The influence of some primary subgroups on the structure of finite groups

Abstract

Suppose that G is a finite group and H is a subgroup of G. H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that $$G=HB$$ and H permutes with every Sylow subgroup of B; H is said to be c-normal in G if G has a normal subgroup T such that $$G=HT$$ and , where $$H_{G}$$ is the normal core of H in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying $$1<|D|<|P|$$ and study the structure of G under the assumption that every subgroup H of P with $$|H|=|D|$$ is either ss-quasinormal or c-normal in G. Some recent results are generalized and unified.