FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2009, VOLUME 15, NUMBER 1, PAGES 3-21

The normalizers of free subgroups in free Burnside groups of odd period $n$³ 1003

V. S. Atabekyan

Abstract

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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 $n$³ 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 $n$³ 1003. For primes £ 997, the mentioned question is still open.

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Last modified: December 2, 2009