Difference Matrices and Orthomorphisms over Non-Abelian Groups

Let \(G\) be a finite group with a normal subgroup \(H\). We prove that if there exist a \((h, r;\lambda, H)\) difference matrix and a \((g/h, r;1, G/H)\) difference matrix, then there exists a \((g, r;\lambda, G)\) difference matrix. This shows in particular that if there exist \(r\) mutually orthogonal orthomorphisms of \(H\) and \(r\) mutually orthogonal orthomorphisms of \(G/H\), then there exist \(r\) mutually orthogonal orthomorphisms of \(G\). We also show that a dihedral group of order \(16\) admits at least \(3\) mutually orthogonal orthomorphisms.