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The normalizers of free subgroups in free burnside groups of odd period n ≥ 1003

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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 ≥ 1003 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 > 1078 to n ≥ 1003. For primes 665 < n ≤ 997, the mentioned question is still open.

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Authors and Affiliations

  1. Erevan State University, Erevan, Armenia

    V. S. Atabekyan

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  1. V. S. Atabekyan
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Correspondence to V. S. Atabekyan.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 15, No. 1, pp. 3–21, 2009.

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Atabekyan, V.S. The normalizers of free subgroups in free burnside groups of odd period n ≥ 1003. J Math Sci 166, 691–703 (2010). https://doi.org/10.1007/s10958-010-9885-1

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