Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups

TL;DR

This paper explores multiplicative and exponential orthomorphisms of cyclic groups, establishing their existence conditions and estimating their quantities, extending previous additive orthomorphism results.

Contribution

It introduces and analyzes multiplicative and exponential orthomorphisms, proving their non-existence or existence under specific conditions, and provides enumeration estimates.

Findings

No multiplicative orthomorphisms exist for n > 2.

Exponential orthomorphisms exist when n is twice a prime p with p-1 squarefree.

Estimated number of exponential orthomorphisms when they exist.

Abstract

An orthomorphism is a permutation $\sigma$ of $\{1, \dots, n-1\}$ for which $x + \sigma(x) \mod n$ is also a permutation on $\{1, \dots, n-1\}$. Eberhard, Manners, Mrazovi\'c, 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 $n > 2$ 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.