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

1. That is, the group acts transitively on the affine plane formed by removing this line and all its points from the projective plane.
2. Hughes & Piper 1973, p. 101
3. Pickert 1975, p. 186
4. 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.
5. Hughes & Piper 1973, p. 153
6. Hughes & Piper 1973, p. 139
7. Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS 54 (10): 1294–1303
8. H. Klein Moufang planes
9. Stevenson 1972, p. 392 Stevenson refers to Moufang planes as alternative planes.
10. 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