CONSTRUCTING SIMPLE GROUPS FOR LOCALIZATIONS

Abstract

ABSTRACT A group homomorphism is called a localization> of if every homomorphism can be extended uniquely' to a homomorphism in the sense that . This categorical concept, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the introduction. For localizations of (almost) commutative structures often resembles properties of , e.g. size or satisfying certain systems of equalities and non-equalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if is a finite (non-abelian) group. Libman showed that for a natural embedding of the alternating group is a localization if is even and . Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [12] that any non-abelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of 's. The new techniques exploit abelian centralizers of free (non-abelian) subgroups of which constitute a rigid system of cotorsion-free abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg–Mac Lane space for many groups . The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13].