### Constructing Simple Groups For Localizations

by Goebel and Shelah. [GbSh:739]

Communication in Algebra, 2002

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 [GRSh:701] 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.

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