Abstract <p> For a finite abelian group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(r\in \mathbb{N}\)</EquationSource> </InlineEquation>, the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(r\)</EquationSource> </InlineEquation>-wise Davenport constant of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation>, denoted by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(D_r(G)\)</EquationSource> </InlineEquation>, is defined to be the least positive integer <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation> such that every sequence of length at least <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation> has <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(r\)</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-groups. On the other hand, for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(r=1\)</EquationSource> </InlineEquation>, the invariant <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(D_r(G)\)</EquationSource> </InlineEquation> is known as the Davenport constant, which is denoted by <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(D(G)\)</EquationSource> </InlineEquation>. A long-standing conjecture is that the Davenport constant of a finite abelian group <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(G =C_{n_1}\times \cdots\times C_{n_d}\)</EquationSource> </InlineEquation> of rank <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(d \in \mathbb{N}\)</EquationSource> </InlineEquation> is <Equation ID="Equi"> <EquationSource Format="TEX">\(1+\sum_{i=1}^d (n_i-1).\)</EquationSource> </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 <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(G = (C_p)^{d-1} \times C_{pq}\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation> is a prime and <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(q\in \mathbb{N}\)</EquationSource> </InlineEquation>, and provide a sufficient condition for the conjecture to hold. </p>

Davenport Constant for Finite Abelian Groups with Higher Rank

期刊论文
发表日期：

2025年8月1日

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A. Biswas,
E. Mazumdar

摘要

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 \(1+\sum_{i=1}^d (n_i-1).\) 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 a sufficient condition for the conjecture to hold.