2000/09/08 by Rüdiger Göbel, Saharon Shelah, Göbel, Rüdiger +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings and Algebras (math.RA) #math.GR #math.LO #math.RA
paper · pdf · doi:10.48550/arxiv.math/0009089
published as Comm. Algebra 30 No. 2 (2002) 809--837
arxiv created 2000/09/08 · openalex publication_date 2000/09/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 An-> SOn-1(R) for a natural embedding of the alternating group An 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.