Questions of the structure of finite near-fields

O. V. Kravtsova ; V. M. Levchuk

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 107-117

Résumé

A semifield is a simple ring in which nonzero elements with respect to multiplication form a loop. Weakening distributivity from two-sided to one-sided yields the more general notion of quasifield (near-field under the condition of associativity). Problems of the structure of finite semifields and quasifields have been studied in various cases for a long time. In recent years, they have been mentioned in a number of papers. These problems were solved earlier for Knuth–Rúa and Hentzel–Rúa semifields, which are counterexamples of orders 32 and 64 to Wene's known hypothesis. The methods of computer algebra were used to describe some quasifields of small orders. It is known that the center of a finite semifield always contains the prime subfield. We show that the center of a finite near-field $Q$ contains the prime subfield $P$ except for Zassenhaus' four near-fields of orders $5^2$, $7^2$, $11^2$, and $29^2$. The kernel of a near-field $Q$ always contains $P$. The maximal subfields of a finite near-field are enumerated under sufficiently general conditions. The automorphism groups of a near-field $Q$ and of its multiplicative group $Q^*$ were found earlier. The group $Q^*$ is metacyclic, which makes it possible to explicitly find the spectrum of group orders of its elements.

Détail

Keywords: quasifield, semifield, near-field, maximal subfield, spectrum.

Bibliographie

[1] Kurosh A.G., Lektsii po obschei algebre, Fizmatgiz, Moskva, 1962, 396 pp.

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

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

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

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

[6] 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

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

[8] Zassenhaus H., “Uber endliche Fastkorper”, Abh. Math. Sem. Hamburg, 11:1 (1936), 187–220 | DOI | MR

[9] Kholl M., Teoriya grupp, Gosinoizdat, M., 1962, 468 pp.

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

[11] Felgner U., “Pseudo-finite near-fields”, Near-rings and near-fields, Ser. North-Holland Mathematics Studies, 137, ed. C. Betrh, Elsevier Science Publisher B. V. (North-Holland), N Y Inc., 1987, 15–29 | DOI | MR

[12] Kokseter G.S.M., Mozer U.O.Dzh., Porozhdayuschie elementy i opredelyayuschie sootnosheniya diskretnykh grupp, Nauka, Moskva, 1980, 240 pp.

[13] Dancs S., “The sub-near-field structure of finite near-fields”, Bull. Austral. Math. Soc., 5 (1971), 275–280 | DOI | MR | Zbl

[14] Dancs Groves S., “Locally finite near-fields”, Abh. Math. Sem. Univ. Hamburg, 48 (1979), 89–107 | DOI | MR | Zbl

[15] Dancs S., “On finite Dickson near-fields”, Abh. Math. Sem. Univ. Hamburg, 37 (1972), 254–257 | DOI | MR | Zbl

[16] Ellers E., Karzel H., “Endliche Inzidenzgruppen”, Abh. Math. Sem. Hamburg, 27:3–4 (1964), 250–264 | DOI | MR | Zbl

[17] Wahling H., Theorie der Fastkorper, Thales Monographs, 1, Thales-Verlag, Essen, 1987, 393 pp. | MR | Zbl

[18] Boykett T., Howell K.-T., “The multiplicative automorphisms of a finite nearfield, with an application”, Commun. Algebra, 44:6 (2016), 2336–2350 | DOI | MR | Zbl