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Commutators and Squares in Free Nilpotent Groups

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  • Mehri Akhavan-Malayeri

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 .

Suggested Citation

  • Mehri Akhavan-Malayeri, 2009. "Commutators and Squares in Free Nilpotent Groups," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2009, pages 1-10, February.
    • Handle: RePEc:hin:jijmms:264150
    DOI: 10.1155/2009/264150
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