Submonoids of integer group determinants

Y. Wakamoto; N. Yamaguchi

doi:10.15421/242612

Submonoids of integer group determinants

Y. Wakamoto (Sadowara Senior High School)
N. Yamaguchi (University of Miyazaki), https://orcid.org/0000-0003-4169-7409

Abstract

For a finite group $$$G$$$, let $$$S(G)$$$ denote the set of all integer values obtained from the group determinant by substituting integers for its variables. Such values are called integer group determinants. Giving a complete description of $$$S(G)$$$ is a difficult problem, known as the Taussky-Todd integer group determinant problem.
In this paper, we introduce a submonoid $$$S'(G)$$$ of the monoid $$$S(G)$$$ arising from a specific specialization of the group determinant, and provide a framework for investigating certain aspects of the lower structure of $$$S(G)$$$. For finite groups $$$G$$$ of order at most $$$5$$$, we determine $$$S'(G)$$$ explicitly and present concrete examples of integer group determinants.
The motivation for considering the submonoid $$$S'(G)$$$ is that specialization patterns can retain meaningful arithmetic information. In particular, this viewpoint may be useful in contexts related to classical number-theoretic problems, such as the search for Wieferich primes.

Keywords

group determinant; integer group determinant; monoid

MSC 2020

11C20; 11E76; 20C15

Full Text:

PDF

References

Bautista Serrano H., Paudel B., Pinner C. "The integer group determinants do not determine the group", Comb. Number Theory, 2024; 13(1): pp. 59-65. doi:10.2140/cnt.2024.13.59

Boerkoel T., Pinner C. "Minimal group determinants and the Lind-Lehmer problem for dihedral groups", Acta Arith., 2018; 186(4): pp. 377-395. doi:10.4064/aa180708-30-8

Frobenius F.G.}: "Über die Primfactoren der Gruppendeterminante", Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1896; pp. 1343-1382. Reprinted in Gesammelte Abhandlungen, Band III, Springer-Verlag, 1968; pp. 38-77. (in German)

Kaiblinger N. "Progress on Olga Taussky-Todd's circulant problem", Ramanujan J., 2012; 28(1): pp. 45-60. doi:10.1007/s11139-011-9354-6

Laquer H.T. "Values of circulants with integer entries", In: A collection of manuscripts related to the Fibonacci sequence, Fibonacci Assoc., 1980; pp. 212-217.

Mahoney M.K. "Determinants of integral group matrices for some nonabelian 2-generator groups", Linear Multilin. Algebra, 1982; 11(2): pp. 189-201. doi:10.1080/03081088208817443

Newman M. "Determinants of circulants of prime power order", Linear Multilin. Algebra, 1980; 9(3): pp. 187-191. doi:10.1080/03081088008817367

Newman M. "On a problem suggested by Olga Taussky-Todd", Illinois J. Math., 1980; 24(1): pp. 156-158. doi:10.1215/ijm/1256047802

Paudel B., Pinner C. "Integer circulant determinants of order 15", Integers, 2022; 22: A4, 21.

Paudel B., Pinner C. "The integer group determinants for the groups of order 18", Ramanujan J., 2025; 68(4): 102. doi:10.1007/s11139-025-01249-6

Paudel B., Pinner C. "The group determinants for $$$\mathbb{Z}_n \times H$$$", Notes Number Theory Discrete Math., 2023; 29(3): pp. 603-619. doi:10.7546/nntdm.2023.29.3.603-619

Paudel B., Pinner C. "The integer group determinants for $$$Q_{16}$$$", arXiv, 2023. doi:10.48550/arxiv.2302.11688

Pinner C., Smyth C. "Integer group determinants for small groups", Ramanujan J., 2020; 51(2): pp. 421-453. doi:10.1007/s11139-018-0092-x

Taussky-Todd O. "Integral group matrices", In: Notices Amer. Math. Soc., 1977; 24: 746-A15; 746th Meeting, Apr. 22-23, 1977.

van der Waerden B.L. A history of algebra. From al-Khwārizmı̄ to Emmy Noether, Springer-Verlag, 1985. doi:10.1007/978-3-642-51599-6

Wieferich A. "Zum letzten Fermatschen Theorem", J. Reine Angewandte Math., 1909; 136: pp. 293-302. doi:10.1515/crll.1909.136.293

Yamaguchi N., Yamaguchi Y. "Generalized Dedekind's theorem and its application to integer group determinants", J. Math. Soc. Japan, 2024; 76(4): pp. 1123-1138. doi:10.2969/jmsj/90539053

Yamaguchi Y., Yamaguchi N. "Integer circulant determinants of order 16", Ramanujan J., 2023; 61(4): pp. 1283-1294. doi:10.1007/s11139-022-00599-9

Yamaguchi Y., Yamaguchi N. "Integer group determinants for $$$\mathrm{C}_2^4$$$", Ramanujan J., 2023; 62(4): pp. 983-995. doi:10.1007/s11139-023-00727-z

Yamaguchi Y., Yamaguchi N. "A necessary and sufficient condition for a prime to be an integer group determinant of certain p-groups", arXiv, 2023. doi:10.48550/arXiv.2310.02297

Yamaguchi Y., Yamaguchi N. "Integer group determinants for abelian groups of order 16", Hiroshima Math. J., 2024; 54(3): pp. 359-373. doi:10.32917/h2023012

Yamaguchi Y., Yamaguchi N. "Integer group determinants for $$$\mathrm{C}_4^2$$$", Integers, 2024; 24: A30, 21.

Yamaguchi Y., Yamaguchi N. "Integer group determinants for three of the non-abelian groups of order 16", Res. Number Theory, 2024; 10(2): 23, 19. doi:10.1007/s40993-024-00508-7

Yamaguchi Y., Yamaguchi N. "Integer group determinants of order 16", Ramanujan J., 2024; 65(3): pp. 1459-1474. doi:10.1007/s11139-024-00946-y