Abstract :
A gyrogroup is a nonassociative grouplike 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.
