Point-primitive Inversive Planes of Odd Order

A famous but still unsolved problem in finite geometries is the question of whether a finite inversive plane J of odd order n is necessarily miquelian, that is, it arises from the plane sections of an elliptic quadric in PG(3,«). The question has been approached from different angles, and it has a positive answer if suitable conditions are added to J. The most classical result in this context is perhaps the one by P. Dembowski and D. R. Hughes [7], stating that if J admits an 'orthogonality' then it is miquelian. J. Kahn proves in [16] that the same conclusion holds if the so-called 'bundle theorem' is assumed to hold universally. J. A. Thas shows in [25, 26] that if the internal affine plane Jx is desarguesian for at least one point X, then J is miquelian except, possibly, when n is 11, 23 or 59; see also [9,17] for earlier work in this respect. The classical possibility offered by Klein's point of view, which imposes restrictions on the automorphisms of J, has been the object of several investigations; see [19, 5,11, 20,13, 4, 22, 2] (some of these papers deal with the even order case as well). A fundamental result in this direction is due to C. Hering [13]: if J is an inversive plane of odd order n admitting an automorphism group G acting 2transitively on its points, then J is miquelian and G contains PSL(2,n). In the present paper we want to continue along this line and prove the following result.