Indexes of long zero-sum sequences over cyclic groups

Abstract

Let G be a cyclic group of order n, and let S∈F(G) be a zero-sum sequence of length |S|≥2⌊n/2⌋+2. Suppose that S can be decomposed into a product of at most two minimal zero-sum sequences. Then there exists some g∈G such that S=(n1g)⋅(n2g)⋅⋯⋅(n|S|g), where ni∈[1,n] for all i∈[1,|S|] and n1+n2+⋯+n|S|=2n. And we also generalize the above result to long zero-sum sequences which can be decomposed into at most k≥3 minimal zero-sum sequences.