Abstract
Let II be a translation plane of orderq 3, with kernel\( \supseteq \) GF(q) forq a prime power, that admits a collineation groupG of orderq 3 in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq 2, whereE is the elation group ofG.
In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.
Furthermore, we completely determined the translation planes of orderp 3, forp a prime, admitting a collineation groupG of orderp 3 in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp 3 or (ii) the Sherk plane of order 27.
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References
Akiyama, K.: On the Sherk plane (to appear inJ. Geom.).
Biliotti, M., Jha, V., Johnson, N. L. and Menichetti, G.: A structure theory for two-dimensional translation planes of orderq 2 that admit collineation groups of orderq 2,Geom. Dedicata 29 (1989), 7–43.
Dempwolff, U.: A characterization of the generalized twisted field planes,Arch. Math. 50 (1988), 477–480.
Dempwolff, U.: Translation planes of order 27 (to appear inGeom. Dedicata).
Fink, J. B., Johnson, N. L. and Wilke, F. W.: A characterization of ‘likeable’ translation planes,Rend. Circ. Mat. Palermo 32 (1983), 76–99.
Hughes, D. R. and Piper, F. C.:Projective Planes, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
Jha, V. and Johnson, N. L.: A note on finite semifield planes that admit affine homologies,J. Geom. 24 (1985), 194–197.
Johnson, N. L. and Wilke, F. W.: Translation planes of orderq 2 that admit collineation group of orderq 2,Geom. Dedicata 15 (1984), 293–312.
Kantor, W. M.: On point-transitive affine planes,Israel J. Math. 42 (1982), 227–234.
Lidl, R. and Niederreiter, H.:Finite Fields, Cambridge Univ. Press, Cambridge, 1983.
Lüneburg, H.:Translation Planes, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
Menichetti, G.: On a Kaplansky conjecture concerning three-dimensional division algebra over a finite field,J. Algebra 47 (1977), 400–410.
Narayana Rao, M. L. and Satyanarayana, K.: On Sherk's plane of order 27,Linear Algebra Appl. 74 (1986), 1–9.
Oyama, T.: On quasifields,Osaka J. Math. 22 (1985), 35–54.
Sherk, F. A.: A translation plane of order 27 with a small translation complement,Geom. Dedicata 9 (1980), 307–316.
Vaughan, T. P.: Polynomials and linear translations over finite fields,J. Reine Angew. Math. 269 (1974), 179–206.
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Akiyama, K., Suetake, C. On translation planes of orderq 3 that admit a collineation group of orderq 3 . Geom Dedicata 55, 1–57 (1995). https://doi.org/10.1007/BF02179084
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DOI: https://doi.org/10.1007/BF02179084