Groups with pronormal deviation

Abstract

The structure of locally soluble groups in which all subgroups are pronormal is known, and many authors have investigated groups which are rich in pronormal subgroups. Here we prove that if a radical group G has pronormal deviation, which means that the set of its non-pronormal subgroups satisfies a very weak chain condition, then either G is minimax or all its subgroups are pronormal. It follows that if a radical group has pronormal deviation, then its pronormal deviation is at most 1.