Powers of a product of commutators as products of squares. (English) Zbl 1070.20038

Summary: We prove that for any odd integer \(N\) and any integer \(n>0\), the \(N\)-th power of a product of \(n\) commutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of \(2n+1\) elements and, for all such odd \(N\) and integers \(n\), there are commutators for which the number \(2n+1\) of squares is the minimum number such that the \(N\)-th power of its product can be written as a product of squares. This generalizes a recent result of M. Akhavan-Malayeri [Int. J. Math. Math. Sci. 31, No. 10, 635-637 (2002; Zbl 1013.20028)].


20F12 Commutator calculus
20E05 Free nonabelian groups
20F05 Generators, relations, and presentations of groups


Zbl 1013.20028
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