Rank Three Affine Planes

Michael J. Kallaher
1972 Proceedings of the American Mathematical Society  
A permutation group has rank 3 if it is transitive and the stabilizer of a point has exactly three orbits. A rank 3 collineation group of an affine plane is one which is a rank 3 permutation group on the points. Several people (see [4], [7], [8], [12]) have characterized different kinds of affine planes using rank 3 collineation groups. In this article we prove the following: Let si be a finite affine plane of nonsquare order having a rank 3 collineation group which acts regularly on one of its
more » ... larly on one of its orbits on the line at infinity, si must be either (i) a Desarguesian plane, (ii) a semifield plane, or (iii) a generalized André plane.
doi:10.2307/2039040 fatcat:z7jy3p53jzf67d5xe2fzkeks74