Finite p-groups, entropy vectors and the Ingleton inequality for nilpotent groups

TL;DR

This paper explores the relationship between finite nilpotent groups and entropy vectors, showing that nilpotent group-characterizable entropy vectors satisfy the Ingleton inequality, thus linking group theory with information theory.

Contribution

It proves that all nilpotent group-characterizable entropy vectors are also Abelian group-characterizable, satisfying the Ingleton inequality, and discusses implications for more complex groups.

Findings

Nilpotent group-characterizable entropy vectors satisfy Ingleton inequality.

All nilpotent groups of bounded class satisfy the inequality.

Exotic entropic vectors require soluble or larger nilpotent groups.

Abstract

In this paper we study the capacity/entropy region of finite, directed, acyclic, multiple-sources, multiple-sinks network by means of group theory and entropy vectors coming from groups. There is a one-to-one correspondence between the entropy vector of a collection of n random variables and a certain group-characterizable vector obtained from a finite group and n of its subgroups. We are looking at nilpotent group characterizable entropy vectors and show that they are all also Abelian group characterizable, and hence they satisfy the Ingleton inequality. It is known that not all entropic vectors can be obtained from Abelian groups, so our result implies that in order to get more exotic entropic vectors, one has to go at least to soluble groups or larger nilpotency classes. The result also implies that Ingleton inequality is satisfied by nilpotent groups of bounded class, depending on the order of the group.