The Burnside Problem on Periodic Groups and Related Topics
by
Sergei Adian
Chair of the Mathematical Logic Dept. at the Steklov Mathematical Institute, Moscow, 119991 ul. Gubkina 8

In 1902, W.Burnside [1] formulated the following problem:

Is every group with a finite number of generators and satisfying an identical relation xⁿ = 1 finite?

This problem is known as the Burnside problem for groups of finite exponent, and it remained open for a considerable time. The negative solution obtained in a joint paper of P.S.Novikov and the author in 1968 [2], where it was shown, that for every odd n >= 4381 and every m > 1, there exists an infinite group \Gamma(m, n) on m generators and satisfying the identical relation xⁿ = 1. Up to then, a positive answer to the Burnside problem had been obtained for n >= 4 and n=6. The existens of a periodic group on two generators having no bound on the orders of its elements was established in 1964 [3].

In order to describe the infinite group \Gamma(m, n), a classification of periodic words in a group alphabet by simultaneous induction was introduced in [2], and a theory constructed of transformation of words corresponding to an identical relation xⁿ = 1 for fixed n >= 4381. It turned out that the group \Gamma(m, n) is isomorphic to so called free Burnside group of exponent n on m generators, usually denoted by B(m, n).

In the book [4] the author has presented an improved version of this theory for odd eponents n >= 665 and proved many other applications of the theory. I follows clearly from the main result of the book [4], that the free Burnside groups B(m, n) for m > 1 are infinite for any exponent of the form n = nk with every odd n >= 4381 and arbitrary natural k. Amongh these applications let us to recall the following interesting properties of the Burnside groups B(m, n) for odd n >= 665 and n > 1, that were established in [4]. They are very close to properties of absolutely free groups: 1). All finite subgroups and all abelian subgroups of B(m, n) are cyclic, 2). The group B(3, n) is isomorphically embedded in B(2, n), 3). The groups B(m, n) do not satisfy minimal and maximal conditions for normal subgroups, 4). The group B(m, n) has an exponential growth and the grouth function is is very close to one for absolutely free group of the same number of generators m > 1.

Later, in 1982, a nonamenability of the groups B(m, n) has been proved as well. It is the only known a case when a nonamenable group is satisfying a nontrivial.

Amongh another group theoretic results proved on the base of the created theory one could mention:

I. The first and very simple examples of infinite irreducible systems of group identitries (1969).

II. First construction of nonabelian analoghues of the additive group of rational numbers, g.e. the groups with an infinite intersection of any two nontrivial subgroups (1971). This construction allows us also give an example of countable infinite group which admits only the discret topology.

II. New commutative and associative operations of periodic product of groups, satisfying the hereditory property for subgroups (1976). The simplicity criterion for the periodic products of groups (1978) gave an opportunity to construct interesting new classes of 2-generated infinite simple groups.

More detailed survey of these results one can find in [5].

Investigations of periodic groups and a solution of several old problems in the group theory using some modifications of our method has been obtained also by some our successors. Amongh interesting results proved by another authors one should mention the work of Yu.Olshansky [6] who proposed a geometric modification of our method by using van Kampen diagrammes (1981). This modification gives some simplification of the classification, but it works only for very large values of exponents (odd n > 10¹0). He also gave a first construction of so called "Tarsky monsters", g.e. infinite periodic groups of prime exponent p > 10⁷⁵ with only proper subgroups having an order p (1982). Later I.G.Lysionok and the author in the joint paper [7] (1991) improved this result to every odd exponent n >= 1003 using the created method on its original form given in [4].

Finilly S.V.Ivanov (1994) and I.G.Lysionok (1996) have independently spreaded the Novikov-Adian theory to the even exponents of the form 2^k for sufficiently large k. In particular I.G.Lysionok proved in his paper [8] that the free Burnside groups B(2, n) are infinite for every n >= 8000. The group B(2, 665) is also infinite, but for B(2, 5) the problem is still open as well as for all 7 >= n >= 664.

References

[1] Burnside W.: On unsetled question in the theory of discontineous groups, Quart. J. Pure and appl. Math. v. 33 (1902), p. 2330-238.

[2] Novikov P.S., Adian S.I.: Infinite periodic groups I, II, III, Izv. Akad. Nauk SSSR Ser. matem., v. 32 , No. 1, 2, 3 (1968), p. 212-244, 251-524, 709-731; English transl. in Math. USSR Izv., 2 (1968)

[3] Golod E.S.: Nil-algebras and residually finite groups, Izv. Akad. Nauk SSSR Ser. matem., v. 28, No. 2 (1964), p. 273-276; English transl. in Amer. Soc. Transl., v. 21 (1965)

[4] Adian S.I.: The Burnside problem and identities in groups, "Nauka", Moscow, 1975; English transl., Springer-Verlag, 1978.

[5] Adian S.I: Investigations on the Burnside problem and questions connected with it, Trudy Matem. Inst. Steklov, v. 168 (1984), p. 171-196.; English transl. in Proc. of the Strklov Inst. of mathem., 1986, No. 3.

[6] Ol'shanski A.Yu.: Geometry of defining relations in groups, "Nauka", Moscow, 1989; English transl., Springer-Verlag, 199?.

[7] Adian S.I., Lysionok I.G.: On groups all of whose proper subgroups are finite cyclic, Izv. Akad. Nauk SSSR Ser. matem., v. 55 , No. 5 (1991), p. 933-990; English transl. in Math. USSR Izv., v. 39, No. 2 (1992)

[8] Infinite Burnside groups of even exponent, Izvestiya RAN Ser. mathem., v. 60, No. 3 (1996); English transl. in Izvestiya: Mathematics, vol. 60, No. 3 (1996).

Date received: July 28, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cajy-42.