Seminversive planes

An H-inversive plane, where H is a set of positive integers, is a pair (Ω,C), where Ω is a set of points and C a family of subsets of Ω called circles such that (i) any three distinct points lie on a unique circle; (ii) given a circle B, a point x∈B and a point y∉B, the number of circles through x and y meeting B just at the point x belongs to H; (iii) there exist at least two circles and every circle contains at least three points. The integer n is defined to be the order of the plane if n+1={|B|:B∈C}. The author investigates the {1,2}-inversive planes called seminversive planes under the assumption that Ω is finite. The main result of the paper under review is the following theorem: Suppose (Ω,C) is a finite seminversive plane of order n>5. Then (Ω,C) is either an inversive plane or a punctured inversive plane of order n. This extends results of M. Oehler [Geom. Dedicata 4 (1975), no. 2-3-4, 419--436; MR0405236 (53 #9030)].