In the paper, automorphisms are studied for free groups of varieties given by a family of identities in the well-known infinite independent system of identities involving two variables that was constructed by S. I. Adian to solve the finite basis problem in group theory. It is proved that every normal automorphism (i.e., an automorphism that stabilizes any normal subgroup) of noncyclic free groups of these varieties is an inner automorphism.

A group is called an n-torsion group if it has a system of defining relations of the form rn = 1 for some elements r, and for any of its finite order element a the defining relation an = 1 holds. It is assumed that the group can contain elements of infinite order. In this paper, we show that for every odd n ≥ 665 for each n-torsion group can be constructed a theory similar to that of constructed in S. I. Adian’s well-known monograph on the free Burnside groups. This allows us to explore the n-torsion groups by methods developed in that work. We prove that every n-torsion group can be specified by some independent system of defining relations; the center of any non-cyclic n-torsion group is trivial; the n-periodic product of an arbitrary family of n-torsion groups is an n-torsion group; in any recursively presented n-torsion group the word and conjugacy problems are solvable.

Сам жанр журнала "Успехи математических наук" предполагает обзор прежде всего математических результатов Владимира Андреевича Успенского.Однако вся его личность и деятельность обязывают к более широкому взгляду на вещи.Владимир Андреевич был выдающимся математиком

Gennadii Semënovich Makanin passed away on 18 May 2017. He was a well-known Russian mathematician, an author of outstanding papers on algorithmic questions of algebra and logic, a Doctor of the Physical and Mathematical Sciences, a leading researcher in the Department of Mathematical Logic at the Steklov Mathematical Institute, and a laureate of the I. M. Vinogradov Prize of the Russian Academy of Sciences. Makanin was born on 19 May 1938 in the town of Orsk of the Orenburg Oblast; his father was an engineer and his mother a teacher in a secondary school. After graduating from secondary school in 1955, he enrolled in the Faculty of Mechanics and Mathematics at Moscow State University (MSU). His older brother Vladimir (1937–2017), who also graduated from the Faculty of Mechanics and Mathematics, became a well-known Russian author. During his studies at the university Makanin became interested in mathematical logic and algorithmic questions of mathematics, which in those years began attracting interest among a wide range of mathematicians from many countries, including the Soviet Union. At the university Makanin studied under the supervision of Professor S. A. Yanovskaya. Although he successfully completed his studies and defended his diploma thesis, he was not recommended for postgraduate

It is proved that any countable abelian group D can be embedded as a centre into a m-generated group A such that the quotient group A/D is isomorphic to the free Burnside group B(m, n) of rank m > 1 and of odd period n ≥ 665.The proof is based on some modification of the method which was used by S.I.Adian in his monograph in 1975 for a positive solution of Kontorovich's famous problem from Kourovka Notebook on the existence of a finitely generated non-commutative analogue of the additive group of rational numbers.More precisely, he proved that the desired analogues in which the intersection of any two non-trivial subgroups is infinite, can be constructed as a central extension of the free Burnside group B(m, n), where m > 1, and n ≥ 665 is an odd number, using as its center the infinite cyclic group.The paper also discusses other applications of the proposed generalization of Adian's technique.In particular, we describe the free groups of the variety defined by the identity [x n , y] = 1 and the Schur multipliers of the free Burnside groups B(m, n) for any odd n ≥ 665.

In this paper we provide an overview of the results relating to the n-periodic products of groups that have been obtained in recent years by the authors of the present paper, as well as some results obtained by other authors in this direction. The periodic products were introduced by S.I. Adian in 1976 to solve the Maltsev's well-known problem. It was shown that the periodic products are exact, associative and hereditary for subgroups. They also possess some other important properties such as the Hopf property, the C*-simplicity, the uniform non-amenability, the SQ-universality, etc. It was proved that the n-periodic products of groups can uniquely be characterized by means of certain quite specific and simply formulated properties. These properties allow to extend to n-periodic products of various families of groups a number of results previously obtained for free periodic groups B(m, n). In particular,we describe the finite subgroups of n-periodic products, Also, we analyze and extend the simplicity criterion of n-periodic products obtained previously by S.I. Adian.

We study the free groups in varieties defined by an arbitrary set of identities in a well-known infinite independent system of identities in two variables constructed by S.I. Adian to solve the finite basis problem in group theory. We prove that the centralizer of any non-identity element in a relatively free group in any of the varieties under consideration is cyclic, and for every m > 1 the set of all non-isomorphic free groups of rank m in these varieties is of the cardinality of the continuum. All these groups have trivial centre, all their abelian subgroups are cyclic, and all their non-trivial normal subgroups are infinite. For any free group in Gamma any of these varieties, we also obtain a description of the automorphisms of the semigroup End(Gamma), answering a question posed by Plotkin in 2000. In particular, we prove that the automorphism group of any such End(Gamma) is canonically embedded in the group Aut( Aut(Gamma)).

On 1 December 2014 the prominent mathematician Vladimir Petrovich Platonov, academician of the Russian Academy of Sciences (RAS) and the National Academy of Sciences of Belarus, and principal research fellow of the Research Institute for System Studies of the RAS and the Steklov Mathematical Institute of the RAS, observed his 75th birthday. Platonov is an internationally known specialist in the area of algebra, algebraic geometry, and number theory. He has solved several major problems which had long defied the efforts of other mathematicians, among them the problem of strong approximation in algebraic groups (posed in 1937), the Kneser–Tits problem, the creation of the reduced K-theory of finite-dimensional division algebras, and the solution on this basis of the Tannaka–Artin problem and the problem of rationality of group algebraic varieties. He has made important contributions to the development of the arithmetic theory of algebraic groups: a deep direction of research that lies at the junction of group theory, algebraic geometry, and number theory. Over the last several years he has developed a theory for finding fundamental units in hyperelliptic fields, and new highly efficient algorithms have been constructed for calculating such units on the basis of this theory. These and many other results brought him wide international recognition and renown, as evidenced by invitations to speak at the International Congresses of Mathematicians in Vancouver (1974) and Helsinki (1978) and at the European Mathematical Congress in Budapest (1996). His results were the subject of a talk by J. Tits at the Bourbaki seminar in 1977, and in 1978 he received the Lenin Prize in the area of science and technology for the cycle of fundamental papers on the arithmetic of algebraic groups and reduced K-theory. In 1993 he was awarded the Humboldt Prize (Germany). Platonov’s mathematical work is multifaceted and is characterized by deep originality. He has written more than 160 research papers, including 2 monographs. The spectrum of problems investigated by him is very broad and includes a detailed study of basic classes of locally compact topological groups, an analysis of the

This article consists of two parts. The first part presents a detailed history of the long-term joint work (1960–1968) of the author and P.S. Novikov on the proof of the infiniteness of the free Burnside groups B ( m , n ) for odd periods n ≥ 4381 and m > 1 generators (Sections 1 and 2). In Sections 3–10 we survey several significant results obtained by the author and his successors using the Novikov-Adian theory and its various modifications. In the second part (Sections 11–15) we outline a new modification of the Novikov-Adian theory. The new modification allows us to decrease to n ≥ 101 the lower bound on the odd periods n for which one can prove the infiniteness of the free periodic groups B ( m , n ). We plan to publish a full proof of this new result in the journal Russian Mathematical Surveys .

We prove that n-periodic products (introduced by the first author in 1976) are uniquely characterized by certain quite specific properties. Using these properties, we prove that if a non-cyclic subgroup H of the n-periodic product of a given family of groups is not conjugate to any subgroup of the product's components, then H contains a subgroup isomorphic to the free Burnside group B(2, n). This means that H contains the free periodic groups B(m, n) of any rank m > 2, which lie in B(2, n) ([1], Russian p. 26). Moreover, if H is finitely generated, then it is uniformly non-amenable. We also describe all finite subgroups of n-periodic products.

LetH be a subgroup of a groupG. A normal subgroupN H ofH is said to be inheritably normal if there is a normal subgroup N G of G such that N H = N G ∩ H. It is proved in the paper that a subgroup \(N_{G_i }\) of a factor G i of the n-periodic product Π i∈I n G i with nontrivial factors G i is an inheritably normal subgroup if and only if \(N_{G_i }\) contains the subgroup G i n . It is also proved that for odd n ≥ 665 every nontrivial normal subgroup in a given n-periodic product G = Π i∈I n G i contains the subgroup G n . It follows that almost all n-periodic products G = G 1 * n G 2 are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.

In this paper, the following system of substitutions in a 3-letter alphabet ∑ = ⟨. a,b,c|a^2 → bc,b^2 → ac,c^2 → ab⟩ is considered. A detailed proof of results that were described briefly in the author’s paper [1] is presented. They give an answer to the specific question on the possibility of giving a polynomial upper bound for the lengths of derivations from a given word in the system Σ stated in the literature. The maximal possible number of steps in derivation sequences starting from a given word W is denoted by D ( W ). The maximum of D ( W ) for all words of length | W | = l is denoted by D ( l ). It is proved that the function D ( W ) on words W of given length | W | = m +2 reaches its maximum only on words of the form W = c 2 b m and W = b m a 2 . For these words, the following precise estimate is established: where ⌌3 m 2 /2⌍ for odd | m | is the round-up of 3 m 2 /2 to the nearest integer.

In a 1959–1975 cycle of papers, P.S. Novikov and S.I. Adian created a new method for studying periodic groups based on the classification of periodic words by means of a complicated simultaneous induction. The method was developed for solving the well-known Burnside problem on periodic groups, but it also enabled the authors to solve a number of other difficult problems of group theory. An extended survey of results contained in the cycle of papers mentioned above and of other significant results obtained after 1975 by Adian and other authors on the basis of the developed theory and its modifications is presented.

Journal Article Proof and Computation Get access Sergei Adian, Sergei Adian Search for other works by this author on: Oxford Academic Google Scholar Lev Beklemishev, Lev Beklemishev Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina str. 8, 119991 Moscow, Russia. E-mail: sia@mi.ras.ru; lbekl@yandex.ru Search for other works by this author on: Oxford Academic Google Scholar Albert Visser Albert Visser Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands. E-mail: albert.visser@phil.uu.nl Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 21, Issue 4, August 2011, Pages 541–542, https://doi.org/10.1093/logcom/exp042 Published: 28 July 2011

DOI: 10.1134/S0081543811060022 We say that a function f(x) is essentially discontinuous if its interval of definition cannot be partitioned into countably many sets on each of which the function is continuous. N.N. Luzin put the following question: Does there exist an essentially discontinuous Borel function? In this paper, we construct a semicontinuous function f(x) defined and essentially discontinuous on the interval [0, 1]. In order to define the function f(x), we need a certain set Fσ1 , which we will now construct. In the interval [0, 1], consider a dense subset that is a countable sum of mutually disjoint perfect sets F1, F2, . . . , Fn, . . . that are nowhere dense in [0, 1], Fσ1 = ∞ ∑