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Geometriae Dedicata

, Volume 55, Issue 1, pp 1–57 | Cite as

On translation planes of orderq3 that admit a collineation group of orderq3

  • Kenzi Akiyama
  • Chihiro Suetake
Article
  • 18 Downloads

Abstract

Let II be a translation plane of orderq3, with kernel\( \supseteq \) GF(q) forq a prime power, that admits a collineation groupG of orderq3 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 orderq2, 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 orderp3, forp a prime, admitting a collineation groupG of orderp3 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 orderp3 or (ii) the Sherk plane of order 27.

Mathematics Subject Classification (1991)

51A40 

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References

  1. 1.
    Akiyama, K.: On the Sherk plane (to appear inJ. Geom.).Google Scholar
  2. 2.
    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.CrossRefGoogle Scholar
  3. 3.
    Dempwolff, U.: A characterization of the generalized twisted field planes,Arch. Math. 50 (1988), 477–480.CrossRefGoogle Scholar
  4. 4.
    Dempwolff, U.: Translation planes of order 27 (to appear inGeom. Dedicata).Google Scholar
  5. 5.
    Fink, J. B., Johnson, N. L. and Wilke, F. W.: A characterization of ‘likeable’ translation planes,Rend. Circ. Mat. Palermo 32 (1983), 76–99.Google Scholar
  6. 6.
    Hughes, D. R. and Piper, F. C.:Projective Planes, Springer-Verlag, Berlin, Heidelberg, New York, 1973.Google Scholar
  7. 7.
    Jha, V. and Johnson, N. L.: A note on finite semifield planes that admit affine homologies,J. Geom. 24 (1985), 194–197.CrossRefGoogle Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    Kantor, W. M.: On point-transitive affine planes,Israel J. Math. 42 (1982), 227–234.Google Scholar
  10. 10.
    Lidl, R. and Niederreiter, H.:Finite Fields, Cambridge Univ. Press, Cambridge, 1983.Google Scholar
  11. 11.
    Lüneburg, H.:Translation Planes, Springer-Verlag, Berlin, Heidelberg, New York, 1980.Google Scholar
  12. 12.
    Menichetti, G.: On a Kaplansky conjecture concerning three-dimensional division algebra over a finite field,J. Algebra 47 (1977), 400–410.CrossRefGoogle Scholar
  13. 13.
    Narayana Rao, M. L. and Satyanarayana, K.: On Sherk's plane of order 27,Linear Algebra Appl. 74 (1986), 1–9.CrossRefGoogle Scholar
  14. 14.
    Oyama, T.: On quasifields,Osaka J. Math. 22 (1985), 35–54.Google Scholar
  15. 15.
    Sherk, F. A.: A translation plane of order 27 with a small translation complement,Geom. Dedicata 9 (1980), 307–316.CrossRefGoogle Scholar
  16. 16.
    Vaughan, T. P.: Polynomials and linear translations over finite fields,J. Reine Angew. Math. 269 (1974), 179–206.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Kenzi Akiyama
    • 1
  • Chihiro Suetake
    • 2
  1. 1.Department of Applied MathematicsFukuoka UniversityFukuokaJapan
  2. 2.Amagasaki-minami High SchoolAmagasaki, HyogoJapan

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