Abstract:
We prove that n-periodic products (introduced by the first author in 1976)
are uniquely characterized by certain quite specific properties.
Using these properties, we prove that if a non-cyclic subgroup H
of the n-periodic product of a given family of groups is not conjugate
to any subgroup of the product's components, then H contains a subgroup
isomorphic to the free Burnside group B(2,n). This means that H
contains the free periodic groups B(m,n) of any rank m>2, which lie
in B(2,n) ([1], Russian p. 26). Moreover, if H is finitely
generated, then it is uniformly non-amenable. We also describe
all finite subgroups of n-periodic products.
This work was carried out with the financial support of the Russian Foundation for Basic Research and the RA MES State Committee of Science in the framework of the joint scientific programme
(projects 15-51-05012-Arm\_a and 15RF-054 respectively).
Citation:
S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of n-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
\Bibitem{AdiAta15}
\by S.~I.~Adian, Varuzhan~Atabekyan
\paper Characteristic properties and uniform non-amenability of $n$-periodic products of groups
\jour Izv. Math.
\yr 2015
\vol 79
\issue 6
\pages 1097--1110
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This publication is cited in the following 7 articles:
V. S. Atabekyan, A. A. Bayramyan, “Probabilistic Identities in n-Torsion Groups”, J. Contemp. Mathemat. Anal., 59:6 (2024), 455
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
S. I. Adian, V. S. Atabekyan, “N-torsion groups”, J. Contemp. Math. Anal.-Armen. Aca., 54:6 (2019), 319–327
V. S. Atabekyan, A. L. Gevorgyan, Sh. A. Stepanyan, “The unique trace property of n-periodic products of groups”, J. Contemp. Math. Anal., 52:4 (2017), 161–165
S. I. Adian, V. S. Atabekyan, “Periodic products of groups”, J. Contemp. Math. Anal., Armen. Acad. Sci., 52:3 (2017), 111–117
S. I. Adian, V. S. Atabekyan, “C∗-Simplicity of n-Periodic Products”, Math. Notes, 99:5 (2016), 631–635
S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71
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