Table of contents

This paper gives a survey of results related to the famous Burnside problem on periodic groups. A negative solution of this problem was first published in joint papers of P.S. Novikov and the author in 1968. The theory of transformations of words in free periodic groups that was created in these papers and its various modifications give a very productive approach to the investigation of hard problems in group theory. In 1950 the Burnside problem gave rise to another problem on finite periodic groups, formulated by Magnus and called by him the restricted Burnside problem. Here it is called the Burnside-Magnus problem. In the Burnside problem the question of local finiteness of periodic groups of a given exponent was posed, but the Burnside-Magnus problem is the question of the existence of a maximal finite periodic group of a fixed period  with a given number  of generators. These problems complement each other. The publication in a joint paper by the author and Razborov in 1987 of the first effective proof of the well-known result of Kostrikin on the existence of a maximal group for any prime , together with an indication of primitive recursive upper bounds for the orders of these groups, stimulated investigations of the Burnside-Magnus problem as well. Very soon other effective proofs appeared, and then Zel'manov extended Kostrikin's result to the case when  is any power of a prime number. These results are discussed in the last section of this paper.

Bibliography: 105 titles.

This is a survey of results related to the Gödel incompleteness theorems and the limits of their applicability. The first part of the paper discusses Gödel's own formulations along with modern strengthenings of the first incompleteness theorem. Various forms and proofs of this theorem are compared. Incompleteness results related to algorithmic problems and mathematically natural examples of unprovable statements are discussed.

Bibliography: 68 titles.

Questions of autostability and algorithmic dimension of models go back to papers by A.I. Malcev and by A. Fröhlich and J.C. Shepherdson in which the effect of the existence of computable presentations which are non-equivalent from the viewpoint of their algorithmic properties was first discovered. Today there are many papers by various authors devoted to investigations of such questions. The present paper deals with the question of inheritance of the properties of autostability and non-autostability relative to strong constructivizations under elementary extensions for almost prime models.

Bibliography: 37 titles.

In this paper, which is from a shorthand report of the author's plenary lecture at the international conference "Malcev Meeting 2009" (Novosibirsk, 24-28 August 2009), the influence of three papers by Anatolii Ivanovich Malcev [1]-[3] on the development of algebra and mathematical logic is discussed.

Bibliography: 75 titles.

New substantial results including the solutions of a number of fundamental problems have been obtained in the last decade or so: the first and rather unexpected examples of arithmetic groups with finite extensions that are not arithmetic were constructed; a criterion for arithmeticity of such extensions was found; deep rigidity theorems were proved for arithmetic subgroups of algebraic groups with radical; a theorem on the finiteness of the number of conjugacy classes of finite subgroups in finite extensions of arithmetic groups was proved, leading to numerous applications, in particular, this theorem made it possible to solve the Borel-Serre problem (1964) on the finiteness of the first cohomology of finite groups with coefficients in an arithmetic group; the problem posed more than 30 years ago on the existence of finitely generated integral linear groups that have infinitely many conjugacy classes of finite subgroups was solved; the arithmeticity question for solvable groups was settled. Similar problems were also solved for lattices in Lie groups with finitely many connected components. This paper is a survey of these results.

Bibliography: 27 titles.