Abstract hyperovals, partial geometries, and transitive hyperovals

dc.contributor.authorCooper, Benjamin C., author
dc.contributor.authorPenttila, Timothy, advisor
dc.contributor.authorBohm, Wim, committee member
dc.contributor.authorCavalieri, Renzo, committee member
dc.contributor.authorDuflot, Jeanne, committee member
dc.descriptionIncludes bibliographical references.
dc.description2015 Summer.
dc.description.abstractA 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
dc.description.abstractG|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G
dc.description.abstract144. 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
dc.description.abstractG|, 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.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.publisherColorado State University. Libraries
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dc.subjectfinite geometry
dc.titleAbstract hyperovals, partial geometries, and transitive hyperovals
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thesis.degree.grantorColorado State University
thesis.degree.nameDoctor of Philosophy (Ph.D.)
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