Localization and Genus in Group Theory

We provide a unifying category theoretical framework to discuss various kinds of local global phenomena. Specializing to localization of groups at sets of primes P , we identify a large class of groups for which localization supports a passage from local information to global information. Local global principles for groups in this class are established and used to calculate certain homomorphism sets as well as splittings of epimorphisms and monomorphisms from local data. Introduction Let J? be a family of localizing functors on a category ^, as described in Adams [1]. We would then like to obtain information abut the objects and morphisms in %? from their localizations by the functors in fH?. Such methods, called local global principles, are now in common use in subjects like number theory, commutative algebra, and homotopy theory of nilpotent spaces. In addition various notions like "genus" [13], "type" [21], "clone" [14], etc., have been introduced to refer to collections of objects which have isomorphic local data. Here we extend the method of local global principles to the category of groups, and we consider the resulting genera of groups. Our approach is guided by a general unifying category theoretical framework for the discussion of local global phenomena. The key ideas underlying this framework are outlined in A, B, C below. Details can be found in §1. A. If SC is a family of localizing functors on a category ^, then the members of 5? are related by natural transformations coming from the following *£intrinsically defined partial ordering: LM > Lv if the class of L^-local objects (objects for which the localizing map X —» L^X is an isomorphism) contains all L„-local objects. We form the category 5? ^ whose objects are diagrams in f modeled on the partial order of 7777? . In particular, applying the functors in 5f to an object X e *W yields an object £?(X) e S? %> . This process is a functor -2": f — Jg'g'. B. The extent to which .2Mocal data actually determine objects in f is measured by the "size" of the fibers of the functor 5C: f -»ff? i?. Thus we define the genus of X_ € 5f ^ to be the collection of all isomorphism classes of objects Received by the editors November 10, 1993. 1991 Mathematics Subject Classification. Primary 18E35, 20E34, 55P60. The author was supported by NSERC of Canada. ©1995 American Mathematical Society 155 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use