Vahagn H. Mikaelian

Yerevan State University | YSU · Faculty of Informatics and Applied Mathematics

Some subnormal embeddings of groups into groups with additional properties are established, and in particular, embeddings of countable groups into two-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, H. Neumann and G. Higman on embeddings of that type. Through the con...

In this note we introduce the class of $\mathcal H$-groups (or Hall groups) related to the class of $\mathcal B$-groups defined by Ph. Hall in 1950's. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that c...

We study the variety generated by cartesian and direct wreath products of arbitrary sets X and Y of abelian groups. In particular, we give a classification of the cases when that variety is equal to the product variety var(X)var(Y). This criterion is a wide generalization of the theorems of Higman and Houghton about the varieties generated by wreat...

We propose a classification of all cases when, for a nilpotent group A of finite exponent and an abelian group B, the variety var(AWrB) generated by their wreath product is equal to the product var(A) var(B) of the varieties var(A) and var(B) generated by A and B. This generalizes some results in the literature concerning the same problem for more...

We consider a specific method for embedding a countable group that is given by generators and relations into some 2-generated group. This embedding enables us to express the images of generators of the countable group in the 2-generated group and explicitly deduce from the defining relations of the latter those of the former which inherit some spec...

This is an extended version of summary of the talk at the International Conference on Group Theory in honor of Victor Mazurov on the occasion of his 80th birthday. The concise version of this report can be found in the talks and communications band at the Conference site. The objective of the current talk is to present our very recent article in wh...

This is an extended version of summary of the talk at the International Conference on Group Theory in honor of Victor Mazurov on the occasion of his 80th birthday. The concise version of this report can be found in the talks and communications band at the Conference site. The objective of the current talk is to present our very recent article in wh...

In the context of Higman embeddings of recursive groups into finitely presented groups, we suggest an approach, termed the 𝐻-machine, which for certain wide classes of groups allows constructive Higman embeddings of recursive groups into finitely presented groups. The approach is based on Higman operations, and it explicitly constructs some specifi...

Доказывается, что для любого конченого симметризованного подмножества S свободной бернсайдовой группы B(m, n) имеет место неравенство |St| ≥ 4 · 2.9t/(400s)3 , где s – наименьший нечетный делитель числа n, удовлетворяющий неравенству s ≥ 1003.

For an arbitrary countable group G = <A|R> given by its generators A and defining relations R we discuss a specific method for embedding of G into a certain 2-generator group T. Our embedding explicitly lists the images of generators from A in the group T, and from the relations R it explicitly deduces defining relations for T inheriting certain sp...

In the context of Higman embeddings of recursive groups into finitely presented groups we suggest an algorithm which uses Higman operations to explicitly constructs the specific recursively enumerable sets of integer sequences arising during the embeddings. This makes the constructive Higman embedding a doable task for certain wide classes of group...

We suggest a modified and briefer version for the proof of Higman's embedding theorem stating that a finitely generated group can be embedded in a finitely presented group if and only if it is recursively presented. In particular, we shorten the main part of original proof establishing characterization of recursive relations in terms of benign subg...

The aim of this communication is to present our recent note in J. Group Theory in which we suggest a method of detection if the given wreath products generate the same variety of groups.

Let A be a nilpotent p-group of finite exponent and B be an abelian p-group of finite exponent for a given prime number p. Then the wreath product A Wr B generates the variety var(A)var(B) if and only if the group B contains a subgroup isomorphic to the direct product Cpv∞ of countably many copies of the cycle Cpv of order pv=exp(B). The obtained t...

We classify certain cases when the wreath products of distinct pairs of groups generate the same variety. This allows us to investigate the subvarieties of some nilpotent-by-abelian product varieties ${\mathfrak U}{\mathfrak V}$ with the help of wreath products of groups. In particular, using wreath products we find such subvarieties in nilpotent-b...

Мы предлагаем классификацию всех случаев, когда для нильпотентной группы конечной экспоненты A и для абелевой группы B многообразие var(AWrB), порожденное их сплетением AWrB, равно произведению var(A) var(B) многообразий var(A) и var(B), порожденных группами A и B. Это обобщает ряд известных в литературе результатов, рассматривающих подобную пробле...

In this talk we announce our current work in which we build subvariety structures in some products of nilpotent and abelian varieties of groups.

This talk 1 consists of two parts, of which the first is a survey of results on product varieties UV generated by wreath products AWr B with A ∈ U and B ∈ V. And the second part displays some very recent, not yet published or submitted applications of wreath products by which we set subvariety structures in some varieties. The material related to t...

These lecture notes reflect the Linear Algebra introductory course at the AUA in spring semester from 2016 to 2022. Short link for downloads: https://bit.ly/LinAlgebra

We present a general criterion under which the equality var(A wr B) = var(A)var(B) holds for finite groups A and B. This generalizes some known results in this direction and continues our previous research [J. Alg., 313, No. 2, 455-458 (2007)] on varieties generated by wreath products of Abelian groups. The classification is based on the techniques...

We present a general criterion under which the equality var(A wr B) = var(A) var(B) holds for finite groups A and B. This continues our previous research on varieties, generated by wreath products of abelian groups, and generalizes some existing results in this direction in literature. The classification is based on criterion of A.L. Shmel'kin for...

The following problem posed in the “Perspectives in Group Theory” section of Advances in Group Theory and Applications, 2 (2016), 125-142: Question: For as wide as possible classes of groups A and B is it possible to classify the cases when the equality var (AWr B) = var (A) var (B) holds or does not hold? Clearly, we assume new cases that were n...

A (D) V Perspectives in Group Theory

The base rank l(V) of a variety of groups V is defined to be the minimal (finite or countable) rank l for which the relatively free group F l (V) generates V. One of the first key facts about base rank was the theorem of R. Burns of 1965: if N c,m = N c ∩ B m is the variety of all nilpotent groups of class at most c and of exponents dividing m, and...

Nilpotent and locally nilpotent subgroups of a group are proved to be very efficient means to study groups. In particular, the Fitting subgroup and the Frattini subgroup help to study finite groups, while the Plotkin-Hirsch radical allows to consider infinite groups. This trend is naturally inherited by varieties of groups: nilpotent varieties are...

Our aim is to present a method of study for the soluble varieties of groups using properties of their nilpotent subvarities. In particular, we study the varieties generated by wreath products AWrB of finite or infinite p-groups A,B.

We consider verbal embedding constructions preserving some residual properties for groups. An arbitrary residually finite countable group H has a V-verbal embedding into a residually finite 2-generator group G for any non-trivial word set V. If in addition H is a residually soluble (residually nilpotent) group, then the group G can be constructed a...

We consider a few modifications of the Big prime modular gcd algorithm for polynomials in Z[x]. Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant and on finding preliminary bounds on degrees of common divisors using auxiliary primes. These modifica...

We present a general criterion, recently established in [4], under which the variety var (AwrB) is equal to the product var(A) var(B) for the given finite groups A and B. This continues our previous research on varieties, generated by standard wreath products of abelian groups, and generalizes some existing results in this direction in literature.

The aim of this talk is to present our recent work, where we suggest a criterion, under which the wreath product $A \Wr B$ of a nilpotent group $A$ of finite exponent and of an abelian group $B$ generates the product of varieties $\var{A}$ and $\var{B}$.

Let $A$ be a nilpotent $p$-group of finite exponent, and $B$ be an abelian $p$-groups of finite exponent. Then the wreath product $A {\rm Wr} B$ generates the variety ${\rm var}(A) {\rm var}(B)$ if and only if the group $B$ contains a subgroup isomorphic to the direct product $C_{p^v}^\infty$ of countably many copies of the cyclic group $C_{p^v}$...

The well-known theorem on embeddings of countable groups into 2-generator groups [1] was a stimulus for wide research on embeddings with additional properties. One of the main directions of research was on embeddings of countable group H into 2-generator group G, where G has a given property P, as soon as H possesses P. Here P may be the property o...

We present a survey of our recent research on varieties, generated by wreath products of groups. In particular, the full classification of all cases, when the (cartesian or direct) wreath product of any abelian groups $A$ and $B$ generates the product variety $\var{A} \var{B}$, is given. The analog of this is given for sets of abelian groups. We al...

Целью сообщения является обзор работ последних лет о многообра- зиях, порождённых сплетениями абелевых групп и множествами абелевых групп [1–4], а также представление некоторых неопубликованных фактов о сплетениях неабелевых групп.

A geometrical construction based on an infinite tree graph is suggested to illustrate the concept of infinite wreath powers of P.~Hall. We use techniques based on infinite wreath powers and on this geometrical constriction to build a 2-generator group which is not soluble, but in which the normal closure of one of the generators is locally soluble.

We obtain full classification of all cases, when for the given sets of abelian groups X and Y their (direct or Cartesian) wreath product X wrY generates the product var(X)var(Y) of varieties, generated by those sets. In particular, when each set consists of one group only: X = {G}, Y = {H}, we get the classification of all cases when var(GwrH) = va...

We present a survey of our recent research on varieties, generated by wreath products of groups. In particular, the full classification of all cases, when the (cartesian or direct) wreath product of any abelian groups $A$ and $B$ generates the product variety $var(A) var(B)$, is given. The analog of this is given for sets of abelian groups. We also...

This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups \( \mathfrak{X} \) Wr \( \mathfrak{Y} \) = { X Wr Y | X ∈ \( \mathfrak{X} \) , Y ∈ \( \mathfrak{Y} \)} a...

abelain subgroups in f.g. metabelian groups Introduction 1. Embeddings of countable groups into 2-generator groups 2. General criterion for all abelian groups 3. Applications of the method

Answering the question of de la Harpe and Bridson in the Kourovka Notebook, Problem 14.10(b), we construct explicit embeddings of the additive group of rational numbers $ \mathbb{Q} $ in a finitely generated group G. In fact, the group G is 2-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubilit...

There is a continuum of 3-generator soluble non-Hopfian groups that generate pairwise distinct varieties of groups. Each countable (soluble) group is subnormally embeddable into a 3-generator (soluble) non-Hopfian group. As an illustration to a problem of Neumann, we find a continuum of nonmetanilpo-tent varieties that contain finitely generated no...

We show that every countable SD-group G can be subnormally embedded into a two-generator SD-group H. This embedding can have additional properties: if the group G is fully ordered then the group H can be chosen to also be fully ordered. For any non-trivial word set V this embedding can be constructed so that the image of G under the embedding lies...

Using methods of embeddings of groups, we prove that there is a continuum of soluble non-metanilpotent varieties which contain 3-generator non-Hopfian groups and non-countably many pairwise non-isomorphic finitely generated groups. This provides us with broad sets of groups for which a problem of Neumann has positive answer.

The aim of this talk is to give an example of application of the method built by Prof. A.Yu. Ol’shanskii in the well known paper "On the problem of finite base of identities in groups" in combination with the methods of verbal embeddings of groups.

In this talk would like to present a rather unexpected application of methods of verbal embeddings we considered recently [HM 00 , M 00 , M 02 , M 03 , M 05 ] to build an illustration of Problem 16 in Hanna Neumann's book [N V G ]. In the literature all known examples of locally soluble varieties contain a finitely generated non-Hopfian group if an...

Answering a question of de la Harpe and Bridson in the Kourovka Notebook, we build the explicit embeddings of the additive group of rational numbers ℚ in a finitely generated group G. The group G in fact is two-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubility or torsion freeness.

We give a new proof to the criterion for the normal verbal embed- dability of groups. The new construction allows us to consider the normal verbal embeddings of soluble groups, of nilpotent groups, and of SN -groups into groups of the same type. We also generalize a theorem of Burnside on embeddings of p-groups into commutator subgroups. MSC 2000:...

Continuing our recent research on embedding properties of generalized soluble and generalized nilpotent groups, we study some embedding properties of SD-groups. We show that every countable SD-group G can be subnormally embedded into a two-generator SD-group H. This embedding can have additional properties: if the group G is fully ordered, then the...

We discuss subnormal ebeddings of countable groups of the given class into two-generator groups of the related class.

Generalizing and strengthening some well-known results of Higman, B. Neumann, Hanna Neumann and Dark on embeddings into two-generator groups, we introduce a construction of subnormal verbal embedding of an arbitrary (soluble, fully ordered or torsion free) ordered countable group into a two- generator ordered group with these properties. Further, w...

The class of those (torsion-free) $SI^*$-groups which are not locally soluble, has the cardinality of the continuum. Moreover, these groups are not only pairwise non-isomorphic, but also they generate pairwise different varieties of groups. Thus, the set of varieties generated by not locally soluble $SI^*$-groups is of the same cardinality as the s...

In this talk we complete our classification of all the cases when the cartesian wreath product $A \Wr B$ (or the direct wreath product $A \Wrr B$) of arbitrary abelian groups $A$ and $B$ generates the product variety $\var A \var B$\,\, by consideration of the case of wreath products of finite groups. Earlier in~\cite{AwrB_paper} we established a c...

Of all kinds of generalized soluble groups SI * -groups (that is, groups with ascending normal series with abelian factors) are the "nearest" to soluble groups. This explains importance of the example of a SI * -group which is not locally soluble, built independently by Hall [1] and by Kovács and Neumann [2]. The mentioned groups constructed in [1]...

Strengthening a theorem of L. G. Kovács and B. H. Neumann on embeddings of countable SN*- and SI*-groups into two-generated SN*- and SI*-groups, we establish embeddability of fully ordered countable SN-, SN*-, SI-, and SI*- groups into appropriate fully ordered two-generated groups of the same type. Moreover, for an arbitrary non-trivial word set V...

We outline results on varieties of groups generated by Cartesian and direct wreath products of abelian groups and pose two problems related to our recent results in that direction. A few related topics are also considered.

Generalizing results of Higman and Houghton on varieties generated by wreath products of finite cycles, we prove that the (direct or cartesian) wreath product of {\it arbitrary} abelian groups $A$ and $B$ generates the product variety $\var A \cdot \var B$ if and only if one of the groups $A$ and $B$ is not of finite exponent, or if $A$ and $B$ are...

For the case of an arbitrary groupH and an arbitrary word setV, we establish a necessary and sufficient condition under which there exists a groupG such thatH is isomorphic to a normal subgroup $$\tilde H$$ ofG such that $$\tilde H$$ lies inV(G). This is a generalization of results of Burnside and Blackburn (concerning the cases of the commutator w...

We study the identities of extensions of groups by abelian groups. The operations of o-products are used to study them.

Some special types of identities of finite groups and of extensions of groups by finite groups are considered.

We use the tool of o-products to study the identities of finite extensions of groups.