Gyrogroup actions: A generalization of group actions

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 Möbius and Einstein gyrogroups.

References in corpus (2)

• Isomorphism Theorems for Gyrogroups and L-Subgyrogroups
• Lagrange's Theorem for Gyrogroups and the Cauchy Property

Cited by in corpus (7)

• Extension of Maschke's theorem
• Construction of New Gyrogroups and the Structure of their Subgyrogroups
• On paratopological gyrogroups
• Special subgroups of gyrogroups: Commutators, nuclei and radical
• Normal Subgyrogroups of Certain Gyrogroups
• The construction of Hartman-Mycielski in topological gyrogroups
• Gyrogroup through its Grothendieck Group Completion and Right gyrogroup action