On an extension of the Frobenius^{prime } theorem about p-nilpotency of a finite group

Suppose that (G) is a finite group and (H) is a subgroup of (G) . (H) is said to be (s) -quasinormally embedded in (G) if for each prime (p) dividing the order of (H) , a Sylow (p) -subgroup of (H) is also a Sylow (p) -subgroup of some (s) -quasinormal subgroup of (G) . We fix in every non-cyclic Sylow subgroup (P) of (G) some subgroup (D) satisfying (1<|D|<|P|) and study the (p) -nilpotency of (G) under the assumption that every subgroup (H) of (P) with (|H|=|D|) is (s) -quasinormally embedded in (G) . Some recent results and the Frobenius (^{prime }) theorem are generalized.