On groups in which irreducible systems of elements form a matroid

Authors

  • Dmytro Bezushchak Taras Shevchenko National University of Kyiv
  • Olexandr Ganyushkin Taras Shevchenko National University of Kyiv https://orcid.org/0009-0001-5439-4610

DOI:

https://doi.org/10.17721/1812-5409.2024/1.2

Keywords:

group, matroid

Abstract

Matroid is defined as a pair $(X,\mathcal{I})$, where $X$ is a non-empty finite set, and $\mathcal{I}$ is a non-empty set of subsets of $X$ that satisfies the hereditary axiom and the augmentation axiom. The paper investigates for which groups (primarily finite) $G$, the pair $(\widehat{G}, \mathcal{I})$ will be a matroid. The obtained criteria of matroidality for finite and infinite abelian groups, for finite nilpotent, finite symmetric, and finite dihedral groups, as well as for certain classes of finite matrix groups, are presented. Additionally, the non-matroidality of a whole range of finite groups has been proven, including Hamiltonian groups, groups of diagonal matrices, general and special linear groups, groups of upper triangular matrices with determinant $1$, and others.

Pages of the article in the issue: 17 - 21

Language of the article: English

References

Aigner, M. (1996). Combinatorial Theory, Springer Verlag.

Bezushchak, D. I. (2023). Matroids related to groups and semigroups, Res. Math., vol. 31, no. 2, pp. 8-13. doi:10.15421/242309

Huppert, B. (1983). Endliche Gruppen I, Springer.

Neel, D. L. M. (2009). Matroids you have known, Mathematics Magazine, vol. 82, no. 1, pp. 26-41. doi:10.4169/193009809X469020

Wilson, R. J. (2010). Introduction to Graph Theory, Longman.

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Published

2024-09-12

How to Cite

Bezushchak, D., & Ganyushkin, O. (2024). On groups in which irreducible systems of elements form a matroid. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, 78(1), 17–21. https://doi.org/10.17721/1812-5409.2024/1.2

Issue

Vol. 78 No. 1 (2024)

Section

Algebra, Geometry and Probability Theory

License

Copyright (c) 2024 Dmytro Bezushchak, Olexandr Ganyushkin

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