An orthomorphism of a finite group \(G\) is a permutation \(\theta\) of the group elements such that \(x\mapsto x^{-1}\theta(x)\) also permutes \(G\). Two orthomorphisms \(\theta_1\), \(\theta_2\) are orthogonal if the map \(x\mapsto \theta_1(x)^{-1}\theta_2(x)\) also permutes \(G\). The orthomorphism graph of a group has the orthomorphisms of that group as its vertices and edges between two orthomorphisms if they are orthogonal. Orthomorphism graphs have been studied because of their application to orthogonal Latin squares.
This paper studies the orthomorphism graph of one group, namely \(\mathbb{Z}_2\times\mathbb{Z}_4\). It is computationally easy to construct and analyse this graph. However, there is some motivation for a theoretical understanding which might then be generalised to larger groups. A. B. Evans [Australas. J. Comb. 80, Part 1, 116–142 (2021; Zbl 1468.05020)] examined the orthomorphism graphs of \(\mathbb{Z}_2\times\mathbb{Z}_4\), \(D_8\) and \(Q_8\) and was able to demonstrate some common features for the three groups. The present paper only does part of what Evans did, and does not provide the kind of insights that he found by comparing several groups of the same order.