Characterizations of Generalized Quadrangles by Generalized Homologies

Let S= (P, B, I) be a generalized quadrangle of order (s, t). For X, y E P, X+ y. let X(x, y) be the group of all collineations of S fixing x and y linewise. If ZE (4 YJL> then the set of all points incident with the line xz (resp. y;) is denoted by z(resp. 3. The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x+ y, if X(x, y) is transitive on each set Z{x, z} and jZ’-{ y, z}, z E {x, y}‘. If S= (P, B, I) is a generalized quadrangle of order (s, t), s z 1 and t z 1, which is (x, y)-transitive for all X, ye P with x+ y, then it is proved that we have one of the following: (i) Sg W(s), (ii) SgQ(4, s), (iii) SE H(4, s), (iv) SgQ(5, s), (v) s = I2 and all points are regular. c 1985 Academic Press, Inc