Cycle structures of orthomorphisms extending partial orthomorphisms of Boolean groups

A partial orthomorphism of a group G (with additive notation) is an injection pi : S -> G for some S subset of G such that pi(x) - x not equal pi(y) - y for all distinct x, y is an element of S. We refer to vertical bar S vertical bar 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, Z(2)(n), where we extend this result to include conjugation by translations in 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 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 Z(2)(n) of size two can be extended to complete orthomorphisms.