ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2014. Vol. 7

Quasifields and Translation Planes of the Smallest Even Order

P. Shtukkert

Constructs of different classes of finite non-Desargues translation planes and quasifields closely related. It used by computer calculations since the middle of last century. We study semifields of order 32 and quasifields of order 16 of corresponding translation planes.
It is known that translation planes of any order pn for a primep can beconstructed by using a coordinatizing set W of order n over the field of order p. Byusingaspread set we providing W of structure of quasifield. The plane is set to be a semifield plane if W is a semifield. The plane is Desargues if W is a field. It is well-known that semifield planes are isomorphic if and only if their semifields are isotopic.
Structure of quasifields of order pn has been studied a few, even for small n. In 1960 Kleinfeld classified quasifields of order 16 with kernel of order 4 and all semifields of order 16 up to isomorphisms. Later Dempwolf and other completed the classification of all translation planes of order 16 and 32. We construct 5 semifields of order 32 and 7 quasifields of order 16 of non-Desargues planes by using their spread sets. For these semifields and for these quasifields (partially) our main results list for them introduced orders of all non-zero elements and all subfields.

translation planes, spread set, quasifield, semifield, order of element of loop

1. Albert A.A. Finite division algebras and finite planes. Proc. Sympos. Appl. Math., vol.10. Amer. Math. Soc., Providence, R.I., 1960, pp. 53-70.

2. Andre J., Uber nicht-Desarguesche Ebenen mit transitiver Translationgruppe. J. Andre. Math. Z. , 1954, vol. 60, pp. 156-186.

3. Dempwolff U., Reifart A. The Classification of the translation planes of order 16, I. Geom. Dedicata, 1983, vol. 15, pp. 137-153.

4. Dempwolff U. File of Translation Planes of Small Order. Available at: www.mathematik.uni-kl.de/dempw.dempw_Plane.html.

5. Dickson L.E. Linear algebras in which division is always uniquely possible. Trans. Amer. Math. Soc., 1906, vol. 7, pp. 370-390.

6. Hall M. Theory of groups. Moscow, 1962.

7. Hughes D.R., Piper F.C. Projective planes. Springer - Verlag: New-York Inc., 1973.

8. Kleinfeld E. Techniques for enumerating Veblen-Wedderburn systems. J. Assoc. Comput. Mach., 1960, vol. 7, pp. 330-337.

9. Knuth D.E. Finite semifields and projective planes. J. Algebra, 1965, vol. 2, pp. 182-217.

10. Kurosh A.A. Lectures on general algebra. Moscow, 1973.

11. Levchuk V.M., Panov S.V., Shtukker P.K. Enumeration of semifield planes and Latin rectangles. Book of scientific articles «Modeling and mechanics*. Krasnoyarsk, Sib. St. Air. Univ., 2012, pp. 56-70.

12. Lorimer P. A Projective Plane of Order . J. Combin. theory (A), 1974, vol. 16, pp. 334-347.

13. Luneburg H. Translation planes. Springer - Verlag: Berlin Heidelberg New-York Inc., 1980.

14. Podufalov N.D. On functions on linear spaces. J. Algebra and Logic, 2002, vol. 41, no. 1, pp. 83-103.

15. Rockenfeller R. Translationsebenen der Ordnung 32. Diploma Thesis, FB Mathematik, Univ. of Kaiserslautern, 2011.

16. Shtukkert P.K. On the properties of semifields of even order. Collection of abstracts, International Conference «Mal'tcev meeting». Novosibirsk, 2013, p. 114.

17. Veblen O. Non-Desarguesian and Non-Pascalian Geometries. J.H. Maclagan-Wedderburn. Trans. Amer. Math. Soc., 1907, vol. 8, no. 3, pp. 379-388.

18. Walker R.J. Determination of Division Algebras with 32 Elements. Proc. Symp. Appl. Math. XV, Amer. Math. Soc., 1962, pp. 83-85.

19. Wesson J.R. On Veblen-Wedderburn Systems. The Amer. Math. Monthly, 1957, vol. 64, no. 9, pp. 631-635.

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