On Weakly ℋ-embedded Subgroups of Finite Groups

Abstract

Let G be a finite group and H a subgroup of G. We say that H is an ℋ-subgroup in G if N_G(H) ∩ H^g ≤ H for all g ∈ G; H is called weakly ℋ-subgroup in G if G has a normal subgroup K such that G = HK and H ∩ K is an ℋ-subgroup in G. We say that H is weakly ℋ -embedded in G if G has a normal subgroup K such that H^G = HK and H ∩ K is an ℋ-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that some subgroups of prime power order are weakly ℋ-embedded in G. Our results improve and generalize several recent results in the literature.

Key Words:

• c-Normal subgroup
• ℋ-Subgroup
• p-Nilpotent group
• Supersolvable group
• Saturated formation
• Weakly ℋ-subgroup

2000 Mathematics Subject Classification:

• 20D10
• 20D15: 20D20