On the Shemetkov problem for Fitting classes.
The paper under review deals with Fitting classes of finite groups, that is, classes which are closed under subnormal subgroups and normal products. More precisely, it answers a question proposed by L. A. Shemetkov about local Fitting classes, which are Fitting classes defined by means of Hartley functions, dualising the concept of local formation. If \(\pi\) is a set of primes and \(F\) is a local Fitting class, it is proved that the class \(K_\pi(F)\) of all finite \(\pi\)-soluble groups with a Hall \(\pi\)-subgroup in \(F\) is a local Fitting class. As a consequence, a description of the \(F\)-radical of a finite \(\pi\)-soluble group is obtained.
scientific article; zbMATH DE number 5152755