By Raynaud M. (Ed), Shioda T. (Ed)
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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.