Normal sequences over finite abelian groups

Guan, Huanhuan; Yuan, Pingzhi; Zeng, Xiaofeng

doi:10.1016/j.jcta.2010.11.009

Cited by 4 publications

(2 citation statements)

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“…Thus, our Theorem 2.5 holds unconditionally, which gives a positive answer to Conjecture 1 for all Abelian groups of rank two. Further progress has since been made on this conjecture [15].…”

Section: Note Added In Proofmentioning

confidence: 99%

On the existence of zero-sum subsequences of distinct lengths

Girard¹

2012

Rocky Mountain J. Math.

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In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland and Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao's conjecture is solved using Alon's Combinatorial Nullstellensatz. To conclude, we show that, assuming every integer satisfies Property B, this conjecture holds in the case of finite Abelian groups of rank two.Comment: 10 pages, to appear in Rocky Mountain Journal of Mathematic

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Section: Note Added In Proofmentioning

confidence: 99%

On the existence of zero-sum subsequences of distinct lengths

Girard¹

2012

Rocky Mountain J. Math.

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“…Moreover, s L (G) has been investigated for various other sets, including: [1, k] for k ≥ exp(G) (see, e.g., [4,2,6]), {k exp(G)} for k ∈ N (see, e.g., [13,26]), N \ kN for k ∤ exp(G) and other unions of arithmetic progressions (see [7,29,21]), and exp(G)N (see, e.g., [3]). And, for recent closely related results, see, e.g., [23,9,12,22,11,31].…”

Section: Introductionmentioning

confidence: 98%

Zero-sum problems with congruence conditions

Geroldinger

Grynkiewicz²,

Schmid³

2011

Acta Math Hung

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International audienceFor a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum subsequence $T$ of length $|T| \equiv 0 \mod d$. We determine $\mathsf s_{d \mathbb N} (G)$ for all $d\geq 1$ when $G$ has rank at most two and, under mild conditions on $d$, also obtain precise values in the case of $p$-groups. In the same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv constant provided that, for the $p$-subgroups $G_p$ of $G$, the Davenport constant $\mathsf D (G_p)$ is bounded above by $2 \exp (G_p)-1$. This generalizes former results for groups of rank two

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