A characterization of the classical generalized quadrangle \(\mathcal Q(5,q)\) and the nonexistence of certain near polygons (Q1399911)

A spread of symmetry in a generalized quadrangle of order \((s,t)\) is a spread whose element-wise stabilizer inside the full collineation group of the quadrangle acts transitively on the \(1+s\) points of each line of the spread. If \(t= s^2\) and the quadrangle contains one elation point, then clearly it contains a line of elation points, and if \(s\) is even, \textit{B. De Bruyn} and \textit{K. Thas} [Ill. J. Math. 46, No. 3, 797-818 (2002; Zbl 1025.51002)] proved that the quadrangle is necessarily isomorphic to \(Q(5,s)\). The paper under review extends this to all values of \(s\). Consequently, some construction of near polygons (using a spread of symmetry) does not yield new examples when applied to the known generalized quadrangles. It should be noted that the author in the meantime generalized the main result of the paper under review in that the hypothesis of the existence of the spread is replaced by the hypothesis that the line every point of which is an elation point is a regular line (unpublished).