Existentially closed locally finite groups

TL;DR

This paper studies existentially closed locally finite groups, establishing their existence, uniqueness, and properties, including automorphism characteristics and parallels to stability theory, within a set-theoretic framework.

Contribution

It introduces a canonical construction of existentially closed locally finite groups of any given cardinality, proving their uniqueness and exploring their automorphism and stability properties.

Findings

Existence of canonical existentially closed locally finite groups for every cardinality.

Many such groups are complete with no non-inner automorphisms.

Parallel development of stability theory concepts for these groups.

Abstract

We investigate this class of groups originally called ulf (universal locally finite groups) of cardinality $\lambda$. We prove that for every locally finite group $G$ there is a canonical existentially closed extention of the same cardinality, unique up to isomorphism and increasing with $G$. Also we get, e.g. existence of complete members (i.e. with no non-inner automorphisms) in many cardinals (provably in ZFC). We also get a parallel to stability theory in the sense of investigating definable types.