Gyrogroup actions: A generalization of group actions

This article explores the novel notion of gyrogroup actions, which is a natural generalization of the usual notion of group actions. As a first step toward the study of gyrogroup actions from the algebraic viewpoint, we prove three well-known theorems in group theory for gyrogroups: the orbit-stabilizer theorem, the orbit decomposition theorem, and the Burnside lemma (or the Cauchy–Frobenius lemma). We then prove that under a certain condition, a gyrogroup G   acts transitively on the set G/HG/H of left cosets of a subgyrogroup H in G in a natural way. From this we prove the structure theorem that every transitive action of a gyrogroup can be realized as a gyrogroup action by left gyroaddition. We also exhibit concrete examples of gyrogroup actions from the Möbius and Einstein gyrogroups.