Kinematic mappings of plane affinities (Q1923487)

The author generalizes the concept ``kinematic space'' to ``general kinematic space \((P,{\mathcal G}, \cdot)\)'', i.e. \((P, {\mathcal G})\) is an incidence space and \((P, \cdot)\) a group such that \(\forall a \in P\), the maps \(a_i\): \(P\to P\); \(x\to ax\) and \(P\to P\); \(x\to x^{-1}\) are collineations of \((P,{\mathcal G})\) [the author, Mitt. Math. Ges. Hamb. 12, No. 3, 785-791 (1991; Zbl 0751.51007)]. Let \(T \subset P\) be invariant, for \(a\in P\) let \(\widetilde a: T\to T\); \(t\to a\) \(ta^{-1}\) and let \(\kappa:P \to\widetilde P\); \(a\to \widetilde a\). Then he calls \((P, {\mathcal G}, \cdot,T)\) a kinematic group and \(\kappa^{-1}\) a kinematic map if \(\kappa\) is injective. Here the author shows that the extension construction of \textit{M. Marchi} and \textit{E. Zizioli} [Ann. Discrete Math. 18, 601-615 (1983; Zbl 0509.51011)] can be applied also on general kinematic spaces \((P, {\mathcal G},+)\): Let \(U\leq \Aut(P, {\mathcal G},+)\) and \({\mathcal W}\) a set of subgroups of \(U\) such that \(\forall V,V' \in {\mathcal W}\), \(\forall \alpha,\beta\in U \backslash\{id\}\): (a) \(\alpha,\beta\in V \Rightarrow\) Fix \(\alpha=\) Fix \(\beta\), (b) \(\alpha V_\alpha^{-1}\in W\), (c) \(V\neq V'\Rightarrow V \cap V'= \{id\}\). Then \((U,\cdot)\) resp. \((M,\cdot): =(P,+) \rtimes U\) can be turned in a 2-sided incidence group \((U,{\mathcal G}_U, \cdot)\) resp. on a general kinematic space \((M, {\mathcal G}_M, \cdot)\) where \({\mathcal G}_U\) resp. \({\mathcal G}_M\) are cosets of subgroups of partial fibrations \({\mathcal W}\) resp. \({\mathcal F}_M\) and suitable 2-sets of \(P\) resp. \(M\). For \(T: =(P,id)\) he studies firstly when \((M,{\mathcal G}_M, \cdot,T)\) is a kinematic group and then connections between \((P, {\mathcal G},+)\) and \((M, {\mathcal G}_M, \cdot)\). Finally the theory is applied on affine collineation groups; the plane case is treated in detail.