Examples of Groups with Collinearity

Abstract

Recently Karzel and Taherian introduced the concept of a group with collinearity $$(G, \kappa )$$ and showed that for $$\alpha \in G $$ the maps $$\begin{aligned} {{\tilde{\alpha }}}{:} \ G \rightarrow G ;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$ of the corresponding reflection structure $$(G , {\widetilde{G}} )$$ , where $$ {\tilde{G}} := \{{{\tilde{\gamma }}} \ | \ \gamma \in G \}$$ are point reflections. We give here a new class of groups with collinearity. This class comes from the kinematic algebra of $$2\times 2$$ -matrices.