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.

中文翻译：

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给定可数中心的准简单群的局部化

组同态$ i \ colon H \ to G $是$ H $的局部化，如果每个同态$ \ varphi \ colon H \ to G $都存在唯一的内同态$ \ psi \ colon G \ to G $，例如$ i \ psi = \ varphi $（地图在右侧起作用）。Göbel和Trlifaj在[18，问题30.4（4），p。1中询问。831]哪个阿贝尔群是简单群本地化的中心。接近这个问题，我们表明，每个可数的阿贝尔群确实是准简单群局部化的中心，即简单群的中心扩展。证明使用Obraztsov和Ol'shanskii的无限简单群的构造，带有特殊的子群格，并且由第二作者以及Scherer，Thévenaz和Viruel扩展了有限简单群的局部化结果。