Affine Hughes groups acting on 4-dimensional compact projective planes

Abstract

Suppose that the real projective plane is contained as a Baer subplane in a compact 4-dimensional plane . Salzmann [15] has shown that if the group of projective collineations of extends to , then is isomorphic to the complex plane. This means that there are no compact 4-dimensional Hughes planes. If the group AGL2ℝ of offine collineations of extends to , then we call the extended group an offine Hughes group. Here, non-classical examples exist. We show that, up to duality, every action of AGL2ℝ on a compact 4-dimensional plane is an offine Hughes group, and we describe explicitly the possible actions on the point space of as transformation groups. In subsequent work (jointly with N. Knarr and H. Klein), this will be used to determine the possible planes , as well. This will complete the classification of 4-dimensional planes admitting a non-solvable group of dimension at least 6.