On Symmetries and Translation Generalized Quadrangles

Abstract

Let S be a generalized quadrangle (GQ) of order (s,t), s,t≠ 1, and suppose p is a point of S. A whorl about p is a collineation of S which fixes p linewise, and a line L is an axis of symmetry if there is a full group of collineations of S of size s of which each element is a whorl about every point on L. Suppose S is a generalized quadrangle of order (s, t), with s, t≠ 1, and suppose that L is a line of S. Then we prove that L is an axis of symmetry if and only if L is regular and if there is a point p on L, a group H of whorls about p, and a line M with M ∼LIp\IM, such that H acts transitively on the points of M\{M ∩ L}. As a corollary, we prove that a line of a generalized quadrangle S through an elation point p is an axis of symmetry if and only if it is a regular line, and that the group of symmetries about such a line is always completely contained in any elation group corresponding to an elation point on this line. This proves the converse of a theorem of [8]. Using these results, we prove new characterizations of translation generalized quadrangles, and we considerably improve some characterizations of translation generalized quadrangles of Chen and Frohardt [2], and Hachenberger [5]; we will prove that an elation generalized quadrangle S with elation point p and elation group G is a translation generalized quadrangle with translation point p and translation group G if and only if p is incident with at least two regular lines, and in the case where there is an odd number of points on a line, one regular line is sufficient. Also, as an application, we prove a new characterization theorem of the classical generalized quadrangles arising from a quadric.