On ℳ-Permutable Maximal Subgroups of Sylow Subgroups of Finite Groups

Abstract

Let d be the smallest generator number of a finite p-group P, and let ℳ _d (P) = {P ₁, P ₂,…, P _d } be a set of maximal subgroups of P such that . Then P is called ℳ-permutable in a finite group G, if there exists a subgroup B of G such that G = PB and P _i B < G for any P _i of ℳ _d (P). In this article, we investigate the influence of ℳ-permutability of some primary subgroups in finite groups. Some new results about supersolvable groups and formations are obtained.

Key Words:

• Finite groups
• ℳ-permutable subgroups
• Supersolvable groups
• Sylow subgroups

2000 Mathematics Subject Classification:

• 20D10
• 20D20

ACKNOWLEDGMENTS

Authors are grateful for the referee for his/her helpful comments.

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.

Notes

Communicated by V. A. Artamonov.