Browsing by Author "Stroppel, Markus (apl. Prof. Dr. rer. nat.)"

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Generalized polygons with doubly transitive ovoids
(2013) Krinn, Boris; Stroppel, Markus (apl. Prof. Dr. rer. nat.)
This thesis studies finite generalized quadrangles and hexagons which contain an ovoid. An ovoid is a set of mutually opposite points of maximum size. The first objective of the present work is to show that a generalized quadrangle or hexagon is a classical polygon if it contains an ovoid in the case of quadrangles or an ovoid-spread pairing in the case of hexagons, such that a group of given isomorphism type acts on the polygon in such a way that it leaves the ovoid invariant. The groups in use here are Suzuki and Ree groups. The second part of the thesis is devoted to the problem of determining all groups which can act on a generalized quadrangle or generalized hexagon in such a way that they operate doubly transitively on an ovoid of this polygon. It will turn out that these groups are essentially the known examples of groups acting on a classical or semi-classical ovoid of a classical polygon. In the first two chapters, meant as an introduction to the problem, constructions of the relevant generalized polygons are given and known results concerning the existence question of ovoids in known generalized quadrangles and hexagons are collected. In the following chapter, the known polarity of the symplectic quadrangle is presented and it is shown that the symplectic quadrangle can be reconstructed from the action of the Suzuki group on the absolute elements of this polarity. Then we show that any generalized quadrangle which contains an ovoid, such that a Suzuki group acts on the ovoid, is isomorphic to the symplectic quadrangle. In Chapter 4, analogous to the approach used for symplectic quadrangles, the known polarity of the split Cayley hexagon is described and it is shown that the split Cayley hexagon can be reconstructed from the action of the Ree group on the absolute points of the polarity. Then we show that any generalized hexagon that contains an ovoid-spread-pairing, on which a Ree group acts, is isomorphic to the classical split-Cayley hexagon. Both of these results are achieved without the use of classification results. The last three chapters are devoted to the problem of determining all groups which can act doubly transitively on an ovoid of a generalized quadrangle or generalized hexagon. This chapter uses the classification of finite simple groups (via the classification of finite doubly transitive groups). For hexagons the result is that only unitary groups and Ree groups are possible. This result, together with the one obtained in the previous chapter and a theorem by Joris De Kaey provides that the generalized hexagon is classical and the ovoid is classical if the ovoid belongs to an ovoid-spread-pairing. The result for quadrangles is less smooth. A further restriction on the order of the quadrangle is needed, namely that the number of points per line and the number of lines per point coincide and that this number is a prime power. This was not necessary in the case of the hexagons. With this additional assumption, we show that only orthogonal groups or Suzuki groups can act on these ovoids.