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A new proof of a theorem of Dembowski

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Abstract

It was shown by P.Dembowski [1;Satz 3] that any finite semiaffine plane(=FSAP) is of the types:

  1. (I)

    A finite affine plane,

  2. (II)

    A finite projective plane with one line and all its points except one deleted,

  3. (III)

    A finite projective plane with one point deleted,

  4. (IV)

    A finite projective plane.

This was established by using the results obtained for natural parallelisms of incidence structures. The purpose of this note is to give a new proof based on purely combinatorial arguments.

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References

  1. P. Dembowski; Semiaffine Ebenen. Arch. Math. XIII (1962), 120–131.

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  2. P. Dembowski; Finite Geometries,Springer-Verlag, New York 1968.

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  3. P. de Witte; Combinatorial Properties of Finite Linear Spaces I. Bull. Soc. Math. de Belgique XVIII (1966), 133–141.

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  4. P. de Witte; A New Property of Non-trivial Finite Linear Spaces. Bull. Soc. Math. de Belgique XVIII (1966), 430–438.

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  5. G. Pickert; Projektive Ebenen, Springer-Verlag, Berlin-Göttingen-Heidelberg 1955.

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Author notes

  1. Bijan Farrahi

    Present address: Dpt. of Pure Mathematics, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

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    1. Bijan Farrahi
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    The author was supported by the National Research Council of Canada.

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    Farrahi, B. A new proof of a theorem of Dembowski. J Geom 5, 185–189 (1974). https://doi.org/10.1007/BF01949681

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    • DOI: https://doi.org/10.1007/BF01949681

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