Finite Groups withS-permutablen-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.

On finite groups with given $ICΦ$-subgroups

“…Other recent results on n-maximal subgroups and their influence on the structure of groups were obtained for example in [3,4,19].…”