On a theorem of Berkovich (Q1852734)

The reviewer [Proc. Am. Math. Soc. 127, No. 9, 2505-2509 (1999; Zbl 0929.20017)] described the nilpotent residual of a finite group \(G\) in terms of the nilpotent residuals of some of its small subgroups. That description required the knowledge of the structure of the minimal nonnilpotent groups. The main goal of this note is to extend the above result replacing the nilpotence by a more general group theoretical property (the groups possessing that property constitute of a subgroup-closed saturated formation). The proof presented is very short and elementary. Next, a family of subgroup-closed saturated formations of nilpotent type is studied.