Two sufficient conditions for a finite group G to be p-supersolvable have been obtained. For example (Theorem 1.1), let N be a normal subgroup of G such that G/N is p-supersolvable for a fixed odd prime p and let Np be a Sylow p-subgroup of N. Suppose that N is p-solvable and Ω1(Np) is generated by the subgroups of order p of Np which are normal in NG(Np). Then G is p-supersolvable.
Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let HsG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that HG = HT and H ∩ T ≦ HsG, where HG is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.