A note on Minkowski planes admitting groups of automorphisms of some large types with respect to homotheties
A Minkowski plane \({\mathcal M}= ({\mathcal P},{\mathcal C},{\mathcal G}_1,{\mathcal G}_2)\) is an incidence structure consisting of a point set \({\mathcal P}\), a circle set \({\mathcal C}\) (the elements of which are non empty subsets of \({\mathcal P}\)) and two different partitions \({\mathcal G}_1\) and \({\mathcal G}_2\) of \({\mathcal P}\) (parallel classes), whose elements are called generators, satisfying a set of defining axioms. Namely: three mutually non-parallel points, i.e., not belonging to the same generator, are joined by a unique circle; each generator meets each circle in a unique point; each generator in \({\mathcal G}_1\) meets each generator in \({\mathcal G}_2\) in exactly one point; there is a circle that contains at least three points; the circles which touch a fixed circle \(K\) at a point \(p\in K\) partition \({\mathcal P}- ([p]_1\cup [p]_2)\). Here \([p]_i\) denotes the parallel class in \({\mathcal G}_i\) that contains \(p\). The incidence structure whose point set is \({\mathcal P}\) minus all points of \([p]_1\) and \([p]_2\), and whose lines are obtained by all circles through \(p\) and by all the generators, except \([p]_1\) and \([p]_2,\) is an affine plane, the so called derived plane at \(p\). A Minkwsi plane is of finite order when it has a finite number of points. Examples of Minkowski planes are obtained over a field \(\mathbb F\) as the geometry of non-trivial plane sections of a ruled quadric in \(3\)-dimensional projective space over \(\mathbb F\). Such planes are referred to as Miquelian planes. An automorphism of a Minkowski plane is a permutation of the point-set mapping circles onto circles and generators onto generators. An automorphism \(\gamma\) is a \(\{p,q\}\)-homothety if \(\gamma\) fixes \(p\) and \(q\) and induces a homothety with centre \(q\) in the derived plane at \(p\). In [\textit{M. Klein}, J. Geom. 43, No. 1--2, 116--128 (1992; Zbl 0746.51009)], a classification of Minkowski planes with respect to linearly transitive subgroups of Minkowski homotheties was given. In particular, 23 possible types were obtained. In this paper, types 12, 13, 14 and 18 are investigated. It is proven that types 12 and 14 can only occur in finite Miquelian Minkowski planes of order 3 and 5. Furthermore it is proven that types 13 and 18 in finite Minkowski planes can only occur in Miquelian planes.
linear transitive group of central automorphisms
scientific article; zbMATH DE number 6151649
scientific article