## Isomorphism theorems for gyrogroups and L-subgyrogroups.(English)Zbl 1318.20060

Summary: 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 $$G$$.

### MSC:

 20N05 Loops, quasigroups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20B35 Subgroups of symmetric groups 20A05 Axiomatics and elementary properties of groups
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