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:
“All the critics who could not make their reputations by discovering you are hoping to make them by predicting hopefully your approaching impotence, failure and general drying up of natural juices. Not a one will wish you luck or hope that you will keep on writing unless you have political affiliations in which case these will rally around and speak of you and Homer, Balzac, Zola and Link Steffens.”
—Ernest Hemingway (1899–1961)
“The writer operates at a peculiar crossroads where time and place and eternity somehow meet. His problem is to find that location.”
—Flannery O’Connor (1925–1964)