On partially s-permutable subgroups of finite groups

Let H be a subgroup of G. H is said to be partially s-permutable in G provided G has a subnormal subgroup T such that G=HT and H∩T≤HsT, where HsT is the subgroup of H generated by all the subgroups of H which permute with all the Sylow subgroups of T. Here, partially s-permutable subgroups were used to study the structure of finite groups and some new criteria of p-nilpotent groups and p-supersoluble groups were obtained.