Let $\varphi\colon\Gamma\to G$ 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 $\Gamma\xrightarrow{ \psi} M\xrightarrow{ n} G$ of $\varphi,$ with $n$ a subnormal map. We search for a universal such factorization. When $\Gamma$ and $G$ are finite we show that such universal factorization exists: $\Gamma\to\Gamma_{ \infty}\to G,$ where $\Gamma_{ \infty}$ is a hypercentral extension of the subnormal closure $\mathcal{ C}$ of $\varphi(\Gamma)$ in $G$ (i.e.~the kernel of the extension $\Gamma_{ \infty}\to { \mathcal C}$ is contained in the hypercenter of $\Gamma_{ \infty}$). This is closely related to the a relative version of the Bousfield-Kan $\mathbb{ Z}$-completion tower of a space. The group $\Gamma_{ \infty}$ is the inverse limit of the normal closures tower of $\varphi$ introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit $\Gamma_{ \infty}$.
Categories: math.GR math.AT