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Hyperovals, Laguerre planes and hemisystems -- an approach via symmetry.

机译：超卵形，Laguerre平面和半系统-通过对称的方法。

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摘要

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,q 2), 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,q 2), r>2, q odd.
机译：1872年，费利克斯·克莱因（Felix Klein）提出了将几何最好地看作是一组变换的不变量的研究的想法。这对几何学的研究产生了深远的影响，最终将对称性提升到了中心位置。本文体现了Klein Erlangen程序在有限几何现代环境中的精神-我们利用有关有限经典群的知识来解决该地区长期存在的问题。；我们首先研究有限Desarguesian投影平面中的超椭圆形。在过去的25年中，已经建立了许多无限的家庭。与羊群，广义四边形和Laguerre飞机等相关联，促使该地区开展了许多活动。确定卵圆及其相关对象的一个​​重要因素是确定它们的类别-实际上，区分它们的唯一方法通常是通过这种计算。我们计算了由Cherowitzo在1998年构建的椭圆族的自同构群，并获得了有关作用于超卵形体的组的一般结果，包括对具有大自同构组的超卵形体的分类。我们将Miquelian Laguerre飞机定性为那些接纳一个包含平凡的兴高采烈并在标志上进行传递的群体，还有一个附加假设–奇数阶飞机在圆上的准原始作用，而偶数阶飞机在组上的不溶性。我们还证明了由伪卵形O引起的翻译广义四边形的平移卵形与由对偶卵形O *引起的兴高采烈的Laguerre平面的平移簇之间的对应关系；我们考虑的最后一个主题是有限半系统的存在厄米空间。半球形是由Segre于1965年引入的-他构造了H（3,32）的半球形，并提出了它们在其他空间中存在的问题。对半系统的大部分兴趣是由于它们与其他组合结构的连接，例如强正则图，部分四边形和关联方案。在2005年，Cossidente和Penttila在q（奇数）H（3，q 2）中构建了一个半系统族；在2009年，同一作者在q（奇数）H（5，q2）中构建了半系统。我们开发了一种新方法，将半系统的先前构造推广为H（2r-1，q 2），r> 2，q奇数。

著录项

  • 作者

    Bayens, Luke.;

  • 作者单位

    Colorado State University.;

  • 授予单位 Colorado State University.;
  • 学科 Applied Mathematics.;Mathematics.
  • 学位 Ph.D.
  • 年度 2013
  • 页码 129 p.
  • 总页数 129
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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