The EGZ-constant and short zero-sum sequences over finite abelian groups

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• 作者：Weidong Gao ; ^{wdgao1963@aliyun.com" class="auth_mail" title="E-mail the corresponding author} ; Dongchun Han ^{han-qingfeng@163.com" class="auth_mail" title="E-mail the corresponding author} ; Hanbin Zhang ^{nkuzhanghanbin@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Zero-sum sequence ; Short zero-sum sequence ; Davenport constant ; EGZ-constant
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：162
• 期：Complete
• 页码：601-613
• 全文大小：308 K

文摘

Let G   be an additive finite abelian group with exponent exp⁡(G)$\exp (G)$. Let η(G)$\eta (G)$ be the smallest integer t such that every sequence of length t   has a nonempty zero-sum subsequence of length at most exp⁡(G)$\exp (G)$. Let s(G)$\mathsf{s}(G)$ be the EGZ-constant of G, which is defined as the smallest integer t such that every sequence of length t   has a zero-sum subsequence of length exp⁡(G)$\exp (G)$. Let p   be an odd prime. We determine η(G)$\eta (G)$ for some groups G   with D(G)≤2exp⁡(G)−1$\mathsf{D}(G)\le 2\exp (G)-1$, including the p-groups of rank three and the p  -groups $G=C_{\exp (G)}\oplus C_{p^m}^r$. We also determine s(G)$\mathsf{s}(G)$ for the groups G   above with more larger exponent than D(G)$\mathsf{D}(G)$, which confirms a conjecture by Schmid and Zhuang from 2010, where D(G)$\mathsf{D}(G)$ denotes the Davenport constant of G.

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