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On abstract ovals with Pascalian secant lines. (English) Zbl 1437.51003

Summary: We prove that an abstract oval is an abstract conic if and only if all of its involutions are regular and all of its secant lines are Pascalian. In doing so, we provide a new characterisation of the rank 1 projective general linear groups over arbitrary fields.

MSC:

51A45 Incidence structures embeddable into projective geometries
51E15 Finite affine and projective planes (geometric aspects)
20G15 Linear algebraic groups over arbitrary fields
20B20 Multiply transitive finite groups
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