Existentially Closed Subgroups of Free Nilpotent Groups

Abstract

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