Abstract
For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of finite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the isomorphism of the corresponding translation planes.
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References
Charnes C., Dempwolff U.: The translation planes of order \(49\) and their automorphism groups. Math. Comput. 67(223), 1207–1224 (1998).
Dempwolff U.: Translation planes of order 27. Des. Codes Cryptogr. 4, 105–121 (1994).
GAP Group: GAP—Groups, Algorithms, and Programming, Version 4r5. University of St Andrews and RWTH Aachen (2012).
Hering C.: Transitive linear groups and linear groups which contain irreducible subgroups of prime order. Geom. Dedicata 2, 425–460 (1974).
Hering C.: Transitive linear groups and linear groups which contain irreducible subgroups of prime order II. J. Algebra 93(1), 151–164 (1985).
Hughes D.R., Piper F.C.: Projective Planes. Springer, New York (1973).
Huppert B.: Endliche Gruppen. Springer, Berlin (1967).
Huppert B., Blackburn N.: Finite Groups III. Springer, New York (1982).
Johnson N.L., Jha V., Biliotti M.: Handbook of Finite Translation Planes. Chapman & Hall, Boca Raton (2007).
Jha V., Kallaher M.J.: On the Lorimer–Rahilly and Johnson–Walker translation planes. Pac. J. Math. 103(2), 409–427 (1982).
Kallaher M.J.: The multiplicative groups of quasifields. Can. J. Math. 39(4), 784–793 (1987).
Liebeck M.W.: The affine permutation groups of rank three. Proc. Lond. Math. Soc. 54(3), 477–516 (1987).
Lüneburg H.: Translation Planes. Springer, New York (1980).
Mathon R., Royle G.F.: The translation planes of order \(49\). Des. Codes Cryptogr. 5(1), 57–72 (1995).
Müller P., Nagy G.P.: On the non-existence of sharply transitive sets of permutations in certain finite permutation groups. Adv. Math. Commun. 5(2), 303–308 (2011).
Nagy G.P.: On the multiplication groups of semifields. Eur. J. Comb. 31, 18–24 (2010).
Niskanen S., Östergård P.R.J.: Cliquer User’s Guide, Version 1.0, Technical Report T48, Communications Laboratory. Helsinki University of Technology, Espoo (2003).
Soicher L.H.: The GRAPE package for GAP, Version 4.6.1. http://www.maths.qmul.ac.uk/~leonard/grape/ (2012). Accessed 11 July 2013.
Zassenhaus H.: Über endliche Fastkörper. Abh. Math. Sem. Univ. Hamburg 11, 187–200 (1936).
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The author is a Janos Bolyai Research Fellow. Supported by TAMOP-4.2.2/B-10/1-2010-0012 project of Hungary.
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This is one of several papers published in Designs, Codes and Cryptography comprising the special topic on “Finite Geometries: A special issue in honor of Frank De Clerck”.
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Nagy, G.P. Linear groups as right multiplication groups of quasifields. Des. Codes Cryptogr. 72, 153–164 (2014). https://doi.org/10.1007/s10623-013-9860-1
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DOI: https://doi.org/10.1007/s10623-013-9860-1