Entropy, partitions and groups

An open problem in information theory is the determination of all points in (2^r-1)-dimensional space which represent the entropy of r random variables and their interactions. Terence Chan showed that, projectively, any such point can be approximated by points where the random variables are described by subgroups of a group: the probability space is the uniform distribution on the group, and the random variable corresponding to a subgroup maps a group element to the right coset of the subgroup which contains it. I will give the simple proof of this theorem. An interesting question is to what extent we can restrict the class of groups allowed.