Classification of span-symmetric generalized quadrangles of order s

A line L of a finite generalized quadrangle S of order ðs; tÞ, s; t > 1, is an axis of symmetry if there is a group of full size s of collineations of S fixing any line which meets L. If S has two non-concurrent axes of symmetry, then S is called a span-symmetric generalized quadrangle. We prove the twenty-year-old conjecture that every span-symmetric generalized quadrangle of order ðs; sÞ is classical, i.e. isomorphic to the generalized quadrangle Qð4; sÞ which arises from a nonsingular parabolic quadric in PGð4; sÞ. 1 Statement of the main result In this paper, we prove the following main result. Theorem 1.1. LetS be a span-symmetric generalized quadrangle of order s, where s0 1. Then S is classical, i.e. isomorphic to Qð4; sÞ. This has the following corollary for groups with a 4-gonal basis (as defined in Section 3). Theorem 1.2. A finite group is isomorphic to SL2ðsÞ for some s if and only if it has a 4-gonal basis. 2 Notation A (finite) generalized quadrangle (GQ) of order ðs; tÞ is an incidence structure S 1⁄4 ðP;B; IÞ, with point set P, line set B and symmetric incidence relation I, where each point is incident with tþ 1 lines (td 1), each line is incident with sþ 1 points (sd 1), and if a point p is not incident with a line L, then there is a unique point-line pair ðq;MÞ such that pIMIqIL. If s 1⁄4 t we say that S has order s. As a general reference we mention the book by S. E. Payne and J. A. Thas [8], see also [10] and [12] for more recent developments, and [11] and [15] for surveys on generalized polygons. Points p and q of S 1⁄4 ðP;B; IÞ are collinear, if they are incident with a common line. For p A P, put p? 1⁄4 fq A P j p; q are collinearg (note that p A p?). More generally, if AJP, we define A? 1⁄4 7fp? j p A Ag. Often we use the dual notion L? 1⁄4 fM A B jL;M are confluentg for lines L, and X ? 1⁄4 7fL? jL A Xg for X JB. If Y is a subset of P or of B, then Y?? denotes ðY ?Þ?. The classical GQ Qðd; qÞ, d A f3; 4; 5g, is the GQ which arises by taking the points and lines of a nonsingular quadric with Witt index 2 (that is, with projective index 1) in the d-dimensional projective space PGðd; qÞ over the Galois field GFðqÞ. Respectively, the orders are ðq; 1Þ, ðq; qÞ and ðq; qÞ. 3 Span-symmetric generalized quadrangles Suppose L is a line of a GQ S of order ðs; tÞ, s; t0 1. A symmetry about L is an automorphism of the GQ which fixes every line of L?. The line L is called an axis of symmetry if there is a group H of symmetries of size s about L. In such a case, if M A L?nfLg, then H acts regularly on the points of M not incident with L. We remark that every line of the classical example Qð4; sÞ is an axis of symmetry (see 8.7.3 of [8]). If L and M are distinct non-concurrent axes of symmetry, then it is easy to see, by transitivity, that every line of fL;Mg?? is an axis of symmetry, and S is called a span-symmetric generalized quadrangle (SPGQ) with base-span fL;Mg??. In this situation, we will use the following notation throughout this paper: the basespan will always be denoted by L. The group which is generated by all the symmetries about the lines of L is G, and we call this group the base-group. This group clearly acts 2-transitively on the lines of L, and fixes every line of L? (see for instance 10.7 of [8]). Theorem 3.1 (S. E. Payne [7]; see also 10.7.2 of [8]). If S is an SPGQ of order s, s0 1, with base-group G, then G acts regularly on the set of ðsþ 1Þsðs 1Þ points of S which are not on any line of L. Note. There is an analogue of Theorem 3.1 for SPGQ’s of order ðs; sÞ, s > 1, see K. Thas [13] and [14]. Let S be an SPGQ of order s0 1 with base-span L, and put L 1⁄4 fU0; . . . ;Usg. The group of symmetries about Ui is denoted by Gi, i 1⁄4 0; 1; . . . ; s, throughout this paper. Then one notes the following properties (see [7] and 10.7.3 of [8]): 1. the groups G0; . . . ;Gs form a complete conjugacy class in G, and are all of order s, sd 2; 2. Gi VNGðGjÞ 1⁄4 f1g for i0 j; 3. GiGj VGk 1⁄4 f1g for i; j; k distinct, and 4. jGj 1⁄4 s s. We say that G is a group with a 4-gonal basis T 1⁄4 fG0; . . . ;Gsg if these four conditions are satisfied. It is possible to recover the GQ S of order s from the base-group G starting from 4-gonal bases, see [7] and 10.7.8 of [8], hence Koen Thas 190