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2010•Quaestiones MathematicaeRequires access

Abelian groups with a minimal generating set

Pavel Růžička

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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 infinite products. We give necessary and sufficient conditions for existence of a minimal generating set providing that the Abelian group is uncountable, torsion, or torsion-free completely decomposable. Quaestiones Mathematicae 33 (2010), 147–159

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What this paper is about

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 infinite products. We give necessary and sufficient conditions for existence of a minimal generating set providing that the Abelian group is uncountable, torsion, or torsion-free completely decomposable. Quaestiones Mathematicae 33 (2010), 147–159

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Available 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 infinite products. We give necessary and sufficient conditions for existence of a minimal generating set providing that the Abelian group is uncountable, torsion, or torsion-free completely decomposable. Quaestiones Mathematicae 33 (2010), 147–159

Key concepts: Mathematics, Abelian group, Uncountable set, Generating set of a group, Rank of an abelian group, Torsion subgroup, Quotient, Torsion (gastropod)

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