Classification of span-symmetric generalized quadrangles of order \(s\) (Q2783474)

Let \(\mathcal S\) be a finite generalized quadrangle of order \((s,t)\), \(s,t > 1\). \(\mathcal S\) is called span-symmetric if there are two disjoint axes of symmetry. The generalized quadrangle \(\mathcal L(4,s)\) arises from non-singular parabolic quadric in \(PG(4,s)\).NEWLINENEWLINENEWLINEThe author gives and proves the following main results:NEWLINENEWLINENEWLINE1. Let \(\mathcal S\) be a span-symmetric generalized quadrangle of order \((s,s)\), \(s\neq 1\). Then \(\mathcal S\) is isomorphic to \({\mathcal L}(4,s)\).NEWLINENEWLINENEWLINE2. A finite group is isomorphic to \(SL_2(s)\) for some \(s\) if and only if it has a 4-gonal basis.