Commutators and Squares in Free Nilpotent Groups

Mehri Akhavan-Malayeri

International Journal of Mathematics and Mathematical Sciences, 2009, vol. 2009, 1-10

Abstract:

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 .

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:264150

DOI: 10.1155/2009/264150

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