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 ![]() ![]() ![]() |
Order 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 ![]() |
![]() ![]() |
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 ![]() ![]() ![]() ![]() |
Definitely true for all primes greater than ![]() ![]() ![]() |