Una proprietà gruppale delle involuzioni planari che mutano in sé un'ovale di un piano proiettivo finito

Summary

The purpose of the present paper is to prove the following theorem: Let Ω be an oval in the projective plane P of odd order n. If P admits a collineation group G wich maps Ω onto itself and is doubly transitive on Ω, then P is desarguesian, Ω is a conic and G contains all collineations in the little projective group PSL(2, n) of P wich leaves Ω invariant.

Bibliografia

  1. [1]

    M. Aschbacher,On doubly transitive permutation groups of degree n = 2 (mod 4), Illinois J. Math.,16 (1969), pp. 276–279.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    H. Bender,Endliche zweifach transitive Permutationsgruppen deren Involutionen keine Fixpunte haben, Math. Zeitschr.,104 (1968), pp. 175–204.

    Article  Google Scholar 

  3. [3]

    P. Dembowski,Finite Geometries, Springer Verlag (1968).

  4. [4]

    J. Cofman,Doubly transitivity in finite affine and projective planes, Proc. Proj. Geom. Conf., Univ. Illinois (1967), pp. 16–19.

  5. [5]

    Ch. Hering -W. M. Kantor -G. M. Seitz,Finite groups with a Split BN-pairs of Rank 1, J. Algebra,20 (1970), pp. 435–475.

    MathSciNet  Article  Google Scholar 

  6. [6]

    Ch. Hering,Zweifach transitive Permutationsgruppen in denen zwei die maximale Anzahl von Fixpunten sind, Math. Zeitschr.,104 (1968), pp. 150–174.

    Article  Google Scholar 

  7. [7]

    Ch. Hering,On 2-groups operating on projective planes, Illinois J. Math.,16 (1972), pp. 581–595.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    B. Huppert,Endliche Gruppen I, Springer Verlag (1967).

  9. [9]

    F. Kárteszi,Introduction to the finite geometries, Akadémiai Kiado (Budapest) (1975).

  10. [10]

    H. Lüneburg, Charakteriseirungen der endlichen desargueschen projektive Ebenen, Math. Zeitschr.,85 (1964), pp.419–450.

    Article  Google Scholar 

  11. [11]

    D. S. Passman,Some 5/2-transitive permutation groups, Pacific J. Math.,13 (1969), pp. 755–1029.

    MathSciNet  MATH  Google Scholar 

  12. [12]

    B. Segre,Lectures on Modern Geometry, Edizioni Cremonese (1961).

  13. [13]

    G. Zappa,Fondamenti di teoria dei gruppi I–II, Edizioni Cremonese (1965, 1970).

  14. [14]

    H. Wielandt,Finite permutation groups, Academic Press (1964).

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Entrata in Redazione il 5 april 1977.

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Korchmáros, G. Una proprietà gruppale delle involuzioni planari che mutano in sé un'ovale di un piano proiettivo finito. Annali di Matematica 116, 189–205 (1978). https://doi.org/10.1007/BF02413875

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