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