Cycle structures of orthomorphisms extending partial orthomorphisms of Boolean groups (Q311569)

Summary: 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.