Colouring bijections of finite $3$-groups

Abstract

Let $G$ be a finite group. A bijection $\sigma\colon G\to G$ is a \emph{colouring bijection} if the three maps \[ \Delta^{+}\colon x\mapsto x\,\sigma(x), \qquad \Delta^{-}\colon x\mapsto x^{-1}\sigma(x), \qquad \Delta^{c}\colon x\mapsto \sigma(x)^{-1}x\,\sigma(x) \] are again bijections of $G$. The first two conditions say that $\sigma$ is a strong complete mapping. The third is a genuinely nonabelian requirement. Our main theorem is that every noncyclic $3$-group not isomorphic to the modular group $M_{3^{r}}$ $(r\ge4)$ admits a colouring bijection. This is the exact analogue, for colouring bijections, of the theorem of Akhtar and Gagola on strong complete mappings. The notion has three equivalent readings. A colouring bijection properly colours the Cayley graph $\mathscr{G}_3(G)=Cay(G^{3},\mathbf S_3)$ with $|G|$ colours. It determines a triple of mutually orthogonal translation Latin squares based on $G$. In the orthomorphism graph of $G$ that triple is a triangle through the Cayley table. It also determines a common transversal of three arrays attached to $G$. These are the multiplication table, the division table, and the operation table of the conjugation quandle. The last of them is not a Latin square. Two consequences follow. Every noncyclic $3$-group $G\not\cong M_{3^{r}}$ $(r\ge4)$ carries three mutually orthogonal Latin squares of order $|G|$ based on $G$. Moreover $\chi(\mathscr{G}_3(G))=|G|$ for every such group.