Ferromagnetic abelian lattice systems are the topic of this paper. Namely, at each site of ZV-invariant lattice is placed a finite abelian group. The interaction is given by any real, negative definite, and translation invariant function on the space of configurations.Algebraic structure of the system is investigated. This allows a complete · description of the family of equilibrium states for given. interaction at low temperatures. At the same time it is proven that the low temperature expansion for Gibbs free energy is analytic. It is also shown that it is not necessary to consider gauge models in the case of Zm on ZV lattice.