Non-simple localizations of finite simple groups

Abstract

Often a localization functor (in the category of groups) sends a finite simple group to another finite simple group. We study when such a localization also induces a localization between the automorphism groups and between the universal central extensions. As a consequence we exhibit many examples of localizations of finite simple groups which are not simple.

On localizations of quasi-simple groups with given countable center

“…Recall that Out G denotes the outer automorphism group Aut G/G of a simple group G (where we identify G and G * ). Lemma 2.1 of [19] holds for infinite groups, too.…”

“…Aut G contains G. In particular, any endomorphism of Aut G is either a monomorphism or contains G in its kernel. Lemma 2.2 of [19] states that any non-abelian finite simple subgroup of Aut G is contained in G. The easy argument is based on the solution of the Schreier conjecture, which ensures that Out G is solvable for every finite non-abelian simple group G. But for infinite groups this is not longer true, because from [2] we know that all groups are outer automorphism groups of simple groups. In order to proceed to infinite groups we will assume that the outer automorphism group is hyperabelian, which extends solvability.…”

“…Libman [13] showed that the natural inclusion A n ֒→ A n+1 of alternating groups (for n ≥ 7) is a localization. This motivated the study of localizations between simple groups in [18,19,17]. First examples of infinite localizations of A n (for n ≥ 10) were also given in [13]; in this case A n ֒→ SO(n) is a localization.…”

“…Then we will show the following claims. The simple groups in Theorem 5.9 can be investigated more thoroughly if results on finite simple groups obtained in [18], [19] are first extended to infinite groups. It is important to observe that localizations H → G between two simple groups often induce localization Aut(H) → Aut(G), as will be discussed next.…”

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.…”