Cycle Structures of Orthomorphisms Extending Partial Orthomorphisms of Boolean Groups

Abstract

A partial orthomorphism of a group $G$ (with additive notation) is an injection $\pi:S \to G$ for some $S \subseteq G$ such that $\pi(x)-x \not= \pi(y)-y$ for all distinct $x,y \in S$. We refer to $|S|$ as the size of $\pi$, and if $S = G$, then $\pi$ is an orthomorphism. Despite receiving a fair amount of attention in the research literature, many basic questions remain concerning the number of orthomorphisms of a given group, and what cycle types these permutations have.

It is known that conjugation by automorphisms of $G$ forms a group action on the set of orthomorphisms of $G$. In this paper, we consider the additive group of binary $n$-tuples, $\mathbb{Z}_2^n$, where we extend this result to include conjugation by translations in $\mathbb{Z}_2^n$ and related compositions. We apply these results to show that, for any integer $n >1$, the distribution of cycle types of orthomorphisms of the group $\mathbb{Z}_2^n$ that extend any given partial orthomorphism of size two is independent of the particular partial orthomorphism considered. A similar result holds for size one. We also prove that the corresponding result does not hold for orthomorphisms extending partial orthomorphisms of size three, and we give a bound on the number of cycle-type distributions for the case of size three. As a consequence of these results, we find that all partial orthomorphisms of $\mathbb{Z}_2^n$ of size two can be extended to complete orthomorphisms.