Minimal zero-sum sequences of length five over finite cyclic groups

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_{1}g)\cdot \dots \cdot (n_{l}g)$ where $g\in G$ and $n_1,\dots ,n_l\in [1,\text{ord}(g)]$, and the index $\text{ind}(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots +n_l)\mathrm{/}\text{ord}(g)$ over all possible $g\in G$ such that $⟨g⟩=G$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=⟨g⟩$ is a cyclic group of a prime order and $S$ has the form $S=g^2(n_{2}g)(n_{3}g)(n_{4}g)$. It is shown that if $G=⟨g⟩$ is a cyclic group of prime order $p \geq 31$,theneveryminimalzero-sumsequence$S$ of the above mentioned form has index 1 except in the case that $S=g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)$.