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:
“Surely one of the peculiar habits of circumstances is the way they follow, in their eternal recurrence, a single course. If an event happens once in a life, it may be depended upon to repeat later its general design.”
—Ellen Glasgow (18731945)
“The problem is that we attempt to solve the simplest questions cleverly, thereby rendering them unusually complex. One should seek the simple solution.”
—Anton Pavlovich Chekhov (18601904)