On Mikheev's construction of enveloping groups

J. I. Hall

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 2, page 245-252
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Hall, J. I.. "On Mikheev's construction of enveloping groups." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 245-252. <http://eudml.org/doc/37756>.

@article{Hall2010,
abstract = {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.},
author = {Hall, J. I.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Bol loop; Moufang loop; autotopism group; group with triality; Bol loops; Moufang loops; enveloping groups; autotopisms; pseudoautomorphisms},
language = {eng},
number = {2},
pages = {245-252},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On Mikheev's construction of enveloping groups},
url = {http://eudml.org/doc/37756},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Hall, J. I.
TI - On Mikheev's construction of enveloping groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 2
SP - 245
EP - 252
AB - 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.
LA - eng
KW - Bol loop; Moufang loop; autotopism group; group with triality; Bol loops; Moufang loops; enveloping groups; autotopisms; pseudoautomorphisms
UR - http://eudml.org/doc/37756
ER -

References

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  1. Bruck R.H., A Survey of Binary Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 20, Springer, Berlin-Göttingen-Heidelberg, 1958. Zbl0141.01401MR0093552
  2. Doro S., 10.1017/S0305004100054669, Math. Proc. Cambridge Philos. Soc. 83 (1978), 377–392. Zbl0381.20054MR0492031DOI10.1017/S0305004100054669
  3. Grishkov A.N., Zavarnitsine A.V., 10.1142/S021949880600182X, J. Algebra Appl. 5 (2006), 441–463. Zbl1110.20023MR2239539DOI10.1142/S021949880600182X
  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. Zbl0806.20059MR1239875
  6. Pflugfelder H.O., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics, 7, Heldermann, Berlin, 1990. Zbl0715.20043MR1125767

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