Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-18T21:40:39.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Pseudo-Homogeneous Coordinates for Hughes Planes

Published online by Cambridge University Press:  20 November 2018

Peter Maier  and
Markus Stroppel
Peter Maier
Affiliation:
Fachbereich Mathematik der TH, Schlossgartenstr. 7, D-64289 Darmstadt
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Among the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

Keywords

Primary 51A35secondary 51A2551A10Hughes planesprojective planesspecial linear groupsnearfiledsorbit decompositions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 3 , 01 September 1996 , pp. 330 - 345
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. André, J., Nichtkommutative Geometrie und verallgemeinerte Hughes-Ebenen, Math. Z. 177(1981), 449— 462.Google Scholar
2. Artin, E., Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, 3, Interscience Publishers, New York (1957).Google Scholar
3. Biliotti, M., Su una generalizzazione di Dembowski dei piani di Hughes, Bolletino U.M.I. (5) 16-B( 1979), 674–693.Google Scholar
4. Cohn, P. M., Quadratic extensions of skewfields, Proc. London Math. Soc. 11(1961), 531–556.Google Scholar
5. Cohn, P. M., Algebra., 3, Wiley, Chichester, (2nd ed.) (1991).Google Scholar
6. Dembowski, P., Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin 44(1968).Google Scholar
7. Dembowski, P., Generalized Hughes planes, Canad. J. Math. 23(1971), 481–494.Google Scholar
8. Dembowski, P., Gruppenerhaltende quadratische Erweiterungen endlicher desarguesscher projektiver Ebenen, Arch. Math. 22(1971), 214–220.Google Scholar
9. Dieudonné, J., La Géométrie des Groupes Classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5, Springer, Berlin, (2nd ed.) (1963).Google Scholar
10. Grundhôfer, T., A synthetic construction of the Figueroa planes, J. Geom. 26(1986), 191—201.Google Scholar
11. Hàhl, H., Charakterisierung der kompakten, zusammenhängenden Moufang-Hughes-Ebenen anhand ihrer Kollineationen, Math. Z. 191(1986), 117–136.Google Scholar
12. Hering, C., Schaeffer, H.-J., On the new projective planes of R. Figueroa, Proceedings Rauischholzhausen, Lecture notes in Mathematics 969(1982), 187–190.Google Scholar
13. Hughes, D. R., A class of non-desarguesian projective planes, Canad. J. Math. 9(1957), 378—388.Google Scholar
14. Karzel, H., Bericht ùber projektive Inzidenzgruppen, Jahresber. Deutsche Math. 67(1965), 58—92.Google Scholar
15. Luneburg, H., Characterizations of the generalized Hughes planes, Canad. J. Math. 28(1976), 376–402.Google Scholar
16. Luneburg, H., Vorlesungen Über Lineare Algebra, versehen mit derzu ihrem Verständnis nötigen Algebra sowie einigen Bemerkungen zu ihrer Didaktik, BI Wissenschaftsverlag, Mannheim, 1993.Google Scholar
17. Maier, P., Die speziellen linearen Gruppen über Schiefkôrpern und Hughes-Ebenen, Thesis, Technische Hochschule Darmstadt, Darmstadt, 1994.Google Scholar
18. Ostrom, T. G., Semi-translation planes, Trans. Amer. Math. Soc. 111(1964), 1–18.Google Scholar
19. Piper, F., Polarities in the Hughes plane, Bull. London Math. Soc. 2(1970), 209–213.Google Scholar
20. Room, T. G., Polarities and ovals in the Hughes plane, J. Austral. Math. Soc. 13(1972), 196–204.Google Scholar
21. Rosati, L. A., Unicità e autodualità dei piani di Hughes, Rendic. Sem. Mat. Padova 30(1960), 316–327.Google Scholar
22. Rosati, L. A., Disegni unitari nei piani di Hughes, Geom. Dedicata 27(1988), 295—299.Google Scholar
23. Salzmann, H., Kompakte, 8-dimensionale projektive Ebenen mit groβer Kollineationsgruppe, Math. Z. 176(1981),345–357.Google Scholar
24. Salzmann, H., Betten, D., T. Grundhöfer, H. Hähl, R. Lôwen, Stroppel, M., Compact projective planes, De Gruyter, Berlin, 1996.Google Scholar
25. Schofield, A. H., Artin's problem for skew field extensions, Math. Proc. Cambridge Phil. Soc. 97(1985), 1—6.Google Scholar
26. Stroppel, M., Reconstruction of incidence geometries from groups of automorphisms, Arch. Math. (Basel) 58(1992), 621–624.Google Scholar
27. Stroppel, M., A categorical glimpse at the reconstruction of geometries, Geom. Dedicata 46(1993), 47–60.Google Scholar
28. Stroppel, M., Locally compact Hughes planes, Canad. Math. Bull. 37(1994), 112–123.Google Scholar