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The integer group determinants for ℤpn
Abstract
We give new conditions on allowable powers of a prime p dividing an integer group determinant for the group ℤpn, and show these conditions are sharp for n≤ 5. The values coprime to p for these groups are already known; determining which multiples of allowable powers of p occur is much more complicated. We show that all multiples of sufficiently large powers of p occur as integer group determinants of ℤpn. We also provide a complete characterization for the group ℤ33, where particular arithmetic conditions are required for multiples of certain powers of $3$.
Related
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- Bernoulli numbers and the integer group determinants of Cp × Cp
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BibTeX
@misc{mossinghoff2026integer,
title = {{The integer group determinants for $\mathbbZ_p^n$}},
author = {Michael J. Mossinghoff and Christopher Pinner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21224},
eprint = {2609.21224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21224},
}
RIS
TY - UNPB
TI - The integer group determinants for ℤ_p^n
AU - Michael J. Mossinghoff
AU - Christopher Pinner
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21224
UR - https://arxiv.org/abs/2609.21224
AB - We give new conditions on allowable powers of a prime p dividing an integer group determinant for the group ℤ_p^n, and show these conditions are sharp for n≤ 5. The values coprime to p for these groups are already known; determining which multiples of allowable powers of p occur is much more complicated. We show that all multiples of sufficiently large powers of p occur as integer group determinants of ℤ_p^n. We also provide a complete characterization for the group ℤ_3^3, where particular arithmetic conditions are required for multiples of certain powers of $3$.
ER -