Isomorphisms of projective planes

Related pages

Involutorial collineations
Homologies and elations
We may think of isomorphisms between partial linear spaces as as special bijections between their sets of points. Given a bijection , 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.


Contributed by Hauke Klein
Version $Id: projiso.html,v 1.2 2001/01/01 18:10:09 hauke Exp $