Sh:739

• Göbel, R., & Shelah, S. (2002). Constructing simple groups for localizations. Comm. Algebra, 30(2), 809–837. arXiv: math/0009089 DOI: 10.1081/AGB-120013184 MR: 1883027
• Abstract:
  A group homomorphism $\eta: A\to H$ is called a localization of $A$ if every homomorphism $\varphi:A\to H$ can be ‘extended uniquely’ to a homomorphism $\Phi:H\to H$ in the sense that $\Phi\eta=\varphi$. 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 $\eta:A\to H$ 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 $A_n\to SO_{n-1}({\mathbb R})$ for a natural embedding of the alternating group $A_n$ is a localization if $n$ even and $n\geq 10$. Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [GRSh:701] 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.
• Version 2001-02-20_11 (25p) published version (31p)

@article{Sh:739,
 author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon},
 title = {{Constructing simple groups for localizations}},
 journal = {Comm. Algebra},
 fjournal = {Communications in Algebra},
 volume = {30},
 number = {2},
 year = {2002},
 pages = {809--837},
 issn = {0092-7872},
 mrnumber = {1883027},
 mrclass = {20F99},
 doi = {10.1081/AGB-120013184},
 note = {\href{https://arxiv.org/abs/math/0009089}{arXiv: math/0009089}},
 arxiv_number = {math/0009089},
 keyword = {localizations of groups, simple groups which are complete, free
products with amalgamation, HNN-extension, Eilenberg--MacLane
spaces}
}