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
Famous quotes containing the word dimensions:
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)