TITLE:   Large localizations of finite simple groups

AUTHORS:  Ruediger Goebel, Jose L. Rodriguez, and Saharon Shelah



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 $H$.