Indexes of unsplittable minimal zero-sum sequences of length [formula omitted

Abstract

Let G be a finite abelian group and S = g 1 ⋯ g l a minimal zero-sum sequence of elements in G . We say that S is unsplittable if there do not exist an element g i ∈ supp ( S ) and two elements x , y ∈ G such that x + y = g i and S a − 1 x y is a minimal zero-sum sequence as well. The notion of the index of a minimal zero-sum sequence in G has been recently addressed in the mathematical literature (see Definition 1.1). Let I ( C n ) be the minimal integer t such that every minimal zero-sum sequence of at least t elements in C n (the cyclic group of order n ) satisfies index ( S ) = 1 . In this paper, all the unsplittable minimal zero-sum sequences of length I ( C n ) − 1 are discovered and their indexes are computed. The results show that Conjecture 1.1 is true when n is odd, and false when n is even.