Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PTo-group (respectively, a PSTo-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PTo-groups and minimal non-PSTo-groups.
Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang  proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and NG(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.