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Mat. Zametki, 2007, Volume 82, Issue 4, Pages 495–500 (Mi mz3810)  

This article is cited in 13 scientific papers (total in 13 papers)

On Periodic Groups of Odd Period $n\ge1003$

V. S. Atabekyan

Yerevan State University

Abstract: In the paper, using the Adyan–Lysenok theorem claiming that, for any odd number $n\ge1003$, there is an infinite group each of whose proper subgroups is contained in a cyclic subgroup of order $n$, it is proved that the set of groups with this property has the cardinality of the continuum (for a given $n$). Further, it is proved that, for $m\ge k\ge2$ and for any odd $n\ge1003$, the $m$-generated free $n$-periodic group is residually both a group of the above type and a $k$-generated free $n$-periodic group, and it does not satisfy the ascending and descending chain conditions for normal subgroups either.

Keywords: periodic group, simple group, group of bounded period, variety of groups of a given exponent, Adyan–Lysenok theorem

DOI: https://doi.org/10.4213/mzm3810

Full text: PDF file (433 kB)
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English version:
Mathematical Notes, 2007, 82:4, 443–447

Bibliographic databases:

UDC: 512.54
Received: 03.02.2006

Citation: V. S. Atabekyan, “On Periodic Groups of Odd Period $n\ge1003$”, Mat. Zametki, 82:4 (2007), 495–500; Math. Notes, 82:4 (2007), 443–447

Citation in format AMSBIB
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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. Atabekyan V. S., “Adian-Lisenok groups and (U) condition”, J. Contemp. Math. Anal., 43:5 (2008), 265–273  crossref  mathscinet  zmath  isi
    2. 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
    3. V. S. Atabekyan, “Nonunitarizable Periodic Groups”, Math. Notes, 87:6 (2010), 908–911  mathnet  crossref  crossref  mathscinet  isi
    4. 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
    5. Atabekyan V.S., “Non-$\phi$-admissible normal subgroups of free Burnside groups”, J. Contemp. Math. Anal., 45:2 (2010), 112–122  crossref  mathscinet  zmath  isi  elib  scopus
    6. H. R. Rostami, “Non-unitarizable groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 3, 40–43  mathnet
    7. 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
    8. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2011, no. 3, 62–64  mathnet
    9. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. 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  mathscinet  zmath  isi  elib  scopus
    11. 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
    12. 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
    13. 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
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