Abstract:
For any odd number n⩾1003, the authors construct an infinite 2-generator group each of whose proper subgroups is contained in a cyclic subgroup of order n. This result strengthens analogous results of Ol'shanskii for prime n>1075 and Atabekyan and Ivanov for odd n>1080. The proof is carried out in the original language of Novikov–Adyan theory.
\Bibitem{AdiLys91}
\by S.~I.~Adian, I.~G.~Lysenok
\paper On groups all of whose proper subgroups of which are finite cyclic
\jour Math. USSR-Izv.
\yr 1992
\vol 39
\issue 2
\pages 905--957
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\crossref{https://doi.org/10.1070/IM1992v039n02ABEH002232}
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This publication is cited in the following 30 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, 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, “Normal Automorphisms of Free Groups of Infinitely Based Varieties”, Math. Notes, 108:2 (2020), 149–154
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
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, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71
S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
V. S. Atabekyan, “Splitting Automorphisms of Order $p^k$ of Free Burnside Groups are Inner”, Math. Notes, 95:5 (2014), 586–589
V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189
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
A. L. Gevorgyan, “On automorphisms of periodic products of groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 2, 3–9
V. S. Atabekyan, “Normal automorphisms of free Burnside groups”, Izv. Math., 75:2 (2011), 223–237
V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24
V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2011, no. 3, 62–64
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