JournalsggdVol. 14, No. 3pp. 1023–1042

On localizations of quasi-simple groups with given countable center

  • Ramón Flores

    Universidad de Sevilla, Spain
  • José L. Rodríguez

    Universidad de Almería, Spain
On localizations of quasi-simple groups with given countable center cover
Download PDF

A subscription is required to access this article.

Abstract

A group homomorphism i ⁣:HGi\colon H \to G is a localization of HH, if for every homomorphism φ ⁣:HG\varphi\colon H\to G there exists a unique endomorphism ψ ⁣:GG\psi\colon G\to G such that iψ=φi \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