The article presents several methods for the arithmetic of finite abelian groups. We introduce a tool - already used by Delsarte in [1] as I found out later - analogous to Dirichlet's convolution to obtain combinatorial results on these groups. Using this convolution and some group actions, we deduce an interesting fact : the number of generating subsets of a finite abelian group is always a multiple of the order of the group. Eventually, we prove a theorem about the generation of the group of permutations of an abelian group G using only transpositions and translations from the group G.