Centre-by-metabelian groups with a condition on infinite subsets. (Q1884092)

Let \(k\) be a positive integer and let \({\mathcal N}_k\) and \(\mathcal F\) denote the class of nilpotent groups of class at most \(k\) and the class of finite groups, respectively. In the paper under review, the class of groups in which the normal closure of each element is nilpotent of class at most \(k-1\), denoted by \({\mathcal L}({\mathcal N}_k)\), is studied. The author and the reviewer [in Bull. Belg. Math. Soc. - Simon Stevin 9, No. 2, 205-215 (2002; Zbl 1041.20022)] studied the following two classes of groups \(E^*_k\) and \(E^\#_k\): The class of groups \(E^*_k\) is defined as the class of all groups \(G\) in which for every infinite subset \(X\) there exist two distinct elements \(x,y\) in \(X\), and integers \(t_0,t_1,\dots,t_k\) depending on \(x,y\) satisfying \([z_0^{t_0},z_1^{t_1},\dots,z_k^{t_k}]=1\), where \(z_i\in\{x,y\}\) for every \(i\in\{0,1,\dots,k\}\) and \(z_0\not=z_1\). The class \(E^\#_k\) is defined as the class of groups \(G\in E^*_k\) for which the integers \(t_0,\dots,t_k\) belong to \(\{-1,1\}\). Finitely generated soluble groups in the classes \(E^*_k\) and \(E^\#_k\) are completely characterized in [loc. cit.]. In the paper under review the following results are proved. Theorem 1.1. A finitely generated centre-by-metabelian group \(G\) is in \(E^*_{k+1}\) if, and only if, \(G\) belongs to \({\mathcal L}({\mathcal N}_k){\mathcal F}\). Theorem 1.2. A finitely generated centre-by-metabelian group \(G\) is in \(E^\#_{k+1}\) if, and only if, \(G\) belongs to \(\mathcal{FL}({\mathcal N}_k)\). In particular, a torsion-free centre-by-metabelian group \(G\) is in \(E^\#_{k+1}\) if, and only if, \(G\) belongs to \({\mathcal L}({\mathcal N}_k)\).