Gyrogroup actions: A generalization of group actions

TL;DR

This paper introduces the concept of gyrogroup actions, extending group actions, and proves foundational theorems, demonstrating their properties and providing concrete examples from M"obius and Einstein gyrogroups.

Contribution

It generalizes classical group action theorems to gyrogroups and establishes a structure theorem for transitive gyrogroup actions.

Findings

Proved orbit-stabilizer, orbit decomposition, and Burnside lemmas for gyrogroups.

Established conditions for transitive gyrogroup actions.

Provided examples from M"obius and Einstein gyrogroups.

Abstract

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 $G/H$ 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 M\"obius and Einstein gyrogroups.