Isomorphisms of partial linear spaces

Related pages

Isomorphisms of linear spaces
Isomorphisms of projective planes
Let and be two partial linear spaces. Given two isomorphisms and from to , it is easily seen that , i.e. an isomorphism is already determined by it's point mapping . Consequently, we may think of an isomorphism between partial linear spaces just as a special bijection Using this point of view, the automorphism group of becomes a subgroup of the symmetric group of .

In fact, given two partial linear spaces and , the line map belonging to an isomorphism is just the induced map on sets of points.

We have the following lemma.

Lemma. Given a bijection , the following statements are equivalent.

  1. is an isomorphism.
  2. For each subset , we have if and only if .
  3. For each three points p,q,r, not necessarily distinct, the three points are collinear if and only if their images are collinear.

Automorphisms of a partial linear space are often called collineations, since the third part of our lemma states that a permutation on the set of points is an automorphism if and only if it respects the relation of collinearity.

Contributed by Hauke Klein
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