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Hyperovals, Laguerre planes and hemisystems - an approach via symmetry




Bayens, Luke, author
Penttila, Tim, advisor
Achter, Jeff, committee member
Bohm, Willem, committee member
Peterson, Chris, committee member

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In 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries -- we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups -- indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting transitively on flags, with an additional hypothesis -- a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order. We also prove a correspondence between translation ovoids of translation generalized quadrangles arising from a pseudo-oval O and translation flocks of the elation Laguerre plane arising from the dual pseudo-oval O*. The last topic we consider is the existence of hemisystems in finite hermitian spaces. Hemisystems were introduced by Segre in 1965 -- he constructed a hemisystem of H(3,32) and rasied the question of their existence in other spaces. Much of the interest in hemisystems is due to their connection to other combinatorial structures, such as strongly regular graphs, partial quadrangles, and association schemes. In 2005, Cossidente and Penttila constructed a family of hemisystems in H(3,q2), q odd, and in 2009, the same authors constructed a family of hemisystem in H(5,q2), q odd. We develop a new approach that generalizes the previous constructions of hemisystems to H(2r - 1,q2), r > 2, q odd.


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