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Fundam. Prikl. Mat., 2009, Volume 15, Issue 1, Pages 3–21 (Mi fpm1204)  

This article is cited in 11 scientific papers (total in 11 papers)

The normalizers of free subgroups in free Burnside groups of odd period $n\ge1003$

V. S. Atabekyan

Yerevan State University, Armenia

Abstract: Let $B(m,n)$ be a free periodic group of arbitrary rank $m$ with period $n$. In this paper, we prove that for all odd numbers $n\ge1003$ the normalizer of any nontrivial subgroup $N$ of the group $B(m,n)$ coincides with $N$ if the subgroup $N$ is free in the variety of all $n$-periodic groups. From this, there follows a positive answer for all prime numbers $n>997$ to the following problem set by S. I. Adian in the Kourovka Notebook: is it true that none of the proper normal subgroups of the group $B(m,n)$ of prime period $n>665$ is a free periodic group? The obtained result also strengthens a similar result of A. Yu. Ol'shanskii by reducing the boundary of exponent $n$ from $n>10^{78}$ to $n\ge1003$. For primes $665<n\leq997$, the mentioned question is still open.

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English version:
Journal of Mathematical Sciences (New York), 2010, 166:6, 691–703

Bibliographic databases:

UDC: 512.54+512.543

Citation: V. S. Atabekyan, “The normalizers of free subgroups in free Burnside groups of odd period $n\ge1003$”, Fundam. Prikl. Mat., 15:1 (2009), 3–21; J. Math. Sci., 166:6 (2010), 691–703

Citation in format AMSBIB
\Bibitem{Ata09}
\by V.~S.~Atabekyan
\paper The normalizers of free subgroups in free Burnside groups of odd period $n\ge1003$
\jour Fundam. Prikl. Mat.
\yr 2009
\vol 15
\issue 1
\pages 3--21
\mathnet{http://mi.mathnet.ru/fpm1204}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2744943}
\elib{http://elibrary.ru/item.asp?id=15461222}
\transl
\jour J. Math. Sci.
\yr 2010
\vol 166
\issue 6
\pages 691--703
\crossref{https://doi.org/10.1007/s10958-010-9885-1}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77952290427}


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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. V. S. Atabekyan, “Nonunitarizable Periodic Groups”, Math. Notes, 87:6 (2010), 908–911  mathnet  crossref  crossref  mathscinet  isi
    2. 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
    3. A. S. Pahlevanyan, “Independent pairs in free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 2, 58–62  mathnet
    4. H. R. Rostami, “Non-unitarizable groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 3, 40–43  mathnet
    5. 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
    6. Pahlevanyan A.S., Rostami H.R., “On automorphisms and embeddings of free periodic groups”, J. Contemp. Math. Anal., 46:2 (2011), 106–112  crossref  mathscinet  zmath  isi  elib
    7. 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
    8. A. L. Gevorgyan, “On automorphisms of periodic products of groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 2, 3–9  mathnet
    9. V. S. Atabekyan, “The automorphism tower problem for free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 3–7  mathnet
    10. 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
    11. A. E. Grigoryan, “Inner automorphisms of non-commutative analogues of the additive group of rational numbers”, Uch. zapiski EGU, ser. Fizika i Matematika, 2015, no. 1, 12–14  mathnet
  • Фундаментальная и прикладная математика
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