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Mat. Sb., 2018, Volume 209, Number 12, Pages 3–16 (Mi msb9007)  

This article is cited in 1 scientific paper (total in 1 paper)

Central extensions of free periodic groups

S. I. Adiana, V. S. Atabekyanb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Yerevan State University, Armenia

Abstract: 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 $n\ge665$. The proof is based on a certain modification of the method that was used by Adian in his monograph in 1975 for a positive solution of Kontorovich's famous problem from the Kourovka Notebook on the existence of a finitely generated noncommutative analogue of the additive group of rational numbers with any number $m>1$ of generators (in contrast to the abelian case). 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 $n\ge665$ is an odd number, using the infinite cyclic group as its centre. The paper also discusses other applications of the proposed generalization of Adian's technique. In particular, 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 $n\ge665$ are described.
Bibliography: 14 titles.

Keywords: free Burnside group, central extension, additive group of rational numbers, Schur multiplier.

Funding Agency Grant Number
Russian Science Foundation 16-11-10252
State Committee on Science of the Ministry of Education and Science of the Republic of Armenia 18RF-109
Russian Foundation for Basic Research 18-51-05006 Арм_а
The work of S. I. Adian was supported by the Russian Science Foundation under grant no. 16-11-10252 and performed in the Steklov Mathematical Institute of Russian Academy of Sciences. The work of V. S. Atabekyan was supported by the Republic of Armenia MES State Committee of Science and the Russian Foundation for Basic Research (RF) within the framework of the joint research project SCS 18RF-109 and RFBR 18-51-05006 Арм_а at Yerevan State University. Sections 1 and 3 of the paper were written by S. I. Adian and sections 2, 4 and 5 by V. S. Atabekyan.


DOI: https://doi.org/10.4213/sm9007

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English version:
Sbornik: Mathematics, 2018, 209:12, 1677–1689

Bibliographic databases:

Document Type: Article
UDC: 512.54+512.543
MSC: 20E22, 20F50
Received: 28.08.2017 and 23.07.2018

Citation: S. I. Adian, V. S. Atabekyan, “Central extensions of free periodic groups”, Mat. Sb., 209:12 (2018), 3–16; Sb. Math., 209:12 (2018), 1677–1689

Citation in format AMSBIB
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    1. Flores R., Scherer J., “Cellular Covers of Local Groups”, Mediterr. J. Math., 15:6 (2018), 229  crossref  mathscinet  zmath  isi
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