Large localizations of finite simple groups
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Publication:4539578
DOI10.1515/CRLL.2002.072zbMATH Open1009.20021arXivmath/9912191OpenAlexW2963762603WikidataQ105978898 ScholiaQ105978898MaRDI QIDQ4539578
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Publication date: 9 July 2002
Published in: Journal für die reine und angewandte Mathematik (Crelles Journal) (Search for Journal in Brave)
Abstract: A group homomorphism eta:H-->G is called a localization of H if every homomorphism phi:H-->G can be `extended uniquely' to a homomorphism Phi:G-->G in the sense that Phi eta=phi. Libman showed that a localization of a finite group need not be finite. This is exemplified by a well-known representation A_n-->SO_{n-1}(R) of the alternating group A_n, which turns out to be a localization for n even and n>9. Dror Farjoun asked if there is any upper bound in cardinality for localizations of A_n. In this paper we answer this question and prove, under the generalized continuum hypothesis, that every non abelian finite simple group H, has arbitrarily large localizations. This shows that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg--Mac Lane space K(H,1) for any non abelian finite simple group H.
Full work available at URL: https://arxiv.org/abs/math/9912191
Simple groups: sporadic groups (20D08) Simple groups (20E32) Automorphisms of infinite groups (20E36) Simple groups: alternating groups and groups of Lie type (20D06)
Cites Work
Related Items (5)
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