Integer group determinants for abelian groups of order 24
Abstract
We determine the integer group determinant sets for C24, C2× C12 and C22× C6, the three abelian groups of order $24$. Character factorization expresses each determinant as a product of cyclotomic norms, whose components must satisfy integral compatibility conditions. The classifications are constructive and reduce each nonzero signed integer to valuation conditions at $2$ and $3$ and a finite test on its remaining prime factors. The tests use conductor quotients and norm-preserving global units. Each prime contribution has a uniform bound independent of its multiplicity, and successful tests yield integer coefficients realizing the prescribed determinant. The critical cyclic strata share one compatibility group. For C22× C6, a normalized rational--Eisenstein pair settles the odd values and most even strata; target stabilizers reduce the remaining tests. We also prove that the odd determinant set of C22× C6 is properly contained in that of C2× C12, exhibit an infinite family in the difference, and determine the signed powers of two in all three sets. Exact arithmetic certificates and coefficient constructors accompany the proofs.
Related
- Integer group determinants for the elementary abelian group of order 49
- The integer group determinants for ℤpn
- Bernoulli numbers and the integer group determinants of Cp × Cp
- Structural taxonomy and complex irreducible representations of non-abelian groups of order p^4
- Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups
BibTeX
@misc{panraksa2026integer,
title = {{Integer group determinants for abelian groups of order 24}},
author = {Chatchawan Panraksa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10423},
eprint = {2609.10423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10423},
}
RIS
TY - UNPB
TI - Integer group determinants for abelian groups of order 24
AU - Chatchawan Panraksa
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10423
UR - https://arxiv.org/abs/2609.10423
AB - We determine the integer group determinant sets for C_24, C_2× C_12 and C_2^2× C_6, the three abelian groups of order $24$. Character factorization expresses each determinant as a product of cyclotomic norms, whose components must satisfy integral compatibility conditions. The classifications are constructive and reduce each nonzero signed integer to valuation conditions at $2$ and $3$ and a finite test on its remaining prime factors. The tests use conductor quotients and norm-preserving global units. Each prime contribution has a uniform bound independent of its multiplicity, and successful tests yield integer coefficients realizing the prescribed determinant. The critical cyclic strata share one compatibility group. For C_2^2× C_6, a normalized rational--Eisenstein pair settles the odd values and most even strata; target stabilizers reduce the remaining tests. We also prove that the odd determinant set of C_2^2× C_6 is properly contained in that of C_2× C_12, exhibit an infinite family in the difference, and determine the signed powers of two in all three sets. Exact arithmetic certificates and coefficient constructors accompany the proofs.
ER -