arXiv subjects

Vahagn H. Mikaelian

Publications and source records attributed to Vahagn H. Mikaelian.

12 recordsLinked to original sources

arXiv2025

Auxiliary free constructions for explicit embeddings of recursive groups

An auxiliary free construction $*_{i=1}^{r}(K_i, L_i, t_i)_M$ based on HNN-extensions and on generalized free product of groups with amalgamated subgroups is suggested, and some of its basic properties are displayed. The proposed construction is a generalization of a series of structures used by Higman for embeddings of recursive groups into finitely presented groups, and also for certain structures we recently applied in the research on embeddings of recursive groups. Usage of this technical tool substantially simplifies some embedding methods for recursive groups. A few technical results on specific subgroups in the suggested $*$-constructions, in the HNN-extensions of groups, and in free products of groups with amalgamated subgroup are proved. The obtained properties are applied to build infinitely generated benign subgroups inside free groups of small rank.

math.GR↗

arXiv2023

Higman operations on integer sequences, and embeddings of recursive groups

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 which we for various types of recursive groups discuss possibility of their explicit embeddings into finitely presented groups.

math.GR↗

arXiv2018

On varieties of groups generated by wreath products of abelian groups

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 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 of finite exponents $m$ and $n$ respectively and for all primes $p$ dividing both $m$ and $n$, the factors $B[p^k]/B[p^{k-1}]$ are infinite, where $B[s]=\langle b\in B|\,b^{s}=1 \rangle$ and where $p^k$ is the highest power of $p$ dividing $n$.

math.GR↗

arXiv2017

On $K_p$-series and varieties generated by wreath products of $p$-groups

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 at least countably many copies of the cyclic group $C_{p^v}$ of order $p^v = \exp{(B)}$. The obtained theorem continues our previous study of cases when ${\rm var}(A {\rm Wr} B ) = {\rm var}(A){\rm var}(B)$ holds for some other classes of groups $A$ and $B$ (abelian groups, finite groups, etc.).

math.GR↗

arXiv2016

A Classification Theorem for Varieties Generated by Wreath Products of Groups

We suggest a criterion under which for a nilpotent group of finite exponent $A$ and for an abelian group $B$ the variety $var(A \,Wr\, B)$ generated by their wreath product $A \,Wr\, B$ is equal to the product of varieties $var(A)$ and $var(B)$ generated by $A$ and $B$. Namely the equality holds if and only if either the group $B$ is not of some non-zero exponent; or if $B$ is of a non-zero exponent $n$, and $B$ contains a subgroup isomorphic to $C_{d}^c \times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, $d$ is the largest divisor of $n$ coprime with $m$, $C_{d}^c$ is the direct power of $c$ copies of the cycle $C_d$ of order $d$, $C_{n/d}^\infty$ is the direct power of countably many copies of the cycle $C_{n/d}$ of order $n/d$. This criterion continues our previous work on cases when the similar criterions were given for wreath products of abelian groups or of finite groups. Also, this is a generalization of known results in literature, which solve the same problem for much more restricted cases. Some applications of the criterion are considered at the end of paper.

math.GR↗

arXiv2015

The Criterion of Shmel'kin and Varieties Generated by Wreath Products of Finite Groups

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 product varieties of groups and on technics developed by R. Burns et al. on critical groups in nilpotent-by-abelian varieties.

math.GR↗

arXiv2015

Varieties generated by wreath products of abelian and nilpotent groups

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 present new, unpublished research on cases when the similar question is considered for non-abelian, finite groups.

math.GR↗

arXiv2014

On degrees of modular common divisors and the Big prime gcd algorithm

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 modifications are used to suggest improved algorithms for $\gcd$ calculation and for coprime polynomials detection. To illustrate the ideas we apply the constructed algorithms on certain polynomials, in particular, on polynomials from Knuth's example of intermediate expression swell.

math.NT↗

arXiv2013

On abelian subgroups of finitely generated metabelian groups

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 can occur as subgroups in finitely-generated metabelian groups. Hall groups allow to give a negative answer to the Baumslag's conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelian groups.

math.GR↗

arXiv2012

On Order-Preserving and Verbal Embeddings of the Group $\mathbb{Q}$

We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the sense that it cannot be chosen to be neither solvable of length 2, nor a nilpotent group. For any non-trivial word set $V \subseteq F_\infty$ there is an order-preserving verbal embedding of $\mathbb{Q}$ into a 2-generator group $G$. The embeddings constructed are subnormal.

math.GR↗

arXiv2012

Infinitely many not locally soluble $SI^*$-groups

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 set of all varieties of groups. It is possible to localize a variety of groups which contains all groups and varieties constructed. The examples constructed here continue the well known example of a not locally soluble $SI^*$-group built by Hall and by Kovács and Neumann.

math.GR↗

arXiv2000

Subnormal embedding theorems for groups

In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal embeddings of finite groups into finite groups with additional properties we bring some illustration to a result of H. Heineken and J. Lennox.

math.GR↗