Low Dimensions
The projective linear group is mostly studied for though it can be defined for low dimensions.
For (or in fact ) the projective space of is empty, as there are no 1-dimensional subspaces of a 0-dimensional space. Thus, PGL(0,K) is the trivial group, consisting of the unique empty map from the empty set to itself. Further, the action of scalars on a 0-dimensional space is trivial, so the map is trivial, rather than an inclusion as it is in higher dimensions.
For the projective space of is a single point, as there is a single 1-dimensional subspace. Thus, PGL(1,K) is the trivial group, consisting of the unique map from a singleton set to itself. Further, the general linear group of a 1-dimensional space is exactly the scalars, so the map is an isomorphism, corresponding to being trivial.
For PGL(2,K) is non-trivial, but is unusual in that it is 3-transitive, unlike higher dimensions when it is only 2-transitive.
Read more about this topic: Projective Linear Group
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