On a Permutation Problem for Finite Abelian Groups

Abstract

Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}$ be elements of $G$. We show that there is a permutation $\sigma\in S_{n-1}$ such that all the elements $sa_{\sigma(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only if
$$\left|\left\{1\leqslant s<n:\ \frac{n}da_s\not=0\right\}\right|\geqslant d-1\ \ \mbox{for any positive divisor}\ d\ \mbox{of}\ n.$$
When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.