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.
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