The Erdős-Ginzburg-Ziv theorem for finite solvable groups.
type: Item
instance of: scholarly article
title: The Erdős-Ginzburg-Ziv theorem for finite solvable groups.
zbMATH Open document ID: 1195.20026
author: Weidong Gao
author: Yuanlin Li
published in: Journal of Pure and Applied Algebra
publication date: 2010-02-19T00:00:00Z
review text: The Davenport constant \(D(G)\) of a group \(G\) is the smallest integer \(d\) such that every sequence of \(d\) elements (repetition allowed) in \(G\) contains a nonempty subsequence (permutation allowed) of product \(=1\). Denote by \(s(G)\) the smallest integer \(t\) such that every sequence of \(t\) elements in \(G\) contains a subsequence of length \(n\) and product \(=1\). In 1961, Erdős, Ginzburg, Ziv showed that \(s(G)\leq 2n-1\) for every finite solvable group \(G\). In this paper the authors substantially improve this, by proving the following Theorem. If \(G\) is a non-cyclic solvable group of order \(n\), then \(s(G)\leq\tfrac{7}{4}n-1\). Moreover, they conjecture \(\tfrac{3}{2}n\) as the best upper bound for \(s(G)\). Reviewer's remark: For a finite Abelian group of order \(n\), the first author proved that \(s(G)=n-1+D(G)\). Another conjecture is that this equality holds for every finite group \(G\) of order \(n\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5673211
zbMATH Keywords: 1-product sequences
zbMATH Keywords: zero sum sequences
zbMATH Keywords: finite solvable groups
zbMATH Keywords: Erdős-Ginzburg-Ziv theorem
zbMATH Keywords: Davenport constant
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.08.003
OpenAlex ID: W2005177185
cites work: Q5841589
cites work: On a combinatorial problem of Erdős, Ginzburg, and Ziv
cites work: Zero-sum problems in finite Abelian groups: a survey
cites work: Q5200682
cites work: Q4895030
cites work: Q5393666
cites work: A combinatorial problem on finite abelian groups
cites work: Erdős-Ginzburg-Ziv theorem for dihedral groups of large prime index.
cites work: Improving the Erdős-Ginzburg-Ziv theorem for some non-Abelian groups.
cites work: The Erdős-Ginzburg-Ziv theorem for dihedral groups.
cites work: A Generalization of an Addition Theorem for Solvable Groups
cites work: Bounds for counter-examples to addition theorems in solvable groups
cites work: Q4333151
cites work: On Subsequence Weighted Products
cites work: Q4124997
cites work: Conditions for a Zero Sum Modulo n
cites work: Long \(n\)-zero-free sequences in finite cyclic groups
cites work: Q3712472
DOI: 10.1016/J.JPAA.2009.08.003
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description: scientific article; zbMATH DE number 5673211