The Integer group determinants for the Heisenberg group of order $p^3$

Abstract

We establish a congruence satisfied by the integer group determinants for the non-abelian Heisenberg group of order $p^3$. 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 ${\mathbb{Z}}_p^2$. 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.