On generalized covering subgroups and a characterisation of ?pronormal?

Abstract

Introduction. The context of this note is the theory of Schunck classes and formations of finite soluble groups. In a 1972 manuscript Fischer [4] generalized the concept of an ~-covering subgroup of a group G to a (P, ~)-covering subgroup, where P is some pronormal subgroup of G, and proved universal existence (for P satisfying a stronger embedding property) in case the class ~ is a saturated formation. The fact tha t the Schunck classes are the classes ~ with the property that every group has an ~-projector [9, 4.3, 4.4; 6] (which coincides with an ~-covering subgroup in the soluble universe | [6, II.15]) raises the question whether it is possible to determine the whole range of universal existence of (P, ~)-covering subgroups. The aim of this paper is to show that such classes are exactly the Schunck classes ~ of form ~ ~ E~ ~ for some formation ~. In a short second part we wish to note that the embedding property "pronormal" can be characterized by the Fratt ini argument.