TY - JOUR
A2 - Bell, Howard
AU - Akhavan-Malayeri, Mehri
PY - 2009
DA - 2010/02/03
TI - Commutators and Squares in Free Nilpotent Groups
SP - 264150
VL - 2009
AB - In a free group no nontrivial commutator is a square. And in thefree 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 class3 freely generated by x1,x2. We prove that in F2,3=〈x1,x2〉, it is possibleto write certain commutators as a square. We denote by Sq(γ) the minimalnumber 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 ofF2,3 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.
SN - 0161-1712
UR - https://doi.org/10.1155/2009/264150
DO - 10.1155/2009/264150
JF - International Journal of Mathematics and Mathematical Sciences
PB - Hindawi Publishing Corporation
KW -
ER -