Pronormal subgroups and homomorphs in finite groups

Abstract

A local version of the theory of homomorphs and Schunck classes is given. It is shown that for any finite soluble group the pronormal subgroups are precisely the covering subgroups with respect to “Schunck sets” in this group. As an application simple proofs of some results on pronormal subgroups of finite soluble groups are obtained. Finally a question of Doerk is answered in the negative: any finite soluble group is a subgroup of a minimal non-trivial pronormal subgroup of some finite soluble group.