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Orthomorphisms of Groups

Orthomorphisms and complete mappings (see Definition 1) have been used in several combinatorial constructions. Cliques of the orthomorphism graph of a group (see Definition 3) have been used in the construction of mutually orthogonal Latin squares (MOLS) by Bose et al. [l], Johnson et al. [9], and Jungnickel [lo]. See Denes and Keedwell [2] for a survey of constructions of MOLS using orthomorphisms and complete mappings. Orthomorphisrns and complete mappings have also been used in the construction of neofields and R-sequencings of groups (see Hsu and Keedwell [S]). Many other applications can be found in the literature. In this paper, we define several orthomorphism graphs and show that there are at most five orthomorphism graphs, up to isomorphism, of a non-Abelian group and only one orthomorphism graph, up to isomorphism, of an Abelian group. We give several classes of orthomorphisms and establish a relationship between cliques of the orthornorphism graph of the cyclic group of order q2 + q + 1 and MOLS of order q + 1.