The minimal extension of P -localization on groups

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 EZP , 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 LP , say, yielding the minimum number of LP -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 LP 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 LP on groups whose lower central series stabilizes. We use our localization and Bousfield’s P -