On ℱ-Quasinormal Primary Subgroups of Finite Groups

Abstract

Let G be a finite group and ℱ a formation. A subgroup H is called ℱ-quasinormal in G if there exists a quasinormal subgroup T of G such that HT is quasinormal in G and (H ∩ T)H _G /H _G is contained in the ℱ-hypercenter of G/H _G . In this article, we study the structure of finite groups by using ℱ-quasinormal subgroups and prove that: Let ℱ be a saturated formation containing 𝒰 and G be a group with a normal subgroup H such that G/H ∈ ℱ. If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is 𝒰-quasinormal in G, then G ∈ ℱ.

Key Words:

• Finite groups
• ℱ-Quasinormal subgroup
• Saturated formation
• Sylow subgroup

AMS Subject Classification:

• 20D10
• 20D20

ACKNOWLEDGMENTS

We thank the referee for the careful reading of the manuscript and for suggestions that have helped to improve our original version.

The first author is supported by the grant of NSFC (Grant #10901133) and sponsored by Qing Lan Project and Natural science fund for colleges and universities in Jiangsu Province. The second author is supported by the grant of NSFC (Grant #10771180; #10926129).

Notes

Communicated by V. A. Artamonov.