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

Abstract

TextLet G be an additive finite abelian group with exponent exp⁡(G). Let η(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). Let 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). Let p be an odd prime. We determine η(G) for some groups G with D(G)≤2exp⁡(G)−1, including the p-groups of rank three and the p-groups G=Cexp⁡(G)⊕Cpmr. We also determine s(G) for the groups G above with more larger exponent than D(G), which confirms a conjecture by Schmid and Zhuang from 2010, where D(G) denotes the Davenport constant of G. VideoFor a video summary of this paper, please visit https://youtu.be/V6yay2i75a0.