International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 7, Pages 373-375
doi:10.1155/S0161171204304047
Powers of a product of commutators as products of squares
Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran
Received 1 April 2003
Copyright © 2004 Alireza Abdollahi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We prove that for any odd integer N and any integer n>0, the Nth power of a product of n commutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of 2n+1 elements and, for all such odd N and integers n, there are commutators for which the number 2n+1 of squares is the minimum number such that the Nth power of its product can be written as a product of squares. This generalizes a recent result of Akhavan-Malayeri.