Über projektive Ebenen, die fast fahnentransitiv sind. (About quasi flag-transitive projective planes) (Q1823465)

Let G be a transitive collineation group of a projective plane \(\pi\) of finite order n such that the stabilizer of a point P fixes some line g through P and is transitive on the other lines \((P)\setminus \{g\}\) through P. Then the following holds: i) G is regular on \((P)\setminus \{g\}\), ii) if \(\pi\) is desarguesian then \(n=3\), iii) \(n\equiv 3 mod 12\) and \(3^ n\equiv 1 mod (n^ 2+n+1)\) where \(n^ 2+n+1\) is prime, iv) if n is a prime power then \(n=3^{(3^ i)}\) for some \(i\in (N\cup \{0\})\setminus \{1,2,3\}\).