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.