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 $G$.

Cited by in corpus (8)

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• Strongly Topological Gyrogroups and Quotient With Respect to L-subgyrogroups
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• Gyrogroup through its Grothendieck Group Completion and Right gyrogroup action
• A class of quotient spaces in strongly topological gyrogroups
• The construction of Hartman-Mycielski in topological gyrogroups
• Feathered gyrogroups and gyrogroups with countable pseudocharacter