Finite Translation Generalized Quadrangles

My talk is a survey on finite translation generalized quadrangles. To each translation generalized quadrangle of order (s, t), with s not equal 1 not equal t, there corresponds a set O(n, m, q) of q(m) + 1 (n - 1)-dimensional subspaces of the projective space PG(2n+m-1, q) satisfying (i) every three subspaces generate a PG (3n - 1, q) and (ii) for every such subspace pi there is a subspace PG(n + m - 1, q) containing pi and having empty intersection with the other elements of O(n, m, q). Conversely, every such O(n, m, q) defines a finite translation generalized quadrangle. For each known example of O(n, m, q) we have m is an element of {n, 2n}, and for q even there are no other examples. Many papers were written on the case m = 2n. Here emphasis is on the case m = n, and besides interesting and useful old results several new theorems are stated.