Burnside's problem

The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. In plain words, if by looking at individual elements of a group we suspect that the whole group is finite, must it indeed be true? The problem has many variants (see Bounded and Restricted below) that differ in the additional conditions imposed on the orders of the group elements.

Brief history

Initial work pointed towards the affirmative answer. For example, if a group "G" is generated by "m" elements and the order of each element of "G" is a divisor of 4, then "G" is finite. Moreover, A. I. Kostrikin (for the case of a prime exponent) and Efim Zelmanov (in general) proved that, among the finite groups with given number of generators and exponent, there exists a largest one.

Nevertheless, the general answer to Burnside's problem turned out to be negative. In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, P. S. Novikov and S. I. Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982, A. Yu. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents (greater than 10¹⁰), and supplied a considerably simpler proof based on geometric ideas.

The case of even exponents turned out to be much harder to settle. In 1992 S.V. Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 (detailed proofs were published in 1994 and occupied some 300 pages). Later joint work of Ol'shanskii and Ivanov established negative solution for an analogue of Burnside's problem for hyperbolic groups, provided the exponent is sufficiently large. By contrast, very little is known when exponents are small, exponents 2,3,4 and 6 excepted.

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 "g"^"n" = 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 is "G" necessarily a finite group?

This question was answered in the negative in 1964 by E.S. Golod and I.R. Shafarevich, who gave an example of an infinite "p"-group which 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.

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", "g"^"n" = 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 B("m", "n") of rank "m" and exponent "n", denoted B("m", "n"), is a group with "m" distinguished generators "x"₁,…,"x"_"m" in which the identity "x"^"n" = 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 "g"₁,…,"g"_"m" and of exponent "n", there is a unique homomorphism from B("m", "n") to "G" that maps the "i"th generator "x"_"i" of B("m", "n") into the "i"th generator "g"_"i" of "G". In the language of group presentations, free Burnside group B("m", "n") has "m" generators "x"₁,…,"x"_"m" and the relations "x"^"n" = 1 for each word "x" in "x"₁,…,"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 "a"² = "b"² = ("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 is not known whether this group is finite.

The breakthrough in Burnside's problem was achieved by P.S. Novikov and S.I. 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. [John Britton proposed a nearly 300 page alternative proof to the Burnside problem in 1973; however, Adian ultimately pointed out a flaw in that proof.] The case of even exponent turned out to be considerably more difficult. It was only in 1992 that S.V. 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.

Restricted Burnside problem

The restricted Burnside problem (formulated in the 1930s) asks another related question:

: If it is known that a group "G" with "m" generators and exponent "n" is finite, can one conclude that the order of "G" is bounded by some constant depending only on "m" and "n"? Equivalently, are there only finitely many "finite" groups with "m" generators of exponent "n", up to isomorphism?

This variant of the Burnside problem can also be stated in terms of certain universal groups with "m" generators and exponent "n". By basic results of group theory, the intersection of two subgroups of finite index in any group is itself a subgroup of finite index. Let "M" be the intersection of all subgroups of the free Burnside group B("m", "n") which have finite index, then "M" is a normal subgroup of B("m", "n") (otherwise, there exists a subgroup "g" ^-1"Mg" with finite index containing elements not in "M"). One can therefore define a group B₀("m","n") to be the factor group B("m","n")/"M". Every finite group of exponent "n" with "m" generators is a homomorphic image of B₀("m","n").The restricted Burnside problem then asks whether B₀("m","n") is a finite group.

In the case of the prime exponent "p", this problem was extensively studied by A.I. Kostrikin in the 1950s (before negative solution of the general Burnside problem). His solution, establishing the finiteness of B₀("m","p"), used a relation with deep questions about identities in Lie algebras in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov, who was awarded the Fields Medal in 1994 for his work.

Notes

Literature

* S.I. Adian (1979) "The Burnside problem and identities in groups". Translated from the Russian by John Lennox and James Wiegold. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] , 95. Springer-Verlag, Berlin-New York. ISBN 3-540-08728-1
* S.V. Ivanov (1994) "The free Burnside groups of sufficiently large exponents," "Internat. J. Algebra Comput. 4".
* S.V. Ivanov, A.Yu. Ol'shanskii (1996) " [http://www.ams.org/tran/1996-348-06/S0002-9947-96-01510-3/home.html Hyperbolic groups and their quotients of bounded exponents,] " "Trans. Amer. Math. Soc. 348": 2091-2138.
* A.I. Kostrikin (1990) "Around Burnside". Translated from the Russian and with a preface by James Wiegold. "Ergebnisse der Mathematik und ihrer Grenzgebiete" (3) [Results in Mathematics and Related Areas (3)] , 20. Springer-Verlag, Berlin. ISBN 3-540-50602-0.
* A.Yu. Ol'shanskii (1989) "Geometry of defining relations in groups". Translated from the 1989 Russian original by Yu. A. Bakhturin (1991) "Mathematics and its Applications" (Soviet Series), 70. Dordrecht: Kluwer Academic Publishers Group. ISBN 0-7923-1394-1.
* E. Zelmanov (1990) "Solution of the restricted Burnside problem for groups of odd exponent". (Russian) "Izv. Akad. Nauk SSSR Ser. Mat. 54", no. 1: 42-59, 221; translation in "Math. USSR-Izv. 36" (1991), no. 1: 41-60.
* E. Zelmanov (1991) "Solution of the restricted Burnside problem for 2-groups". (Russian) "Mat. Sb. 182", no. 4: 568-592. Translation in "Math. USSR-Sb. 72" (1992), no. 2, 543-565.

External links

* [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Burnside_problem.html History of the Burnside Problem] at MacTutor History of Mathematics archive