Cycle Structures of Orthomorphisms Extending Partial Orthomorphisms of Boolean Groups

A partial orthomorphism of a group G (with additive notation) is an injection π : S → G for some S ⊆ G such that π(x) − x 6= π(y) − y for all distinct x, y ∈ S. We refer to |S| as the size of π, and if S = G, then π 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, Z2 , where we extend this result to include conjugation by translations in Z2 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 Z2 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 cycletype distributions for the case of size three. As a consequence of these results, we find that all partial orthomorphisms of Z2 of size two can be extended to complete orthomorphisms.