Canonical universal locally finite groups

Abstract

We prove that for lambda = beta_omega or just lambda strong limit singular of cofinality aleph_0, if there is a universal member in the class K^lf_lambda of locally finite groups of cardinality lambda, then there is a canonical one (parallel to special models for elementary classes, which is the replacement of universal homogeneous ones and saturated ones in cardinals lambda = lambda^ < lambda). For this, we rely on the existence of enough indecomposable such groups, as proved in "Density of indecomposable locally finite groups". We also more generally deal with the existence of universal members in general classes for such cardinals.