On weakly s-semipermutable or ss-quasinormal subgroups of finite groups

Abstract

Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly s-semipermutable in G if there are a subnormal subgroup T of G and an s-semipermutable subgroup $$H_{ssG}$$ of G contained in H such that $$G=HT$$ and $$H\cap T\le H_{ssG}$$; 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. 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 weakly s-semipermutable or ss-quasinormal in G. Some recent results are generalized and unified.