A group homomorphism eta:A-> H is called a localization of A if every homomorphism phi:A-> H can be `extended uniquely' to a homomorphism Phi:H-> H in the sense that Phi eta = phi. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory. For localizations eta:A-> H of (almost) commutative structures A often H resembles properties of A, e.g. size or satisfying certain systems of equalities and non-equalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if A is a finite (non-abelian) group. Libman showed that A_n-> SO_{n-1}(R) for a natural embedding of the alternating group A_n is a localization if n even and n >= 10 . Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in math.LO/9912191 that any non-abelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of A 's.
A. K. Bousfield的群的$H\mathbb Z$ -局部化同调反转群的两连通同态。J. P. Levine的群的代数闭包反转了同调二连通的群和群表示的群之间的同态。我们解决了一个老问题有关Bousfield $H\mathbb Z$ -局部化的群体，并回答了两个问题Levine有关代数闭包的群体。特别地，我们证明了从一个群$G$到它的Bousfield $H\mathbb Z$ -局部化的自然同态的核并不总是$G$ -完美子群。在群的代数闭包的情况下，我们证明了类似的结果，即这个核不总是一个不可见子群。
Beijing Institute of Mathematical Sciences and Applications（北京应用数学与计算物理研究所）；Tsinghua University（清华大学）；National Research University Higher School of Economics（国立研究大学高等经济学院）
