Papers published in the English version of the journal
Davenport Constant for Finite Abelian Groups with Higher Rank
A. Biswasa,
E. Mazumdarb a Department of Mathematics, National Institute of Technology Silchar, India
b Mathematical and Physical Sciences, School of Arts and Sciences, Ahmedabad University, India
Abstract:
For a finite abelian group
$G$ and
$r\in \mathbb{N}$, the
$r$-wise Davenport constant of
$G$, denoted by
$D_r(G)$, is defined to be the least positive integer
$k$ such that every sequence of length at least
$k$ has
$r$ disjoint nontrivial zero-sum subsequences. Several mathematicians have studied the behavior of this invariant. In this paper, we examine its value for any finite abelian group, specifically for
$p$-groups. On the other hand, for
$r=1$, the invariant
$D_r(G)$ is known as the Davenport constant, which is denoted by
$D(G)$. 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
\begin{equation*} 1+\sum_{i=1}^d (n_i-1). \end{equation*}
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 a sufficient condition for the conjecture to hold.
Keywords:
zero-sum problem, Davenport constant. Received: 31.10.2024
Revised: 29.04.2025
Language: English
