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Homologies 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.