Abelian Groups with a Minimal Generating Set

Abstract

We study the existence of minimal generating sets in Abelian groups. We prove that Abelian groups with minimal generating sets are not closed under quotients, nor under subgroups, nor under in.nite products. We give necessary and su.cient conditions for existence of a minimal generating set providing that the Abelian group is uncountable, torsion, or torsion-free completely decomposable.