International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 264150, 10 pages
doi:10.1155/2009/264150

Commutators and squares in free nilpotent groups

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.