  A group homomorphism eta:A-> H is called a localization of A if every
homomorphism phi:A-> H can be `extended uniquely' to a homomorphism Phi:H-> H
in the sense that Phi eta = phi. This categorical concepts, 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. For localizations eta:A-> H of (almost) commutative structures
A often H resembles properties of A, 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 A is a finite (non-abelian) group. Libman showed that A_n->
SO_{n-1}(R) for a natural embedding of the alternating group A_n is a
localization if n even and n >= 10 . Answering an immediate question by Dror
Farjoun and assuming the generalized continuum hypothesis GCH we recently
showed in math.LO/9912191 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 A 's.
