On groups with finitely many derived subgroups. (Q2874709)

If \(G\) is a locally graded group for which the set of derived subgroups is finite, it was proved by \textit{D. J. S. Robinson} and the reviewer [J. Lond. Math. Soc., II. Ser. 71, No. 3, 658-668 (2005; Zbl 1084.20026)] that the derived subgroup \(G'\) of \(G\) is finite. Recall here that a group \(G\) is said to be locally graded if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index.NEWLINENEWLINE In the paper under review, the author obtains some new informations on the structure of such groups. In fact, he proves that if \(G\) is a locally graded group with only \(k\) derived subgroups (where \(k\) is a positive integer), then the derived subgroup \(G'\) of \(G\) contains a nilpotent-by-Abelian subgroup with index at most \(k!\). Moreover, he shows that any locally graded group with at most \(22\) derived subgroups must be soluble, but the alternating group \(A_5\) has precisely \(23\) derived subgroups.