On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ℋ-subgroup in G if N G (H) ∩ H x ≤ H for all x ∈ G. A subgroup H of G is called weakly ℋ-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ℋ-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ℋ-subgroups in G. Some recent results are extended and generalized.