Abstract
A group is called an n-torsion group if it has a system of defining relations of the form rn = 1 for some elements r, and for any of its finite order element a the defining relation an = 1 holds. It is assumed that the group can contain elements of infinite order. In this paper, we show that for every odd n ≥ 665 for each n-torsion group can be constructed a theory similar to that of constructed in S. I. Adian’s well-known monograph on the free Burnside groups. This allows us to explore the n-torsion groups by methods developed in that work. We prove that every n-torsion group can be specified by some independent system of defining relations; the center of any non-cyclic n-torsion group is trivial; the n-periodic product of an arbitrary family of n-torsion groups is an n-torsion group; in any recursively presented n-torsion group the word and conjugacy problems are solvable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. I. Adian, The Burnside problem and identities in the groups (Nauka, Moscow, 1975).
A. Karrass, W. Magnus, D. Solitar, “Elements of finite order in groups with a single defining relation”, Comm. Pure Appl. Math., 13, 57–66, 1960.
S. I. Adian, “On the word problem for groups defined by periodic relations”, Burnside groups (Proc. Workshop, Univ. Bielefeld, 1977), Lecture Notes in Math., 806, Springer, Berlin, 41–46, 1980.
S. I. Adian, “Groups with Periodic Defining Relations”, Math. Notes, 83(3), 293–300, 2008.
S. I. Adian, “New Estimates of Odd Exponents of Infinite Burnside Groups”, Proc. Steklov Inst. Math., 289, 33–71, 2015.
S. I. Adyan, “Groups with periodic commutators”, Dokl. Math., 62(2), 174–176, 2000.
A. Yu. Olshanskii, The Geometry of Defning Relations in Groups (Kluwer, Amsterdam, 1991).
S. V. Ivanov, “The free Burnside groups of sufficiently large exponents”, Int. J. of Algebra and Computation, 4, 1–307, 1994.
I. G. Lysenok, “Infinite Burnside groups of even exponent”, Izv. Math., 60(3), 453–654, 1996.
S. V. Ivanov, A. Yu. Olshanskii, “On finite and locally finite subgroups of free Burnside groups of large even exponents, J. Algebra, 195(1), 241–284, 1997.
A. Yu. Ol’shanskii, “Self-normalization of free subgroups in the free Burnside groups, Groups, rings, Lie and Hopf algebras” Math. Appl., 555, 179–187, 2003.
A. Yu. Ol’shanskii, D. Osin, “C*-simple groups without free subgroups”, Groups, Geometry and Dynamics, 8, 933–983, 2014.
S. I. Adian, “Periodic products of groups”, Proc. Steklov Inst. Math., 142, 1–19, 1979.
S. I. Adian, V. S. Atabekyan, “The Hopfian Property of n-Periodic Products of Groups”, Math. Notes, 95(4), 443–449, 2014.
V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274, 9–24, 2011.
S. I. Adian, V. S. Atabekyan, “Characteristic properties and uniform non-amenability of n-periodic products of groups”, Izv. RAN. Ser. Mat., 79(6), 3–17, 2015.
V. S. Atabekyan, A. L. Gevorgyan, Sh. A. Stepanyan, “The unique trace property of n-periodic product of groups”, Journal of contemporary mathematical analysis, 52(4), 161–165, 2017.
S. I. Adian, V. S. Atabekyan, “Periodic product of groups”, Journal of contemporary mathematical analysis, 52(3), 111–117, 2017.
S. I. Adian, V. S. Atabekyan, “On free groups in the infinitely based varieties of S. I. Adian”, Izv. RAN. Ser. Mat., 81(5), 3–14, 2017.
C. Delizia, H. Dietrich, P. Moravec, C. Nicotera, “Groups in which every non-abelian subgroup is self-centralizing”, Journal of Algebra, 462, 23–36, 2016.
V. S. Atabekyan, “The automorphisms of endomorphism semigroups of relatively free groups”, Internat. J. Algebra Comput., 28(1), 207–215, 2018.
S. I. Adian, I. G. Lysenok, “On groups all of whose proper subgroups of which are finite cyclic”, Math. USSR-Izv., 39(2), 905–957, 1992.
V. L. Shirvanjan, “Embedding the group B(∞, n) in the group B(2, n)”, Math. USSR-Izv., 10(1), 181–199, 1976.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2019, No. 6, pp. 3–18.
The research was supported by the Russian Foundation for Basic Research (Project 18-01-00822 A) in Steklov Mathematical Institute.
The research was supported by the RA MES State Committee of Science, the project 18T-1A306.
Sections 1, 4 are written by S. Adian; sections 2, 3 are written by V. Atabekyan.
About this article
Cite this article
Adian, S.I., Atabekyan, V.S. n-torsion Groups. J. Contemp. Mathemat. Anal. 54, 319–327 (2019). https://doi.org/10.3103/S1068362319060013
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1068362319060013