Davenport constant for finite abelian groups with higher rank

Abstract

For a finite abelian group $G,$ the Davenport Constant, denoted by $D(G)$, is defined to be the least positive integer $k$ such that every sequence of length at least $k$ has a non-trivial zero-sum subsequence. A long-standing conjecture is that the Davenport constant of a finite abelian group $G =C_{n_1}\times\cdots\times C_{n_d}$ of rank $d \in \mathbb{N}$ is $1+\displaystyle\sum_{i=1}^d (n_i-1) $. This conjecture is false in general, but it remains to know for which groups it is true. In this paper, we consider groups of the form $G = (C_p)^{d-1} \times C_{pq},$ where $p$ is a prime and $q\in \mathbb{N}$ and provide sufficient condition when the conjecture holds true.