Linearly small elation quadrangles (Q2380487)

A generalized quadrangle is called \textit{small}, if it has order \((p,p)\) for a prime \(p\). Let \((Q,c,E)\) be an elation (generalized) quadrangle. With each line \(m\) through a point \(o\) opposite \(c\) we associate two subgroups: its stabilizer \(E_m\) and the stabilizer \(E_m^{*}\) of the projection \(\pi(c,m)\) of \(c\) onto \(m\). The system of all stabilizers \(E_m\) and \(E_m^{*}\) forms the so-called \textit{fourgonal family}. Put \(S:=\{E_m|m\in{\mathcal L}_o\}\) where \({\mathcal L}_o\) denotes the set of all lines through \(o\). The authors speak of a \textit{linearly small} elation quadrangle \((Q,c,E)\), if E is either isomorphic to the Heisenberg group \(H_F\) over a field \(F\) or \(E\cong\,F^3\) and if \(S\) consists of one-dimensional \(F\)-subgroups resp. \(F\)-subspaces of \(F^3\). Each small elation quadrangle is a linearly small one. Elation quadrangles of order \(s\) with \(s\in\{3^2,5^2,7^2\}\) and \(E\cong\,H_{GF(s)}\) are linearly small. The authors determine all linearly small elation quadrangles and get two types: 1. \(E\cong\,H_F\) and \(Q\) is isomorphic to the symplectic quadrangle \(W(F)\) 2. \(E\cong\,F^3\) where \(F\) is a finite field with \(char\not=2\) and \(Q\) is isomorphic to the orthogonal quadrangle \(Q(4,F)\). Furthermore, the authors characterize symplectic quadrangles over algebraic closures of real closed fields and prove that every root elation quadrangle \((Q,c,H_F)\) is a skew translation quadrangle. Finally, they show: If \((Q,c,H_{{\mathbb C}})\) (\(\mathbb C\) denotes the field of complex numbers) is a compact elation quadrangle, then \(Q\cong\,W({\mathbb C})\).