A property of the variety of \(2\)-Engel groups
Let \(V\) be a variety of groups defined by a single law in \(n\) variables, and define \(V^*\) to be the class of groups with the property that for every set of \(n\) infinite sets \(X_ 1,\dots, X_ n\) of elements there exist elements \(x_ i \in X_ i\) that generate a subgroup in \(V\). This definition, and the question ``for which varieties \(V\) is every infinite \(V^*\)-group a \(V\)-group?'', are due to \textit{P. S. Kim, A. H. Rhemtulla}, and \textit{H. Smith} [Houston J. Math. 17, 429-437 (1991; Zbl 0744.20033)]. The author answers the question positively for 2-Engel groups by proving the theorem: If for every pair \(X\), \(Y\) of infinite sets of elements of a group there exist elements \(x \in X\) and \(y \in Y\) such that \([x, y, y] = 1\), then the group is a 2-Engel group. The proof uses a special case of a lemma of the author's [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 3, No. 3, 177-183 (1992; Zbl 0791.20038)], namely that in a group of the theorem the centraliser of every element is infinite.
scientific article; zbMATH DE number 627431