Isomorphism Theorems for Gyrogroups and L-Subgyrogroups

Abstract

We extend well-known results in group theory to gyrogroups, especially the isomorphism theorems. We prove that an arbitrary gyrogroup $G$ induces the gyrogroup structure on the symmetric group of $G$ so that Cayley's Theorem is obtained. Introducing the notion of L-subgyrogroups, we show that an L-subgyrogroup partitions $G$ into left cosets. Consequently, if $H$ is an L-subgyrogroup of a finite gyrogroup $G$, then the order of $H$ divides the order of $G$

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

“…Further, we show that several well-known results in group theory continue to hold for gyrogroups. In particular, we extend the results of Suksumran and Wiboonton [60,61] by including more details and adding some new results.…”

“…Further, we prove that if G is an arbitrary gyrogroup, then the set of left gyrotranslations of G admits the gyrogroup structure induced by G. The gyrogroup of left gyrotranslations is isomorphic to the underlying gyrogroup G. This results in a version of Cayley's theorem for gyrogroups [61]. …”

“…In the case k D 0, the statement is clear, so assume that k ¤ 0. If m 0 and k > 0, the statement holds by (61). Furthermore, if m < 0 and k < 0, then…”

“…As we will see, gyrogroups are a natural generalization of groups. Gyrogroups that generalize abelian groups are given a name: Gyrogroups share algebraic properties with groups, and many of group-theoretic theorems are extended to the case of gyrogroups with the aid of gyroautomorphisms [61,67,70]. For example, the gyrogroup counterpart of the group identity .g 1 h/.h 1 k/ D g 1 k is described by…”

“…This expository paper grew from our three research papers [60][61][62]. Let c be a positive constant representing the speed of light in vacuum, and let R where u is the Lorentz factor or gamma factor given by u D 1 q 1…”