Characterizations of generalized quadrangles by generalized homologies
Let \(S=(P,B,I)\) be a finite generalized quadrangle of order (s,t) and \({\mathcal H}(x,y)\) the group of all collineations of S fixing the noncollinear points x and y linewise. S is called (x,y)-transitive if \({\mathcal H}(x,y)\) is transitive on each set \(^{\sim}-\{x,z\}\) and \(^{\sim}-\{y,z\},\) where z is any point collinear with both x and y and \(^{\sim}\) denotes the set of all points incident with the line xz. The author proves the following theorem: If \(s>1\), \(t>1\) and S is (x,y)- transitive for all noncollinear x,y\(\in P\), then one of the following cases occurs: S arises from (i) a symplectic polarity, (ii) a nonsingular hermitian variety in PG(4,s), (iii)-(iv) a nonsingular quadric in PG(4,s) or PG(5,s), respectively, (v) \(s=t^ 2\) and all points are regular.
scientific article; zbMATH DE number 3935866
scientific article