A group $G$ is *Noetherian* (or *slender*) if all its subgroups are finitely generated. Does this imply that the minimal number of generators of subgroups of $G$ is bounded above?

For example, if $G$ is a polycyclic group that admits a polycyclic series of length $n$ then every subgroup of $G$ can be generated by $n$ (or fewer) elements. This idea also applies for virtually-polycyclic groups.

It is unknown whether all finitely *presented* Noetherian groups are virtually-polycyclic. On the other hand, there are finitely *generated* Noetherian groups that are not virtually-polycyclic, for example the Tarski monster. However, all proper subgroups of the Tarski monster are cyclic and hence there is a bound on the minimal number of generators of its subgroups.

(See this post for a related question.)

*Edit:* What if we restrict ourselves to finitely presented Noetherian groups?

"Are all finitely presented Noetherian groups virtually polycyclic?is (an open) Question 11.38 (due to S.V.Ivanov, 1990) from theKourovka Notebook. $\endgroup$ – Anton Klyachko Nov 5 '13 at 18:59