Conjugacy in nilpotent groups

1. Statement of the theorem. The aim of the present note is to investigate possible generalizations of the well-known fact that if a is a nonidentity element of a finitely-generated nilpotent group G, there exists an epimorphism 4 of G onto a finite group such that acu P 1. The generalization that we consider is the following. Let G be a finitely-generated nilpotent group, and let w(xi, * * *, Xn; a,, * * *, am) be a word in variables x1, * * *, xn and elements a1, * * *, am of G. If w = 1 has no solution in G, does there exist an epimorphism 4 of G onto a finite group H such that w(x1, * , Xn; a14, * * * , am+) =1 has no solution in H? The answer in general is in the negative, as is shown by a counterexample constructed below. However, we shall prove, in answer to a question posed by A. W. Mostowski, that the answer is in the affirmative if w = x-laxb-1. Our aim then is to prove the following.