By J a Thas;K Thas;H Van Maldeghem
Translation generalized quadrangles play a key function within the idea of generalized quadrangles, such as the function of translation planes within the concept of projective and affine planes. The thought of translation generalized quadrangle is a neighborhood analog of the extra worldwide "Moufang Condition", an issue of serious curiosity, additionally as a result type of all Moufang polygons. recognition is therefore paid to fresh ends up in that path, but in addition some of the most vital ends up in the final conception of generalized quadrangles that seemed for the reason that 1984 are handled. "Translation Generalized Quadrangles" is basically self-contained, because the reader is barely anticipated to be acquainted with a few uncomplicated proof on finite generalized quadrangles. Proofs which are both too lengthy or too technical are passed over, or simply sketched. the 3 commonplace works on generalized quadrangles are (co-)authored through the writers of this publication: "Finite Generalized Quadrangles" (1984) by means of S E Payne and J A Thas, "Generalized Polygons" (1998) by way of H Van Maldeghem, and "Symmetry in Finite Generalized Quadrangles" (2004) via ok Thas.
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T}; (iii) a symbol (oo). • LINES are of two kinds: (a) right cosets Hig, g e G, i e {0,1,.. • ,t}; (b) symbols [Hi], i e { 0 , 1 , . . , t}. • INCIDENCE. A point g of Type (i) is incident with each line Hig, 0 < i < t. A point H*g of Type (ii) is incident with [Hi] and with each line Hih contained in H*g. The point (oo) is incident with each line [Hi] of Type (b). There are no further incidences. 32 Chapter 3. Elation and Translation Generalized Quadrangles It is straightforward to check that the incidence structure S(G, J) is a GQ of order (s,t).
Chapter 2. Regularity, Antiregularity and 3-Regularity 14 The number of triads {x, y, Zi} is equal to (s + l)(s 2 + 1) - 2(s + l)s + (s + 1) - 2 = s 3 - s2. Hence Y^(U-l)(U-(s + l)) = 0. i As ti < s + 1 and t; is odd, we necessarily have U £ {1, s + 1}. 3 the pair {a;, y} is regular. Hence the point x is regular. Conversely, assume that x is regular. 3 we have 1 \{x, y, z}- ] e {1, s + 1} for any triad {x, y, z}. 4, s is even. 6 The point (oo) of any GQ T 2 (C) of order q, with q even, is regular.
Nothing is known about £ = 11 or £ = 12. In the other cases unique examples are known, but the uniqueness question is settled only in the case t = 4. The proof of this uniqueness that appears in Payne and Thas [128] is that of Payne [110, 111], with a gap filled in by Tits. Chapter 2 Regularity, Antiregularity and 3-Regularity In this chapter the important notions of regularity, antiregularity and 3regularity are introduced, and the connections with planes, nets, Laguerre planes, inversive planes and subquadrangles are described.