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.

Abstract

In a free group no nontrivial commutator is a square. And in the free group F2=F(x1,x2) freely generated by x1,x2 the commutator [x1,x2] is never the product of two squares in F2, although it is always the product of three squares. Let F2,3=〈x1,x2〉 be a free nilpotent group of rank 2 and class 3 freely generated by x1,x2. We prove that in F2,3=〈x1,x2〉, it is possible to write certain commutators as a square. We denote by Sq(γ) the minimal number of squares which is required to write γ as a product of squares in group G. And we define Sq(G)=sup{Sq(γ);γ∈G′}. We discuss the question of when the square length of a given commutator of F2,3 is equal to 1 or 2 or 3. The precise formulas for expressing any commutator of F2,3 as the minimal number of squares are given. Finally as an application of these results we prove that Sq(F′2,3)=3.