Pronormal subgroups and homomorphs in finite groups (Q1084176)

The author gives a local version of the theory of homomorphs and Schunck classes, defining the notions of G-homomorph and covering subgroup with respect to a single arbitrary finite group G. It is shown that a subgroup of a finite soluble group G is pronormal if, and only if, it is a covering subgroup with respect to some G-homomorph. As an application simple proofs of some basic 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.