2023 The Integer Group Determinants for the Heisenberg Group of Order p3
Michael J. Mossinghoff, Christopher Pinner
Michigan Math. J. Advance Publication 1-19 (2023). DOI: 10.1307/mmj/20216124


We establish a congruence satisfied by the integer group determinants for the non-Abelian Heisenberg group of order p3. We characterize all determinant values coprime to p, give sharp divisibility conditions for multiples of p, and determine all values when p=3. We also provide new sharp conditions on the power of p dividing the group determinants for Zp2.

For a finite group, the integer group determinants can be understood as corresponding to Lind’s generalization of the Mahler measure. We speculate on the Lind–Mahler measure for the discrete Heisenberg group and for two other infinite non-Abelian groups arising from symmetries of the plane and 3-space.


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Michael J. Mossinghoff. Christopher Pinner. "The Integer Group Determinants for the Heisenberg Group of Order p3." Michigan Math. J. Advance Publication 1 - 19, 2023. https://doi.org/10.1307/mmj/20216124


Received: 11 August 2021; Revised: 27 April 2022; Published: 2023
First available in Project Euclid: 15 June 2023

Digital Object Identifier: 10.1307/mmj/20216124

Keywords: 11B83 , 11C08 , 11C20 , 11G50 , 11R06 , 11R09 , 11T22 , 15B36 , 20H25 , 20H30 , 43A40

Rights: Copyright © 2023 The University of Michigan


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