The Inverse Problem Associated to the Davenport Constant for C2+C2+C2n, and Applications to the Arithmetical Characterization of Class Groups

The inverse problem associated to the Davenport constant for some finite abelian group is the problem of determining the structure of all minimal zero-sum sequences of maximal length over this group, and more generally of long minimal zero-sum sequences. Results on the maximal multiplicity of an element in a long minimal zero-sum sequence for groups with large exponent are obtained. For groups of the form C 2 ⊕ C2n the results are optimal up to an absolute constant. And, the inverse problem, for sequences of maximal length, is solved completely for groups of the form C2 2 ⊕ C2n. Some applications of this latter result are presented. In particular, a characterization, via the system of sets of lengths, of the class group of rings of algebraic integers is obtained for certain types of groups, including C2 2 ⊕ C2n and C3 ⊕ C3n; and the Davenport constants of groups of the form C2 4 ⊕ C4n and C 2 6 ⊕ C6n are determined.