The Ingleton inequality holds for metacyclic groups and fails for supersoluble groups

The Ingleton inequality first appeared in matroid theory, where Ingleton proved in 1971 that every rank function coming from a representable matroid on four subsets satisfies a particular inequality. Because this inequality is not implied by submodularity, Shannon-type axioms alone, it and various analogues play a central role in separately linear and non-linear phenomena in a variety of areas of mathematics. The Ingleton inequality for finite groups concerns the various intersections of four subgroups. It holds for many quadruples of subgroups of finite groups, but not all, the smallest example being four subgroups of $S_5$, of order 120. Open questions are whether the Inlgeton inequality always holds for metacycle and nilpotent groups. (There is a proof in the literature due to Oggier and Stancu, but there is an already known issue with their proof, which we address in this article.) In this paper we prove that the Ingleton inequality always holds for metacycle groups, but that it fails for supersoluble groups, a class of groups only a little larger than nilpotent groups. Although we do not resolve the nilpotent case here we do make some reductions, and also prove that there are no nilpotent violators of the Ingleton inequality of order less than 1024. We end with a list of Ingleton inequality violating groups of order at most 1023. The article comes with a Magma package that allows reproduction of all results in the paper and for the reader to check the Ingleton inequality for any given finite group.