Finite Groups with S-permutable n-maximal Subgroups

Abstract

Let G be a finite group and M _n (G) be the set of n-maximal subgroups of G, where n is an arbitrary given positive integer. Suppose that M _n (G) contains a nonidentity member and all members in M _n (G) are S-permutable in G. Then any of of the following conditions guarantees the supersolvability of G: (1) M _n (G) contains a nonidentity member whose order is not a prime; (2) all nonidentity members in M _n (G) are of prime order, and all cyclic members in M _n−1(G) of order 4 are S-permutable in G.

Key Words:

• Finite group
• n-Maximal subgroup
• S-Permutable subgroup
• Supersolvable group

2010 Mathematics Subject Classification:

• 20D30
• 20E15

Notes

Communicated by A. Turull.