Quanfu Yan - Department of Mathematical Sciences, Kent State University, Kent, USA
Zhencai Shen - Department of Mathematics, College of Science, China Agricultural University, Beijing, China.

Let G be a finite group. A subgroup H of G is an \mathscr{H}-subgroup in G if N_G(H)\cap H^g \leq H for any g \in G. In this article, by using the concept of \mathscr{H}-subgroups, we study the influence of the intersection of O^p(G_p^*) and the members of some fixed \mathcal{M}_d(P) on the structure of the group G, where P is a Sylow p-subgroup of G. Some new criteria for a group to be p-nilpotent and p-supersolvable are given and some recent results are extended and generalized.

\mathscr{H}-subgroup, weakly \mathscr{H}-subgroup, p-supersolvablility, p-nilpotency