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Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be writtenin the form $S=(n_1g)\cdot...\cdot(n_kg)$ where $g\in G$ and$n_1,\cdots,n_k\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind S$ of $S$ isdefined to be the minimum of $(n_1+\cdots+n_k)/\hbox{\rm ord}(g)$ over allpossible $g\in G$ such that $\langle g\rangle=G$. A conjecture says that if $G$is finite such that $\gcd(|G|,6)=1$, then $\ind(S)=1$ for every minimalzero-sum sequence $S$. In this paper, we prove that the conjecture holds if$|G|$ has two prime factors.

Minimal zero-sum sequences of length four over cyclic group with order $n=p^\alpha q^\beta$