{"id":{"repo_id":"colostate","oai_identifier":"oai:mountainscholar.org:10217/167107"},"canonical_url":"https://search.dev.ndltd.org/etd/colostate/oai:mountainscholar.org:10217/167107","repository":{"repo_id":"colostate","name":"Colorado State University","base_url":"https://api.mountainscholar.org/server/oai/request"},"display":{"title":"Abstract hyperovals, partial geometries, and transitive hyperovals","abstract":"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","abstract_html":"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","abstract_has_math":false,"creators":["Cooper, Benjamin C., author","Penttila, Timothy, advisor","Bohm, Wim, committee member","Cavalieri, Renzo, committee member","Duflot, Jeanne, committee member"],"institution":"Colorado State University. Libraries","degree_name":"Doctor of Philosophy (Ph.D.)","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015","date_published":"2015","updated_at":"2026-07-27T19:13:20Z","subjects":["finite geometry","combinatorics"],"languages":["eng","English"],"rights":["Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://doi.org/10.25675/3.018898"],"render_values":[{"text":"https://doi.org/10.25675/3.018898","href":"https://doi.org/10.25675/3.018898","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10217/167107","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Cooper, Benjamin C., author","Penttila, Timothy, advisor","Bohm, Wim, committee member","Cavalieri, Renzo, committee member","Duflot, Jeanne, committee member"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-08-28T14:35:09Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-08-28T14:35:09Z"]},{"key":"dc:date.issued","label":"Date","values":["2015"]},{"key":"dc:publisher","label":"Institution","values":["Colorado State University. Libraries"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (Ph.D.)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Colorado State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["finite geometry","combinatorics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["Cooper_colostate_0053A_13076.pdf"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10217/167107","https://doi.org/10.25675/3.018898"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["born digital","doctoral dissertations"]},{"key":"dc:title","label":"Title","values":["Abstract hyperovals, partial geometries, and transitive hyperovals"]}]}],"canonical_facts":{"dc:creator":["Cooper, Benjamin C., author","Penttila, Timothy, advisor","Bohm, Wim, committee member","Cavalieri, Renzo, committee member","Duflot, Jeanne, committee member"],"dc:date.accessioned":["2015-08-28T14:35:09Z"],"dc:date.available":["2015-08-28T14:35:09Z"],"dc:date.issued":["2015"],"dc:description.abstract":["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."],"dc:format.medium":["born digital","doctoral dissertations"],"dc:identifier":["Cooper_colostate_0053A_13076.pdf"],"dc:identifier.uri":["http://hdl.handle.net/10217/167107","https://doi.org/10.25675/3.018898"],"dc:language":["English"],"dc:language.iso":["eng"],"dc:publisher":["Colorado State University. Libraries"],"dc:rights":["Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright."],"dc:subject":["finite geometry","combinatorics"],"dc:title":["Abstract hyperovals, partial geometries, and transitive hyperovals"],"dc:type":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy (Ph.D.)"],"thesis:institution_name":["Colorado State University"]},"updated_at":"2026-07-27T19:13:20Z"}