2026/03/20 by Piotr Grzeszczuk
#math.CO #math.GR
Let G be a finite group. A bijection σ\colon G→ G is a colouring bijection if the three maps Δ+\colon x↦ x σ(x), Δ-\colon x↦ x-1σ(x), Δc\colon x↦ σ(x)-1x σ(x) are again bijections of G. The first two conditions say that σ 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 M3r (r≥4) 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 \mathscrG3(G)=Cay(G3,\mathbf S3) 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≅ M3r (r≥4) carries three mutually orthogonal Latin squares of order |G| based on G. Moreover χ(\mathscrG3(G))=|G| for every such group.