The Lind-Lehmer constant for certain 𝑝-groups

We establish some new congruences satisfied by the Lind Mahler measure on p p -groups, and use them to determine the Lind-Lehmer constant for many finite groups. First, we determine the minimal nontrivial measure of p 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 p -group of the form Z p × Z p k Z_p× Zp^k with k ≥ 2 k≥ 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 8 million additional p p -groups.