New characterizations of p-nilpotency of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this note, we fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < vertical bar D vertical bar < vertical bar P vertical bar and study the p-nilpotency of G under the assumption that every subgroup H of P with vertical bar H vertical bar = vertical bar D vertical bar is SS-quasinormal in G. The Frobenius theorem is generalized.