Classification of projective translation planes of order \(q^{2}\) admitting a two-transitive orbit of length \(q + 1\) (Q1021321)

It is considered a translation plane \(\pi\) of order \(q^2\). For the associated projective extension \(\pi^+\) it is supposed that it admits a collineation group \(G\) inducing a two-transitive group on a set \(\Gamma\) of \(q+1\) points of \(\pi^+\). Then it is proved that \(\pi\) is one of the following types of planes: (1) Desarguesian; (2) Hering; (3) Ott-Schaeffer; (4) the Hall plane of order 9. When the plane is known, it is determined the action of the group \(G\) as a permutation group.