Distance transitive generalized quadrangles of prime order (Q1591223)

Using the classification of finite simple groups, \textit{F. Buekenhout} and \textit{H. Van Maldeghem} [Geom. Dedicata 52, No. 1, 41-51 (1994; Zbl 0809.51008)] classified all generalized quadrangles \(\Gamma\) for which the collinearity graph of \(\Gamma\) is distance transitive, i.e., the automorphism group of the collinearity graph of \(\Gamma\) (which coincides with the automorphism group of \(\Gamma)\) has rank 3 on the graph. It seems to be a very difficult problem to achieve this without the classification of finite simple groups. In the present paper the author gives such a proof of this in the case that \(\Gamma\) has finite parameters \((s,t)= (p,p)\), \(p\) a prime. He credits W. M. Kantor with having found such a proof somewhat earlier, but the one under review is apparently the first one to be published. Specifically, the main result of the paper is that a finite generalized quadrangle \(\Gamma\) of order \((p,p)\), \(p\) a prime, whose collinearity graph admits a rank 3 automorphism group is isomorphic to the symplectic quadrangle \(W(p)\) or to the orthogonal quadrangle \(Q(4,q)\). The proof uses a blend of group theoretic and geometric arguments.