Copyright © 2009 Mehri Akhavan-Malayeri. 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.
In a free group no nontrivial commutator is a square. And in the
free group freely generated by the commutator is never the product of two squares in , although it is always the product of three squares. Let be a free nilpotent group of rank 2 and class
3 freely generated by . We prove that in , it is possible
to write certain commutators as a square. We denote by the minimal
number of squares which is required to write as a product of squares in group . And we define .
We discuss the question of when the square length of a given commutator of
is equal to 1 or 2 or 3. The precise formulas for expressing any commutator of as the minimal number of squares are given. Finally as an application of these results we prove that .