On partial cap-subgroups of finite groups

Abstract

Let G be a finite group, P∈Sylp(G), and pd be a given prime power such that |P|>pd>1. Assume that all subgroups of order pd, and all cyclic group of order 4 when pd=2 and P is nonabelian, are partial CAP-subgroups of G. We show that G/Op′,p(G) is supersolvable.