Moufang plane
Encyclopedia
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 plane
s.
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:
Ruth Moufang
Ruth Moufang was a German mathematician.Born to a German chemist Dr. Eduard Moufang and Else Fecht Moufang, she studied mathematics at the University of Frankfurt. In 1931 she received her Ph.D. on projective geometry under the direction of Max Dehn, and in 1932 spent a fellowship year in Rome...
, is a type of projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
, 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
Desarguesian plane
In projective geometry a Desarguesian plane, named after Gérard Desargues, is a plane in which Desargues' theorem holds. The ordinary real projective plane is a Desarguesian plane. More generally any projective plane over a division ring is Desarguesian, and conversely Hilbert showed that any...
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 plane
Non-Desarguesian plane
In mathematics, a non-Desarguesian plane, named after Gérard Desargues, is a projective plane that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane...
s.
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 a is 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.
External links
- H. Klein Moufang planes