On planes of order \(p^2\) in which every quadrangle generates a subplane of order \(p\)
According to \textit{A. M. Gleason} [Am. J. Math. 78, 797-807 (1956; Zbl 0072.38001)], a finite projective plane is desarguesian if every quadrangle is contained in a subplane of order 2. The authors prove the following beautiful result: let \(p\) be a prime; if every quadrangle of a finite projective plane of order \(p^2\) is contained in a subplane of order \(p\), then the plane is desarguesian. The proof relies on a theorem of \textit{J. Cofman} [Geom. Dedicata 2, 195-199 (1973; Zbl 0279.50015)] and on the enumeration of all doubly transitive permutation groups of degree \(p^2\) (which seems to require the classification of all finite simple groups).
scientific article; zbMATH DE number 1455309