Moufang plane
In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically it is a special type of translation plane. A translation plane is a projective plane that has a translation line, that is, a line with the property that the group of automorphisms that fixes every point of the line acts transitively on the points of the plane not on the line.[1] A translation plane is Moufang if every line of the plane is a translation line.[2]
Characterizations
A Moufang plane can also be described as a projective plane in which the little Desargues Theorem holds.[3] This theorem states that a restricted form of Desargues' theorem holds for every line in the plane.[4] Every Desarguesian plane is a Moufang plane.[5]
In algebraic terms, a projective plane over any alternative division ring is a Moufang plane,[6] and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and Moufang planes. As a consequence of the algebraic Artin–Zorn theorem, that every finite alternative division ring is a field, every finite Moufang plane is Desarguesian, but some infinite Moufang planes are non-Desarguesian planes. In particular, the Cayley plane, an infinite Moufang projective plane over the octonions, is one of these because the octonions do not form a division ring.[7]
Properties
The following conditions on a projective plane P are equivalent:[8]
- 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.
- Some ternary ring of the plane is an alternative division ring.
- P is isomorphic to the projective plane over an alternative division ring.
Also, in a Moufang plane:
- The group of automorphisms acts transitively on quadrangles.[9][10]
- Any two ternary rings of the plane are isomorphic.
Notes
- ↑ That is, the group acts transitively on the affine plane formed by removing this line and all its points from the projective plane.
- ↑ Hughes & Piper 1973, p. 101
- ↑ Pickert 1975, p. 186
- ↑ This restricted version states that if two triangles are perspective from a point on a given line, and two pairs of corresponding sides also meet on this line, then the third pair of corresponding sides meet on the line as well.
- ↑ Hughes & Piper 1973, p. 153
- ↑ Hughes & Piper 1973, p. 139
- ↑ Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS, 54 (10): 1294–1303
- ↑ H. Klein Moufang planes
- ↑ Stevenson 1972, p. 392 Stevenson refers to Moufang planes as alternative planes.
- ↑ If transitive is replaced by sharply transitive, the plane is pappian.
References
- Hughes, Daniel R.; Piper, Fred C. (1973), Projective Planes, Springer-Verlag, ISBN 0-387-90044-6
- Pickert, Günter (1975), Projektive Ebenen (Zweite Auflage ed.), Springer-Verlag, ISBN 0-387-07280-2
- Stevenson, Frederick W. (1972), Projective Planes, W.H. Freeman & Co., ISBN 0-7167-0443-9
Further reading
- 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