Long unsplittable zero-sum sequences over a finite cyclic group

摘要

Let G be an additively written finite cyclic group of order n and let S be a minimal zero-sum sequence with elements of G, i.e. the sum of elements of S is zero, but no proper nontrivial subsequence of S has sum zero. S is called unsplittable if there do not exist an element g in S and two elements x, y in G such that g = x + y and the new sequence Sg(-1) xy is still a minimal zero-sum sequence. In this paper, we investigate long unsplittable minimal zero-sum sequences over G. Our main result characterizes the structures of all such sequences S and shows that the index of S is at most 2, provided that the length of S is greater than or equal to [n/3] + 8 where n > 20,585 is a positive integer with least prime divisor greater than 13.