On infinitely generated groups whose proper subgroups are solvable. (Q397899)

This very technical paper is motivated by the following difficult problem. If \(G\) is a group that is not finitely generated and if every proper subgroup of \(G\) is soluble, is \(G\) necessarily soluble? (If \(G\) is a Tarski monster, then \(G\) is finitely generated, so \(G\) is not a counterexample.) Here the author presents at least half a dozen partial solutions, most of which are too complex to easily summarize. We confine ourselves to stating some of the simpler ones, to give something of the flavour of this paper.NEWLINENEWLINE Thus let \(G\) be a group that is not finitely generated, all of whose proper subgroups are soluble. If in each (homomorphic) image of \(G\) every finitely generated subgroup has its normal closure residually (finite and nilpotent), then \(G\) is soluble. If in each image of \(G\) every finitely generated subgroup has its normal closure residually nilpotent, then \(G\) is periodic if and only if \(G\) is locally nilpotent.NEWLINENEWLINE Perhaps motivated by the second result quoted above, the author poses the following more specific problem. Is a torsion-free group \(G\) soluble if each of its proper subgroups is soluble and if in each image of \(G\) every finitely generated subgroup has its normal closure residually nilpotent?