On class-regular projective Hjelmslev planes

Summary

Regular projective Hjelslev planes (PH-planes), i.e. PH-planes admitting an automorphism group which is regular on the points and blocks (i.e. a Singer group) were introduced and studied by Jungnickel. In this paper, we study (t,r)-PH-planes admitting an automorphism group which is regular on each point and block class (neighbourhood). We prove that this notion is equivalent to the existence of a projective plane of order r and a (t,r)- Hjelmslev matrix in an abelian group of order t² as defined by Jungnickel. This enables us to construct many class-regular PHplanes which do not admit a Singer group. Further, all the results of Jungnickel (on regular PH-planes) also hold for class-regular PH-planes.

REMARKS: Projective Hjelmslev planes (H-planes) (more generally, projective Klingenberg planes or K-planes) are generalizations of projective planes in which two points (resp. two lines) are allowed to have more than one line passing through them (resp. more than one point of intersection) and which admit an epimorphism onto a projective plane. For a nice introduction to H-planes (and K-planes) and their automorphism groups, we refer to [3], [4] and [5] ; [1] gives an extensive bibliography of the literature on H-planes and related structures.