian, so any group of exponent 2 must be an elementary abelian 2-group | |

3 | Yes | 1 for 2 for 3 for |
where | exponent three implies class three: this follows from exponent three implies 2-Engel for groups, 2-Engel implies class three for groups |

4 | Yes | 1 for PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
? | |

5 | Unknown | 1 for Unknown for |
? | |

6 | Yes | 1 for Not nilpotent for |
? |

### Large exponent cases

Statement | Best known bounds |
---|---|

For very large values of , the group is infinite for every . In particular, is infinite. |
Definitely true for all primes greater than , because of the existence of Tarski monsters. This property also holds for all odd greater than or equal to 665, and for all even greater than or equal to 8000. |