A pair of orthogonal orthomorphisms of finite nilpotent groups

A bijection \(\theta :G\rightarrow G\) $\theta :G\to G$ of a finite group G is an orthomorphism of G if the mapping \(x\mapsto x^{-1}\theta (x)\) $x\mapsto x^{-1}\theta (x)$ is also a bijection. Two orthomorphisms \(\theta \) $\theta$ and \(\phi \) $\varphi$ of a finite group G are orthogonal if the mapping \(x\mapsto \theta (x)^{-1}\phi (x)\) $x\mapsto \theta {(x)}^{-1}\varphi (x)$ is also bijective. We show that there is a pair of orthogonal orthomorphisms of a finite nilpotent group G if and only if the Sylow 2-subgroup of G is either trivial or noncyclic with the definite exceptions of \(G\cong G'\) $G\cong G^{\prime }$ where \(G'\in \{D_8,Q_8,{\mathbb {Z}}_3,{\mathbb {Z}}_9\}\) $G^{\prime }\in \{D_8,Q_8,{\mathbb{Z}}_3,{\mathbb{Z}}_9\}$ and except possibly for \(G\cong Q_8\times {\mathbb {Z}}_9\) $G\cong Q_8\times {\mathbb{Z}}_9$ or \(G\cong SD_{2^n}\times {\mathbb {Z}}_3\) $G\cong S{D}_{2^n}\times {\mathbb{Z}}_3$ for any \(n\geqslant 4\) $n⩾4$ . This result yields the existence of difference matrices over finite nilpotent groups with four rows.