The Lind-Lehmer constant for certain $p$-groups

Abstract

We establish some new congruences satisfied by the Lind Mahler measure on $p$-groups, and use them to determine the Lind-Lehmer constant for many finite groups. First, we determine the minimal nontrivial measure of $p$-groups where one component has particularly high order. Second, we describe an algorithm that determines a small set of possible values for the minimal nontrivial measure of a $p$-group of the form $\mathbb {Z}_p\times \mathbb {Z}_{p^k}$ with $k\geq 2$. This algorithm is remarkably effective: applying it to more than 600000 groups the minimum was determined in all but six cases. Finally, we employ the results of our calculations to compute the Lind-Lehmer constant for nearly $8$ million additional $p$-groups.