On Abelian Group Representability of Finite Groups

TL;DR

This paper investigates whether certain non-abelian finite groups can be represented by abelian groups in terms of entropic vectors, with implications for network coding and information theory.

Contribution

It characterizes abelian representability for specific non-abelian groups and establishes a criterion for when groups are abelian representable for two subgroups.

Findings

Dihedral, quasi-dihedral, and dicyclic groups are characterized regarding abelian representability.

A group is abelian representable for n=2 if and only if it is nilpotent.

Complete characterization of abelian representability for certain non-abelian groups.

Abstract

A set of quasi-uniform random variables $X_1,...,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in more detail different families of non-abelian groups with respect to the entropic vectors they yield. In particular, we address the question of whether a given non-abelian group $G$ and some fixed subgroups $G_1,...,G_n$ end up giving the same entropic vector as some abelian group $A$ with subgroups $A_1,...,A_n$, in which case we say that $(A, A_1,..., A_n)$ represents $(G, G_1, ..., G_n)$. If for any choice of subgroups $G_1,...,G_n$, there exists some abelian group $A$ which represents $G$, we refer to $G$ as being abelian (group) representable for $n$. We completely characterize dihedral, quasi-dihedral and dicyclic groups with respect to their abelian representability, as well as the case when $n=2$, for which we show a group is abelian representable if and only if it is nilpotent. This problem is motivated by understanding non-linear coding strategies for network coding, and network information theory capacity regions.