Geodesic
Problems on structure for quasifields of orders $16$ and $32$
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 3, pp. 362-372.

Voir la notice de l'article provenant de la source Math-Net.Ru

Well-known method of the construction of finite projective translation planes (analogously, semifield planes) uses their correspondence with quasifields (resp., semifields). We distinguish certain questions on the structure of any finite quasifield (possible maximal subfields, the property of cyclicity of multiplicative loop of non-zero elements and possible orders of elements). In the present paper we discover some anomalous properties of finite quasifields of small even orders.
Keywords: projective translation plane, quasifield, semifield, orders of elements.
Mots-clés : multiplicative loop 
@article{JSFU_2014_7_3_a10,
     author = {Vladimir M. Levchuk and Polina K. Shtukkert},
     title = {Problems on structure for quasifields of orders $16$ and $32$},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {362--372},
     publisher = {mathdoc},
     volume = {7},
     number = {3},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2014_7_3_a10/}
}
TY  - JOUR
AU  - Vladimir M. Levchuk
AU  - Polina K. Shtukkert
TI  - Problems on structure for quasifields of orders $16$ and $32$
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2014
SP  - 362
EP  - 372
VL  - 7
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2014_7_3_a10/
LA  - en
ID  - JSFU_2014_7_3_a10
ER  - 
%0 Journal Article
%A Vladimir M. Levchuk
%A Polina K. Shtukkert
%T Problems on structure for quasifields of orders $16$ and $32$
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2014
%P 362-372
%V 7
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2014_7_3_a10/
%G en
%F JSFU_2014_7_3_a10
Vladimir M. Levchuk; Polina K. Shtukkert. Problems on structure for quasifields of orders $16$ and $32$. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 3, pp. 362-372. http://geodesic.mathdoc.fr/item/JSFU_2014_7_3_a10/

[1] A. G. Kurosh, Lectures on general algebra, Chelsea Publishing, New-York, 1965 | MR

[2] D. R. Hughes, F. C. Piper, Projective planes, Springer-Verlag, New-York Inc., 1973 | MR | Zbl

[3] H. Lüneburg, Translation planes, Springer-Verlag, Berlin–New-York, 1980 | MR | Zbl

[4] N. L. Johnson, V. Jha, M. Biliotti, Handbook of finite translation planes, London–New-York, 2007 | MR

[5] G. P. Wene, “On the multiplicative structure of finite division rings”, Aequationes Math., 41 (1991), 791–803 | DOI | MR

[6] I. F. Rúa, “Primitive and non primitive finite semifields”, Commun. Algebra, 32:2 (2004), 793–803 | DOI | MR | Zbl

[7] N. D. Podufalov, “About functions on linear spaces, which are connected with finite projective planes”, J. Algebra and Logic, 41:1 (2002), 83–103 | MR | Zbl

[8] The Kourovka Notebook (unsolved problems in group theory), 15-th ed., Inst. of Math., SB RAN, Novosibirsk, 1992 | MR

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

[10] J. R. Wesson, “On Veblen-Wedderburn Systems”, Amer. Math. Monthly, 64:9 (1957), 631–635 | DOI | MR | Zbl

[11] E. Kleinfeld, “Techniques for enumerating Veblen–Wedderburn systems”, J. Assoc. Comput. Mach., 7 (1960), 330–337 | DOI | MR | Zbl

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

[13] U. Dempwolff, A. Reifart, “The Classification of the translation planes of order 16, Part I”, Geom. Dedicata, 15 (1983), 137–153 | DOI | MR | Zbl

[14] N. L. Johnson, T. G. Östrom, “Tangentially transitive planes of order 16”, J. Geometry, 10:2 (1977), 146–163 | DOI | MR

[15] N. L. Johnson, “Elations in translation planes of order 16”, J. Geometry, 20 (1983), 101–110 | DOI | MR | Zbl

[16] U. Dempwolff http://www.mathematik.uni-kl.de/\allowbreakd̃empw/dempw_Plane.html

[17] P. K. Shtukkert, “Quasifields and projective translation planes of even small orders”, Izvestiya Irkut. Gos. Univ., 7:1 (2014), 144–159 (in Russian)

[18] M. Hall, Theory of groups, Macmillan, 1959 | MR | Zbl

[19] T. Oyama, “On quasifields”, Osaka J. Math., 22:1 (1985), 35–54 | MR | Zbl

[20] A. I. Mal'cev, Foundations of linear algebra, W. H. Freeman, 1956 | MR

[21] O. V. Kravcova, P. K. Kurshakova (Shtukkert), “On the isomorphism of semifield planes”, Vestnik Krasnoyar. Gos. Technic. Univ., 42 (2006), 13–19 (in Russian)

[22] R. Rockenfeller, Translationsebenen der Ordnung 32, Diploma Thesis, FB Mathematik, University of Kaiserslautern, 2011

[23] R. J. Walker, “Determination of division algebras with 32 elements”, Proc. Symp. Appl. Math., XV, Amer. Math. Soc., 1962, 83–85 | MR