Geometry of gyrogroups via Klein's approach

Abstract

Using Klein's approach, geometry can be studied in terms of a space of points and a group of transformations of that space. This allows us to apply algebraic tools in studying geometry of mathematical structures. In this article, we follow Klein's approach to study the geometry $(G,\mathcal{T})$, where $G$ is an abstract gyrogroup and $\mathcal{T}$ is an appropriate group of transformations containing all gyroautomorphisms of $G$. We focus on $n$-transitivity of gyrogroups and also give a few characterizations of coset spaces to be minimally invariant sets. We then prove that the collection of open balls of equal radius is a minimally invariant set of the geometry $(G,{\Gamma }_m)$ for any normed gyrogroup $G$, where ${\Gamma }_m$ is a suitable group of isometries of $G$.

20 pages

References in corpus (1)

• Isomorphism Theorems for Gyrogroups and L-Subgyrogroups