On group violations of inequalities in five subgroups

Abstract

In this paper we use group theoretic tools to obtain random variables which violate linear rank inequalities, that is inequalities which always hold on ranks of subspaces. We consider ten of the 24 (non-Shannon type) generators of linear rank inequalities in five variables and look at them as group inequalities. We prove that for primes $p,q$, groups of order $pq$ always satisfy these ten group inequalities. We give partial results for groups of order $p^2q$, and find that the symmetric group $S_4$ is the smallest group to yield violations for two among the ten group inequalities.