Remarks on the proof of the representation theorem for Miquelian Möbius planes by A. Lenard (Q1969667)

\textit{B. L.\ van der Waerden} and \textit{L. J.\ Smid} [Math. Ann. 110, 753-776 (1935; Zbl 0010.26803)] showed that a Miquelian Möbius plane \({\mathcal M}\) can be described as the geometry of plane sections of an elliptic quadric in 3-dimensional projective space. Their proof involved showing that every derived affine plane at a point \(p\) of \({\mathcal M}\) is Pappian, i.e., can be coordinatised over a field \({\mathbb K}\), and that all circles of \({\mathcal M}\) not passing through the distinguished point \(p\) are conic sections with the same associated quadratic form in the derived affine plane at \(p\). In [Abh. Math. Sem. Univ. Hamburg 65, 57-82 (1995; Zbl 0851.51007)] \textit{A. Lenard} gave a new proof of this representation theorem for Miquelian Möbius planes using the description of these planes as chain geometries \(\Sigma({\mathbb L},{\mathbb K})\) where \({\mathbb L}\) is a quadratic extension field of \({\mathbb K}\). Circles are the images of a fixed circle under the group of Möbius transformations \(z\mapsto {az+b\over cx+d}\) with \(a,b,c,d\in{\mathbb L}\), \(ad-bc\neq 0\). Applying Miquel's configuration and various of its degenerations Lenard geometrically obtains the field operations of \({\mathbb L}\). Whereas the addition of \({\mathbb L}\) is readily constructed multiplication is more involved and is based on the four-cycle relation where a 6-tuple \((a_{12},a_{34},a_{13},a_{24},a_{14},a_{23})\) of at least 5 distinct points is in the four-cycle relation if and only if there are four circles \(K_1, K_2,K_3,K_4\) and a point \(q\) such that \(K_i\cap K_j=\{a_{ij},q\}\) for \(1\leq i<j\leq 4\). This relation is used to construct automorphisms of the Möbius plane: Given four points \(0,\infty,a,a'\) not on a circle a point \(x\neq 0,\infty,a,a'\) is taken to the unique point \(x'\) such that \((0,\infty,x,a',a,x')\) is in the four-cycle relation. These automorphisms represent \((\infty,0)\)-similarities in the derived affine plane at \(\infty\). In his paper A. Lenard also considers the case that \(0,\infty,a,a'\) are concircular and the associated automorphism becomes a homothety. However, the inclusion of this degenerated case requires to know that derived affine planes are Pappian, so this is verified first. In the note under review the authors show that the preliminary step of verifying that derived affine planes are Pappian can be avoided by a clever and consequent use of properties of proper \((\infty,0)\)-similarities. Although the collection of all proper \((\infty,0)\)-similarities no longer is a group, the authors show that the group \(\Delta_{\infty,0}\) generated by it is commutative and operates regularly on the points \(\neq \infty,0\). Furthermore, the group \(\Sigma_{\infty,0}\) of all \((\infty,0)\)-homotheties forms a subgroup of \(\Delta_{\infty,0}\) that operates regularly on \(X\setminus\{0,\infty\}\) for each circle \(X\) through 0 and \(\infty\). The multiplication in \({\mathbb L}\) is then obtained using the group \(\Delta_{\infty,0}\) and the subfield \({\mathbb K}\) is obtained from \(\Sigma_{\infty,0}\).