Finite groups with S-supplemented p-subgroups

Abstract

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem. Theorem. Let E be a normal subgroup of a group G and let p be a prime divisor of |E| such that (p − 1, |E|) = 1. Take a Sylow p-subgroup P of E. Suppose that either all maximal subgroups of P lacking p-supersoluble supplement in G or all order p subgroups and quaternionic order 4 subgroups of P lacking p-supersoluble supplement in G are S-supplemented in G. Then E is p-nilpotent and all its G-chief p-factors are cyclic.