Finite simple groups and localization

Abstract

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.

Non-simple localizations of finite simple groups

“…We get many other examples of this type using [RST,Corollary 2.2]. All sporadic groups appearing in this corollary which have trivial Schur multiplier (that is M 11 , M 23 , M 24 , J 1 , J 4 , Co 2 , Co 3 , He, Fi 23 , HN, and Ly) admit the double cover of an alternating group as localization (as Mult(A n ) is cyclic of order 2 for n > 7).…”

“…Remark 1.9. The inclusion Fi 23 → B of the Fischer group into the baby monster is a localization by [RST,Section 3(vi)]. This yields a localization Fi 23 →B.…”

“…Note that M 11 is not maximal inM 12 (the maximal subgroup is M 11 × C 2 ), so this does not contradict Proposition 1.1. The following inclusions are localizations: Co 2 → Co 1 and Co 3 → Co 1 by [RST,Section 4]. As the smaller group is superperfect we get localizations Co 2 →Co 1 and Co 3 →Co 1 .…”

“…Naturally, the second part of the paper deals with the effect of a localization on the outer automorphism group, which roughly speaking is dual to the Schur multiplier as it encodes the information about the "super-group" of all automorphisms. We first find a general criterion telling when an inclusion of automorphism groups is a localization (in the spirit of [RST,Theorem 1.4]). If we assume, moreover, that we start with a localization of finite simple groups, the conditions become quite elementary, see Theorem 2.4.…”

On localizations of torsion abelian groups

“…For instance, the effect on the fundamental group can often be described by means of group-theoretical localization functors. Motivated by this relationship, important advances have recently been achieved in the study of group localization functors, especially related to their behavior on certain classes of groups, like for example finite or nilpotent groups ( [6], [19], [20]), simple groups ( [16], [17], [22], [23], [26]) or perfect groups ( [1], [24]). The papers by Casacuberta [6] and Libman [19] are good starting points for non-expert readers.…”