TITLE: Large localizations of finite simple groups
AUTHORS: Ruediger Goebel, Jose L. Rodriguez, and Saharon Shelah
R.Goebel@uni-essen.de, jlrodri@mat.uab.es, shelah@math.huji.ac.il
ABSTRACT:
A group homomorphism $\eta: H\to G$ is called a localization of $H$
if every homomorphism $\varphi : H\to G$ can be
`extended uniquely' to a homomorphism $\Phi :G\to G$
in the sense that $\Phi \eta = \varphi$.
Libman showed that a localization of a finite group need not be finite.
This is exemplified by a well-known representation
$A_n\to SO_{n-1}(\R)$ of the alternating group $A_n$,
which turns out to be a localization for $n$ even and $n\geq 10$.
Dror Farjoun asked if there is any upper bound in cardinality for
localizations of $A_n$. In this paper we answer this question and
prove, under the generalized continuum hypothesis,
that every non abelian finite simple group $H$,
has arbitrarily large localizations.
This shows that there is a proper class of distinct homotopy types
which are localizations of a given Eilenberg--Mac Lane space $K(H,1)$
for any non abelian finite simple group $H$.