The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.

Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.

In this paper, we give a positive answer to this question.