10,513 research outputs found

    The group fixing a completely regular line-oval

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    We prove that the action of the full collineation group of a symplectic translation plane of even order on the set of completely regular line–ovals is transitive. This provides us with a complete description of the group of collineations fixing a completely regular line–oval

    Singer quadrangles

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    Two-transitive ovals

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    An oval script O sign of a projective plane is called two-transitive if there is a collineation group G fixing script O sign and acting 2-transitively on its points. If the plane has odd order, then the plane is desarguesian and the oval is a conic. In the present paper we prove that if a plane has order a power of two and admits a two-transitive oval, then either the plane is desarguesian and the oval is a conic, or the plane is dual to a Lüneburg plane. © de Gruyter 2006

    Association schemes from the action of PGL(2,q)PGL(2,q) fixing a nonsingular conic in PG(2,q)

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    The group PGL(2,q)PGL(2,q) has an embedding into PGL(3,q)PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q)PG(2,q). This action affords a coherent configuration R(q)R(q) on the set L(q)L(q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions R+(q)R_{+}(q) and R−(q)R_{-}(q) to the sets L+(q)L_{+}(q) of secant lines and to the set L−(q)L_{-}(q) of exterior lines, respectively, are both association schemes; moreover, we show that the elliptic scheme R−(q)R_{-}(q) is pseudocyclic. We further show that the coherent configuration R(q2)R(q^2) with qq even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme R+(q2)R_{+}(q^2), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes R+(q2)R_{+}(q^2) and $R_{-}(q^2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.Comment: 33 page

    Abstract hyperovals, partial geometries, and transitive hyperovals

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    Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠ FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries
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