A problem on generalized quadrangles (Q1299087)

Let \({\mathcal S}=(P,B,I)\) be a generalized quadrangle of order \((s,t)\), \(s>1\), \(t>1\). Let \(p\) be a distinguished point of \(\mathcal S\) and \(G\) a group of collineations of \(\mathcal S\) acting regularly on the \(s^2t\) points of \(P-p^\bot\). In this case \(({\mathcal S}_p,G)\) is called a homogeneous generalized quadrangle (HGQ). If \(G\) also fixes \(p\) linewise, \(G\) is called a group elations about \(p\) and we say that \(({\mathcal S}_p,G)\) is an elation generalized quadrangle (EGQ). A main purpose of this note is to stimulate the search for some description of two infinite families of examples for which \(({\mathcal S}_p,G)\) is an (HGQ) but not an (EGQ) based somehow on relatively simple properties of the action of \(G\) on \(P-p^\bot\). Let \(F=GF(q)\), \(f: F^2\times F^2 \to F\) be a symmetric nonsingular biadditive map. Put \(G=\{(\alpha,c,\beta):\alpha,\beta\in F^2,c\in F\}\). Define a binary operation on \(G\) by: \((\alpha,c,\beta)(\alpha',c',\beta')= (\alpha+\alpha',c+c'+f(\beta,\alpha'), \beta+\beta')\). This makes \(G\) into a group whose center is \(C=\{(0,c,0)\in G:c\in F\}\). Suppose that for each \(t\in F\) there is an additive map \(\delta_t: F^2\to F^2\) and a map \(g_t:F^2 \to F\) for which \(g_t(\alpha+\beta)-g_t(\alpha)-g_t(\beta)=f(\alpha^{\delta_t},\beta)= f(\beta^{\delta_t},\alpha)\). We can define a family of subgroups of \(G\) by \(A(t)=\{(\alpha,g_t(\alpha),\alpha^{\delta_t}):\alpha\in F^2\}\), \(A(\infty)=\{(0,0,\beta)\in G:\beta\in F^2\}\). Then put \({\mathcal F}=\{A(t):t\in \{\infty\}\cup F\}\). And for each \(A\in {\mathcal F}\), put \(A^*=AC\). Then \(\mathcal F\) be a 4-gonal family. Suppose that there is also a map \(\sigma:(F,+)\to GL(2,F)\) (\(x\mapsto \sigma_x\)) for which \(\sigma_x\sigma_y=\sigma_{x+y}\) and \(g_{t+x}(\alpha^{\sigma_x})-g_x(\alpha^{\sigma_x})=g_t(\alpha)\) (\(t,x\in F, \alpha\in F^2\)). And we may let \(\sigma_x^T\) denote the unique adjoint of \(\sigma_x\) with respect to \(f\) over \(Z_p\): \(f(\alpha,\beta^{\sigma_x})=f(\alpha^{\sigma_x^T},\beta)\) (\(x\in F, \alpha,\beta\in F^2\)). For each \(x\in F\) define \(\theta_x:G\to G\) (\(\theta_x(\alpha,c,\beta)= (\alpha^{\sigma_x},c+g_x(\alpha^{\sigma_x}),\alpha^{\sigma_x \delta_x}+ \beta^{\sigma_x^{-T}})\). Theorem 2.1. The map \(\theta_x\) is an automorphism of \(G\) fixing \(A(\infty)\) and mapping \(A(t)\) to \(A(t+x)\), for each \(t,x\in F\). \({\mathcal T}=\{\theta_x\cdot (0,c,\beta)\;:\;c\in F,\beta\in F^2\}\) is a group of order \(q^4\) of collineations of the associated GQ \(\mathcal S\) of order \((q^2,q)\) acting regularly on the set of lines of \(\mathcal S\) not meeting \([A(\infty)]\) (Theorem 2.2, 3.1). Payne (1990) give examples with nonabelian group \(\mathcal T\). Let \[ {\mathcal C}=\{A_t=\left(\begin{smallmatrix} x_t&y_t\\ 0&z_t\end{smallmatrix}\right):t\in F\} \] be a \(q\)-clan. Define \(g_t(\alpha)\) to be \(\alpha A_t\alpha^T\), etc. In example 1 \(q\equiv 2\) (mod 3) and \[ A_t=\left(\begin{smallmatrix} t&3t^2 \\ 0&3t^3\end{smallmatrix}\right), \sigma_x:\alpha \mapsto \alpha\left(\begin{smallmatrix} 1&0 \\ 3x&1\end{smallmatrix}\right). \] In example 2 \(q=5^e>5\) and \(k\) is any nonsquare of \(F\). Then put \[ A_t=\left(\begin{smallmatrix} t&t^2\\ 0&k^{-1}t(1+kt^2)^2\end{smallmatrix}\right), \sigma_x:\alpha \mapsto \alpha\left(\begin{smallmatrix} 1&0\\ -x&1\end{smallmatrix}\right). \]