University of Kashan Mathematics Interdisciplinary Research 2538-3639 1 1 2016 01 01 Special Subgroups of Gyrogroups: Commutators, Nuclei and Radical 53 68 13907 10.22052/mir.2016.13907 EN Teerapong Suksumran Department of Mathematics, North Dakota State University, Fargo, ND 58105, USA Journal Article 2016 01 29 ‎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. https://mir.kashanu.ac.ir/article_13907_7d64c578f99c83315fe22b9317d61813.pdf