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Keywords:
Bol loop; Moufang loop; autotopism group; group with triality
Bol loop; Moufang loop; autotopism group; group with triality
Summary:
Mikheev, starting from a Moufang loop, constructed a groupoid and reported that this groupoid is in fact a group which, in an appropriate sense, is universal with respect to enveloping the Moufang loop. Later Grishkov and Zavarnitsine gave a complete proof of Mikheev's results. Here we give a direct and self-contained proof that Mikheev's groupoid is a group, in the process extending the result from Moufang loops to Bol loops.
References:
[1] Bruck R.H.: A Survey of Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 20, Springer, Berlin-Göttingen-Heidelberg, 1958. MR 0093552 | Zbl 0141.01401
[2] Doro S.: Simple Moufang loops. Math. Proc. Cambridge Philos. Soc. 83 (1978), 377–392. DOI 10.1017/S0305004100054669 | MR 0492031 | Zbl 0381.20054
[3] Grishkov A.N., Zavarnitsine A.V.: Groups with triality. J. Algebra Appl. 5 (2006), 441–463. DOI 10.1142/S021949880600182X | MR 2239539 | Zbl 1110.20023
[4] Hall J.I.: Moufang loops and groups with triality are essentially the same thing. submitted.
[5] Mikheev P.O.: Enveloping groups of Moufang loops. Uspekhi Mat. Nauk 48 (1993), 191–192; translation in Russian Math. Surveys 48 (1993), 195–196. MR 1239875 | Zbl 0806.20059
[6] Pflugfelder H.O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics, 7, Heldermann, Berlin, 1990. MR 1125767 | Zbl 0715.20043
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