Abstract
In a free group no nontrivial commutator is a square. And in the free groupF2=F(x1,x2)freely generated byx1,x2the commutator[x1,x2]is never the product of two squares inF2, although it is always the product of three squares. LetF2,3=〈x1,x2〉be a free nilpotent group of rank 2 and class 3 freely generated byx1,x2. We prove that inF2,3=〈x1,x2〉, it is possible to write certain commutators as a square. We denote bySq(γ)the minimal number of squares which is required to writeγas a product of squares in groupG. And we defineSq(G)=sup{Sq(γ);γ∈G′}. We discuss the question of when the square length of a given commutator ofF2,3is equal to 1 or 2 or 3. The precise formulas for expressing any commutator ofF2,3as the minimal number of squares are given. Finally as an application of these results we prove thatSq(F′2,3)=3.
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