Isomorphism Theorems for Gyrogroups and L-Subgyrogroups
Teerapong Suksumran, Keng Wiboonton

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 induces the gyrogroup structure on the symmetric group of so that Cayley's Theorem is obtained. Introducing the notion of L-subgyrogroups, we show that an L-subgyrogroup partitions into left cosets. Consequently, if is an L-subgyrogroup of a finite gyrogroup , then the order of divides the order of .
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