Planes in which every quadrangle lies on a unique Baer subplane (Q438903)

In a projective plane of order \(n^2\) a \textit{Baer subplane} is a subplane of order \(n\). A \textit{quadrangle} is a set of four points, no three of which are collinear. In 2000 Blokhuis and Sziklai showed that a projective plane of order \(p^2\), \(p\) prime, in which every quadrangle lies on a unique Baer subplane is Desarguesian. In this paper the authors reach the same conclusion with a weaker hypothesis on the order of the plane, showing that a projective plane of square order is Desarguesian if and only if every quadrangle lies on a unique Baer subplane. The proof of the result is dependent upon the classification of finite simple groups.