Some Generalized Quadrangles with Parameters q 2 , q

To Professor Helmut Wielandt, to commemorate his seventy-fifth birthday 1. Introduction In [2] a new generalized quadrangle with parameters q2, q was constructed using the G 2(q) generalized hexagon (q == 2 (mod 3), q > 2). In addition, an elementary group theoretic technique was presented for constructing generalized quadrangles. This technique was refined by Payne [3] in order to simplify calculations and search for new quadrangles. All the known generalized quadrangles with parameters q2, q are described in [2] and [3], but only the aforementioned new family was found using the method in [2]. In this note we will use the formulation in [3] in order to obtain additional quadrangles: (1.1) Theorem. Let q be a power pe of an odd prime p. Then (i) If e> 1 there are [(e-1)/2] pairwise non isomorphic generalized quadrangles with parameters q2, q not isomorphic to any previously known generalized quadrangle; and (ii) If q> 3 and q == ± 2 (mod 5) then there is a generalized quadrangle with parameters q2, q not isomorphic to any quadrangle in (i) nor to any previously known generalized quadrangle. Each of the quadrangles in (1.1) admits an automorphism group of order q5 fixing one point x and transitive on the q5 points not collinear with x. The quadrangles in (Ll i) have two interesting features. One is their number. The other is the fact that, for every point y collinear with x, there are q automor-phisms acting as "elations with center y": automorphisms fixing every point collinear with y. One other interesting aspect of these quadrangles and of those in [2] is simply their parameters. A generalized quadrangle with parameters s, q necessarily has s;£ q2, with a great deal of combinatorial information implied by equality (see, e.g., [1]). This tightness makes the number of examples in (1.1 i) seem somewhat unexpected. The general results contained in [2] and [3] are summarized in § 2. After some preliminary remarks in § 3, we construct the quadrangles in (1.1) in the remaining sections of the paper.