Longobardi, Patrizia; Maj, Mercede A finiteness condition concerning commutators in groups. (English) Zbl 0813.20026 Houston J. Math. 19, No. 4, 505-512 (1993). In a paper by P. S. Kim, A. H. Rhemtulla and H. Smith [Houston J. Math. 17, 429-437 (1991; Zbl 0744.20033)], the following question was asked. Let \(V\) be a variety of groups defined by means of a law \(v(x_ 1,\dots,x_ n) = 1\). Let \(G\) be an infinite group with the property that for all choices of infinite subsets \(X_ 1,\dots,X_ n\) of \(G\) there exist elements \(x_ 1,\dots,x_ n\), with \(x_ i \in X_ i\) for all \(i\), such that \(v(x_ 1,\dots,x_ n) = 1\). Is it true that \(G \in V\)? In the paper under review the authors prove that the answer is yes for the variety defined by the law \([x,y]^ 2 = 1\). This is one of the simplest cases where the answer was previously unknown. Reviewer: R.M.Bryant (Manchester) Cited in 1 ReviewCited in 3 Documents MSC: 20E10 Quasivarieties and varieties of groups 20F12 Commutator calculus Keywords:laws in groups; variety of groups; infinite group; infinite subsets Citations:Zbl 0744.20033 PDFBibTeX XMLCite \textit{P. Longobardi} and \textit{M. Maj}, Houston J. Math. 19, No. 4, 505--512 (1993; Zbl 0813.20026)