Isomorphisms of partial linear spaces
partial linear spaces.
Given two isomorphisms
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
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.
is an isomorphism.
- For each subset
if and only if
- For each three points p,q,r, not necessarily distinct, the
three points are collinear if and only if their images
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
Version $Id: pliniso.html,v 1.2 2000/12/28 20:51:15 hauke Exp $