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.
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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.
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Adian, S.I., Atabekyan, V.S. n-torsion Groups. J. Contemp. Mathemat. Anal. 54, 319–327 (2019). https://doi.org/10.3103/S1068362319060013
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DOI: https://doi.org/10.3103/S1068362319060013