The Lind–Lehmer Constant for $\mathbb{Z}_2^r \times \mathbb{Z}_4^s$

Abstract

For a finite abelian group the Lind–Lehmer constant is the minimum positive logarithmic Lind–Mahler measure for that group. Finding this is equivalent to determining the minimal nontrivial group determinant when the matrix entries are integers.

For a group of the form $G={\mathbb{Z}}_2^r\times {\mathbb{Z}}_4^s$ with $\left|G\right|\ge 4$ we show that this minimum is always $\left|G\right|-1$, a case of sharpness in the trivial bound. For $G={\mathbb{Z}}_2\times {\mathbb{Z}}_{2^n}$ with $n\ge 3$ the minimum is $9$, and for $G={\mathbb{Z}}_3^{}\times {\mathbb{Z}}_3^n$ the minimum is $8$. Previously the minimum was only known for $2$- and $3$-groups of the form $G={\mathbb{Z}}_p^k$ or ${\mathbb{Z}}_{p^k}$. We also show that a congruence satisfied by the group determinant when $G={\mathbb{Z}}_p^r$ generalizes to other abelian $p$-groups.