# 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?

1. Every group of exponent $n$ is locally finite
2. Every finitely generated group of exponent $n$ is finite
3. For every positive integer $d$, the Burnside group $B(d,n)$ is a finite group.

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 $d$.

## Facts

### Small exponent cases

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 $d = 1$
5 Unknown 1 for $d = 1$
Unknown for $d \ge 2$
6 Yes 1 for $d = 1$
Not nilpotent for $d \ge 2$
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.