## Related pagesInvolutorial collineationsHomologies and elations |

**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.

Version