On Short Zero-sum Subsequences of Zero-sum Sequences

Abstract. Let G be a finite abelian group, and let η(G) be the smallest integer d such that every sequence over G of length at least d contains a zero-sum subsequence T with length |T | ∈ [1, exp(G)]. In this paper, we investigate the question whether all non-cyclic finite abelian groups G share with the following property: There exists at least one integer t ∈ [exp(G)+1, η(G)− 1] such that every zero-sum sequence of length exactly t contains a zero-sum subsequence of length in [1, exp(G)]. Previous results showed that the groups C n (n ≥ 3) and C 3 3 have the property above. In this paper we show that more groups including the groups Cm ⊕Cn with 3 ≤ m | n, C 3a5b , C 3×2a , C 4 3a and C r 2b (b ≥ 2) have this property. We also determine all t ∈ [exp(G) + 1, η(G) − 1] with the property above for some groups including the groups of rank two, and some special groups with large exponent.