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The minimal extension of P-localization on groups

Published online by Cambridge University Press:  24 October 2008

A. J. Berrick
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Kent Ridge Crescent, Singapore0511
G. C. Tan
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Kent Ridge Crescent, Singapore0511

Extract

Let P be a fixed set of primes, the category of all groups and group-homomorphisms, and the full subcategory of nilpotent groups. In [9], an idempotent functor called P-localization, was defined so as to extend the ℤ-module-theoretic localization of abelian groups. There are two well-known extensions of e to , namely, Bousfield's P-localization [2], [4], denoted by EZP, and Ribenboim's P-localization [13], usually denoted by ( )P. Ribenboim's P-localization is the maximal extension among localizations extending e to in that it maximizes the number of groups in its image [7]. The localized groups obtained after applying Ribenboim's P-localization are precisely the P-local groups, that is, the groups having unique nth-root for every n whichis co-prime to P, [13]. Being maximal is equivalent to this class of P-localized groups being the saturated class of groups generated by e–equivalences, that is, group homomorphisms between nilpotent groups which become isomorphisms after applying e.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1996

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