Electronic Journal of Combinatorics
Boolean algebra, Permutation groups, Combinatorial analysis
A partial orthomorphism of a group GG (with additive notation) is an injection π:S→G for some S⊆G such that π(x)−x ≠ π(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, Z, where we extend this result to include conjugation by translations in Z2n, where we extend this result to include conjugation by translations in Z2n 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 Z2n 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 Z2n of size two can be extended to complete orthomorphisms.
Schimanski, N. L., & Caughman IV, J. S. (2016). Cycle structures of orthomorphisms extending partial orthomorphisms of Boolean groups. The Electronic Journal of Combinatorics, 23(3), #P3.41.