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 Citation[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 Citation[5]. Like Citation[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 Citation[13].
ACKNOWLEDGMENT
The authors are supported by the project No. G 0545-173, 06/97 of the German-Israeli Foundation for Scientific Research & Development. GbSh:739 in Shelah's list of publications.