A coordinatization of generalized quadrangles of order \((s,s+2)\)
Let \(S=(P,B,I)\) be a generalized quadrangle of order (s,t), \(s\geq 1\), \(t\geq 1\), where P and B are the sets of points and lines respectively, and I is a point-line incidence relation. Then, by definition, each point is incident with \(1+t\) lines and each line is incident with \(1+s\) points. In this paper, the case \(t=s+2\) is considered. Let \({\mathcal R}\) be a spread of S and let \(G_{{\mathcal R}}\) be the group of all automorphisms of S which fix all lines of \({\mathcal R}\). If \(G_{{\mathcal R}}\) acts transitively on each line of \({\mathcal R}\), then \({\mathcal R}\) is called a spread of symmetry. The authors prove that a generalized quadrangle S of order \((s,s+2)\) containing a spread of symmetry induces a generalized quadrangle S' of order \((s+1,s+1)\) with a center of symmetry (\(\infty)\). The converse theorem also holds. Then, S' are coordinatized by using the method due to \textit{S. E. Payne} [J. Algebra 31, 367-391 (1974; Zbl 0284.50013)]. Thus, the authors give explicitly a coordinatization of S of order \((s,s+2)\) containing a spread of symmetry \({\mathcal R}\) in terms of a planar ternary ring R, \(| R| =s+1,\) satisfying some conditions, a group \(G_{{\mathcal R}}\) with \(| G_{{\mathcal R}}| =s+1,\) and a certain function of \(R^ 3\) into \(G_{{\mathcal R}}\). Examples are also discussed.
scientific article; zbMATH DE number 4042694