Solution of a Classical Problem on Finite Inversive Planes

Abstract

Abstract Let Ibe a finite inversive plane of odd order q, qr/.{11, 23,59). If for at least one point pof Ithe internal affine plane Ipis Desarguesian, then Iis Miquelian. Other formulation: for q (/.{11,23,59) the finite Desarguesian affine plane of odd order qhas a unique extension; this extension is the Miquelian inversive plane of order q.As a direct corollary we obtain a computer-free proof of the uniqueness of the inversive plane of order 7.Let Cbe a circle of the inversive plane I,let pE Cand let pbe a point not on C.Since Ipis an affine plane it follows that there is a unique circle C’ through pand pwhich has only pin common with C.We say that the circles C1 and C2 are tangentif C1 = C2 or IC1 n C2I = l. Now it is clear that tangency is an equivalence relation on the set of all circles. If Ois an ovoid[7] of PG(3, q), then the points of Otogether with the intersections 1rnO, with ,r a non-tangent plane of O,form an inversive planeof order q.Such an inversive plane is called egglike.For an egglike inversive plane Ieach internal plane Ipis Desarguesian. By a celebrated theorem of P. Dembowski [7] each finite inversive plane of even order is egglike. If the ovoid Ois an elliptic quadric, then the corresponding inversive plane is called classicalor Miquelian.By a celebrated theorem of A. Barlotti [14] each egglike inversive plane of odd order is Miquelian.