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 $1+\sum _{i=1}^d(n_i-1)$. 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.

11 pages