Subnormal closure of a homomorphism

Let $φ\colonΓ\to G$ 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 $Γ\xrightarrowψ M\xrightarrow{n} G$ 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: $Γ\toΓ_{\infty}\to G,$ where $Γ_{\infty}$ is a hypercentral extension of the subnormal closure $\mathcal{C}$ of $φ(Γ)$ in $G$ (i.e.~the kernel of the extension $Γ_{\infty}\to {\mathcal C}$ is contained in the hypercenter of $Γ_{\infty}$). This is closely related to the a relative version of the Bousfield-Kan $\mathbb{Z}$-completion tower of a space. The group $Γ_{\infty}$ 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 $Γ_{\infty}$.

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