Polycyclic-by-finite groups and strong Carter subgroups (Q1568179)

A subgroup \(C\) of a group \(G\) is called a strong Carter subgroup, if \(C\) is nilpotent and if for \(C\leq X\leq G\) we have \(N_G(X)=X\). Let \(\mathcal L\) denote the class of groups in which every residually nilpotent section is nilpotent. The main result of this paper is following Theorem: For every polycyclic-by-finite group \(G\) the following statements are equivalent: (a) \(G\in{\mathcal L}\). (b) In every soluble subgroup \(H\) of \(G\) there exists a strong Carter subgroup. (c) In every nilpotent-by-cyclic subgroup \(H\) of \(G\) there exists a strong Carter subgroup. (d) \(H\in{\mathcal L}\) for every nilpotent-by-cyclic subgroup \(H\) of \(G\). (e) If \(H\) is a nilpotent-by-cyclic subgroup of \(G\) and \(t(H)\) is the maximal normal finite subgroup of \(H\), then \(H/t(H)\) is nilpotent. (f) If \(t(G)\) is the maximal normal finite subgroup of \(G\), then \(G/t(G)\) is nilpotent. (g) There is a nilpotent subgroup \(C\) such that \(G=t(G)C\). If \(Z\) is the upper hypercenter of \(G\) then \(G/Z\) is finite.