Canonical universal locally finite groups

Abstract

We prove that the existence of universally locally finite groups implies that there is a canonical one in any strong limit singular cardinality of countable cofinality. Moreover, those canonical groups are parallel to the special models for complete first order theories. For showing the existence we rely on the existence of enough indecomposable such groups as we proved in [Rend. Sem. Mat. Univ. Padova 144 (2020), 253–270]. More generally, we also deal with the existence of a universal member in general classes for such cardinals.