Constructing Simple Groups for Localizations

Abstract

Abstract. A group homomorphism η : A → H is called a localization of A if every homomorphism ϕ : A → H can be 'extended uniquely' to a homomorphism Φ : H → H in the sense that Φη = ϕ. This categorical concept, 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, see the introduction.For localizations η : 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 is even and n ≥ 10. Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [12] 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. The new techniques exploit abelian centralizers of free (non-abelian) subgroups of H which constitute a rigid system of cotorsion-free abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg-Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13].

Cellular covers of groups

“…Both concepts are considered in [8]. We notice, however, that statements similar to the above are either open questions or patently false for the localizations L(M) of M. See, for example, the results of Libman [14,15], of Göbel and Shelah [12] and of Aschbacher [2], who considered localizations of finite p-groups; see also [16]. The main open question for localization is whether or not an arbitrary localization L(P) of a finite p-group P is always a quotient of P.…”

E(R)-algebras that are sharply transitive modules

“…The crucial definition and the basic properties of E-rings (for R = Z) come from the seminal paper of Schultz [27] from 1973. Some years later it became clear that E-rings are not only useful in module theory, but also have fruitful applications in homotopy theory and in non-commutative group theory; see [8,9,14,16,23,26]. They are useful tools in the work of Pierce (see [25]), and the existence of arbitrarily large indecomposable E-rings was first shown in [5].…”

Envelopes and covers for groups

“…The use of infinite combinatorial principles, like Shelah's Black Box and its relatives, has allowed one to produce either arbitrarily large localizations or cellular covers for certain groups. For instance, in [14] and [16] the authors constructed large localizations of finite simple groups. Countable as well as arbitrarily large cellular covers of cotorsion-free abelian groups with given ranks have also been constructed (see [5], [12] [15]).…”

“…For envelopes we have the following characterization. Göbel-Rodríguez-Shelah [14] and Göbel-Shelah [16] found out that every finite simple group admits arbitrarily large localizations.…”