Existentially Closed Subgroups of Free Nilpotent Groups

Let be a variety of all nilpotent groups of class at most c, and let N r,c be a free nilpotent group of finite rank r and nilpotency class c. It is proved that a subgroup N of N r,c for c ≥ 3 is existentially closed in N r,c iff N is a free factor of the group N r,c with respect to the variety . Consequently, N ≃ N m,c , 1 ≤ m ≤ r, and m ≥ c − 1.