On a permutation problem for finite abelian groups

TL;DR

This paper characterizes when a permutation exists in a finite abelian group such that all scaled permuted elements are nonzero, confirming a conjecture for cyclic groups and providing a complete criterion.

Contribution

It establishes a necessary and sufficient condition for the existence of such permutations in finite abelian groups, generalizing previous conjectures.

Findings

Provides a complete criterion for permutation existence in finite abelian groups.

Confirms Z.-W. Sun's conjecture for cyclic groups.

Extends understanding of permutation problems in algebraic structures.

Abstract

Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}\in G$. 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 $$\left|\left\{1\le s<n:\ \frac{n}{d}a_s\ne 0\right\}\right|\ge d-1\ \ \textrm{ for every positive divisor }\ d\ \textrm{ of }\ n.$$ When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.