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Izv. Akad. Nauk SSSR Ser. Mat., 1991, Volume 55, Issue 5, Pages 933–990 (Mi izv977)  

This article is cited in 21 scientific papers (total in 22 papers)

On groups all of whose proper subgroups of which are finite cyclic

S. I. Adian, I. G. Lysenok


Abstract: For any odd number $n\geqslant 1003$, 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 $n>10^{75}$ and Atabekyan and Ivanov for odd $n>10^{80}$. The proof is carried out in the original language of Novikov–Adyan theory.

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English version:
Mathematics of the USSR-Izvestiya, 1992, 39:2, 905–957

Bibliographic databases:

Document Type: Article
UDC: 510.6
MSC: Primary 20E07, 20F05; Secondary 20E34
Received: 04.01.1991

Citation: S. I. Adian, I. G. Lysenok, “On groups all of whose proper subgroups of which are finite cyclic”, Izv. Akad. Nauk SSSR Ser. Mat., 55:5 (1991), 933–990; Math. USSR-Izv., 39:2 (1992), 905–957

Citation in format AMSBIB
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\by S.~I.~Adian, I.~G.~Lysenok
\paper On groups all of whose proper subgroups of which are finite cyclic
\jour Izv. Akad. Nauk SSSR Ser. Mat.
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\vol 55
\issue 5
\pages 933--990
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992IzMat..39..905A}
\transl
\jour Math. USSR-Izv.
\yr 1992
\vol 39
\issue 2
\pages 905--957
\crossref{https://doi.org/10.1070/IM1992v039n02ABEH002232}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. I. Adian, “The Burnside Problem on Periodic Groups, and Related Problems”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S2–S12  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. L. D. Beklemishev, I. G. Lysenok, A. A. Mal'tsev, S. P. Novikov, M. R. Pentus, A. A. Razborov, A. L. Semenov, V. A. Uspenskii, “Sergei Ivanovich Adian (on his 75th birthday)”, Russian Math. Surveys, 61:3 (2006), 575–588  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. V. S. Atabekyan, “On Periodic Groups of Odd Period $n\ge1003$”, Math. Notes, 82:4 (2007), 443–447  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. V. S. Atabekyan, “Uniform Nonamenability of Subgroups of Free Burnside Groups of Odd Period”, Math. Notes, 85:4 (2009), 496–502  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. V. S. Atabekian, “On subgroups of free Burnside groups of odd exponent $n\ge 1003$”, Izv. Math., 73:5 (2009), 861–892  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. V. S. Atabekyan, “Monomorphisms of Free Burnside Groups”, Math. Notes, 86:4 (2009), 457–462  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. V. S. Atabekyan, “The normalizers of free subgroups in free Burnside groups of odd period $n\ge1003$”, J. Math. Sci., 166:6 (2010), 691–703  mathnet  crossref  mathscinet  elib
    8. S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Atabekyan V.S., “Non-Phi-Admissible Normal Subgroups of Free Burnside Groups”, J. Contemp. Math. Anal.-Armen. Aca., 45:2 (2010), 112–122  crossref  isi
    10. A. S. Pahlevanyan, “Independent pairs in free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 2, 58–62  mathnet
    11. V. S. Atabekyan, “Normal automorphisms of free Burnside groups”, Izv. Math., 75:2 (2011), 223–237  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24  mathnet  crossref  mathscinet  isi
    13. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2011, no. 3, 62–64  mathnet
    14. A. L. Gevorgyan, “On automorphisms of periodic products of groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 2, 3–9  mathnet
    15. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. 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  crossref  isi
    17. V. S. Atabekyan, “The automorphism tower problem for free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 3–7  mathnet
    18. V. S. Atabekyan, “Splitting Automorphisms of Order $p^k$ of Free Burnside Groups are Inner”, Math. Notes, 95:5 (2014), 586–589  mathnet  crossref  crossref  mathscinet  isi  elib
    19. V. S. Atabekyan, “Automorphism groups and endomorphism semigroups of groups $B(m,n)$”, Algebra and Logic, 54:1 (2015), 58–62  mathnet  crossref  crossref  mathscinet  isi
    20. S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71  mathnet  crossref  crossref  isi  elib
    21. S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    22. S. I. Adian, V. S. Atabekyan, “$C^*$-Simplicity of $n$-Periodic Products”, Math. Notes, 99:5 (2016), 631–635  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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