Geometriae Dedicata

, Volume 36, Issue 2–3, pp 389–404 | Cite as

On the structure of the automorphism group of 2-dimensional Laguerre planes

  • Günter F. Steinke
Article
Received:
Revised:

Abstract

In this note we consider 2-dimensional Laguerre planes and prove structure theorems on their automorphism group Г. In particular, we look at connected locally simple Lie subgroups of Г and the factor group Σ/Δ of a connected closed subgroup Σ of Г over the kernel Δ of the action of Σ on the set of parallel classes. The informations obtained will be useful in the later classification of 2-dimensional Laguerre planes having a 4-dimensional automorphism group.

Keywords

Automorphism Group Factor Group Closed Subgroup Structure Theorem Parallel Classis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brouwer, L. E. J., ‘Die Theorie der endlichen kontinuierlichen Gruppen, unabhängig von den Axiomen von Lie’, Math. Ann. 67 (1909), 246–267.Google Scholar
  2. 2.
    Buchanan, T., Hähl, H. and Löwen, R., ‘Topologische Ovale’, Geom. Dedicata 9 (1980), 401–424.Google Scholar
  3. 3.
    Groh, H., ‘Topologische Laguerre-Ebenen I’, Abh. Math. Sem. Hamburg 32 (1968), 216–231.Google Scholar
  4. 4.
    Groh, H., ‘Topologische Laguerre-Ebenen II’, Abh. Math. Sem. Hamburg 34 (1970), 11–21.Google Scholar
  5. 5.
    Groh, H., ‘Characterizations of ovoidal Laguerre planes’, Arch. Math. 20 (1969), 219–224.Google Scholar
  6. 6.
    Groh, H., ‘1-dimensional orbits in flat projective planes’, Math. Z. 122 (1971), 117–124.Google Scholar
  7. 7.
    Groh, H., ‘Point homogeneous flat affine planes’, J. Geom. 8 (1976), 145–165.Google Scholar
  8. 8.
    Halder, H. R., ‘Dimension der Bahnen lokalkompakter Gruppen’, Arch. Math. 22 (1971) 302–303.Google Scholar
  9. 9.
    Löwen, R. and Pfüller, U., ‘Two-dimensional Laguerre planes with large automorphism groups’, Geom. Dedicata 23 (1987), 87–96.Google Scholar
  10. 10.
    Montgomery, D. and Zippin, L., Topological Transformation Groups, Wiley, Interscience, New York, 1955.Google Scholar
  11. 11.
    Pfüller, U., ‘Topologische Laguerreebenen’, Dissertation, Erlangen-Nürnberg, 1986.Google Scholar
  12. 12.
    Salzmann, H., ‘Kompakte zweidimensionale projektive Ebenen’, Math. Ann. 145 (1962), 401–428.Google Scholar
  13. 13.
    Salzmann, H., ‘Topological planes’, Adv. Math. 2 (1967), 1–60.Google Scholar
  14. 14.
    Steinke, G. F., ‘The automorphism group of locally compact connected topological Benz planes’, Geom. Dedicata 16 (1984), 351–357.Google Scholar
  15. 15.
    Steinke, G. F., ‘The automorphism group of Laguerre planes’, Complement to: ‘The automorphism group of locally compact connected topological Benz planes’, Geom. Dedicata 21 (1986), 55–58.Google Scholar
  16. 16.
    Steinke, G. F., ‘Semiclassical topological flat Laguerre planes obtained by pasting along two parallel classes’, J. Geom. 32, (1988), 133–156.Google Scholar
  17. 17.
    Steinke, G. F., ‘4-dimensional point-transitive groups of automorphisms of 2-dimensional Laguerre planes’, (Preprint).Google Scholar
  18. 18.
    Varadarajan, V. S., Lie Groups, Lie Algebras, and their Representation, Prentice-Hall, Englewood Cliffs, 1974.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Günter F. Steinke
    • 1
  1. 1.Department of Mathematics & StatisticsUniversity of AucklandAuckland 1New Zealand

Personalised recommendations