Minimal zero-sum sequences of length four over cyclic group with order n=pαqβ

Let G be a finite cyclic group. Every sequence S over G can be written in the form S=(n1g)⋅⋯⋅(nkg) where g∈G and n1,…,nk∈[1,ord(g)], and the index ind S of S is defined to be the minimum of (n1+⋯+nk)/ord(g) over all possible g∈G such that 〈g〉=G. A conjecture says that if G is finite such that gcd(|G|,6)=1, then ind(S)=1 for every minimal zero-sum sequence S. In this paper, we prove that the conjecture holds if |G| has two prime factors.