  In this paper, we study the asymptotic performance of Abelian group codes for
the lossy source coding problem for arbitrary discrete (finite alphabet)
memoryless sources as well as the channel coding problem for arbitrary discrete
(finite alphabet) memoryless channels. For the source coding problem, we derive
an achievable rate-distortion function that is characterized in a single-letter
information-theoretic form using the ensemble of Abelian group codes. When the
underlying group is a field, it simplifies to the symmetric rate-distortion
function. Similarly, for the channel coding problem, we find an achievable rate
characterized in a single-letter information-theoretic form using group codes.
This simplifies to the symmetric capacity of the channel when the underlying
group is a field. We compute the rate-distortion function and the achievable
rate for several examples of sources and channels. Due to the non-symmetric
nature of the sources and channels considered, our analysis uses a synergy of
information theoretic and group-theoretic tools.
