SOME OPEN PROBLEMS IN THE THEORY OF INFINITE DIMENSIONAL ALGEBRAS

Abstract

We will discuss some very old and some new open problems concerning infinite dimensional algebras. All these problems have been inspired by combinatorial group theory. I. The Burnside and Kurosh problems In 1902 W. Burnside formulated his famous problems for torsion groups: (1) let G be a finitely generated torsion group, that is, for an arbitrary element g ∈ G there exists n = n(g) > 1, such that g = 1. Does it imply that G is finite? (2) Let a group G be finitely generated and torsion of bounded degree, that is, there exists n > 1 such that for an arbitrary element g ∈ G g = 1. Does it imply that G is finite? W. Burnside [7] and I. Schur [43] proved (1) for linear groups. The positive answer for (2) is known for n = 2, 3 (W. Burnside, [6]), n = 4 (I. N. Sanov, [42]) and n = 6 (M. Hall, [17]). In 1964 E. S. Golod and I. R. Shafarevich ([12], [13]) constructed a family of infinite finitely generated p–groups (for an arbitrary element g there exists n = n(g) > 1 such that gpn = 1) for an arbitrary prime p. This was a negative answer to the question (1). Other finitely generated torsion groups were constructed by S. V. Alyoshin [1], R. I. Grigorchuk [14], N. Gupta –S. Sidki [16], V. I. Sushchansky [48]. In 1968 P. S. Novikov and S. I. Adian constructed infinite finitely generated groups of bounded odd degree n > 4381. In 1994 S. Ivanov [19] extended this to n = 2, k > 32, so now we can say that the question (2) has negative solution for all sufficiently large n. Remark though that all the counterexamples above are not finitely presented. The following important problem still remains open. Problem 1. Do there exist infinite finitely presented torsion groups? Received October 28, 2006. 2000 Mathematics Subject Classification. 17, 20.