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This article is cited in 12 scientific papers (total in 12 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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Journal of Mathematical Sciences (New York), 2010, 166:6, 691–703
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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
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\jour J. Math. Sci.
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\vol 166
\issue 6
\pages 691--703
\crossref{https://doi.org/10.1007/s10958-010-9885-1}
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This publication is cited in the following articles:
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V. S. Atabekyan, “Nonunitarizable Periodic Groups”, Math. Notes, 87:6 (2010), 908–911
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S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855
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A. S. Pahlevanyan, “Independent pairs in free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 2, 58–62
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H. R. Rostami, “Non-unitarizable groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 3, 40–43
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V. S. Atabekyan, “Normal automorphisms of free Burnside groups”, Izv. Math., 75:2 (2011), 223–237
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Pahlevanyan A.S., Rostami H.R., “On automorphisms and embeddings of free periodic groups”, J. Contemp. Math. Anal., 46:2 (2011), 106–112
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V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24
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A. L. Gevorgyan, “On automorphisms of periodic products of groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 2, 3–9
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V. S. Atabekyan, “The automorphism tower problem for free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 3–7
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V. S. Atabekyan, “Automorphism groups and endomorphism semigroups of groups $B(m,n)$”, Algebra and Logic, 54:1 (2015), 58–62
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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
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S. I. Adian, V. S. Atabekyan, “Normal Automorphisms of Free Groups of Infinitely Based Varieties”, Math. Notes, 108:2 (2020), 149–154
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