Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G . If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G . We establish tests for a finite group G to be p -supersoluble provided that every maximal subgroup of a Sylow p -subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G , X = O p',p ( H ), and X = F *( H ) where H is a normal subgroup of G .