Abstract: We supposed that a subgroup D of a Sylow p-subgroup P of a group G satisfied 1<D≤P, where p was a prime divisor of |G|. The new sufficient conditions for the characterization of p-nilpotent groups and p-supersolvable groups were obtained by using Msupplemented property of each order of |D| subgroup in the normalizer N_G(P) of P and combining the weakly ssupplemented property of H (P)={H
≤P|P′≤H≤Φ(P)}.

Key words: normalizer, weakly s-supplemented subgroup, Sylow p-subgroup, supplemented subgroup