Burnside problem

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

Statement

For what values of $n$ are the following equivalent conditions true?

Note that, at least prima facie, it is possible, for a given $n$ , that $B(d,n)$ is finite for small $d$ but infinite for large $d$ .

Value of $n$	Answer to Burnside's question	Nilpotency class of $B(d,n)$ in terms of $n$ (assume $d \ge 1$ )	Order of $B(d,n)$ in terms of $n,d$	Explanation and comments
1	Yes	0	1	The only possible group is the trivial group
2	Yes	1	$2^d$	exponent two implies abelian, so any group of exponent 2 must be an elementary abelian 2-group
3	Yes	1 for $d = 1$ 2 for $d = 2$ 3 for $d \ge 3$	$3^r$ where $r = d + \binom{d}{2}+ \binom{d}{3}$	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 $d = 1$ PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	?
5	Unknown	1 for $d = 1$ Unknown for $d \ge 2$	?
6	Yes	1 for $d = 1$ Not nilpotent for $d \ge 2$	?

Statement	Best known bounds
For very large values of $n$ , the group $B(d,n)$ is infinite for every $d > 1$ . In particular, $B(2,n)$ is infinite.	Definitely true for all primes greater than $10^{75}$ , because of the existence of Tarski monsters. This property also holds for all odd $n$ greater than or equal to 665, and for all even $n$ greater than or equal to 8000.