The structure of finite quasi-fields with associative degrees is investigated. These are, above all, associative quasifields, called near-fields. These also include the Moufang quasifields which have loops of nonzero elements are, by definition, loops introduced by Ruth Moufang in 1935.

The paper presents the main definitions associated with quasifields. It is shown that identity element of any finite (right) quasifield Q generates a simple subfield P, and Q is always a one-sided module over P , and a two-way — is not always. As a result, found new proof of a well-known statement: a simple subfield of a finite semifield always lies in the center. At the same time, the finite near-fields with a simple subfield that does not lie in the center. Famous the question of maximal subfields of finite quasifields is completely solved for a class of finite near-fields of order p^r with prime numbers p and r.

In solving the questions about maximal subfields and spectra of group orders of nonzero elements of finite Moufang quasifields, it is proposed to use the well-known analogues of the group-theoretic theorems of Lagrange and Sylow. All possible two-digit orders of the proper Moufang quasifields are listed.

Tatiyana Yakovleva, Postgraduate Student, Siberian Federal University, 79, Svobodniy Avenue, Krasnoyarsk, 660041, Russian Federation, tel.: 89659139701, e-mail: tnyakovleva@sfu-kras.ru

Yakovleva T.N. Questions of Construction of Quasifields with Associative Powers. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 29, pp. 107-119. (in Russian) https://doi.org/10.26516/1997-7670.2019.29.107

1. Kurosh A.G. Lectures on general algebra [Lekcii po obshhej algebre]. Saint-Petersburg, Lan Publ., 2007, 560 p.
2. Holl M. Theory of group [Teorija grupp]. Moscow, Foreign Literature Publishing House, 1962, 460 p.
3. Bruck R.H. A survey of binary systems. Springer-Verlag, 1971, 188 p.
4. Chein O. Moufang loops of small order. Transactions of the American mathematical society, 1974, vol.188, issue 2, pp. 31-51. https://doi.org/10.1090/S0002-9947-1974-0330336-3
5. Dancs S. The sub-near-field structure of finite near-fields. Bull. Austral. Math. Soc., 1971, vol. 5, pp. 275–280. https://doi.org/10.1017/S000497270004716X
6. Dancs S. On finite Dickson near-fields. Abh. Math. Sem. Univ. Hamburg, 1972, vol. 37, pp. 254–257. https://doi.org/10.1007/BF02999702
7. Dancs S. Groves Locally finite near-fields. Abhandl.Math.Seminar Hamburg, 1979, vol. 48, pp. 89-107. https://doi.org/10.1007/BF02941292
8. Dickson L.E. Linear algebras in which division is always uniquely possible. Trans. Amer. Math. Soc. 1906, vol. 7, pp. 370-390.
9. Felgner U. Pseudo-finite near-field. textit Proc. Conf. Tubingen, North-Holland, Amsterdam, 1987, pp. 15-29. https://doi.org/10.1016/S0304-0208(08)72282-5
10. Gagola S.M. The conjugacy of triality subgroups of Sylow subloops of Moufang loops. Journal of Group Theory., 2010, vol. 13, issue 6, pp. 821-840. https://doi.org/10.1515/jgt.2010.026
11. Grishkov A.N., Zavarnitsine A.V. Lagrange’s theorem for Moufang Loops. Math. Proc. Camb. Phil. Soc., 2005, no. 1(139), pp. 41-57. https://doi.org/10.1017/S0305004105008388
12. Grishkov A.N., Zavarnitsine A.V. Sylow’s theorem for Moufang loops. Journal of Algebra, 2009, vol. 321, issue 7, pp. 1813-1825. https://doi.org/10.1016/j.jalgebra.2008.08.035
13. Hughes F.S., Piper F.C. Projective planes. New-York Inc, Springer-Verlag, 1973, 291 p.
14. Johnson N.L., Jha V., Biliotti M. Handbook of finite translation planes. London New York, Chapman Hall/CRC, 2007, 861 p.
15. Kallaher M.J. Right Bol Quasi-Fields. Canadian journal of mathematics, 1969, vol. 21, pp. 1409-1420. https://doi.org/10.4153/CJM-1969-155-8
16. Knuth D.E. Finite semifields and projective planes. J. Algebra, 1965, vol. 2, pp. 182-217. https://doi.org/10.1016/0021-8693(65)90018-9
17. Levchuk V.M., Panov S.V., Shtukkert P.K. The structure of finite quasifields and their projective translation planes. Proceed. XII Intern. Conf. on Algebra and Number Theory. Tula, 2014, pp. 106-108.
18. Levchuk V.M., Shtukkert P.K. The structure of quasifields of small even orders. Proceed. Steklov Inst. Math. and Mechanics Ural Branch of RAS. Pleiades Publishing, 2015, vol. 21 (3), pp. 197-212.
19. Levchuk V.M., Kravtsova O.V. Problems on structure of finite quasifields and projective translation planes. Lobachevskii Journal of Mathematics, 2017, vol. 38, no. 4, pp. 688-698.
20. Luneburg H. Translation planes. Berlin, Heidelberg, New York, Springer-Verlag, 1980, 278 p. https://doi.org/10.1007/978-3-642-67412-9
21. Moufang R. Zur Struktur von Alternativkorpern. Math. Ann., 1935, vol. 110, no. 1, pp. 416-430. https://doi.org/10.1007/BF01448037
22. Zassenhaus H. Uber endliche Fastkorper. Abh. Math. Sem. Hamburg, 1935, vol. 11, issue 1, pp. 187-220. https://doi.org/10.1007/BF02940723