Relative Schur multipliers and universal extensions of group homomorphisms

TL;DR

This paper constructs a universal central extension related to a given group homomorphism, using relative Schur multipliers, providing a homological framework for lifting solutions and hypercentral factorizations.

Contribution

It introduces a universal extension construction for homomorphisms with a surjective abelianization, utilizing relative Schur multipliers to analyze lifting problems.

Findings

Defines the kernel as the relative Schur multiplier group.

Establishes a universal property for the constructed extension.

Provides homological obstructions for solving equations in groups.

Abstract

In this note, starting with any group homomorphism $f\colon\Gamma\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $\Gamma$ with the same surjective property, such that for any central extension $m\colon M\twoheadrightarrow G,$ under $f,$ there is a unique homomorphism $U\to M$ with the obvious commutation condition. The kernel of $u$ is the relative Schur multiplier group $H_2(G,\Gamma;\mathbb{Z})$ as defined in the paper. The case where $G$ is perfect corresponds to $\Gamma=1$. This yields homological obstructions to lifting solution of equations in $G.$ Upon repetition, for finite groups, this gives a universal hypercentral factorization of the map $f\colon\Gamma\to G$.