Finite simple groups and localization

Jose L. Rodriguez, Jerome Scherer and Jacques Thevenaz

20D06, 20D08, 55P60 

Departamento de Geometria, Topologia y Quimica Organica
Universidad de Almeria
E--04120 Almeria
Spain

Institut de Mathematiques
Universite de Lausanne
CH--1015 Lausanne
Switzerland


The purpose of this paper is to explore the concept of localization, which
comes from homotopy theory, in the context of finite simple groups. We give an
easy criterion for a finite simple group to be a localization of some simple
subgroup and we apply it in various cases. Iterating this process allows us to
connect many simple groups by a sequence of localizations. We prove that all
sporadic simple groups (except possibly the Monster) and several groups of Lie
type are connected to alternating groups. The question remains open whether or
not there are several connected components within the family of finite simple
groups. In some cases, we also consider automorphism groups and universal
covering groups and we show that a localization of a finite simple group may
not be simple.