The minimal extension of P-localization on groups

Abstract

Let P be a fixed set of primes, G the category of all groups and grouphomomorphisms, and N the full subcategory of nilpotent groups. In [9], an idempotent functor e: N → N , called P -localization, was defined so as to extend the Z-module-theoretic localization of abelian groups. There are two wellknown extensions of e to G, namely, Bousfield's P -localization [2], [4], denoted by EZ P , and Ribenboim's P -localization [13], usually denoted by ( ) P . Ribenboim's P -localization is the maximal extension among localizations extending e to G 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 which is coprime 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.This suggests the problem of whether e has a minimal extension to G, a localization L P , say, yielding the minimum number of L P -local groups. A general attack on this problem is to be found in [6]. The solution below is arguably more concrete, concentrating attention on groups of a given cardinality, where L P takes on a more recognizable form. The reason for this strategy is to overcome technical difficulties that arise because the collection of all (P -local nilpotent) groups fails to form a set.In the final section, we discuss the effect of the localization L P on groups whose lower central series stabilizes. We use our localization and Bousfield's P -1

P-nilpotent completion is not idempotent

“…It is the "smallest" (in the sense that it provides the least local objects) idempotent monad which extends P -localization on the category of nilpotent groups to the category of groups. This minimal P -localization coincides with the P -nilpotent completion on groups which have finitely generated abelianization [3] and groups with stable lower central series [2].…”

“…This turns out to be the minimal P -localization, which is also obtained in [2]. It is the "smallest" (in the sense that it provides the least local objects) idempotent monad which extends P -localization on the category of nilpotent groups to the category of groups.…”