structure of finite groups with some weakly s-semipermutable subgroups

Let g be a finite group. if a ≤ g, recall that a is weaklys-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.