Editing Möbius transformation (section)

=== Other groups ===

For any [[field (mathematics)|field]] ''K'', one can similarly identify the group {{nowrap|PGL(2, ''K'')}} of the projective linear automorphisms with the group of fractional linear transformations. This is widely used; for example in the study of [[homography|homographies]] of the [[real line]] and its applications in [[optics]]. 

If one divides <math>\mathfrak{H}</math> by a square root of its determinant, one gets a matrix of determinant one. This induces a surjective group homomorphism from the [[special linear group]] {{nowrap|SL(2, '''C''')}} to {{nowrap|PGL(2, '''C''')}}, with <math>\pm I</math> as its kernel.

This allows showing that the Möbius group is a 3-dimensional complex [[Lie group]] (or a 6-dimensional real Lie group), which is a [[Semisimple Lie group|semisimple]] and  non-[[Compact group|compact]], and that SL(2,'''C''') is a [[Double covering group|double cover]] of {{nowrap|PSL(2, '''C''')}}.  Since {{nowrap|SL(2, '''C''')}} is [[simply-connected]], it is the [[universal cover]] of the Möbius group, and the [[fundamental group]] of the Möbius group is&nbsp;'''Z'''<sub>2</sub>.