Characterizations of translation generalized quadrangles (Q5937091)

If \(x\) is a regular point of the generalized quadrangle \({\mathcal S}\) of order \((s,t)\), \(s\neq 1\neq t\), then \(x\) defines a dual net \({\mathcal N}^*_x\). In this paper a particular class of collineations, called transvections with axis \(x\), of the point-line dual of \({\mathcal N}^*_x\), has been introduced. If \({\mathcal S}\) contains a line \(L\) of regular points and \(x\) is any point of \(L\), then each transvection with axis \(x\) of the point-line dual of \({\mathcal N}^*_x\) can be extended to a symmetry about \(x\) of \({\mathcal S}\). Hence, if the group of all transvections with axis \(x\) of the point-line dual of \({\mathcal N}^*_x\) satisfies certain transitivity properties, then \({\mathcal S}\) is a translation generalized quadrangle. This interesting result has many applications in the theory of generalized quadrangles, some of which are discussed in detail in the paper. In particular the following characterization of the classical generalized quadrangle \(W(3,q)\) has been achieved. Suppose \(s=t\neq 1\). Then \({\mathcal N}^*_x\) is a dual affine plane. If \({\mathcal S}\) contains a line \(L\) of regular points and for a point \(x\) of \(L\) the plane \({\mathcal N}^*_x\) is desarguesian, then \({\mathcal S}\) is isomorphic to the generalized quadrangle \(W(3,q)\) associated with a symplectic polarity of \(\text{PG} (3,q)\).