In this paper, the authors made a study of minimal zero-sum sequences over a finite cyclic group \(C_n\). Let \(S\) be a minimal zero-sum sequence over \(C_n\). Define \(\text{Index}(S)=\min_{(m,n)=1} \{\sum_{i=1}^k| mg_i|\},\) where \(|x|\) denotes the least positive inverse image under homomorphism from the additive group of integers \(\mathbb{Z}\) onto \(C_n\). Let \(I(C_n)=\max_{S} \{\text{Index}(S)\}\), where \(S\) runs over all minimal zero-sum sequences of elements in \(C_n\). We say \(S\) is insplittable if for any \(g\) of \(S\) and any two elements \(x,y\in C_n\) satisfying \(x+y=g\), \(Sg^{- 1}xy\) is not a minimal zero-sum sequence any more. The authors discovered that for any \(k\in [1,\frac{I(C_n)}{n}]\), there exists a minimal zero-sum sequences over \(C_n\) with index \(kn\). They also made a investigation of the problem proposed by Gao: what is the least integer \(\ell(C_n)\) such that every minimal zero-sum sequence \(S\) of length at least \(\ell(C_n)\) has index \(n\), and obtained that for an insplittable minimal zero-sum sequences \(S\), if \(\text{Index}(S)=2n\), then \(S\) has a length at most \(\lfloor\frac{n}{2}\rfloor+1\).
Reviewer’s remark: The index of sequences is a hot topic in zero-sum theory, which was investigated by several authors recently. The most significant work concerned with this topic is due to Svetoslav Savchev, Fang Chen and Pingzhi Yuan. They completely determined the constant \(\ell(C_n)\). The main contribution of this paper is discovering that indexes of minimal zero-sum sequences over \(C_n\) distribute continuously between \(n\) and \(I(C_n)\).