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Journal of Mathematical Sciences

, Volume 166, Issue 6, pp 691–703 | Cite as

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

  • V. S. AtabekyanEmail author
Article

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 ≥ 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.

Keywords

Normal Subgroup Quotient Group Cyclic Subgroup Prime Period Elementary Period 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Erevan State UniversityErevanArmenia

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