A pair of orthogonal orthomorphisms of finite nilpotent groups

Abstract

A bijection \(\theta :G\rightarrow G\) of a finite group G is an orthomorphism of G if the mapping \(x\mapsto x^{-1}\theta (x)\) is also a bijection. Two orthomorphisms \(\theta \) and \(\phi \) of a finite group G are orthogonal if the mapping \(x\mapsto \theta (x)^{-1}\phi (x)\) 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 \(G\cong G'\) where \(G'\in \{D_8,Q_8,{\mathbb {Z}}_3,{\mathbb {Z}}_9\}\) and except possibly for \(G\cong Q_8\times {\mathbb {Z}}_9\) or \(G\cong SD_{2^n}\times {\mathbb {Z}}_3\) for any \(n\geqslant 4\). This result yields the existence of difference matrices over finite nilpotent groups with four rows.