A nilpotency condition for finitely generated soluble groups (Q1293086)

Assume that \(G\) is a finitely generated soluble group such that every infinite set of elements of \(G\) contains a pair that generates a nilpotent subgroup of class at most \(k\), a fixed integer. The author shows for this situation that \(G\) is an extension of a finite group by a torsionfree \(k\)-Engel group, and that there is an integer \(n\) depending on \(k\) and the derived length of \(G\) such that \(| G:Z_n(G)|\) is finite. For \(k\leq 3\) the integer \(n\) does no longer depend on the derived length of \(G\).