A permutation group acting transitively on certain collections of models

TL;DR

The work addresses whether a proper dense subgroup of the symmetric group on a countable set can act transitively on the isomorphism class of inductively flexible relational structures. It combines a construction of a dense strongly independent set in $\operatorname{Sym}(\Omega)$ with a MA$(\sigma$-centered$)$-based forcing argument to build a proper dense subgroup $G$ that acts transitively on $\overline{\mathbb{M}}$ for all inductively flexible $\mathbb{M}$, yielding a consistency result relative to ZFC. The approach yields concrete corollaries: $G$ is transitive on $\aleph_0$-sections and on $\mathbb{Q}$-type orderings, thus giving consistent positive answers to Kourovka Notebook questions of Peter M. Neumann. This demonstrates that transitivity on isomorphism classes can be achieved by a dense subgroup under MA$(\sigma$-centered$)$, highlighting interactions between set-theoretic axioms and model-theoretic flexibility.

Abstract

It is shown, from $σ$-centered Martin's Axiom, that there exists a proper dense subgroup of the symmetric group on a countably infinite set whose natural action on sufficiently flexible relational structures is transitive. This allows us to give consistent positive answers to some questions of Peter M. Neumann from the 1980s.

Theorems & Definitions (16)

• Definition 1
• Theorem 1
• Definition 2
• Lemma 2
• Definition 3
• Lemma 4
• proof
• Definition 4
• Lemma 5
• proof