Difference matrices and orthomorphisms over non-abelian groups (Q2761011)

Let \(G\) be a finite group and \(r,\lambda \) positive integers. A difference matrix \(D\) (corresponding to \(G,r,\lambda \)) is an \(r\times \lambda |G|\) matrix with entries in \(G\) such that, for every two different rows \(A\) and \(B\) of \(D\), \(A\cdot B^{-1}\) (termwise) contains every element of \(G\) exactly \(\lambda \) times. K. A. S. Quinn proves that if \(G\) is a finite group and difference matrices exist for \(H,r,\lambda \) and \(G/H,r,1\) where \(H\) is a normal subgroup of \(G\), then there exists a difference matrix for \(G,r,\lambda \). This gives immediately (setting \(\lambda =1\)) an analogous result for the existence of orthomorphisms. It is shown that the dihedral group of order 16 admits at least 3 mutually orthogonal orthomorphisms.