Burnside's Problem - Bounded Burnside Problem

Bounded Burnside Problem

Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give very little information about the possible structure of a group. 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. Burnside problem for groups with bounded exponent asks:

If G is a finitely generated group with exponent n, is G necessarily finite?

It turns out that this problem can be restated as a question about the finiteness of groups in a particular family. The free Burnside group of rank m and exponent n, denoted B(m, n), is a group with m distinguished generators x₁,…,x_m in which the identity xn = 1 holds for all elements x, and which is the "largest" group satisfying these requirements. More precisely, the characteristic property of B(m, n) is that, given any group G with m generators g₁,…,g_m and of exponent n, there is a unique homomorphism from B(m, n) to G that maps the ith generator x_i of B(m, n) into the ith generator g_i of G. In the language of group presentations, free Burnside group B(m, n) has m generators x₁,…,x_m and the relations xn = 1 for each word x in x₁,…,x_m, and any group G with m generators of exponent n is obtained from it by imposing additional relations. The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if G is any finitely generated group of exponent n, then G a homomorphic image of B(m, n), where m is the number of generators of G. Burnside's problem can now be restated as follows:

For which positive integers m, n is the free Burnside group B(m,n) finite?

The full solution to Burnside's problem in this form is not known. Burnside considered some easy cases in his original paper:

• For m = 1 and any positive n, B(1, n) is the cyclic group of order n.
• B(m, 2) is the direct product of m copies of the cyclic group of order 2. The key step is to observe that the identities a2 = b2 = (ab)2 = 1 together imply that ab = ba, so that a free Burnside group of exponent two is necessarily abelian.

The following additional results are known (Burnside, Sanov, M. Hall):

• B(m,3), B(m,4), and B(m,6) are finite for all m.

The particular case of B(2, 5) remains open: as of 2005 it was not known whether this group is finite.

The breakthrough in Burnside's problem was achieved by Pyotr Novikov and Sergei Adian in 1968. Using a complicated combinatorial argument, they demonstrated that for every odd number n with n > 4381, there exist infinite, finitely generated groups of exponent n. Adian later improved the bound on the odd exponent to 665. The case of even exponent turned out to be considerably more difficult. It was only in 1992 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an even n ≥ 248, n divisible by 29, the group B(m, n) is infinite. Both Novikov–Adian and Ivanov established considerably more precise results on the structure of the free Burnside groups. In the case of the odd exponent, all finite subgroups of the free Burnside groups were shown to be cyclic groups. In the even exponent case, each finite subgroup is contained in a product of two dihedral groups, and there exist non-cyclic finite subgroups. Moreover, the word and conjugacy problems were shown to be effectively solvable in B(m, n) both for the cases of odd and even exponents n.

A famous class of counterexamples to Burnside's problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group, the so-called Tarski Monsters. First examples of such groups were constructed by A. Yu. Ol'shanskii in 1979 using geometric methods, thus affirmatively solving O. Yu. Schmidt's problem. In 1982 Ol'shanskii was able to strengthen his results to establish existence, for any sufficiently large prime number p (one can take p > 1075) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group of order p. In a paper published in 1996, Ivanov and Ol'shanskii solved an analogue of Burnside's problem in an arbitrary hyperbolic group for sufficiently large exponents.