and
,
a bijection
is an isomorphism of projective planes
if and only if each line l of
is mapped to a collinear set of points of
,
i.e.
.
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,
we will have to check some criterion in order to verify whether
is an isomorphism.
For projective planes, this
criterion is pretty simple.
Lemma. Given two projective planes
and
,
a bijection
is an isomorphism of projective planes
if and only if each line l of
is mapped to a collinear set of points of
,
i.e.
.
In fact, there is also a more sophisticated criterion for isomorphism.
Theorem. Given two projective planes
and
,
and two maps
and
with
for all
,
then
is an isomorphism of projective planes
if and only if
and there exists a point q of
such that
is finite.
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