Localizations of Groups

TL;DR

This paper explores the concept of localizations in group theory, extending classical notions to non-commutative groups, with applications in homotopy theory and results on finite simple groups without relying on GCH.

Contribution

It generalizes the theory of localizations for groups, removes the GCH assumption, and extends results to a broader class of groups.

Findings

Localizations of finite abelian groups remain finite abelian groups.

Non-abelian finite simple groups can have arbitrarily large localizations.

The results are established within standard set theory, removing GCH dependency.

Abstract

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.