Subnormal closure of a homomorphism

Abstract

Let $$\varphi :\Gamma \rightarrow G$$ 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, namely a finite composition of the underlying maps of crossed modules. We search for a universal such factorization. When $$\Gamma $$ and $$G$$ are finite we show that such universal factorization exists: $$\Gamma \rightarrow \Gamma _{\infty }\rightarrow G,$$ 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 $$\Gamma _{\infty }\twoheadrightarrow \mathcal {C}$$ 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 $$\Gamma _\infty $$ .