The classification of generalized quadrangles with two translation points (Q1856587)

A generalized quadrangle \({\mathcal S}\) is called span-symmetric if there are non-concurrent axes of symmetry. Let \(\mathcal S\) be a span-symmetric generalized quadrangle of order \((s,t)\), \(s\neq 1\neq t\). Then \(s=t\) or \(t=s^2\), and \(s\) and \(t\) are powers of the same prime (Theorem 1). If \(t=s^2\), then \(\mathcal S\) contains at least \(s+1\) subquadrangles isomorphic to the classical \(Q(4,s)\) (Theorem 2). The main result of this paper is Theorem 3. Suppose \({\mathcal S}\) is a generalized quadrangle of order \((s,t)\), \(s\neq 1\neq t\) with two distinct collinear translation points. Then (i) \(s=t\), \(s\) is a prime power and \({\mathcal S}\simeq Q(4,s)\); (ii) \(t=s^2\), \(s=2^a\) and \({\mathcal S}\simeq Q(5,s)\); (iii) \(t=s^2\), \(s=q^n\) with \(q\) odd, where \(GF(q)\) is the kernel of the TGQ \({\mathcal S}={\mathcal S}^{(\infty)}\) with \((\infty)\) an arbitrary translation point, \(q\geq 4n^2-8n+2\) and \({\mathcal S}\) is the point-line dual of a flock GQ \({\mathcal S}({\mathcal F})\) where \(\mathcal F\) is a Kantor flock; (iv) \(t=s^2\), \(s=q^n\) with \(q\) odd, where \(GF(q)\) is the kernel of the TGQ \({\mathcal S}={\mathcal S}^{(\infty)}\) with \((\infty)\) an arbitrary translation point, \(q< 4n^2-8n+2\) and \({\mathcal S}\) is the point-line dual of a flock GQ \({\mathcal S}({\mathcal F})\) fore some flock \(\mathcal F\). If a thick GQ \({\mathcal S}\) has two non-collinear translation points then \({\mathcal S}\) is isomorphic \(Q(4,s)\) or \(Q(5,s)\).