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 Tr. Mat. Inst. Steklova, 2011, Volume 274, Pages 15–31 (Mi tm3325)

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

V. S. Atabekyan

Faculty of Mathematics and Mechanics, Yerevan State University, Yerevan, Armenia

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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 274, 9–24

Bibliographic databases:

UDC: 512.54

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, Tr. 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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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. L. Gevorgyan, “On automorphisms of periodic products of groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 2, 3–9
2. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189
3. Atabekyan V.S., “The Groups of Automorphisms Are Complete for Free Burnside Groups of Odd Exponents N >= 1003”, Int. J. Algebr. Comput., 23:6 (2013), 1485–1496
4. Zusmanovich P., “On the Utility of Robinson-Amitsur Ultrafilters”, J. Algebra, 388 (2013), 268–286
5. V. S. Atabekyan, “The automorphism tower problem for free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 3–7
6. S. I. Adian, V. S. Atabekyan, “The Hopfian Property of $n$-Periodic Products of Groups”, Math. Notes, 95:4 (2014), 443–449
7. V. S. Atabekyan, “Automorphism groups and endomorphism semigroups of groups $B(m,n)$”, Algebra and Logic, 54:1 (2015), 58–62
8. S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
9. Atabekyan V.S. Gevorgyan A.L. Stepanyan Sh.A., “The Unique Trace Property of N-Periodic Product of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:4 (2017), 161–165
10. Adian S.I., Atabekyan V.S., “Periodic Products of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117
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