Normal closure and injective normalizer of a group homomorphism

Let ϕ: Γ→. G be a homomorphism of groups. We consider factorizations Γ→fM→gG of ϕ having certain universal properties. First we continue the investigation (see [4]) of the case where g is a universal normal map (our term for a crossed module). Then we introduce and investigate a seemingly new dual case, where f is a universal normal map. These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup.Iteration of these universal factorizations yields certain towers associated with the map ϕ we prove stability results for these towers. In one of the cases we get a generalization of the stability of the automorphisms tower of a center-less group. The case where g is a universal normal map is closely related to hyper-central group extensions, Bousfield's localizations, and the relative Schur multiplier H₂(G, Γ)=H₂(BG∪_BϕCone(BΓ)).Although our constructions here have strong ties to topological constructions, we take here a group theoretical point of view.