Groups with all proper subgroups (finite rank)-by-nilpotent (Q1293176)

Let \(c\) be a positive integer. It has been proved by \textit{J. Otal} and \textit{J. M. Peña} [Arch. Math. 51, No. 3, 193-197 (1988; Zbl 0632.20018)] that a locally graded group is Chernikov-by-(nilpotent of class at most \(c\)) if and only if all its proper subgroups have the same property (here a group \(G\) is said to be locally graded if every finitely generated non-trivial subgroup of \(G\) has a finite non-trivial homomorphic image). The authors prove that, if \(G\) is a locally soluble-by-finite group whose proper subgroups are extensions of a group with finite Prüfer rank by a nilpotent group with class at most \(c\), then \(G\) itself is an extension of the same type.