By J a Thas;K Thas;H Van Maldeghem

Translation generalized quadrangles play a key position within the concept of generalized quadrangles, resembling the function of translation planes within the conception of projective and affine planes. The inspiration of translation generalized quadrangle is an area analog of the extra international "Moufang Condition", a subject of serious curiosity, additionally end result of the class of all Moufang polygons. realization is hence paid to contemporary leads to that course, but in addition a number of the most vital leads to the overall conception of generalized quadrangles that seemed in view that 1984 are handled. "Translation Generalized Quadrangles" is basically self-contained, because the reader is just anticipated to be accustomed to a few easy proof on finite generalized quadrangles. Proofs which are both too lengthy or too technical are passed over, or simply sketched. the 3 usual works on generalized quadrangles are (co-)authored by way of the writers of this e-book: "Finite Generalized Quadrangles" (1984) by means of S E Payne and J A Thas, "Generalized Polygons" (1998) through H Van Maldeghem, and "Symmetry in Finite Generalized Quadrangles" (2004) through ok Thas.

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Ii) Any three distinct points, no two of which are incident with a common line, are incident with exactly one common circle. (iii) If x and y are distinct noncollinear points of £ and if C is a circle incident with x but not with y, then there is just one circle C incident with x, y and having only x in common with C (two circles having exactly one point x in common will be called tangent circles at x). (iv) There exist a point x and a circle C with x IC; each circle is incident with at least three points.

Hence the point x is antiregular. Conversely, assume that x is antiregular. 1, s is odd. 3 The point (oo) of any GQ T 2 (C) of order q, with q odd, is antiregular. Proof. 1 the point (oo) of T 2 (C) is coregular, hence by the previous theorem it is antiregular. 4 Antiregularity and Laguerre Planes This section will be about the connection between antiregular points, affine planes and Laguerre planes. 1 Let x be an antiregular point of the GQ ~~
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~~Generalized quadrangles were introduced by Tits [217] in the appendix of his celebrated work on triality. If x and L are as in (iii), then we will denote the unique point y on L collinear with x also by projLa;, and call it the projection ofx onto L. Dually we define proj x L •= M. We generalize this to points z on L by putting z = proj L z, if zIL, and to lines M concurrent with L by denoting the intersection point proj L M. The dual notation is also used. l 2 Chapter 1. Generalized Quadrangles The integers s and t are the parameters of the GQ and S is said to have order (s, t); if s = t, S is said to have order s. ~~