Translation Generalized Quadrangles (Series in Pure by J a Thas;K Thas;H Van Maldeghem

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By J a Thas;K Thas;H Van Maldeghem

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Example text

For any h e G, define 0h by

Hence, we assume without further notice that the dual of a given theorem or definition has also been given. If the point x is not incident with the line L, in a GQ S = (V, B, I), we write plL. Let S = (P,B, I) be a (finite) GQ of order (s,t). 1 of Payne and Thas [128]. 3 of Payne and Thas [128]. Given two (not necessarily distinct) points x,y of S, we write x ~ y and say that x and y are collinear, provided that there is some line L for which xlLly. And x ^ y means that x and y are not collinear.

1, if zi is not incident with a line uu', with u e T1- and u' e T-1-1-, then the point Zi is either collinear with two points of T1- or two points of T-1-1. 6 of Payne and Thas [128]; the results are due to Thas [171]. As the main result plays an important role in the theory of translation GQs, we will give proofs. 1 Let {x, y, z) be a 3-regular triad of the GQ S = (P, B, I) of order (s, s2), s / 1 , and let V be the set of all points incident with lines of the form uu', u 6 {x, y, z}1- = U and u' g {x, y, z } ± J - = U'.

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Pomme Pidou Library > Mathematics > Translation Generalized Quadrangles (Series in Pure by J a Thas;K Thas;H Van Maldeghem
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