The Integer Group Determinants for the Heisenberg Group of Order p3

Abstract

We establish a congruence satisfied by the integer group determinants for the non-Abelian Heisenberg group of order ${\mathit{p}}^3$. We characterize all determinant values coprime to p, give sharp divisibility conditions for multiples of p, and determine all values when $\mathit{p}=3$. We also provide new sharp conditions on the power of p dividing the group determinants for ${\mathbb{Z}}_{\mathit{p}}^2$.

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.