Density of indecomposable locally finite groups

Abstract

We prove that for any locally finite group there is an extension of the same cardinality which is indecomposable for almost all regular cardinals smaller than its cardinality. Note that a group $G$ is called $\theta$-indecomposable when for every increasing sequence $⟨G_i:i<\theta ⟩$ of subgroups with union $G$ there is $i<\theta$ such that $G=G_i$.