Permutations over cyclic groups

TL;DR

This paper generalizes a finite field result to cyclic groups, proving the existence of permutations that sum to zero under certain conditions, with a classification of exceptions.

Contribution

It extends a known finite field theorem to cyclic groups, identifying conditions for permutation sums equaling zero and classifying exceptions.

Findings

Existence of permutations summing to zero in cyclic groups under certain conditions

Classification of exceptions where such permutations do not exist

Generalization of finite field results to cyclic group context

Abstract

Generalizing a result in the theory of finite fields we prove that, apart from a couple of exceptions that can be classified, for any elements $a_1,...,a_m$ of the cyclic group of order $m$, there is a permutation $\pi$ such that $1a_{\pi(1)}+...+ma_{\pi(m)}=0$.