  Inspired by an experimental study of energy-minimizing periodic
configurations in Euclidean space, Cohn, Kumar and Sch\"urmann proposed the
concept of formal duality between a pair of periodic configurations, which
indicates an unexpected symmetry possessed by the energy-minimizing periodic
configurations. Later on, Cohn, Kumar, Reiher and Sch\"urmann translated the
formal duality between a pair of periodic configurations into the formal
duality of a pair of subsets in a finite abelian group. This insight suggests
to study the combinatorial counterpart of formal duality, which is a
configuration named formally dual pair. In this paper, we initiate a systematic
investigation on formally dual pairs in finite abelian groups, which involves
basic concepts, constructions, characterizations and nonexistence results. In
contrast to the belief that primitive formally dual pairs are very rare in
cyclic groups, we construct three families of primitive formally dual pairs in
noncyclic groups. These constructions enlighten us to propose the concept of
even sets, which reveals more structural information about formally dual pairs
and leads to a characterization of rank three primitive formally dual pairs.
Finally, we derive some nonexistence results about primitive formally dual
pairs, which are in favor of the main conjecture that except two small
examples, no primitive formally dual pair exists in cyclic groups.
