# On localizations of quasi-simple groups with given countable center

### Ramón Flores

Universidad de Sevilla, Spain### José L. Rodríguez

Universidad de Almería, Spain

## Abstract

A group homomorphism $i\colon H \to G$ is a *localization* of $H$, if for every homomorphism $\varphi\colon H\to G$ there exists a unique endomorphism $\psi\colon G\to G$ such that $i \psi=\varphi$ (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] 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, Thévenaz and Viruel.

## Cite this article

Ramón Flores, José L. Rodríguez, On localizations of quasi-simple groups with given countable center. Groups Geom. Dyn. 14 (2020), no. 3, pp. 1023–1042

DOI 10.4171/GGD/573