Finite Minkowski planes of type 20 with respect to homotheties (Q268475)

In [J. Geom. 43, No. 1--2, 116--128 (1992; Zbl 0746.51009)], \textit{M. Klein} classified Minkowski planes with respect to \(\{p,p'\}\)-homotheties. If \({\mathcal M}\) is a Minkowski plane and \(p\) and \(p'\) are non-parallel points of \({\mathcal M}\), then a \(\{p,p'\}\)-homothety is an automorphism of \({\mathcal M}\) fixing both \(p\) and \(p'\) and inducing, in the derived plane at \(p\), a homology with centre \(p'\) and axis the line at infinity.NEWLINENEWLINEA group \(\Gamma\) of \(\{p,p'\}\)-homotheties is called \(\{p,p'\}\)-transitve if it acts transitively on each circle through \(p\) and \(p'\) minus these two points. M. Klein [loc. cit.] investigated the configurations \({\mathcal H}(\Gamma)\) of all unordered pairs of points \(\{p,p'\}\) for which \(\Gamma\) is \(\{p,p'\}\)-transitve and obtained \(23\) possible types of groups of automorphisms, where the type refers to the full automorphism group of \({\mathcal M}\).NEWLINENEWLINEIn the present paper, the author only deals with the unresolved case of type \(20\) and proves that there is no finite Minkowski plane of this type. This, together with previous known results, gives us the following state of art: there is no finite Minkowski plane of type \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(12\), \(13\), \(14\), \(16\), \(18\), \(19\), \(20\), \(21\) or \(22\). There are examples of finite Minkowski planes of type \(1\) and \(23\).