Via representation of vertex-transitive graphs on groupoids, we show that left loops with units are factors of groups, i.e., left loops are transversals of left cosets on which it is possible to define a binary operation which allows left cancellation.
@article{bwmeta1.element.doi-10_2478_BF02479200,
author = {Eric Mwambene},
title = {Multiples of left loops and vertex-transitive graphs},
journal = {Open Mathematics},
volume = {3},
year = {2005},
pages = {245-250},
zbl = {1117.05052},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02479200}
}
Eric Mwambene. Multiples of left loops and vertex-transitive graphs. Open Mathematics, Tome 3 (2005) pp. 245-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02479200/
[1] A.A. Albert: “Quasigroups I”, Trans. Amer. Math. Soc., Vol. 54, (1943), pp. 507–520. http://dx.doi.org/10.2307/1990259
[2] W. Dörfler: “Every regular graph is a quasi-regular graph”, Discrete Math., Vol. 10, (1974), pp. 181–183. http://dx.doi.org/10.1016/0012-365X(74)90031-4
[3] E. Mwambene: Representing graphs on Groupoids: symmetry and form, Thesis (PhD), University of Vienna, 2001.
[4] G. Gauyacq: “On quasi-Cayley graphs”, Discrete Appl. Math., Vol. 77, (1997), pp. 43–58. http://dx.doi.org/10.1016/S0166-218X(97)00098-X | Zbl 0881.05057
[5] C. Praeger: “Finite Transitive permutation groups and finite vertex-transitive graphs”, In: G. Sabidussi and G. Hahn (Eds.): Graph Symmetry: Algebraic Methods and Applications, NATO ASI Series, Vol. 497, Kluwer Academic Publishers, The Netherlands, Dordrecht, 1997. | Zbl 0885.05072
[6] G. Sabidussi: “Vertex-transitive graphs”, Monatsh. Math., Vol. 68, (1964), pp. 426–438. http://dx.doi.org/10.1007/BF01304186 | Zbl 0136.44608