Functional degrees and arithmetic applications I: The set of functional degrees

We give a further development of the Aichinger-Moosbauer calculus of functional degrees of maps between commutative groups. For any finitely generated commutative groups A and B, we compute the complete set D(A,B) of functional degrees of all maps between A and B. In particular, we see how big the functional degree of a function with finite functional degree can get, in which cases there is a maximal finite functional degree, and when there are functions of infinite functional degree between A and B. This yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form Z_p^β[Z_p^α1×Z_p^α2×⋯×Z_p^αn] with p,β,n,α₁,…,α_n∈Z⁺, p prime.