Nilpotence relations in products of groups (Q831198)

Let \(G=AB\) be a group which is factorized by the two subgroups \(A\) and \(B\). Suppose further that \(A\) and \(B\) are \textit{\(\mathfrak{N}\)-connected} (that is, \(\langle a,b\rangle\) is nilpotent for all \(a\in A\) and \(b\in B\)). The aim of the paper under review (Theorems A and~B) is to show that if \(A\) and \(B\) are both supersoluble (Chernikov, respectively), then~\(G\) is supersoluble (Chernikov, respectively). The supersoluble case is known when~\(G\) is finite, so, in his proof, the author is able to apply induction on the sum \(m\) of the Hirsch lengths of \(A\) and \(B\) (if \(m=0\), the group \(G\) is finite); in this case one only need to show that the group is polycyclic (see Lemma 3.1). Observe also that one can replace ``supersoluble'' by ``finitely generate nilpotent''. The Chernikov case proceeds by induction on the sum of the total ranks of the finite residuals of \(A\) and \(B\).