Subnormal closure of a homomorphism

TL;DR

This paper introduces a universal factorization of group homomorphisms through subnormal maps, constructing a hypercentral extension that generalizes the subnormal closure and relates to Bousfield-Kan completions.

Contribution

It defines the subnormal closure of a homomorphism and constructs a universal factorization via a hypercentral extension, extending previous work on normal closures.

Findings

Existence of a universal factorization for finite groups

Construction of a hypercentral extension as a limit of normal closures

Proved stability and finiteness properties of the tower and its inverse limit

Abstract

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 $\Gamma_{\infty}$.