Abstract:
We prove that for any odd number n⩾1003, every non-cyclic subgroup of the 2-generator free Burnside group of exponent n contains a subgroup isomorphic to the free Burnside group of exponent n and infinite rank. Various families of relatively free n-periodic subgroups are constructed in free periodic groups of odd exponent n⩾665. For the same groups, we describe a monomorphism τ such that a word A is an elementary period of rank α if and only if its image τ(A) is an elementary period of rank α+1.
Keywords:
free Burnside group, variety of periodic groups, group with cyclic subgroups, periodic word, reduced word.
\Bibitem{Ata09}
\by V.~S.~Atabekian
\paper On subgroups of free Burnside groups of odd exponent $n\geqslant 1003$
\jour Izv. Math.
\yr 2009
\vol 73
\issue 5
\pages 861--892
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\crossref{https://doi.org/10.1070/IM2009v073n05ABEH002466}
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https://doi.org/10.1070/IM2009v073n05ABEH002466
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This publication is cited in the following 21 articles:
Atabekyan V.S. Gevorkyan G.G., “Central Extensions of N-Torsion Groups”, J. Contemp. Math. Anal.-Armen. Aca., 57:1 (2022), 26–34
V. S. Atabekyan, H. T. Aslanyan, S. T. Aslanyan, “Powers of subsets in free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 56:2 (2022), 43–48
V. S. Atabekyan, V. G. Mikaelyan, “On the Product of Subsets in Periodic Groups”, J. Contemp. Mathemat. Anal., 57:6 (2022), 395
V. S. Atabekyan, V. G. Mikaelyan, “O proizvedenii podmnozhestv v pereodicheskikh gruppakh”, Proceedings of NAS RA. Mathematics, 2022, 12
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
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
Button J.O., “Groups and Embeddings in Sl(2, C)”, Commun. Algebr., 44:1 (2016), 265–278
S. I. Adian, V. S. Atabekyan, “C∗-Simplicity of n-Periodic Products”, Math. Notes, 99:5 (2016), 631–635
V. S. Atabekyan, “Automorphism groups and endomorphism semigroups of groups B(m,n)”, Algebra and Logic, 54:1 (2015), 58–62
S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of n-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
Atabekyan V.S., “the Automorphisms of Endomorphism Semigroups of Free Burnside Groups”, Int. J. Algebr. Comput., 25:4 (2015), 669–674
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
V. S. Atabekyan, “The automorphism tower problem for free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 3–7
V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24
Atabekyan V.S., “On CEP-subgroups of n-periodic products”, J. Contemp. Math. Anal., 46:5 (2011), 237–242
V. S. Atabekyan, “Nonunitarizable Periodic Groups”, Math. Notes, 87:6 (2010), 908–911
S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855
A. S. Pahlevanyan, “Independent pairs in free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 2, 58–62
H. R. Rostami, “Non-unitarizable groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 3, 40–43
V. S. Atabekyan, “Monomorphisms of Free Burnside Groups”, Math. Notes, 86:4 (2009), 457–462
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