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 $d\geq 1$)	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 $d\geq 3$	$3^{r}$ 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 $d\geq 2$	?
6	Yes	1 for $d=1$ Not nilpotent for $d\geq 2$	?

Large exponent cases

Statement

Best known bounds

For very large values of $n$, the group $B(d,n)$ is infinite for every $d>1$. In particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} greater than or equal to 665, and for all even $n$ greater than or equal to 8000.

External links

• Open Problems in Group Theory discussion