Characterization of Abelian groups with a minimal generating set

Abstract

We characterize Abelian groups with a minimal generating set: Let τ A denote the maximal torsion subgroup of A. An infinitely generated Abelian group A of cardinality ϰ has a minimal generating set iff at least one of the following conditions is satisfied:
1. dim(A/pA) = dim(A/qA) = ϰ for at least two different primes p, q.
2. dim(τ A/pτA) = ϰ for some prime number p.
3.Σ {dim(A/(pA+B)) | dim(A/(pA+B)) < ϰ} = ϰ for every finitely generated
subgroup B of A.
Moreover, if the group A is uncountable, property (3) can be simplified to
(3’) Σ{dim(A/pA) | dim(A/pA) < ϰ} = ϰ,
and if the cardinality of the group A has uncountable cofinality, then A has a minimal generating set iff any of properties (1) and (2) is satisfied.

Keywords: Minimal generating, Abelian group, torsion, torsion free, countable, uncountable, divisible.