On s-semipermutable or s-quasinormally Embedded Subgroups of Finite Groups

Abstract Suppose that G is a finite group and H is a subgroup of G. H is said to be s-semipermutable in G if HGp = GpH for any Sylow p-subgroup Gp of G with (p, |H|) = 1; H is said to be s-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G. In every non-cyclic Sylow subgroup P of G we fix 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 s-semipermutable or s-quasinormally embedded in G. Some recent results are generalized and unified.