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The affine planeAG(2,q),q odd, has a unique one point extension

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LetJ be a finite inversive plane of odd orderq. If for at least one pointp ofJ the internal affine planeJ p is Desarguesian, thenJ is Miquelian. Other formulation: the finite Desarguesian affine plane of odd orderq has a unique one point extension; this extension is the Miquelian inversive plane of orderq. It follows that there is a unique inversive plane of orderq, withq∈{3, 5, 7}.

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Authors and Affiliations

  1. Department of Pure Mathematics and Computer Algebra, University of Gent, Krijgslaan 281, B-9000, Gent, Belgium

    J. A. Thas

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  1. J. A. Thas
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Oblatum 23-X-1992 & 24-I-1994

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Thas, J.A. The affine planeAG(2,q),q odd, has a unique one point extension. Invent Math 118, 133–139 (1994). https://doi.org/10.1007/BF01231529

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