Coordinatizing projective planes

Let be a projective plane. Let (o,e,u,v) be a quadrangle in . We will call the line at infinity and the diagonal, respectively. Obviously,

is a coordinate frame of the affine derivative . Using the coordinatization of affine planes, we get the following theorem.

Theorem. The map T defined by

yields a ternary field (K,T), and the map

defines an isomorphism of onto the projective plane over (K,T).

The ternary field (K,T) in this theorem will be called the ternary field of with respect to the quadrangle (o,e,u,v). Note that distinct quadrangles usually lead to non isomorphic ternary fields. In fact, the following lemma holds.

Lemma. Let (o,e,u,v) and (o',e',u',v') be two quadrangles in projective planes and , respectively, and denote the corresponding ternary fields by (K,T) and (K',T'), respectively. Then

defines a one-to-one correspondence between isomorphisms with , , and on one side and and isomorphisms on the other side.

In particular, the stabilizer of the automorphism group of a projective plane on a quadrangle corresponds to the automorphism group of the corresponding ternary field.

Moreover, all ternary fields belonging to a projective plane are isomorphic if and only if the automorphism group of is transitive on the set of all quadrangles of , and this happens if and only if is a Moufang plane.

Contributed by Hauke Klein
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