Inverse zero-sum problems II

TL;DR

This paper studies the structure of minimal zero-sum sequences of maximal length in finite abelian groups of rank two, providing new insights and conditional results based on a conjecture for groups of the form C_m ⊕ C_m.

Contribution

It characterizes the structure of these sequences for rank two groups, assuming a conjecture, and offers unconditional results for specific cases.

Findings

Determined the structure of maximal length minimal zero-sum sequences for certain rank two groups.

Provided conditional results based on a conjecture for groups of the form C_m ⊕ C_m.

Achieved unconditional results for specific groups of rank two.

Abstract

Let $G$ be an additive finite abelian group. A sequence over $G$ is called a minimal zero-sum sequence if the sum of its terms is zero and no proper subsequence has this property. Davenport's constant of $G$ is the maximum of the lengths of the minimal zero-sum sequences over $G$. Its value is well-known for groups of rank two. We investigate the structure of minimal zero-sum sequences of maximal length for groups of rank two. Assuming a well-supported conjecture on this problem for groups of the form $C_m \oplus C_m$, we determine the structure of these sequences for groups of rank two. Combining our result and partial results on this conjecture, yields unconditional results for certain groups of rank two.