Burnside problem

Remarks

The group $B(m, n)$ is the free Burnside group: every group with $m$ generators and exponent dividing $n$ is a quotient of it. So the bounded problem asks whether all groups with $m$ generators and exponent $n$ are finite of bounded order. Burnside posed the problem in 1902.

The general problem has a negative answer. Golod and Shafarevich built infinite finitely generated $p$-groups in 1964, and the Grigorchuk group is a later explicit example.

For the bounded problem, $B(m, n)$ is finite for $n = 2, 3, 4, 6$ and all $m$. For instance $B(m, 2) \cong (\Z/2)^m$, since $x^2 = 1$ for all $x$ forces the group to be abelian. Novikov and Adian proved that $B(m, n)$ is infinite for $m \ge 2$ and all odd $n$ large enough, and later work extended this to large even $n$. Tarski monster groups are infinite $2$-generated groups of large prime exponent. Whether $B(2, 5)$ is finite is open.