A note on quasi-uniform distributions and Abelian group representability

In this note, we study quasi-uniform distributions that are obtained from finite groups. We derive a few simple properties of entropic vectors obtained from Abelian groups, and consider the problem of determining when non-Abelian groups can provide richer entropic vectors than Abelian groups. We focus in particular on the family of dihedral groups D_2n, and show that when 2_n is not a power of 2, the induced entropic vectors for two variables cannot be obtained from Abelian groups, contrarily to the case of D₈ which does not provide more than Abelian groups.