On the Compressibility of non-Abelian-Group Sources in a Distributed Source Coding Setting

We consider the problem of compression of a non-Abelian source.This is motivated by the problem of distributed function computation,where it is known that if one is only interested in computing a function of several sources, then one can often improve upon the compression rate required by the Slepian-Wolf bound. Let G be a non-Abelian group having center Z(G). We show here that it is impossible to compress a source with symbols drawn from G when Z(G) is trivial if one employs a homomorphic encoder and a typical-set decoder.We provide achievable upper bounds on the minimum rate required to compress a non-Abelian group with non-trivial center. Also, in a two source setting, we provide achievable upper bounds for compression of any non-Abelian group, using a non-homomorphic encoder.