A simplified proof of Moufang’s theorem
HTML articles powered by AMS MathViewer
- by Aleš Drápal PDF
- Proc. Amer. Math. Soc. 139 (2011), 93-98 Request permission
Abstract:
Moufang’s theorem states that if $Q$ is a Moufang loop with elements $x$, $y$ and $z$ such that $x\cdot yz = xy \cdot z$, then these three elements generate a subgroup of $Q$. The paper contains a new proof of this theorem that is shorter and more transparent than the standardly used proof of Bruck.References
- V. D. Belousov, Osnovy teorii kvazigrupp i lup, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0218483
- G. Bol, Gewebe und gruppen, Math. Ann. 114 (1937), no. 1, 414–431 (German). MR 1513147, DOI 10.1007/BF01594185
- R. H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), 245–354. MR 17288, DOI 10.1090/S0002-9947-1946-0017288-3
- R. H. Bruck, On a theorem of R. Moufang, Proc. Amer. Math. Soc. 2 (1951), 144–145. MR 41839, DOI 10.1090/S0002-9939-1951-0041839-3
- R. H. Bruck, Pseudo-automorphisms and Moufang loops, Proc. Amer. Math. Soc. 3 (1952), 66–72. MR 47635, DOI 10.1090/S0002-9939-1952-0047635-6
- Richard Hubert Bruck, A survey of binary systems, Reihe: Gruppentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0093552
- A. Drápal and P. Jedlička, On loop identities that can be obtained by a nuclear identification, Europ. J. Comb. (in press). http://dx.doi.org/10.1016/j.ejc.2010.01.007.
- Ruth Moufang, Zur Struktur von Alternativkörpern, Math. Ann. 110 (1935), no. 1, 416–430 (German). MR 1512948, DOI 10.1007/BF01448037
- Hala O. Pflugfelder, Quasigroups and loops: introduction, Sigma Series in Pure Mathematics, vol. 7, Heldermann Verlag, Berlin, 1990. MR 1125767
Additional Information
- Aleš Drápal
- Affiliation: Department of Mathematics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
- Email: drapal@karlin.mff.cuni.cz
- Received by editor(s): December 31, 2009
- Received by editor(s) in revised form: March 12, 2010
- Published electronically: July 21, 2010
- Additional Notes: The author was supported by the Grant Agency of Czech Republic, grant 201/09/0296.
- Communicated by: Jonathan I. Hall
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 93-98
- MSC (2010): Primary 20N05; Secondary 08A05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10501-4
- MathSciNet review: 2729073