Large localizations of finite simple groups

A group homomorphism η: H → G is called a localization of H if every homomorphism φ: H Φ G can be 'Extended uniquely' to a homomorphism Φ: G → G in the sense that Φη = φ. Libman showed that a localization of a finite group need not be finite. This is exemplified by a well-known representation A_n → SO_n-1 (ℝ) of the alternating group A_n, which turns out to be a localization for n even and n ≥ 10. Emmanuel 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.