Burnsides problem

One of the oldest open problems in group theory was first posed by William Burnside in a paper published in 1902. Some variations of the problem which were also stated in this paper have been resolved; but a full solution to the basic problem is still open as of 2004.

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: if G is a periodic group, and G is finitely generated, then is G necessarily a finite group?

This question was answered in the negative in 1964, when Golod and Shafarevich gave an example of an infinite p-group that can be finitely generated.

Burnside problem

As a related question which seems as if it might have an easier answer, consider a periodic group G with the additional property that there exists a single integer n such that for all g in G, gn = 1. A group with this property is said to be periodic with bounded exponent n, or just a group with exponent n.

Then the Burnside problem is stated as, if G is a finitely generated group with exponent n, is G finite?

There are exponents n for which this problem also has a negative answer. S.I. Adian and P.S. Novikov showed in 1968 that for every odd n with n > 4381, there are finitely generated exponent n groups that are infinite. (John Britton proposed a nearly 300 page alternative proof to the Burside problem in 1973, however Adian ultimately pointed out a flaw in that proof.) A more famous class of counterexamples (given in 1982) are the Tarski Monsters - finitely generated infinite groups in which every subgroup is cyclic of order p, where p is a prime greater than 1075.

The problem of completely determining for which particular exponents n the answer to the Burnside Problem is positive has turned out to be more intractable.

To summarize the results to date, let Fr be the free group of rank r; and given a fixed integer n, let Frn be the subgroup of Fr generated by the set {gn : g in Fr}. Frn is a normal subgroup of Fr; since if h = a1na2n...amn is in Frn, then

g -1hg = (g -1a1ng)(g -1a2ng)...(g -1amng) = (g -1a1g)n(g -1a2g)n...(g -1amg)n

is also in Frn.

We then define the Burnside free group B(r, n) to be the factor group Fr/(Frn).

If G is any finitely generated group of exponent n, then G has a presentation including relations {gn = 1} for all g in G, plus some additional relations. G is then a homomorphic image of B(r, n) for some r; so the Burnside problem can be re-stated as: for which positive integers r, n is B(r,n) finite?

Burnside proved some easy cases in his original paper:

• Clearly if r = 1, then for all n, B(1, n) = Cn, the cyclic group of order n.
• B(r, 2) is the direct product of r copies of C2.
• This follows since, for any a, b in B(r, 2), we have that (ab)(ab) = 1, and a = a -1; therefore ab = ba, and so B(r,2) is abelian, and so every element of B(r,2) is of the form a1n1a2n2...arnr, where ni is either 0 or 1 and {ai} is the set of r generators.

In addition, Burnside gave finite upper bounds on the order of B(r, 3) and B(2,4).

One hundred years later, the following additional results have been established:

• B(r,3), B(r,4), and B(r,6) are finite for all r.
• B(r, n) is infinite if r > 2 and n > 12.

The particular case of B(r, 5) is still an area of intense research. It is not even known whether B(2,5) is finite.

Restricted Burnside problem

The restricted Burnside problem (formulated in the 1930s) asks another related question: are there only finitely many finite r-generator groups of exponent n, up to isomorphism? (An r-generator group is group which can be generated by r elements.)

If this holds for a given r and n, then consider subgroups H and K of B(r, n), where both H and K have finite index. The intersection of H and K then also has finite index. Let M be the intersection of all subgroups of B(r, n) which have finite index. M is a normal subgroup of B(r, n) (otherwise, there exists a subgroup g -1Mg with finite index containing elements not in M). We can then define B0(r,n) to be the factor group formed by B(r,n)/M. B0(r,n) is a finite group; and every finite r-generator group of exponent n is a homomorphic image of B0(r,n).

The restricted Burnside problem was answered in the affirmative in 1991 by Efim Zelmanov, for which he was awarded the Fields Medal in 1994. The solution relates the problem to deep questions about Lie algebras.