arXiv subjects

V. S. Atabekyan

Publications and source records attributed to V. S. Atabekyan.

3 recordsLinked to original sources

arXiv2026

Mazurov's question on fixed points of automorphisms of free Burnside groups

In the Kourovka Notebook, V.\,D.~Mazurov asked whether an automorphism $α$ of prime order $m$ of the infinite free Burnside group $G=B(m,q)$ of prime period $q$ that cyclically permutes the free generators of $G$ must have a nontrivial fixed point. In this paper, we show that for $m\ge3$ the answer is negative for all sufficiently large prime periods $q$. Precisely, we prove that for any odd $m\ge3$, there exists $q_m>0$ such that for every prime period $q>q_m$, the automorphism of $B(m,q)$ of order $m$ that cyclically permutes its free generators is fixed-point-free.

math.GR↗

arXiv2025

A continuum of non-isomorphic 3-generator groups with probabilistic law $x^n=1$

In this paper we construct a continuum family of non-isomorphic 3-generator groups in which the identity $x^n = 1$ holds with probability 1, while failing to hold universally in each group. This resolves a recent question about the relationship between probabilistic and universal satisfaction of group identities. Our construction uses $n$-periodic products of cyclic groups of order $n$ and two-generator relatively free groups satisfying identities of the form $[x^{pn}, y^{pn}]^n = 1$. We prove that in each of these products, the probability of satisfying $x^n = 1$ is equal to 1, despite the fact that the identity does not hold throughout any of these groups.

math.GR↗

arXiv2015

The Hopfian Property of $n$-Periodic Products of Groups

Let $H$ be a subgroup of a group $G$. A normal subgroup $N_H$ of $H$ is said to be inheritably normal if there is a normal subgroup $N_G$ of $G$ such that $N_H=N_G\cap H$. It is proved in the paper that a subgroup $N_{G_i}$ of a factor $G_i$ of the $n$-periodic product $\prod_{i\in I}^nG_i$ 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 $n\ge 665$ every nontrivial normal subgroup in a given $n$-periodic product $G=\prod_{i\in I}^nG_i$ contains the subgroup $G^n$. It follows that almost all $n$-periodic products $G=G_1\overset{n}{\ast}G_2$ 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.

math.GR↗