Orthomorphisms of dihedral groups
An orthomorphism \(\phi\) of a finite group \(G\) is a permutation of \(G\) such that the mapping \(x\mapsto x^{-1}\phi (x)\) is also a permutation. Orthomorphisms \(\phi_1,\phi_2\) of \(G\) are orthogonal if the mapping \(x\mapsto \phi_1(x)^{-1}\phi_2(x)\) is a permutation of \(G.\) Denote by \(\omega (G)\) the maximum cardinality of a set of pairwise orthogonal orthomorphisms of \(G.\) It can be shown that the group table of \(G\) can be embedded in a set of \(\omega (G)+1\) pairwise orthogonal Latin squares. This explains the significance of the concept (an equivalent concept is that of a complete mapping). Main theorem: If \(\omega (Z_n)\geq 2,\) then \(\omega (D_{4n})\geq 2.\)\ Here \(Z_n\) is the cyclic group of order \(n,\) and \(D_{4n}\) is the dihedral group of order \(4n.\) If \(n\) is odd and not divisible by 3 (equivalently \(n\equiv 1, 5 (\text{mod }6)\)) then it is easy to show that \(\omega (Z_n)\geq 2.\) This yields the most important corollary of the paper:\ If \(n\equiv 1, 5\pmod 6,\) then \(\omega (D_{4n})\geq 2.\)
scientific article; zbMATH DE number 1018508
scientific article