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

Suksumran, Teerapong; Wiboonton, Keng

J. Geom. Symmetry Phys. 37, 67-83 (2015).

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

Keywords:

finite gyrogroups; L-subgyrogroups; Cayley theorem; Lagrange theorem; isomorphism theorems; Bol loops; \(A_\ell\)-loops