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 $665\; <\; n$£ 997, the mentioned
question is still open.

Location: http://mech.math.msu.su/~fpm/eng/k09/k091/k09101h.htm

Last modified: December 2, 2009