Abstract: Let G be a finite group. We prove that if G is a group which is an extension of a n-central group by a n-nilpotent group, then G is a n-nilpotent group. Furthermore, it is also proved that if G is a product of A and B, where A and B are n-nilpotent normal subgroups of G, and every element of A is n-commutative with every element of B, then G is a n-nilpotent group. The main theorems generalize the corresponding results on nilpotent groups and also extend Baer's results on n-nilpotent groups.