On the index conjecture in zero-sum theory: singular case

TL;DR

This paper proves that for singular minimal zero-sum sequences over finite cyclic groups with four elements and gcd condition, the index is always 1, confirming a special case of the index conjecture.

Contribution

It establishes that singular minimal zero-sum sequences of length four have index 1 under the given conditions, advancing understanding of the index conjecture.

Findings

Confirmed the index is 1 for singular sequences with length 4

Extended the validity of the index conjecture to singular cases

Provided new proof techniques for zero-sum sequence analysis

Abstract

Let $S=(a_1)\cdots(a_k)$ be a minimal zero-sum sequence over a finite cyclic group $G$. The index conjecture states that if $k=4$ and $\gcd(|G|,6)=1$, then $S$ has index $1$. In this paper we prove that if $S$ is singular then the index of $S$ is $1$.