On localizations of quasi-simple groups with given countable center

TL;DR

This paper demonstrates that every countable abelian group can be realized as the center of a localization of a quasi-simple group, expanding understanding of group localizations and centers.

Contribution

It shows that all countable abelian groups are centers of localizations of quasi-simple groups, answering a question about centers of localizations of simple groups.

Findings

Every countable abelian group is the center of some localization of a quasi-simple group.

Utilizes constructions of infinite simple groups with specific subgroup lattices.

Extends previous results on localizations of finite simple groups.

Abstract

A group homomorphism $i: H \to G$ is a localization of $H$ if for every homomorphism $\varphi: H\rightarrow G$ there exists a unique endomorphism $\psi: G\rightarrow G$, such that $i \psi=\varphi$ (maps are acting on the right). G\"{o}bel and Trlifaj asked in \cite[Problem 30.4(4), p. 831]{GT12} which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Th\'{e}venaz and Viruel.