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 $G$.

References in corpus (1)

• Distances of qubit density matrices on Bloch sphere

Cited by in corpus (17)

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• Strongly Topological Gyrogroups and Quotient With Respect to L-subgyrogroups
• Separability in (strongly) topological gyrogroups
• On paratopological gyrogroups
• Suitable sets for strongly topological gyrogroups
• Special subgroups of gyrogroups: Commutators, nuclei and radical
• Geometry of gyrogroups via Klein's approach
• Embedding into connected, locally connected topological gyrogroups
• A class of quotient spaces in strongly topological gyrogroups
• The construction of Hartman-Mycielski in topological gyrogroups
• Gyrogroup through its Grothendieck Group Completion and Right gyrogroup action
• Feathered gyrogroups and gyrogroups with countable pseudocharacter
• Gyrogroup extensions and semi cross product
• Quotient with respect to admissible $L$-subgyrogroups