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 $[gcolon h]$ is the smallest prime dividing $|g|$, is normal in $g$.