Special subgroups of gyrogroups: Commutators, nuclei and radical

A gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by Einstein’s velocity addition law. In this article, we present a few of groups sitting inside a gyrogroup $G$, including the commutator subgyrogroup, the left nucleus, and the radical of $G$. The normal closure of the commutator subgyrogroup, the left nucleus, and the radical of $G$ are in particular normal subgroups of $G$. We then give a criterion to determine when a subgyrogroup $H$ of a finite gyrogroup $G$, where the index $[G\text{\hspace{0.17em}⁣}:H]$ is the smallest prime dividing $\mathrm{\mid }G\mathrm{\mid }$, is normal in $G$.