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
$\leftG\right\ge 4$ we show that this
minimum is always
$\leftG\right1$,
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.
