On left-gyrotranslation groups of gyrogroups

Abstract

A gyrogroup is an algebraic structure whose operation is, in general, non-associative that shares some common properties with groups. In this paper, we prove that every gyrogroup induces a permutation group, called the left-gyrotranslation group, that can be used to understand the algebraic structure of the gyrogroup itself. We also show several connections between gyrogroups and their left-gyrotranslation groups and give a few related examples, especially the left-gyrotranslation group of the famous Möbius gyrogroup in the complex plane.