The spread set method for the construction of finite quasifields

O. V. Kravtsova   ; D. S. Skok

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 1, pp. 164-181

Résumé

The weakening of the field axioms leads to more general algebraic systems such as near-fields, semifields, and quasifields. The tools for studying these systems are more difficult to use. The spread set method is based on recording multiplication in a quasi-field as a linear transform in the associated linear space. The transition to matrix operations enables the effective application of the method for studying the finite translation planes and their coordinatizing quasifields. We obtain a characteristic property of a spread set for a near-field of dimension two over the kernel. The result is applied to two non-isomorphic near-fields of order 25 and quasifields of order 9. The existence of quasifields with a multiplicative Moufang loop is also discussed. It is proved by the spread set method that a non-associative Moufang quasifield of order 25 does not exist. We list some questions of the theory of finite semifields and semifield projective planes where the spread set method may be useful. This method is also effective in computer constructions of quasifields and translation planes.

Détail

Keywords: quasifield, near-field, semifield, spread set, translation plane.

Bibliographie

[1] Dickson L.E., “Linear algebras in which division is always uniquely possible”, Trans. Amer. Math. Soc., 7:3 (1906), 370–390 | DOI | MR | Zbl

[2] Veblen O., Maclagan-Wedderburn J.H., “Non-Desarguesian and Non-Pascalian Geometries”, Trans. Amer. Math. Soc., 8:3 (1907), 379–388 | DOI | MR | Zbl

[3] Kholl M., Teoriya grupp, Izd-vo inostr. lit., M., 1962, 468 pp.

[4] Hughes D.R., Piper F.C., Projective planes, Springer-Verlag, NY, 1973, 292 pp. | MR | Zbl

[5] Dickson L.E., “On finite algebras”, Nachr. Akad. Wiss. Göttingen, Math.-Phys, II (1905), 358–393 URL: https://eudml.org/doc/58621

[6] Zassenhaus H., “Uber endliche Fastkörper”, Abh. Math. Sem. Hamburg, 11 (1936), 187–220 | DOI | MR

[7] Hall M., Jr., “Projective planes”, Trans. Amer. Math. Soc., 54 (1943), 229–277 | DOI | MR | Zbl

[8] Johnson N.L., Jha V., Biliotti M., Handbook of finite translation planes, Chapman Hall/CRC, London; NY, 2007, 888 pp. | MR | Zbl

[9] Kravtsova O.V., “On automorphisms of semifields and semifield planes”, Sib. Elektron. Mat. Izv., 13 (2016), 1300–1313 | DOI | MR | Zbl

[10] Levchuk V.M., Kravtsova O.V., “Problems on structure of finite quasifields and projective translation planes”, Lobachevskii J. Math., 38:4 (2017), 688–698 | DOI | MR | Zbl

[11] Maduram D.M., “Matrix representation of translation planes”, Geom. Dedicata, 4 (1975), 485–492 | DOI | MR | Zbl

[12] Podufalov N.D., “On spread sets and collineations of projective planes”, Contem. Math., 131:1 (1992), 697–705 | DOI | MR | Zbl

[13] Biliotti M., Jha V., Johnson N.L., Foundations of translation planes, Marcel Dekker Inc., NY, Basel, 2001, 542 pp. | MR | Zbl

[14] Mäurer H., “Die affine Projektivitätengruppe der Hallebenen [The affine group of projectivities of the Hall planes]”, Aequationes Math, 32 (1987), 271–273 URL: https://eudml.org/doc/137191 | MR | Zbl

[15] Nesbitt-Stobert S.B., Garner C.W.L., “A direct proof that all Hall planes of the same finite order are isomorphic”, Riv. Mat. Univ. Parma, 12:4 (1986), 241–247 URL: http://rivista.math.unipr.it/fulltext/1986-12/1986-12-241.pdf | MR | Zbl

[16] Wähling H., Theorie der Fastkörper, Thales Monographs, 1, Thales-Verlag, Essen, 1987, 393 pp. | MR | Zbl

[17] Kravtsova O.V., Levchuk V.M., “Voprosy stroeniya konechnykh pochti-polei”, Tr. In-ta matematiki i mekhaniki UrO RAN, 25:4 (2019), 107–117 | DOI | MR

[18] Grishkov A.N., Zavarnitsyn A.V., “Lagrange's theorem for Moufang loops”, Math. Proc. Phil. Soc., 139 (2005), 41–57 | DOI | MR | Zbl

[19] Grishkov A.N., Zavarnitsyn A.V., “Sylow's theorems for Moufang loops”, J. Algebra, 321:7 (2009), 1813–1825 | DOI | MR | Zbl

[20] Yakovleva T.N., “Voprosy stroeniya kvazipolei s assotsiativnymi stepenyami”, Izv. Irkut. gos. un-ta. Ser. “Matematika”, 29 (2019), 107–119 | DOI | Zbl

[21] Chein O., “Moufang loops of small order. I”, Trans. of the Amer. Math. Soc., 188:2 (1974), 31–51 | DOI | MR | Zbl

[22] Knuth D.E., “Finite semifields and projective planes”, J. Algebra, 2 (1965), 182–217 | DOI | MR | Zbl

[23] Kantor W.M., “Commutative semifields and symplectic spreads”, J. Algebra, 270:1 (2003), 96–114 | DOI | MR | Zbl

[24] Lavrauw M., Polverino O., “Finite semifields”, Current research topics in Galois Geometry, Chapter 6, eds. L. Storme and J. De Beule, NOVA Acad. Publ., NY, 2011, 131–160 URL: http://hdl.handle.net/1854/LU-2152960

[25] Wene G.P., “On the multiplicative structure of finite division rings”, Aequationes Math., 41 (1991), 222–233 | DOI | MR | Zbl

[26] Hentzel I.R., Rua I.F., “Primitivity of finite semifields with 64 and 81 elements”, Internat. J. Algebra and Computation, 17:7 (2007), 1411–1429 | DOI | MR | Zbl

[27] Kravtsova O.V., “On alternating subgroup $A_5$ in autotopism group of finite semifield plane”, Sib. Elektron. Mat. Izv., 17 (2020), 47–50 | DOI | MR | Zbl

[28] Oyama T., “On quasifields”, Osaka J. Math., 22 (1985), 35–54 URL: https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-22/issue-1/On-quasifields/ojm/1200778033.full | MR | Zbl