Skip to main content
Log in

Metodi di teoria dei gruppi nello studio delle ovali dei piani proiettivi finiti

  • Conferenze
  • Published:
Rendiconti del Seminario Matematico e Fisico di Milano Aims and scope Submit manuscript

Sunto

La presente relazione è una rassegna di risultati sui gruppi di collineazioni che mutano in sé un'ovale di un piano proiettivo di ordine finito.

Summary

We give an account of the subject of collineation groups of a finite projective plane which fixes an oval. Almost all of the results require deep theorems on finite groups. We will mention these and, in some cases, also outline how they come into play in the proofs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Bibliografia

  1. Abatangelo V., doubly transitive (n+2)-arcs in a projective plane of even order n,J. Comb. Theory Ser. A 42 (1986) 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  2. Biliotti M.Korchmáros G., Collineation groups strongly irreducible on an oval,Annals of Discrete Math. 30 (1986) 85–98.

    Google Scholar 

  3. Biliotti M.Korchmáros G., Collineation groups which are primitive on an oval of a projective plane of odd order,J. London Math. Soc. (2)33 (1986) 525–534.

    Article  MathSciNet  MATH  Google Scholar 

  4. Biliotti M.Korchmáros G., Hyperovals with a transitive collineantion group,Geometriae Dedicata 24 (1987) 269–281.

    Article  MathSciNet  MATH  Google Scholar 

  5. Biliotti M.Korchmáros G., Collineation groups preserving an oval in a projective plane of odd order,J. Austral. Math. Soc. A 48 (1990) 156–170.

    MATH  Google Scholar 

  6. Biscarini P.Korchmáros G., Ovali di un piano di Galois di ordine pari dotate di un gruppo di collineazioni transitivo sui punti,Rend. Sem. Mat. Brescia 7 (1984) 125–135.

    MATH  Google Scholar 

  7. Beth Th.Jungnickel D.Lenz H., Design Theory (Wissenschaftsverlag, Mannheim-Wien-Zürich, 1985).

    MATH  Google Scholar 

  8. Cherowitzo W., Hyperovals in desarguesian planes of even order,Annals of Discrete Math. 30 (1986) 87–94.

    Google Scholar 

  9. Cherowitzo W.Kield D. I.Killgrowe R. B., Ovals and other configurations in the known planes of order nine,Congr. Numerantium 55 (1986) 167–179.

    Google Scholar 

  10. Cherowitzo W., Ovals in Figueroa Planes,J. Geometry 37 (1990) 84–86.

    Article  MathSciNet  MATH  Google Scholar 

  11. Cofman J., Doubly transitivity in finite affine and projective planes,Proc. Proj. Geometry Conference, Univ. of Illinois, Chicago; pp. 16–19.

  12. Dembowski P., Finite geometries (Springer Verlag, Berlin-Heidelberg-New York, 1968).

    MATH  Google Scholar 

  13. Gorenstein D., The classification of the finite simple groups (Plenum Press, New York 1983).

    MATH  Google Scholar 

  14. Hall M. Jr., Ovals in the desarguesian plane of order 16,Annali Mat. Pura Appl. 102 (1975) 159–176.

    Article  MATH  Google Scholar 

  15. Hartman A. andRosa A., Cyclic one-factorization of the complete graph, Eur. J. Combinatorics6 (1985) 45–49.

    MathSciNet  MATH  Google Scholar 

  16. Hering C., On subgroups with trivial normalizer intersection,J. Algebra 20 (1972) 622–29.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hering C., On the structure of finite collineation groups of projective planes,Abh. Math. Sem. Hamburg 49 (1979) 155–182.

    MathSciNet  MATH  Google Scholar 

  18. Hering C., On Beweglichkeit in affine planes, in Finite geometries,Lecture Notes Pure Appl. Math. 82 (1983) 197–209.

    MathSciNet  Google Scholar 

  19. Hering C. andKantor W. M. andSeitz G., Finite groups with a split BN-pairs of rank 1,J. Algebra 20 (1970) 435–475.

    Article  MathSciNet  Google Scholar 

  20. Hirschfeld J. W. P., Projective geometrics over finite fields (Clarendon Press, Oxford, 1979).

    Google Scholar 

  21. Jungnickel D., Design Theory: an Update,Ars Combinatoria 28 (1989) 129–199.

    MathSciNet  MATH  Google Scholar 

  22. Kantor W. M., On unitary polarities of finite projective planes,Canad. J. Math. 23 (1971) 1060–1077.

    MathSciNet  MATH  Google Scholar 

  23. Kantor W. M., Symplectic groups, symmetric designs and line ovals,J. Algebra 33 (1975) 43–58.

    Article  MathSciNet  MATH  Google Scholar 

  24. Kirkpatrick P. B., Collineation group which are sharply transitive on an oval,Bull. Austral. Math. Soc. 11 (1974) 197–211.

    MathSciNet  MATH  Google Scholar 

  25. Korchmáros G., Una proprietà gruppale delle involuzioni planari che mutano in sé un'ovale di un piano proiettivo,Annali Mat. Pura Appl. (4)116 (1978) 189–205.

    Article  MATH  Google Scholar 

  26. Korchmáros G., Gruppi di collineazioni transitivi sui punti di una ovale [(q+2)-arco] d iS2,q, q pari,Atti Sem. Mat. Fis. Univ. Modena 27 (1978), 89–105.

    MathSciNet  MATH  Google Scholar 

  27. Korchmáros G., Le ovali di linea del piano di Lüneburg d'ordine 22r che possono venir mutate in sé da un gruppo di collineazioni isomorfo al gruppo semplice Sz(2r) di Suzuki,Atti Accad. Naz. Lincei, Memor. (8)15 (1979) 295–315.

    Google Scholar 

  28. Korchmáros G., Collineation groups doubly transitive on the points at infinity in an affine plane of order 2r,Arch. Math. 37 (1981) 572–576.

    Article  MATH  Google Scholar 

  29. Korchmáros G., Inherited arcs in affine planes,J. Combin. Theory-A 42 (1986) 140–143.

    Article  MATH  Google Scholar 

  30. Korchmáros G., Cyclic one-factorization with an invariant one-factor of the complete graph,Ars Combinatoria 27 (1989) 133–138.

    MathSciNet  MATH  Google Scholar 

  31. Lüneburg H., Charakterisierungen der endlichen desarguesschen projektiven Ebenen,Math. Z. 85 (1964) 419–450.

    Article  MathSciNet  MATH  Google Scholar 

  32. O'Keefe C. M., Ovals in desarguesian planes,Australas. J. Combin. 1 (1990) 149–159.

    MathSciNet  MATH  Google Scholar 

  33. O'Keefe C. M.—Pentilla T., Symmetries of arcs,J. Comb. Theory Sez. A in corso di stampa.

  34. O'Keefe C. M.Pentilla T., Hyperovals in PG(2,16),Europ. J. Combinatorics 12 (1990) 51–59.

    Google Scholar 

  35. O'Keefe C. M.—Pentilla T.—Praeger C. H., Stabilizers of hyperovals in PG(2,32) inAdvances in Finite Geometries and Designs, Oxford University Press (1991) 337–351.

  36. Lunelli L.Sce M., Considerazioni aritmetiche e risultati sperimentali sui {K; n}-archi,Ist. Lomb. Accad. Sci. Rend. A 98 (1964) 3–52.

    MathSciNet  MATH  Google Scholar 

  37. Ostrom T. G., Ovals, dualities, and Desargues's theorem,Canad. J. Math. 7 (1955) 417–431.

    MathSciNet  MATH  Google Scholar 

  38. Ostrom T. G., Concoids: Cone-like figures in non-apappian planes. Geometry von Staudt's point of view (Reider Publishing Co. London 1981) pp. 175–198.

    Google Scholar 

  39. Payne S. E. andConklin J. E., An unusual generalized quadrangle of order sixteen,J. Comb. Theory Ser. A 24 (1978) 50–74.

    Article  MathSciNet  MATH  Google Scholar 

  40. Room T. G., Polarities and ovals in the Hughes plane,J. Austral. Math. Soc. 13 (1972) 196–204.

    Article  MathSciNet  MATH  Google Scholar 

  41. Segre B., Sulle ovali nei piani lineari finiti,Atti Accad. Naz. Lincei, Rend. 17 (1954) 1–2.

    MathSciNet  Google Scholar 

  42. Segre B., Ovals in a finite projective plane,Canad. J. Math. 7 (1955) 414–416.

    MathSciNet  MATH  Google Scholar 

  43. Segre B., Ovali e curve σ nei piani di Galois di caratteristica due.Atti Accad. Naz. Lincei. Rend. 34 (1962) 785–790.

    MathSciNet  Google Scholar 

  44. Stroh G., Über Gruppen mit 2-Sylow-Durchschnitten vom Rang ≤3. I,Journal of Algebra 43 (1976) 398–456.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

(Conferenza tenuta ill'11 giugno 1990)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korchmáros, G. Metodi di teoria dei gruppi nello studio delle ovali dei piani proiettivi finiti. Seminario Mat. e. Fis. di Milano 60, 93–111 (1990). https://doi.org/10.1007/BF02925080

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02925080

Navigation