Kim, Pan Soo; Rhemtulla, Akbar; Smith, Howard A characterization of infinite metabelian groups. (English) Zbl 0744.20033 Houston J. Math. 17, No. 3, 429-437 (1991). If \(\mathfrak X\) is a variety of groups defined by a law \(\theta(x_ 1,\dots,x_ n)=1\), with \(n\) minimal, define \({\mathfrak X}^*\) to be the class of all groups \(G\) such that, given infinite subsets \(X_ 1,\dots,X_ n\) of \(G\), there exist elements \(x_ i\in X_ i\) such that \(\langle x_ 1,\dots,x_ n\rangle\in{\mathfrak X}\). The authors raise the general question of when every infinite \({\mathfrak X}^*\)-group is in \({\mathfrak X}\), and they prove that this is true for the variety \({\mathfrak A}^ 2\) of metabelian groups. Reviewer: D.J.S.Robinson (Urbana) Cited in 4 ReviewsCited in 6 Documents MSC: 20F16 Solvable groups, supersolvable groups 20E10 Quasivarieties and varieties of groups Keywords:variety of groups; metabelian groups PDFBibTeX XMLCite \textit{P. S. Kim} et al., Houston J. Math. 17, No. 3, 429--437 (1991; Zbl 0744.20033)