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

Suppose that G is a finite group and H is a subgroup of G. H is said to be s-semipermutable in G if HG(p) = G(p)H for any Sylow p-subgroup G(p) of G with (p, vertical bar H vertical bar) = 1; 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 will study finite groups G saisfying the following: for each noncyclic Sylow subgroup P of G, there exists a subgroup D of P such that 1 < vertical bar D vertical bar < vertical bar P vertical bar and every subgroup H of P with order vertical bar D vertical bar is s-semipermutable or ss-quasinormal in G. Some recent results are generalized and unified.