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 fixed given commutative groups $A$ and $B$, we compute the largest possible finite functional degree that a map $f: A \longrightarrow B$ can have. We also determine the set of all possible degrees of such maps. This also yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form $Z_{p^\beta}[Z_{p^{\alpha_1}}\times Z_{p^{\alpha_2}}\times\dotsm\times Z_{p^{\alpha_n}}]$ with $p,\beta,n,\alpha_1,\dotsc,\alpha_n\in\mathbb{Z}^+$, $p$ prime.