Determination of generalized quadrangles with distinct elation points

J Algebr Comb (2006) 24:5–22 DOI 10.1007/s10801-006-9098-3 Determination of generalized quadrangles with distinct elation points Koen Thas Received: 31 March 2005 / Accepted: 7 November 2005  C Springer Science + Business Media, LLC 2006 Abstract In this paper, we classify the finite generalized quadrangles of order (s, t), s, t > 1, which have a line L of elation points, with the additional property that there is a line M not meeting L for which {L , M} is regular. This is a first fundamental step towards the classification of those generalized quadrangles having a line of elation points. Keywords Generalized quadrangle . Elation generalized quadrangle . Translation generalized quadrangle . Moufang condition . Symmetry . Regularity . Classification Mathematics Subject Classification (2000): 51E12, 51E20, 20B25, 20E42 1. Introduction and statement of the main result A (finite) generalized quadrangle (GQ) of order (s,t) is an incidence structure S = (P, B, I ) in which P and B are disjoint (nonempty) sets of objects called points and lines respectively, and for which I is a symmetric point-line incidence relation satisfying the following axioms. (1) Each point is incident with t + 1 lines (t ≥ 1) and two distinct points are incident with at most one line. (2) Each line is incident with s + 1 points (s ≥ 1) and two distinct lines are incident with at most one point. (3) If p is a point and L is a line not incident with p, then there is a unique point-line pair (q, M) such that pIMIqIL. K. Thas () Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281, S22, B-9000 Ghent, Belgium e-mail: Springer 6 J Algebr Comb (2006) 24:5–22 If s = t, then S is also said to be of order s. There is a point-line duality for GQ’s of order (s, t) for which in any definition or theorem the words “point” and “line” are interchanged and also the parameters. Normally, we assume without further notice that the dual of a given theorem or definition has also been given. Also, sometimes a line will be identified with the set of points incident with it without further notice. This will be done frequently. For notation and definitions not explicitly mentioned here, we refer to the monograph FINITE GENERALIZED QUADRANGLES by S.E. Payne and J.A. Thas [25]. For an extensive survey on recent results on automorphisms and characterizations of GQ’s, see [34]. Let S = (P, B, I ) be a GQ of order (s, t), s, t > 1. An elation about the point p is either the identity, or a collineation of S that fixes p linewise and no point of P\ p ⊥ . By definition, the identity is an elation (about every point). If p is a point of the GQ S for which there exists a group G of elations about p which acts regularly on the points of P\ p ⊥ , then S is said to be an elation generalized quadrangle (EGQ) with elation point p and elation group (or base-group) G, and we often write (S ( p) , G) for S, or (S p , G). The natural models of finite generalized quadrangles for which each point is an elation point are the so-called ‘classical’ and ‘dual classical’ examples, as defined by J. Tits in [10]. Those are constructed as follows. (a) Consider a nonsingular quadric of Witt index 2, that is, of projective index 1, in PG(3, q), PG(4, q), PG(5, q), respectively. The points and lines of the quadric form a generalized quadrangle which is denoted by Q(3, q), Q(4, q), Q(5, q), respectively, and has order (q, 1), (q, q), (q, q 2 ), respectively. (b) The points of PG(3, q) together with the totally isotropic lines with respect to a symplectic polarity form a GQ W (q) of order q. (c) Let H be a nonsingular Hermitian variety in PG(3, q 2 ), respectively PG(4, q 2 ). The points and lines of H form a generalized quadrangle H (3, q 2 ), respectively H (4, q 2 ), which has order (q 2 , q), respectively (q 2 , q 3 ). Vice versa, using the CLASSIFICATION OF FINITE SIMPLE GROUPS (CFSG), see e.g. [9, 15], F. Buekenhout and H. Van Maldeghem were the first to obtain the converse [4]: if a finite generalized quadrangle has the property that each point is an elation point, then it is one of the classical or dual classical examples. Recently, the author of this paper and H. Van Maldeghem found a classification-free proof of that result. Now consider an EGQ S of order (s, t), s = 1 = t. Then, by transitivity, we clearly have the following possibilities for S: (a) S has precisely one elation point; (b) S has a line L of elation points; (c) each point of S is an elation point, and S is classical or dual classical. In this paper, we will be concerned with the classification of GQ’s of Type (b). Without additional hypotheses, this case seems completely hopeless at present, even if one allows the use of CFSG. In [37], such a classification was obtained with the following additional hypothesis: (AB) There is a point on L so that the corresponding elation group is abelian. Springer J Algebr Comb (2006) 24:5–22 7 If pIL is such a point, then p is called a translation point. If (AB) is satisfied, all points incident with L are translation points. The result we obtained was the following (for notions not explicitly defined yet, see Section 2, or [25] and [34]): Theorem 1.1. (K. Thas [37]) Supposes S is a generalized quadrangle of order (s, t), s = 1 = t, with two distinct collinear translation points. Then we have one of the following: (i) s = t and S ∼ = Q(4, s); (ii) t = s 2 , s is even and S ∼ = Q(5, s); (iii) t = s 2 , s is odd, and S is the translation dual of the point-line dual of a flock GQ S(F). If a GQ S has two non-collinear translation points, then S is always of classical type, i.e. isomorphic to one of Q(4, s), Q(5, s). In [38], the converse of (iii) was (unexpectedly) obtained: Theorem 1.2. (K. Thas [38]) The non-classical GQ’s of order (s, t), where 1 < s < t, which have distinct translation points are precisely those GQ’s S which are the translation dual of the point-line dual of a flock GQ S(F), where F is nonlinear. Each of these examples has the following essential properties (which characterizes the examples by Theorem 1.2, see [38]): (a) they have some line L each point of which is an elation point (in fact, each of these points is a translation point); (b) each line M of the GQ which meets L is a regular line (including L). Therefore, we propose the (theoretically much more general) problem to classify the finite generalized quadrangles of order (s, t), s, t > 1, having a line L of elation points, satisfying the following additional assumption: (R) There exists a line M not concurrent with L so that {L , M} is a regular pair of lines. The following main result will be obtained: Main Theorem. Suppose S is a generalized quadrangle of order (s, t), s = 1 = t, which has a line L each point of which is an elation point. Furthermore, suppose that Property (R) is satisfied for L. Then we have one of the following: (i) s = t and S ∼ = Q(4, s); (ii) t = s 2 , s is even and S ∼ = Q(5, s); (iii) S is the (...truncated)