Gyrogroup actions: A generalization of group actions

JOURNAL OF ALGEBRA

卷:454

Article

Suksumran, Teerapong¹ 

[1] N Dakota State Univ, Dept Math, Fargo, ND 58105 USA

关键词: Permutation representation;    Gyrogroup action;    Orbit-stabilizer theorem;    Burnside lemma;    Gyrogroup;    Left gyroaddition;   

DOI  :  10.1016/j.jalgebra.2015.12.033

来源: Elsevier

【 摘 要 】

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 Mobius and Einstein gyrogroups. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free