Isomorphism Theorems for Gyrogroups and L-Subgyrogroups

TL;DR

This paper extends classical group theory results to gyrogroups, establishing isomorphism theorems, Cayley's Theorem, and properties of L-subgyrogroups, including coset partitioning and order divisibility.

Contribution

It introduces the concept of L-subgyrogroups and extends fundamental theorems like isomorphism and Cayley's Theorem to gyrogroups.

Findings

Gyrogroups induce structures on symmetric groups, leading to Cayley's Theorem.

L-subgyrogroups partition gyrogroups into cosets.

Order of L-subgyrogroups divides the order of finite gyrogroups.

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 $G$.