Moufang loop

PropositionPlanetmathPlanetmathPlanetmath: Let Q be a nonempty quasigroup.

I) The following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

(x(yz))x = (xy)(zx)  for all x,y,zQ (1)
((xy)z)y = x(y(zy))  for all x,y,zQ (2)
(xz)(yx) = x((zy)x)  for all x,y,zQ (3)
((yz)y)x = y(z(yx))  for all x,y,zQ (4)

II) If Q satisfies those conditions, then Q has an identity elementMathworldPlanetmath (i.e., Q is a loop).

For a proof, we refer the reader to the two references. Kunen in [1] shows that that any of the four conditions implies the existence of an identity element. And Bol and Bruck [2] show that the four conditions are equivalent for loops.

Definition: A nonempty quasigroup satisfying the conditions (1)–(4) is called a Moufang quasigroup or, equivalently, a Moufang loop (after Ruth Moufang, 1905–1977).

The 16-element set of unit octonions over is an example of a nonassociative Moufang loop. Other examples appear in projective geometry, coding theory, and elsewhere.


[1] Kenneth Kunen, Moufang Quasigroups, J. AlgebraPlanetmathPlanetmath 83 (1996) 231–234. (A preprint in PostScript format is available from Kunen’s website: kunen/moufang.psMoufang Quasigroups.)

[2] R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1958.

Title Moufang loop
Canonical name MoufangLoop
Date of creation 2013-03-22 13:50:29
Last modified on 2013-03-22 13:50:29
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Definition
Classification msc 20N05