Journal of Geometry

, Volume 20, Issue 1, pp 101–110 | Cite as

Elations in translation planes of order 16

  • N. L. Johnson
Article

Abstract

In this article we prove the following: Let E denote the group generated by the set of all elations in a non-Desarguesian translation plane π of order 16. Then E is elementary abelian or dihedral of order 6 or 10. If ¦E¦=10 then the set of elation axes defines a derivable net contained in an E-invariant Desarguesian net of degree 7 and in this case π is the Hall plane. Thus, the only translation planes of order 16 to admit at least four elations with distinct axes are the Desarguesian and Hall planes.

Keywords

Translation Plane Hall Plane Distinct Axis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D.A. Foulser. Subplanes of partial spreads in translation planes. Bull. London Math. Soc. 4(1972), 32–38.Google Scholar
  2. [2]
    D.A. Foulser, N.L. Johnson, T.G. Ostrom. Characterization of the Desarguesian planes of order q2 by SL(2,q). (To appear) Inter. J. Math.Google Scholar
  3. [3]
    Ch. Hering. On shears of translation planes. Abh. Math. Sem. Hamburg 37(1972), 258–268.Google Scholar
  4. [4]
    Ch. Hering. On finite line transitive affine planes. Geom. Ded. 1(1973), 387–398.Google Scholar
  5. [5]
    N.L. Johnson. The translation planes of order 16 that admit nonsolvable collineation groups. (To appear).Google Scholar
  6. [6]
    N.L. Johnson, T.G. Ostrom. The translation planes of order 16 that admit PSL(2,7). J. Comb. Theory, Ser. A 26/2(1979), 127–134.Google Scholar
  7. [7]
    M.J. Kallaher. A note on finite Bol quasifields. Arch. Math. 23(1972), 161–166.CrossRefGoogle Scholar
  8. [8]
    E. Kleinfeld. Techniques for enumerating Veblen-Wedderburn systems. J. Assoc. Comput. Mach. 7(1960), 330–337.Google Scholar
  9. [9]
    T.G. Ostrom. “Finite Translation Planes,” Lecture Notes in Mathematics, No. 158, Springer-Verlag, Berlin-New York, 1970.Google Scholar
  10. [10]
    T.G. Ostrom. Linear transformations and collineations of translation planes. J. Algebra 14(1970), 405–416.CrossRefGoogle Scholar
  11. [11]
    T.G. Ostrom. Semi-translation planes. Trans. Amer. Math. Soc. 111(1964), 1–18.Google Scholar
  12. [12]
    D.S. Passman. “Permutation Groups,” W.A. Benjamin Pub. Co., New York, 1968.Google Scholar

Copyright information

© Birkhäuser Verlag 1983

Authors and Affiliations

  • N. L. Johnson
    • 1
  1. 1.Department of MathematicsThe University of IowaIowa City

Personalised recommendations