On the dimension of an Abelian group

Timo Tossavainen and Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 267–275
DOI: 10.7546/nntdm.2021.27.4.267-275
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Authors and affiliations

Timo Tossavainen
Department of Health, Education and Technology, Lulea University of Technology
SE-97187 Lulea, Sweden

Pentti Haukkanen
Faculty of Information Technology and Communication Sciences, Tampere University
FI-33014 Tampere University, Finland

Abstract

We introduce a measure of dimensionality of an Abelian group. Our definition of dimension is based on studying perpendicularity relations in an Abelian group. For G ≅ ℤⁿ, dimension and rank coincide but in general they are different. For example, we show that dimension is sensitive to the overall dimensional structure of a finite or finitely generated Abelian group, whereas rank ignores the torsion subgroup completely.

Keywords

• Abelian group
• Dimension
• Perpendicularity

2020 Mathematics Subject Classification

• 11A41
• 20K25
• 52C99

References

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