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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 5, Pages 3–14 (Mi izv8631)  

This article is cited in 10 scientific papers (total in 10 papers)

On free groups in the infinitely based varieties of S. I. Adian

S. I. Adiana, V. S. Atabekyanb

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

Abstract: We study the free groups in varieties defined by an arbitrary set of identities in a well-known infinite independent system of identities in two variables constructed by S. I. Adian to solve the finite basis problem in group theory. We prove that the centralizer of any non-identity element in a relatively free group in any of the varieties under consideration is cyclic, and for every $m>1$ the set of all non-isomorphic free groups of rank $m$ in these varieties is of the cardinality of the continuum. All these groups have trivial centre, all their abelian subgroups are cyclic, and all their non-trivial normal subgroups are infinite. For any free group $\Gamma$ in any of these varieties, we also obtain a description of the automorphisms of the semigroup $\operatorname{End}(\Gamma)$, answering a question posed by Plotkin in 2000. In particular, we prove that the automorphism group of any such $\operatorname{End}(\Gamma)$ is canonically embedded in the group $\operatorname{Aut}(\operatorname{Aut}(\Gamma))$.

Keywords: infinitely based variety, self-centralizing subgroup, semigroup of endomorphisms, automorphism group, free Burnside group.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
Ministry of Education and Science of the Russian Federation
State Committee on Science of the Ministry of Education and Science of the Republic of Armenia 15T-1A258
§§ 1 and 3 of the paper were written by the first author and §§ 2 and 4 by the second. The first author was supported by a grant from the Russian Science Foundation (project no. 14-50-00005) at the Steklov Mathematical Institute of the Russian Academy of Sciences. The second author was supported at the Russian-Armenian University by funds allocated within the framework of the MES of RF subsidy for the financial support of the research activities of RAU and by the financial support of the State Science Committee of the Ministry of Education and Science of the Armenian Republic within the framework of the scientific project no. 15T-1A258.


DOI: https://doi.org/10.4213/im8631

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English version:
Izvestiya: Mathematics, 2017, 81:5, 889–900

Bibliographic databases:

UDC: 512.54
MSC: Primary 20F50; Secondary 20F05, 20E22, 20E26
Received: 21.11.2016

Citation: S. I. Adian, V. S. Atabekyan, “On free groups in the infinitely based varieties of S. I. Adian”, Izv. RAN. Ser. Mat., 81:5 (2017), 3–14; Izv. Math., 81:5 (2017), 889–900

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. S. I. Adian, V. S. Atabekyan, “Periodic products of groups”, J. Contemp. Math. Anal., Armen. Acad. Sci., 52:3 (2017), 111–117  mathnet  crossref  isi  elib  scopus
    2. V. S. Atabekyan, H. T. Aslanyan, “The automorphisms of endomorphism semigroups of relatively free groups”, Internat. J. Algebra Comput., 28:2 (2018), 207–215  crossref  mathscinet  zmath  isi  scopus
    3. H. T. Aslanyan, “On automorphisms and endomorphisms of $CC$ groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 52:1 (2018), 60–63  mathnet
    4. S. I. Adian, V. S. Atabekyan, “Central extensions of free periodic groups”, Sb. Math., 209:12 (2018), 1677–1689  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. H. T. Aslanyan, A. L. Gevorgyan, H. A. Grigoryan, “Finite subgroups of the free groups of the infinitely based varieties of s. I. Adian”, Armen. J. Math., 11:6 (2019), 1–6  mathscinet  isi
    6. S. I. Adian, V. S. Atabekyan, “N-torsion groups”, J. Contemp. Math. Anal.-Armen. Aca., 54:6 (2019), 319–327  crossref  mathscinet  zmath  isi  scopus
    7. S. I. Adian, V. S. Atabekyan, “Normal Automorphisms of Free Groups of Infinitely Based Varieties”, Math. Notes, 108:2 (2020), 149–154  mathnet  crossref  crossref  mathscinet  isi  elib
    8. V. S. Atabekyan, “The set of $2$-genereted $C^*$-simple relatively free groups has the cardinality of the continuum”, Uch. zapiski EGU, ser. Fizika i Matematika, 54:2 (2020), 81–86  mathnet  crossref
    9. A. L. Gevorgyan, G. G. Gevorgyan, “Finite subgroups of the relatively free n-torsion groups”, J. Contemp. Math. Anal.-Armen. Aca., 55:1 (2020), 1–4  crossref  mathscinet  zmath  isi
    10. V. S. Atabekyan, L. D. Beklemishev, V. S. Guba, I. G. Lysenok, A. A. Razborov, A. L. Semenov, “Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian”, Russian Math. Surveys, 76:1 (2021), 1–27  mathnet  crossref  crossref  mathscinet  isi  elib
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