On a permutation problem for finite abelian groups

Abstract

Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\dots ,a_{n-1}\in G$. We show that there is a permutation $\sigma \in S_{n-1}$ such that all the elements $s{a}_{\sigma (s)}(s=1,\dots ,n-1)$ are nonzero if and only if $$\left|\left\{1\le s<n:\frac{n}{d}{a}_s\ne 0\right\}\right|\ge d-1\text{ for every positive divisor }d\text{ of }n.$$ When $G$ is the cyclic group $\mathbb{Z}/n\mathbb{Z}$, this confirms a conjecture of Z.-W. Sun.

arXiv comment7 pages, final published version