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2010•Quaestiones MathematicaeRequires access

Abelian Groups with a Minimal Generating Set

Pavel Růžička

Open publisher page 8 citations

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.

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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 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.

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OpenAlex reports 8 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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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 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.

Key concepts: Mathematics, Abelian group, Generating set of a group, Set (abstract data type), Pure mathematics, Discrete mathematics, Combinatorics, Algebra over a field

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