Inversive planes of odd order (Q914687)

There is a big number of characterization problems for t-(v,k,\(\lambda\)) designs. [See for example the book: \textit{T. Beth, D. Jungnickel} and \textit{H. Lenz}, Design Theory (1985; Zbl 0569.05002)]. In this paper the authors are concerned with the characterization of inversive planes of odd order q or so called \(3-(q^ 2+1,q+1,1)\) design, defining a graph, the vertices of which are the blocks of the inversive plane and two blocks are adjacent if they meet in one point. The following main theorem is proved: An inversive plane \({\mathcal P}\) of odd order q is isomorphic to \({\mathcal M}(q)\) iff \(\Gamma\) (\({\mathcal P})\) has 2 connected components and there is some point P such that the residue of \({\mathcal P}\) at P is isomorphic to AG(2,q). As a corollary of this theorem it is shown that for odd order classical inversive planes each of the components is a 2- design. Furthermore, for odd \(q>3\) each of the components of \(\Gamma\) (\({\mathcal M}(q))\) forms a 3-class association scheme, whereas for even \(q>2\) all the circles of an inversive plane form such a scheme.