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Projectivities in Free-Like Geometries

  • A. Barlotti
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 70)

Abstract

In a projective plane all the projectivities of a line onto itself form a group II with respect to the composition of mappings. This group is an invariant for the plane since different lines have groups which are isomorphic (also as permutation groups).

Keywords

Irreducible Representation Projective Plane Permutation Group Incidence Structure Partial Plane 
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.

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References

  1. [1]
    A. Barlotti; La determinazione del gruppo delle proiettività di una retta in se in alcuni particolari piani grafici finiti non desarguesiani; Boll. Un. Mat. Ital. 14 (1959), pp. 182–187.MathSciNetGoogle Scholar
  2. [2]
    A. Barlotti; Sul gruppo delle proiettività di una retta in sè nei piani liberi e nei piani aperti; Rend. Sem. Mat. Padova, 34 (1964), 135–159.MathSciNetMATHGoogle Scholar
  3. [3]
    A. Barlotti; Configurazioni k-chiuse e piani k- aperti; Rend. Sem. Mat. Padova, 35 (1965), 56–64.MathSciNetMATHGoogle Scholar
  4. [4]
    A. Barlotti; Sulle m-strutture di Möbius; Rend. 1st. Mat. Univ. Trieste 1 (1969), 35–46MathSciNetMATHGoogle Scholar
  5. [5]
    A. Barlotti, E. Schreiber, K. Strambach; The group of projectivities in free-like geometries; Rend. Sem. Mat. Univ. Padova 60 (1978), 183–200.MathSciNetMATHGoogle Scholar
  6. [6]
    A. Barlotti, K. Strambach; The geometry of binary systems, to appear.Google Scholar
  7. [7]
    W. Benz; Vorlesungen über Geometrie der Algebren; Berlin-Heidelberg-New York, Springer Verlag, 1973.MATHGoogle Scholar
  8. [8]
    P. Dembowski; Freie und offene projektive Ebenen; Math. Z., 72 (1960), 410–438.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    P. Dembowski; Finite Geometries; Berlin-Heidelberg- New York, Springer Verlag, 1968.MATHGoogle Scholar
  10. [10]
    H. Freudenthal, K. Strambach; Schlieβungssätze und Projektivitäten in der Möbius- und Laguerregeo- metrie; Math. Z., 143 (1975), 213–234.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    M. Funk; Regularität in Benz-Ebenen; Ph. D. Thesis, Erlangen (1980).Google Scholar
  12. [12]
    H.R. Halder, W. Heise; Einführung in die Kombinatorik; München-Wien, Hanser-Verlag (1 976).Google Scholar
  13. [13]
    M. Hall Jr.; Projective planes; Trans. Amer. Math. Soc., 54 (1943), 229–277.MATHCrossRefGoogle Scholar
  14. [14]
    W. Heise, H. Seybold; Das Existenzproblem der Möbius-, Laguerre- und Minkowski-Erweiterungen endlicher affiner Ebenen; Sitz. Ber. Bayer. Akad. Wiss., Math. Nat. Kl. 1975, 43–58.Google Scholar
  15. [15]
    W. Heise, K. Sörensen; Freie Minkowski-Ebenenerwei- terungen, J. Geom. 3, (1973) 1–4.MATHCrossRefGoogle Scholar
  16. [16]
    R.D. Hughes, F.C. Piper; Projective planes; New York- Heidelberg-Berlin, Springer Verlag, 1973.MATHGoogle Scholar
  17. [17]
    J. Joussen; Die Anordnungsfähigkeit der freien Ebenen; Abh. Math. Sem. Univ. Hamburg 29 (1966), 137–184.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    H.J. Kroll; Die Gruppe der eigentlichen Projektivitäten in Benz-Ebenen; Geometriae Dedicata 6, 407–413 (1977).MathSciNetMATHGoogle Scholar
  19. [19]
    H.J. Kroll; Perspektivitäten in Benz-Ebenen; in Beiträge zur geometrischen Algebra, 203–207, Basel-Stuttgart, Birkhäuser 1977.Google Scholar
  20. [20]
    G. Pickert; Projektive Ebenen; Berlin-Heidelberg- New York, zweite Auflage, Springer Verlag, 1975.MATHGoogle Scholar
  21. [21]
    A. Schleiermacher; Bemerkungen zum Fundamentalsatz der projektiven Geometrie; Math. Z. 99, 299–304 (1976).MathSciNetCrossRefGoogle Scholar
  22. [22]
    A. Schleiermacher, K. Strambach; Über die Gruppe der Projektivitäten in nichtgeschlossenen Ebenen; Arch. Math. 18, 299–307 (1967).MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    A. Schleiermacher, K. Strambach; Freie Erweiterungen in der affinen Geometrie und der Geometrie der Kreise (I u. II); Abh. Math. Sem. Univ. Hamburg 34, 22–37 and 209–226 (1969–70).Google Scholar
  24. [24]
    E. Schreiber; Freie Strukturen und die Gruppe der affinen Projektivitäten; Dissertation, Erlangen, 1979.Google Scholar
  25. [25]
    E.C. Siebenmann; A characterization of free projective planes; Pac. Journ. 15 (1965) pp. 293–298.MathSciNetMATHGoogle Scholar
  26. [26]
    K. Strambach; Die Gruppe der Projektivitäten in projektiven und affinen Ebenen; Erlangen 1976.Google Scholar
  27. [27]
    K. Strambach; unpublished lectures noted by H. Krauβ; Erlangen (1978); Vorlesungsausarbeitung.Google Scholar
  28. [28]
    K.G.Ch. v. Staudt; Geometrie der Lage; Nürnberg, Verlag von Bauer und Raspe 1847.Google Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • A. Barlotti
    • 1
  1. 1.Università di BolognaBolognaItaly

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