[Paper Review] Localizations of Groups

Rüdiger Göbel, Saharon Shelah|ArXiv.org|Sep 8, 2000

Homotopy and Cohomology in Algebraic Topology8 references3 citations

TL;DR

This paper establishes that in ZFC (without assuming the Generalized Continuum Hypothesis), any finite non-abelian simple group admits localizations of arbitrarily large cardinality. Using techniques from combinatorial group theory and the existence of rigid families of cotorsion-free abelian groups, the authors construct a simple group H of successor cardinality λ = µ⁺ that universally embeds all groups in a given family of suitable groups, ensuring all monomorphisms are inner automorphisms and all localizations are trivially extendable.

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.

Motivation & Objective

• To remove the dependence on the Generalized Continuum Hypothesis (GCH) in prior results on localizations of finite non-abelian simple groups.
• To generalize the existence of large localizations beyond simple groups to a broader class of 'suitable' finite groups with trivial center and complete automorphism groups.
• To construct a group H of arbitrarily large cardinality λ = µ⁺ that universally embeds all groups in a given family of suitable groups via monomorphisms induced by inner automorphisms.
• To show that all monomorphisms H → H are inner automorphisms, ensuring the localization property holds universally.

Proposed method

• Constructs a group H of cardinality λ = µ⁺ for any infinite cardinal µ using a direct limit process over a hierarchy of groups Hαj(ε) indexed by ordinals.
• Employs a rigid family of cotorsion-free abelian groups {Ui : i < κ} to ensure uniqueness of embeddings and control over centralizers.
• Uses the existence theorem from [5] on large rigid families of abelian groups to avoid reliance on GCH or combinatorial games like Hart–Laflamme–Shelah.
• Applies a model-theoretic Black Box (Black Box 5.1) to predict embeddings and submodels within a stationary subset of λ, ensuring universal embedding properties.
• Imposes conditions on centralizers: for any isomorphic copy A′ ⊆ H of a suitable group A, the centralizer cHA′ is trivial, enforcing rigidity.
• Proves that any monomorphism ϕ: A → H for A ∈ A arises from conjugation by some h ∈ H, ensuring the localization is universal.

Experimental results

Research questions

• RQ1 Can the existence of arbitrarily large localizations for finite non-abelian simple groups be established in ZFC without assuming GCH?
• RQ2 What structural properties must a family of finite groups satisfy to admit universal localizations of arbitrarily large cardinality?
• RQ3 How can the rigidity of embeddings and trivial centralizers be enforced in a large group H to ensure all monomorphisms are inner automorphisms?
• RQ4 To what extent can the construction be generalized from simple groups to the broader class of 'suitable' groups (finite, trivial center, complete automorphism group)?
• RQ5 Can model-theoretic prediction principles (like the Black Box) be used to construct universal localizations in group theory without combinatorial set-theoretic assumptions?

Key findings

• For any infinite cardinal µ, there exists a simple group H of cardinality λ = µ⁺ such that every monomorphism H → H is an inner automorphism.
• Any group A in a family of suitable groups embeds into H via monomorphisms induced by conjugation in H, and any two distinct copies of such groups intersect trivially.
• The centralizer of any isomorphic copy A′ ⊆ H of a suitable group A is trivial, ensuring strong rigidity in the embedding.
• The construction avoids GCH and combinatorial games, relying instead on the existence of rigid families of cotorsion-free abelian groups and a model-theoretic Black Box.
• The result implies that for any finite non-abelian simple group A, there are localizations of arbitrarily large cardinality in ZFC, answering a question of Dror Farjoun.
• The Main Theorem 1.3 confirms that the Eilenberg–MacLane space K(A,1) has a proper class of distinct homotopy types that are localizations, valid in ZFC.