Finite Groups with Certain Weakly S-permutable Subgroups

Let G be a finite group. A subgroup H of G is said to be weakly S-permutable in G if G has a subnormal subgroup T such that G = HT and T ∩ H ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are S-permutable in G. In this paper, we prove the following: For a Sylow p-subgroup P of G (p > 2), suppose that P has a subgroup D such that 1 < |D| < |P| holds and all subgroups H of P with |H| = |D| are weakly S-permutable in G. Then, the commutator subgroup G ′ is p-nilpotent. We certainly belive that this result will improve and extend a current and classical theories in the literature.