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The projective group *PGL*(2, **C**).

This is the group that acts on the complex projective line *P*(**C**^{2}) = **C***P*^{1}.

As in the last example the elements of this group can be regarded as the set of those *rational analytic functions* of the form:

*z* (*az* + *b*)/(*cz* + *d*) with *a*, *b*, *c*, *d* ∈ **C** with *ad* - *bc* ≠0,

from **C** ∪ {∞} to itself.

**Remarks**

- These transformations are sometimes called Möbius functions (after the German mathematician August Möbius (1790 to 1868) best known for his results in Number Theory and for the so-called Möbius band).
- Complex variable theory shows that these mappings are
*conformal*(angle preserving) transformations at all points. - As in the last example, we may define the cross-ratio of four points (a complex number this time) and this is preserved by these transformations.

The projective group

This is the group that acts on the real projective plane *P*(**R**^{3}) = **R***P*^{2}.

One may show that there is a unique element of this group taking any four points in **R***P*^{2} (no three on a line) into any other four points in **R***P*^{2}.

As in the projective line case, one may take standard reference points to be: [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]. These are the points at infinity on the *x* and *y* axes, the origin and the point (1, 1). We may manipulate these as in the earlier case.

We may define the cross-ratio of four points on a line and these transformations preserve that.

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