Minimal zero-sum sequences in \(C_{n}\oplus C_{n}\)
type: Item
instance of: scholarly article
title: Minimal zero-sum sequences in \(C_{n}\oplus C_{n}\)
zbMATH Open document ID: 1120.11013
author: Günter Lettl
author: Wolfgang A. Schmid
published in: European Journal of Combinatorics
publication date: 2007-03-27T00:00:00Z
review text: By a result of \textit{J. E. Olson} [``A combinatorial problem on finite abelian groups. II'', J. Number Theory 1, 195--199 (1969; Zbl 0167.28004)], any sequence of elements of \(G=C_n\oplus C_n\) with length at least \(2(n-1)+1\) contains a nonempty zero-sum subsequence, where \(C_n\) is the additive cyclic group of order \(n\). Let \(\{g_i\}_{i=1}^{2n-1}\) be a minimal zero-sum sequence with \(g_i\in G\). \textit{W. D. Gao} and \textit{A. Geroldinger} [``On zero-sum sequences in \(\mathbb Z/n\mathbb Z\oplus\mathbb Z/n\mathbb Z\),'' Integers 3, Paper A08, 45 p., electronic only (2003; Zbl 1078.11010)] conjectured that the exact value of \(M=\max_{g\in G}| \{1\leq i\leq 2n-1:\, g_i=g\}| \) is \(n-1\); when \(n\) is an odd prime \(p\), they showed that \(3\leq| \{g_1,\ldots,g_{2p-1}\}| \leq p+1\), which is a consequence of the conjectured equality \(M=p-1\). In the paper under review, the authors obtain the further inequality \(| \{g_1,\ldots,g_{2p-1}\}| \leq p\).
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 5137580
zbMATH Keywords: zero-sum sequences
reviewed by: Zhi-Wei Sun
MaRDI profile type: Publication
full work available at URL: j.ejc.2005.12.008
OpenAlex ID: W2010796394
cites work: Zero-sum problems -- a survey
cites work: On the Addition of Residue Classes
cites work: Cyclic Spaces for Grassmann Derivatives and Additive Theory
cites work: On long minimal zero sequences in finite abelian groups
cites work: Q4430614
cites work: Zero-sum problems and coverings by proper cosets
cites work: On the structure of \(p\)-zero-sum free sequences and its application to a variant of Erdős-Ginzburg-Ziv theorem
cites work: Sequences not containing long zero-sum subsequences
cites work: Q4462638
cites work: Q3856819
cites work: La fonction de Brakemeier dans le probléme d'Erdős–Ginzburg–Ziv
cites work: Q4895030
cites work: A combinatorial problem on finite Abelian groups. II
cites work: Q4004140
DOI: 10.1016/J.EJC.2005.12.008
description: scientific article; zbMATH DE number 5137580