STRUCTURE OF FINITE GROUPS WITH SOME WEAKLY S-SEMIPERMUTABLE SUBGROUPS

Let G be a finite group. If A ≤ G, recall that A is weakly S-semipermutable in G provided there is K  G such that KA is S-permutable in G, and K ∩ A is S-semipermutable in G. The purpose of this paper is to demonstrate that weakly S-semipermutability of special types of subgroups in a finite group G can help us to determine structural properties of G. For example, given a prime p, a p-soluble finite group G and a Sylow p-subgroup Gp of G, we will show that G is p-supersoluble if the maximal subgroups of Gp are weakly S-semipermutable in G. More-over, we use the concept of weakly S-semipermutability to prove new criteria for p-nilpotency of finite groups.