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Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and $n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind S$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\hbox{\rm ord}(g)$ over all possible $g\in G$ 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 minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $S$ is reduced and the (A1) condition is satisfied(see [19])., arXiv admin note: text overlap with arXiv:1303.1682, arXiv:1303.1676 by other authors

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = ( n 1 g ) ⋅ ⋯ ⋅ ( n k g ) where g ∈ G and n 1 , … , n k ∈ [ 1 , ord ( g ) ] , and the index ind S of S is defined to be the minimum of ( n 1 + ⋯ + n k ) / 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.

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n1g)⋅…⋅(nlg) where g ∈ G and n1,…,nl ∈ [1, ord (g)], and the index ind (S) of S is defined to be the minimum of (n1 +⋯+ nl)/ 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 S is reduced and at least one ni coprime to |G|.

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$. A conjecture on the index of length four sequences says that every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\gcd(|G|, 6)=1$ has index 1. The conjecture was confirmed recently for the case when $|G|$ is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. Based on earlier work on this problem, we show that if $G=\langle g\rangle$ is a finite cyclic group of order $|G|=n$ such that $\gcd(n,6)=1$ and $S=(x_1g)(x_2g)(x_3g)(x_4g)$ is a minimal zero-sum sequence over $G$ such that $x_1,\cdots,x_4\in[1,n-1]$ with $\gcd(n,x_1,x_2,x_3,x_4)=1$, and $\gcd(n,x_i)>1$ for some $i\in[1,4]$, then $\ind(S)=1$. By using an innovative method developed in this paper, we are able to give a new (and much shorter) proof to the index conjecture for the case when $|G|$ is a product of two prime powers.