Weakly s-semipermutable subgroups of finite groups

In this paper, we introduce the concept of weakly s-semipermutable subgroups. Let G be a finite group. Using the condition that the minimal subgroups or subgroups of order p ² of a given Sylow p-subgroup of G are weakly s-semipermutable in G, we give a criterion for p-nilpotency of G and get some results about formation.