On Weakly ℋ-Subgroups of Finite Groups

Abstract

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.

Key Words:

• Fitting subgroup
• Generalized Fitting subgroup
• p-Nilpotent group
• c-Normal subgroup
• Saturated formation
• ℋ-Subgroup
• Supersolvable group
• Sylow subgroup

2010 Mathematics Subject Classification:

• 20D10
• 20D15
• 20D20
• 20F16

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) represented by the Unit of Research Groups through the grant number (G/31/01) for the group entitled “Abstract Algebra and its Applications”.

Notes

Communicated by A. Turull.