A group homomorphism is a localization of , if for every homomorphism there exists a unique endomorphism such that (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. 1023–1042DOI 10.4171/GGD/573