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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Volume 274, Pages 15–31 (Mi tm3325)  

This article is cited in 11 scientific papers (total in 11 papers)

On normal subgroups in the periodic products of S. I. Adian

V. S. Atabekyan

Faculty of Mathematics and Mechanics, Yerevan State University, Yerevan, Armenia
References:
Abstract: A subgroup $H$ of a given group $G$ is called a hereditarily factorizable subgroup (HF subgroup) if each congruence on $H$ can be extended to some congruence on the entire group $G$. An arbitrary group $G_1$ is an HF subgroup of the direct product $G_1\times G_2$, as well as of the free product $G_1*G_2$ of groups $G_1$ and $G_2$. In this paper a necessary and sufficient condition is obtained for a factor $G_i$ of Adian's $n$-periodic product $\prod_{i\in I}^nG_i$ of an arbitrary family of groups $\{G_i\}_{i\in I}$ to be an HF subgroup. We also prove that for each odd $n\geq1003$ any noncyclic subgroup of the free Burnside group $B(m,n)$ contains an HF subgroup isomorphic to the group $B(\infty,n)$ of infinite rank. This strengthens the recent results of A. Yu. Ol'shanskii and M. Sapir, D. Sonkin, and S. Ivanov on HF subgroups of free Burnside groups. This result implies, in particular, that each (noncyclic) subgroup of the group $B(m,n)$ is $SQ$-universal in the class of all groups of period $n$. Moreover, it turns out that any countable group of period $n$ is embedded in some $2$-generated group of period $n$, which strengthens the previously obtained result of V. Obraztsov. At the end of the paper we prove that the group $B(m,n)$ is distinguished as a direct factor in any $n$-periodic group in which it is contained as a normal subgroup.
Received in August 2010
English version:
Proceedings of the Steklov Institute of Mathematics, 2011, Volume 274, Pages 9–24
DOI: https://doi.org/10.1134/S0081543811060034
Bibliographic databases:
Document Type: Article
UDC: 512.54
Language: Russian
Citation: V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Algorithmic aspects of algebra and logic, Collected papers. Dedicated to Academician Sergei Ivanovich Adian on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 274, MAIK Nauka/Interperiodica, Moscow, 2011, 15–31; Proc. Steklov Inst. Math., 274 (2011), 9–24
Citation in format AMSBIB
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\by V.~S.~Atabekyan
\paper On normal subgroups in the periodic products of S.\,I.~Adian
\inbook Algorithmic aspects of algebra and logic
\bookinfo Collected papers. Dedicated to Academician Sergei Ivanovich Adian on the occasion of his 80th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2011
\vol 274
\pages 15--31
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
     
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