The Restricted Burnside Problem

In 1902 William Burnside [5] wrote 'A still undecided point in the theory of discontinuous groups is whether the order of a group may be not finite, while the order of every operation it contains is finite'. In modern terminology the most general form of the problem is 'can a finitely generated group be infinite while every element in the group has finite order?'. This question was answered in 1964 by Golod [8], who constructed finitely generated infinite /^-groups. The version of the problem which has attracted most interest can be stated in the following way. Let Fd be the free group of rank d, and let N be the subgroup of Fd generated by {g :gsFd}. Then let B(d, n) = FJN. The group B(d, n) is known as the d generator Burnside group of exponent n, and any d generator group of exponent n is a homomorphic image of B(d,n). The Burnside problem (as it has come to be known) is the following: