Multiplicative and exponential variations of orthomorphisms of cyclic groups

An orthomorphism is a permutation $\sigma$ of $\lbrace 1, \dotsc , n - 1\rbrace$ for which $x + \sigma (x) \operatorname{mod} n$ is also a permutation on $\lbrace 1, \dotsc , n-1\rbrace$. Eberhard, Manners, Mrazović showed that the number of such orthomorphisms is $(\sqrt{e} + o(1)) \cdot \frac{n!^2}{n^n}$ for odd $n$ and zero otherwise.

In this paper we prove two analogs of these results where $x + \sigma (x)$ is replaced by $x \sigma (x)$ (a “multiplicative orthomorphism”) or with $x^{\sigma (x)}$ (an “exponential orthomorphism”). Namely, we show that no multiplicative orthomorphisms exist for $n \gt 2$, but that exponential orthomorphisms exist whenever $n$ is twice a prime $p$ such that $p - 1$ is squarefree. In the latter case we then estimate the number of exponential orthomorphisms.

Full Text (PDF format)

This research was funded by NSF grant 1659047, as part of the 2017 Duluth Research Experience for Undergraduates (REU).

Received 10 October 2017

Published 14 January 2020