A criterion for p-supersolvability of finite groups

Let G be a finite group and H a subgroup of G. We say that H is an 7-l-subgroup of G if NG(H) Pi H-g <= H for all g is an element of G; H is called weakly HC-embedded in G if G has a normal subgroup T such that H-G = HT and N-T (H) Pi H-g <= H for all g is an element of G, where H-G is the normal closure of H in G. Let p be a prime dividing the order of G and let P be a Sylow p-subgroup of G. We fix a p-power integer d with 1 < d < |P| and study the p-supersolvability criterion of a finite group G under the assumption that each subgroup of P of order d and pd is weakly HC-embedded in G. Some recent results about the supersolvability of finite groups are obtained.