THE AFFINE PLANE AG(2,Q), Q-ODD, HAS A UNIQUE ONE-POINT EXTENSION

Let I be a finite inversive plane of odd order q. If for at least one point p of I the internal affine plane I(p) is Desarguesian, then I is Miquelian. Other formulation: the finite Desarguesian affine plane of odd order q has a unique one point extension; this extension is the M iquelian inversive plane of order q. It follows that there is a unique inversive plane of order q, with q is-an-element-of{3, 5, 7}.