Finite nilpotent and metacyclic groups never violate the Ingleton inequality

Abstract

In [5], Mao and Hassibi started the study of finite groups that violate the Ingleton inequality. They found through computer search that the smallest group that does violate it is the symmetric group of order 120. We give a general condition that proves that a group does not violate the Ingleton inequality, and consequently deduce that finite nilpotent and metacyclic groups never violate the inequality. In particular, out of the groups of order up to 120, we give a proof that about 100 orders cannot provide groups which violate the Ingleton inequality.