On Formations of Finite Groups with the Wielandt Property for Residuals

Abstract

Given two subgroups U, V of a finite group which are subnormal subgroups of their join <U, V> and a formation F, in general it is not true that (U, V)F = <UF , VF>. A formation is said to have the Wielandt property if this equality holds universally. A formation with the Wielandt property must be a Fitting class. Wielandt proved that the most usual Fitting formations (e.g., nilpotent groups and π-groups) have the Wielandt property. At present, neither a general satisfactory result on the universal validity of the Wielandt property nor a counterexample is known. In this paper a criterion for a Fitting formation to have the Wielandt property is given. As an application, it is proved that many of the known Fitting formations have the Wielandt property. Show more