# 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