Grid-symmetric generalized quadrangles

Abstract

A generalized quadrangle is classical if it has a grid of axes of symmetry. In a finite generalized quadrangle Q of order (s, t) with s, t > 1, a line L is called an axis of symmetry if the group T (L) of all automorphisms (“symmetries”) that fix every line meeting L has the maximal possible order s. Moreover, Q is called span– symmetric if there are two disjoint axes of symmetry; we will call Q grid–symmetric if there are two further disjoint axes of symmetry, each of which meets L and M . Span–symmetric generalized quadrangles were first studied in [Pa] (cf. [PT1]), in view of the known examples Q(4, q) and Q(5, q), arising respectively from quadrics in 4– and 5–dimensional projective spaces. More than 20 years ago it was shown that the generalized quadrangles Q(4, q) are the only span–symmetric ones with t 6= s (cf. [Ka, Th1]). While nonclassical examples exist if t = s, this is not so in the grid–symmetric case: Theorem. Any grid–symmetric generalized quadrangle of order (s, t) is isomorphic to Q(4, s) or Q(5, s). Proof. By the result just noted, we may assume that t = s. There are sets Λ and Λ⊥, each consisting of s + 1 lines of symmetry, where each line in Λ meets each line in Λ⊥. Let A and B be the groups generated by the symmetries corresponding to Λ and Λ⊥, respectively. By [Th2, 12.5.5], A ∼= B ∼= SL(2, s). If L ∈ Λ and M ∈ Λ⊥ then T (L) fixes M and hence normalizes T (M). Also T (M) normalizes T (L), so that these two groups commute since T (L) ∩ T (M) = 1. Thus, A and B are commuting groups each of which is isomorphic to SL(2, s). ∗This research was supported in part by the National Science Foundation. Received by the editors August 2002. Communicated by J. Thas. 2000 Mathematics Subject Classification : Primary 51E12.