Abstract
We develop an approach to constructing and classification of semifield projective planes with the use of a linear space and a spread set. We construct a matrix representation of the spread set of a semifield plane of odd order that admits a Baer involution in the translation complement or a subgroup of autotopisms isomorphic to the alternating group A 4. We give examples of semifield planes of order 81 satisfying the above indicated condition.
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Biliotti, M., Jha, V., Johnson, N. L., and Menichetti, G. “A Structure Theory for Two-dimensional Translation Planes of Order q 2 that Admit Collineation Group of Order q 2”, Geom.Dedic. 29, No. 1, 7–43 (1989).
Huang, H. and Johnson, N. L. “8 Semifield Planes of Order 82”, DiscreteMath. 80, No. 1, 69–79 (1990).
Cordero, M. “Matrix Spread Sets of p-Primitive Semifield Planes”, Int. J. Math. and Math. Sci. 20, No. 2, 293–298 (1997).
Podufalov, N. D., Durakov, B. K., Kravtsova, O. V., and Durakov, E. B. “On Semifield Planes of Order 162”, Sib. Math. J. 37, No. 3, 535–541 (1996).
Kravtsova, O. V. “Some Subgroups of Automorphisms of Semifield Planes”, in Algebra and Logic: Theory and Applications. Proceedings of Internat. Conf. Dedicated to V. P. Shunkov, Krasnoyarsk, July 21–27, 2013 (Krasnoyarsk, 2013), pp. 78–80.
Kravtsova, O. V. “A Subgroup of Autotopisms of anOddOrder Semifield Plane Isomorphic to the Alternating Group A 4”, in Proceedings of Conf. ‘Algebra and Math. Logic: Theory and Applications’ (Kazan, June, 2–6, 2014) and the satellite Youth Scientific School ‘Computability and Computable Structures’ (Kazan University, Kazan, 2014), p. 89 [in Russian].
Hughes, D. R. and Piper, F. C. Projective Planes (Springer-Verlag, New York, 1973).
Podufalov, N. D. “On Spread Sets and Collineations of Projective Planes”, Contem.Math. 131, No. 1, 697–705 (1992).
Unsolved Problems of the Theory of Groups. Kourov Notebook. 16-th Edition, complemented and including the archive of solved problems. Ed. by V. D. Mazurov and E. I. Khukhro (Novosibirsk, 2006).
Levchuk, V. M., Panov, S. V., and Stukkert, P. K. “Questions of Classification of Projective Planes and Latin Rectangles”, in Mechanics and Modelling (SibGAU, Krasnoyarsk, 2012), pp. 56–70 [in Russian].
Kravtsova, O. V. “Semifield Planes of Even Order that Admit the Baer Involution”, Izv. Irkutsk.Gos. Univ., Ser. Mat. 6, No. 2, 26–37 (2013).
Kravtsova, O. V., Panov, S. V., and Shevelyova, I. V. “Some Results on Isomorphisms of Finite Semifield Planes”, J. Siberian Federal University.Mathematics & Physics 6, No. 1, 33–39 (2013).
Kravtsova, O. V. and Kurshakova, P. K. “On the Question of Isomorphity of Semifield Planes”, Vestnik KGTU.Matem.Metody iModelir., No. 42, 13–19 (2006) [in Rusian].
Podufalov, N. D., Busarkina, I. V., and Durakov, B. K. “On the Autotopism Group of a Semifield p-Primitive Plane”, in Proceedings of the Interregional Scientific Conference ‘Investigations on Analysis and Algebra’ (TGU, Tomsk, 1998), pp. 190–195.
Kravtsova, O. V. “On Some Translation Planes Admitting A 4”, in Abstracts of III All-Siberian Congress of Women-Mathematicians, Krasnoyarsk, January 15–18, 2004 (Krasnoyarsk, 2004), pp. 38–39 [in Russian].
Kravtsova, O. V. and Pramzina, V. O. “On a Subgroup of Collineations of a Semifield Plane Isomorphic to A 4”, J. Siberian Federal University. Mathematics & Physics 4, No. 4, 498–504 (2011) [in Russian].
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Original Russian Text © O.V. Kravtsova, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 9, pp. 10–25.
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Kravtsova, O.V. Semifield planes of odd order that admit a subgroup of autotopisms isomorphic to A 4 . Russ Math. 60, 7–22 (2016). https://doi.org/10.3103/S1066369X16090024
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DOI: https://doi.org/10.3103/S1066369X16090024