Orthomorphisms of dihedral groups
type: Item
instance of: scholarly article
title: Orthomorphisms of dihedral groups
zbMATH Open document ID: 0868.05013
DOI: 10.1016/S0012-365X(96)00222-1
author: Andrew Bowler
published in: Discrete Mathematics
publication date: 1997-08-04T00:00:00Z
review text: 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 \(\omega (D_{4n})\geq 2.\)
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1018508
zbMATH Keywords: orthomorphisms
zbMATH Keywords: complete mappings
zbMATH Keywords: orthogonal Latin squares
zbMATH Keywords: cyclic groups
zbMATH Keywords: dihedral groups
reviewed by: Jürgen Bierbrauer
MaRDI profile type: Publication
cites work: Q4893923
cites work: On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I
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cites work: Q3870911
Wikidata QID: Q127015286
description: scientific article; zbMATH DE number 1018508