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.
Read more about this topic: Burnside's Problem
Famous quotes containing the words general and/or problem:
“Everyone confesses in the abstract that exertion which brings out all the powers of body and mind is the best thing for us all; but practically most people do all they can to get rid of it, and as a general rule nobody does much more than circumstances drive them to do.”
—Harriet Beecher Stowe (1811–1896)
“The general public is easy. You don’t have to answer to anyone; and as long as you follow the rules of your profession, you needn’t worry about the consequences. But the problem with the powerful and rich is that when they are sick, they really want their doctors to cure them.”
—Molière [Jean Baptiste Poquelin] (1622–1673)