Half Moufang implies Moufang for finite generalized quadrangles
Let \({\mathcal S}=(P,B,I)\) be a finite generalized quadrangle of order \((s,t)\), \(s>1\), \(t>1\). A panel of \({\mathcal S}\) is an ordered triple \((x,L,y)\) where \(x\) and \(y\) are distinct points incident with the line \(L\), or an ordered triple \((L,p,M)\) where \(L\) and \(M\) are distinct lines incident with the point \(P\). For a given panel \((x,L,y)\), let \(H(x,L,y)\) be the group of all collineations of \({\mathcal S}\) which fix \(x\) and \(y\) linewise and \(L\) pointwise. We say the panel \((x,L,y)\) is Moufang provided \(| H(x,L,y)| =s\). For a panel \((L,p,M)\), the group \(H(L,p,M)\) is defined in a dual manner. And \((L,p,M)\) is Moufang provided \(| H(L,p,M)| =t\). If every panel of \({\mathcal S}\) is Moufang, then by a theorem of Fong and Seitz, \({\mathcal S}\) must be isomorphic to one of the so-called classical or dual classical generalized quadrangles. The main result of this paper is that if every panel of one type is Moufang, then all panels are Moufang.
scientific article; zbMATH DE number 4163621