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
Download Full paper: PDF, 192 Kb
Download Corrigendum: PDF, 110 Kb


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


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.


  • Abelian group
  • Dimension
  • Perpendicularity

2020 Mathematics Subject Classification

  • 11A41
  • 20K25
  • 52C99


  1. Burness, T. C., Garonzi, M., & Lucchini, A. (2020). On the minimal dimension of a finite simple group. Journal of Combinatorial Theory, Series A, 171, Article ID 105175.
  2. Davis, G. (1975). Orthogonality relations on Abelian groups. Journal of the Australian Mathematical Society. Series A, 19, 173–179.
  3. Fernando, R. (2015). On an inequality of dimension-like invariants for finite groups. Preprint, available online at https://arxiv.org/abs/1502.00360.
  4. Haukkanen, P., Mattila, M., Merikoski, J. K., & Tossavainen, T. (2013). Perpendicularity in an Abelian group. International Journal of Mathematics and Mathematical Sciences, Article ID 983607.
  5. Haukkanen, P., Merikoski, J. K., & Tossavainen, T. (2011). Axiomatizing perpendicularity and parallelism. Journal of Geometry and Graphics, 15, 129–139.
  6. Mattila, M., Haukkanen, P., Merikoski, J. K., & Tossavainen, T. (2017). Maximal perpendicularity in certain Abelian groups. Acta Universitatis Sapientiae, Mathematica, 9, 235–247.
  7. Tossavainen, T., & Haukkanen, P. (2021). Exploring perpendicularity and parallelism through play. To appear in Mathematics Magazine.
  8. Veksler, A. I. (1967). Linear spaces with disjoint elements and their conversion into vector lattices. Leningradskiı Gosudarstvennyı Pedagogiceskiı Institut imeni AI Gercena. Ucenye Zapiski, 328, 19–43 (Russian).


Related papers

Cite this paper

Tossavainen, T., & Haukkanen, P. (2021). On the dimension of an Abelian group. Notes on Number Theory and Discrete Mathematics, 27(4), 267-275, doi: 10.7546/nntdm.2021.27.4.267-275.

Comments are closed.