Subnormal closure of a homomorphism

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Let φΓ→ G be a homomorphism of groups. In this paper we introduce the notion of a subnormal map (the inclusion of a subnormal subgroup into a group being a basic prototype). We then consider factorizations Γψ Mn G of φ, with n a subnormal map. We search for a universal such factorization. When Γ and G are finite we show that such universal factorization exists: Γ→Γ∞→ G, where Γ∞ is a hypercentral extension of the subnormal closure C of φ(Γ) in G (i.e. the kernel of the extension Γ∞→ C is contained in the hypercenter of Γ∞). This is closely related to the a relative version of the Bousfield-Kan Z-completion tower of a space. The group Γ∞ is the inverse limit of the normal closures tower of φ introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit Γ∞.