Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum

TL;DR

This paper investigates the index conjecture in zero-sum theory for cyclic groups, linking it to Dedekind-type sums and providing a proof for the case when the group order is prime.

Contribution

It establishes a connection between the index conjecture and Dedekind-type sums and re-proves a special case for prime group orders.

Findings

Confirmed the index conjecture for prime cyclic groups.

Connected zero-sum sequences to Dedekind-type sums.

Provided a new proof for a specific case of the conjecture.

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 note we study the index conjecture and connect it to a Dedekind-type sum. In particular we reprove a special case of the conjecture when $|G|$ is prime.