On Finite Quasi-ℱ-Groups

Abstract

Let G be a finite group and ℱ a saturated formation of finite groups. We say that G is quasi-ℱ-group if for every ℱ-eccentric chief factor H/K of G and every x ∈ G, x induces an inner automorphism on H/K. In particular, we have the concepts of quasisoluble, quasisupersoluble, quasimetanilpotent groups, and so on. In this article, we obtain some results about the quasi-ℱ-groups and use them to give the conditions under which a group is quasisoluble, quasimetanilpotent, or quasisupersoluble.

Key Words:

• Finite groups
• Quasi-ℱ-subgroup
• Quasimetanilpotent group
• Quasisoluble group
• Quasisupersoluble group
• Saturated formations

2000 Mathematics Subject Classification:

• 20D10
• 20D15
• 20D20

ACKNOWLEDGMENT

The author is grateful for the helpful suggestions of the referee. In fact, some results and their proofs were modified by the referee.

The project is supported by the National Natural Science Foundation of China (Grant 10771180).

Notes

Communicated by V. A. Artamonov. Dedicated to our teacher Prof. Dr. Leonid A. Shemetkov on the occasion of his 70th birthday.