×
Evans, Anthony B.

Orthomorphism graphs of groups. (English) Zbl 0678.20012

J. Geom. 35, No. 1-2, 66-74 (1989).
The orthomorphism graph of a group G is the set of all bijections \(\alpha\) : \(G\to G\) fixing the identity such that \(g\mapsto g^{- 1}g^{\alpha}\) is bijective, with \(\alpha\), \(\beta\) adjacent precisely if \(g\mapsto (g^{\alpha})^{-1}g^{\beta}\) is bijective. An r-clique in this graph corresponds to a net on \(G^ 2\) with \(r+2\) parallel classes admitting G as a group of translations. The main part of the paper is concerned with upper bounds \(\omega\) (G) for cliques consisting of automorphisms or antiautomorphisms of a finite group G. It turns out that \(\omega (G)=| G| -2\) if and only if G is elementary abelian.
Reviewer: Th.Grundhöfer

Cited in 3 Documents

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
05B15 Orthogonal arrays, Latin squares, Room squares
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20D45 Automorphisms of abstract finite groups
20N05 Loops, quasigroups

Keywords:

fixed-point free permutation; latin square; orthomorphism graph; r- clique; net; group of translations; automorphisms; antiautomorphisms; finite group
Cite Review PDF
Full Text: DOI

References:

[1] BAUMERT, L. and HALL, M. Jr.: Nonexistence of Certain Planes of Order 10 and 12. J. Combinatorial Theory Ser. A14 (1973), 273-280. · Zbl 0261.05014 · doi:10.1016/0097-3165(73)90002-2
[2] BEHZAD, M. and CHARTRAND, G. and LESNIAK-FOSTER, L.: Graphs and Digraphs. Wadsworth, California 1979. · Zbl 0403.05027
[3] BOSE, R. C. and CHAKRAVARTI, I. M. and KNUTH, D. E.: On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I. Technometrics2 (1960), 507-516. · Zbl 0096.12306 · doi:10.2307/1266458
[4] DEMBOWSKI, P.: Finite Geometries. Springer-Verlag, New York 1968. · Zbl 0159.50001
[5] Evans, A. B.: Orthomorphisms of GF(q)+. ARS Combinatoria, to appear.
[6] -: Orthomorphisms of Groups. Annals of NYAS, to appear. · Zbl 0713.05015
[7] EVANS, A. B., and McFARLAND, R. L.: Planes of Prime Order with Translations. Proc. 15th. S-E Conf on Combinatorics, Graph Theory and Computing (Baton Rouge, Louisiana, March 1984). Congressus Numerantium44 (1984), 41-46. · Zbl 0575.51005
[8] GORENSTEIN, D.: Finite Groups. Harper and Row, New York-Evanston-London 1968.
[9] HALL, M. and PAIGE, L. J.: Complete Mappings of Finite Groups. Pacific J. Math.5 (1955), 541-549. · Zbl 0066.27703
[10] HSU, D. F. and KEEDWELL, A. D.: Generalized Complete Mappings, Neofields, Sequenceable Groups and Block Designs I. Pacific J. Math.111 (1984), 317-322. · Zbl 0557.05019
[11] ?: Generalized Complete Mappings, Neofields, Sequenceable Groups and Block Designs II. Pacific J. Math.117 (1985), 291-312. · Zbl 0575.05011
[12] HUPPERT, B.: Endliche Gruppen I. Springer-Verlag, Berlin ? Heidelberg - New York 1967. · Zbl 0217.07201
[13] JOHNSON, D. and DULMAGE, A. K. and MENDELSOHN, N. S.: Orthomorphisms of Groups and Orthogonal Latin Squares. I. Can. J. Math.13 (1961), 356-372. · Zbl 0097.25102 · doi:10.4153/CJM-1961-031-7
[14] JUNGNICKEL, D.: On Difference Matrices and Regular Latin Squares. Abh. Math. Sem. Univ. Hamburg50 (1980), 219-231. · doi:10.1007/BF02941430
[15] KALLAHER, M. J.: Affine Planes with Transitive Collineation Groups. North Holland, New York ? Amsterdam - Oxford 1982. · Zbl 0485.51006
[16] MANN, H. B.: The Construction of Orthogonal Latin Squares. Ann. Math. Statist.13 (1942), 418-423. · Zbl 0060.02706 · doi:10.1214/aoms/1177731539
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.