Integer Group Determinants for All Groups of Order 49
Year: 2026
Type: preprint
Abstract: We determine the integer group-determinant spectrum of the cyclic group C₄₉. Every integer coprime to 7 and every multiple of 7⁴ occurs, and the remaining values have the form 7³m with 7 ∤ m. We characterize these values by explicit conditions on three weighted prime-factor counts and four activity indicators defined in ℚ(ζ₇). The proof reduces the critical valuation to a norm-divisor condition involving two odd logarithmic characters modulo 7 and gives an integral construction for every admissible cofactor, including mixed and repeated prime factors and either sign. The construction avoids any class-group computation in ℚ(ζ₄₉). We prove that an admissible cofactor has a witnessing ideal with at most four prime-ideal factors, counted with multiplicity, and that the least positive determinant of 7-adic valuation exactly three is 7³·379 = 129997. We also obtain a componentwise split-prime restriction on the two cyclotomic norm factors. Together with Panraksa's classification of C₇ × C₇, this determines the integer group-determinant spectra of both groups of order 49. Exact verification programs and certificates accompany the proofs.
Author: Radolović, Alen
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212445719