Universal factorization property of certain polycyclic groups

Let H be a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group. Let G be any group with maximal condition. We show that there exists a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group image and an epimorphism image such that for any homomorphism phi:G→H, it factors through image, i.e., there exists a homomorphism image such that image. We show that this factorization property cannot be extended to any finitely generated group G. As an application of factorization, we give necessary and sufficient conditions for image to hold for maps f,g:X→Y between closed orientable n-manifolds where π1(X) has the maximal condition, Y is an infra-solvmanifold, N(f,g) and image denote the Nielsen and Reidemeister coincidence numbers, respectively.