A Course on Elation Quadrangles doc

[...]... Moufang conditions became obvious – not only numerous characterizations of known classes of generalized quadrangles came out; also the theory of translation generalized quadrangles essentially arose from it, and the abstraction to elation generalized quadrangles eventually led to many new classes of generalized quadrangles In this chapter, we review some basic facts concerning the Moufang condition 2.1... culmination of work by S E Payne and J A Thas [44] (Chapter 9), W M Kantor [28] and the author [61] We refer to the author and H Van Maldeghem [83] for a survey on old and new results on Moufang generalized quadrangles We also refer to Chapter 11 of [59], and especially [58] on that matter In the aforementioned work of S E Payne and J A Thas (and the references therein), the importance of local Moufang... Figure 1.1 A grid of order 3; 1/ (a) there are no ordinary digons and triangles contained in Ã; (b) every two elements of P [ B are contained in an ordinary quadrangle; (c) there exists an ordinary pentagon In (a) , (b), (c), ordinary digons, triangles, quadrangles and pentagons are meant to be induced subgeometries Exercise Show that if à satisfies (a) –(c), there exist constants s and t such that each line... also called a grid, while a thin GQ of order 1; t / is a dual grid A GQ of order 1; 1/ is both a grid and a dual grid – it is an ordinary quadrangle If s D t, then à is also said to be of order s There is a parameter-free way to introduce generalized quadrangles, as follows A rank 2 geometry à D P ; B; I/ is a thick generalized quadrangle if the following axioms are satisfied: 2 1 Generalized quadrangles. .. identify them as generalized quadrangles – see Dembowski [15].) Their point-line duals are called the dual classical generalized quadrangles The classical quadrangles are characterized by the fact that they are fully embedded in finite projective space – see Chapter 4 in [44] for details Recall that a full embedding of a rank 2 geometry € D P ; B; I/ in a projective space P , is an injection à W P ,!... [59], Chapter 11 For reasons of convenience, we will call a generalized quadrangle admitting an automorphism group that acts transitively on its ordered ordinary 4-gons, a Tits quadrangle Two flags F D fx; Lg and F 0 D fy; M g are opposite if x ¦ y and L ¦ M Proposition 2.1 Let à D P ; B; I/ be a thick Tits generalized quadrangle and let G be a collineation group of à such that G acts transitively on the... By the previous paragraph, we may also assume that the flag fx2 ; L2 g is contained in †, a contradiction Suppose that ÃG;B;N contains a triangle x1 I L3 I x2 I L1 I x3 I L2 I x1 , with the xi points and the Li lines, i D 1; 2; 3 We may again assume x1 D x and L2 D L, and by the previous paragraph, we may assume that the flag fx2 ; L1 g belongs to † Since † is a quadrangle, we must have x2 D x sL sx... 4.4 Special and extra-special p-groups 4.5 Another approach 4.6 Lie algebras 4.7 Lie algebras from p-groups 31 31 32 35 37 38 39 41 Parameters of elation quadrangles and structure of elation groups 5.1 Parameters of elation quadrangles 5.2 Skew translation quadrangles ... are regular, and all points are antiregular In particular, all spans of noncollinear points have size 2 (ii) Let q be even Then all lines and points of Q.4; q/ are regular (iii) Any line of Q.5; q/ is regular and any span of noncollinear points has size 2 All points are 3-regular (iv) For any pair fx; yg of noncollinear points of H.4; q 2 /, we have jfx; yg?? j D q C 1, while all spans of nonconcurrent... contained in that hyperplane Exercise Show that each nontrivial elation of PG.n; q/, n 2 N and n 2, has a unique center, that is, a point which is fixed linewise (and necessarily contained in the axis) 1.4 Finite classical examples and their duals In this section we will introduce some classes of finite rank 2 geometries which are known as the finite “classical generalized quadrangles (Tits was the first . [68] and “Translation Generalized Quadrangles [59] (on elation quadrangles with an abelian elation group). These works will only have a small overlap with. p-groups 41 5 Parameters of elation quadrangles and structure of elation groups 44 5.1 Parameters of elation quadrangles 44 5.2 Skew translation quadrangles 46 5.3