On joint short minimal zero-sum subsequences over finite abelian groups of rank two

Abstract

Let (G,+,0) be a finite abelian group and let ηN(G) be the smallest integer ℓ such that every sequence over G∖{0} of length ℓ has two joint short minimal zero-sum subsequences. In 2013, Gao et al. obtained that ηN(Cn⊕Cn)=3n+1 for every n≥2 and solved the corresponding inverse problem for groups Cp⊕Cp, where p is a prime. In this paper, we determine the precise value of ηN(G) for all finite abelian groups of rank 2 and resolve the corresponding inverse problem for groups Cn⊕Cn, where n≥2, which confirms a conjecture of Gao, Geroldinger and Wang for all n≥2 except n=4.