Symmetry in generalized quadrangles

In this paper, we describe some aspects of a Lenz(-Barlotti)-type classification of finite generalized quadrangles, which is being prepared by the author. Some new points of view are given. We also prove that each span-symmetric generalized quadrangle of order s > 1 with s even is isomorphic to Q(4, s), without using the canonical connection (obtained by S. E. Payne in [15]) between groups of order s(3) - s with a 4-gonal basis and span-symmetric generalized quadrangle of order s. ( The latter result was obtained for general s independently by W. M. Kantor in [10], and the author in [30].) Finally, we obtain a classification program for all finite translation generalized quadrangles, which is suggested by the main results of [27], [30], [32], [35], [38] and [37].