Let Fn, Mn and Un denote the free group of rank n, the free metabelian group of rank n, and the free abelian-by-(nilpotent of class 2) group of rank n, respectively. Thus Mn≅ Fn/F″″n and Un≅Fn/ [γ3(Fn), γ3(Fn)], where γ3(Fn) =[F′n, Fn], the third term of the lower central series of Fn.

Consider an arbitrary epimorphism θ[ratio ]Fn[Rarr ]Fk, where k<n. A routine argument (see [10, Theorem 3.3]) using the Nielsen reduction process shows that there exists a free basis {y1, y2, …, yn} of Fn such that kerθ is the normal closure in Fn of {y1, y2, …, yn−k}. A similar result has been obtained for free metabelian groups by C. K. Gupta, N. D. Gupta and G. A. Noskov [5]. Indeed, it follows immediately from [5, Theorem 3.1] that if k<n and θ[ratio ]Mn[Rarr ]Mk is an epimorphism, then there exists a free basis {m1, m2, …, mn} of Mn such that kerθ is the normal closure in Mn of {m1, m2, …, mn−k}.