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: h] is the smallest prime dividing |g|, is normal in g.