On the Ingleton-Violations in Finite Groups

Given $n$ discrete random variables, its entropy vector is the $2^n-1$ dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a one-to-one correspondence between such an entropy vector and a certain group-characterizable vector obtained from a finite group and $n$ of its subgroups [3]. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al [4] that linear network codes cannot achieve capacity in general network coding problems. All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. It is therefore of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group $S5$ to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family $PGL(2,q)$ with a prime power $q \geq 5$, i.e., the projective group of $2\times 2$ nonsingular matrices with entries in $\mathbb{F}q$. We further interpret this family using the theory of group actions. We also extend the construction to more general groups such as $PGL(n,q)$ and $GL(n,q)$. The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes.