Moufang plane

In mathematics, a Moufang plane, named for Ruth Moufang, is a type of projective plane, characterised by the property that the group of automorphisms fixing all points of any given line acts transitively on the points not on the line. In other words, symmetries fixing a line allow all the other points to be treated as the same, geometrically. Every Desarguesian plane is a Moufang plane, and (as a consequence of the Artin–Zorn theorem) every finite Moufang plane is Desarguesian, but some infinite Moufang planes are non-Desarguesian planes.

The projective plane over any alternative division ring is a Moufang plane, and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and Moufang planes.

The following conditions on a projective plane P are equivalent:

• P is a Moufang plane.
• The group of automorphisms fixing all points of any given line acts transitively on the points not on the line.
• The group of automorphisms acts transitively on quadrangles.
• Any two ternary rings of the plane are isomorphic.
• Some ternary ring of the plane is an alternative division ring.
• P is isomorphic to the projective plane over an alternative division ring.

References

• Tits, Jacques; Weiss, Richard M. (2002), Moufang polygons, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43714-7, MR 1938841 
• Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS 54 (10): 1294–1303 

External links

• H. Klein Moufang planes