Indexes of unsplittable minimal zero-sum sequences of length I(Cn)−1I(Cn)−1

Let GG be a finite abelian group and S=g1⋯glS=g1⋯gl a minimal zero-sum sequence of elements in GG. We say that SS is unsplittable if there do not exist an element gi∈supp(S) and two elements x,y∈Gx,y∈G such that x+y=gix+y=gi and Sa−1xySa−1xy is a minimal zero-sum sequence as well. The notion of the index   of a minimal zero-sum sequence in GG has been recently addressed in the mathematical literature (see Definition 1.1). Let I(Cn)I(Cn) be the minimal integer tt such that every minimal zero-sum sequence of at least tt elements in CnCn (the cyclic group of order nn) satisfies index(S)=1. In this paper, all the unsplittable minimal zero-sum sequences of length I(Cn)−1I(Cn)−1 are discovered and their indexes are computed. The results show that Conjecture 1.1 is true when nn is odd, and false when nn is even.