Permutations over cyclic groups

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 $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_m$ of the cyclic group of order $m$, there is a permutation $\pi$ such that $1a_{\pi (1)}+\mathrm{.}\mathrm{.}\mathrm{.}+m{a}_{\pi (m)}=0$.