Powers of commutators as products of squares. (English) Zbl 1013.20028

Akhavan-Malayeri, M.

Int. J. Math. Math. Sci. 31, No. 10, 635-637 (2002).

Let \(F\) be a non-Abelian free group and \(x,y\) be two distinct elements of a free generating set. It is proved that \([x,y]^n\neq a^2b^2\), for any odd integer \(n>0\) and any two elements \(a,b\in F\), and there exist three elements \(v\), \(w\) and \(z\) in \(F\) such that \([x,y]^n=u^2v^2w^2\). However it is mentioned that there are commutators in \(F\) which can be expressed as a product of squares of two elements in \(F\).

Reviewer: Alireza Abdollahi (Isfahan)

MSC:

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

Keywords:

commutators in free groups; products of squares of elements; commutators as products of squares