Burnside's Problem - General Burnside Problem

General Burnside Problem

A group G is called periodic if every element has finite order; in other words, for each g in G, there exists some positive integer n such that gn = 1. Clearly, every finite group is periodic. There exist easily defined groups such as the p∞-group which are infinite periodic groups; but the latter group cannot be finitely generated.

The general Burnside problem can be posed as follows:

If G is a periodic group, and G is finitely generated, then must G necessarily be a finite group?

This question was answered in the negative in 1964 by Evgeny Golod and Igor Shafarevich, who gave an example of an infinite p-group that is finitely generated (see Golod-Shafarevich theorem). However, the orders of the elements of this group are not a priori bounded by a single constant.