Some remarks about groups of finite special rank (Q2818391)

A group \(G\) is said to have finite rank \(r\) if every finitely generated subgroup of \(G\) can be generated by at most \(r\) elements, and \(r\) is the smallest positive integer with such property. It was proved by \textit{R. Baer} and \textit{H. Heineken} [Ill. J. Math. 16, 533--580 (1972; Zbl 0248.20052)] that if all abelian subgroups of a radical group \(G\) have finite rank, then \(G\) itself has finite rank. Recall here that a group \(G\) is radical if it admits an ascending (normal) series with locally nilpotent factors. Moreover, \(G\) is called generalized radical if it has a an ascending (normal) series whose factors are either locally nilpotent or locally finite; in this case, if the locally finite factors have finite rank at most \(k\) (for some fixed positive integer \(k\)), \(G\) is said to be a \(k\)-generalized radical group.NEWLINENEWLINEIn the first part of this very interesting paper, the authors provide a satisfactory description of locally generalized radical groups of finite rank, and prove that if all locally soluble subgroups of a locally generalized radical group \(G\) have finite rank, then \(G\) itself has finite rank. Moreover, they extend the above quoted theorem of Baer and Heineken [loc. cit.] to the case of locally \(k\)-generalized radical groups. In the final section of the paper, similar results for other types of rank are obtained.