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Moufang planes
Lenz classification
A projective plane is called a translation plane if there exists a line l such that the group of elations with axis l is transitive on the affine plane . Each such line l is called a translation axis. Usually, the translation axis l is unique. In fact, a projective plane admits two distinct translation axes if and only if the plane is a Moufang plane.

The plane P(K,T) over a ternary field (K,T) is a translation plane with respect to the line at infinity if and only if (K,T) is a quasifield.

An affine plane is called an affine translation plane if it's projective closure is a translation plane with respect to the line at infinity.

Given a vector space V over a skewfield K, and a spread S, then we may define an affine translation plane as follows. Points of are the points of V, and the lines of are defined to be the subsets of the form t+U where t is an element of V and All affine translation planes are of this form, up to isomorphism.

See also

Contributed by Hauke Klein
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