Group homomorphisms hihi such that h1(x1)…hn(xn)=yh1(x1)…hn(xn)=y is solvable

Let X1,…,Xn,YX1,…,Xn,Y be not necessarily abelian multiplicative groups. We identify a class of group homomorphisms hi:Xi→Yhi:Xi→Y for i=1,n¯ such that both the solvability and the set of solutions of h1(x1)…hn(xn)=y,h1(x1)…hn(xn)=y, can be characterized. This generalizes a result of Cain [B. Cain, Group homomorphisms hihi such that h1(x1)+⋯+hn(xn)=yh1(x1)+⋯+hn(xn)=y is solvable, Linear Algebra Appl. 360 (2003) 191–195].