any field, one can similarly identify the group 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 homographies of the real line and its applications in optics. If one divides 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 SL(2, C) to PGL(2,C), with 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 and non-compact, and that SL(2,C) is a double cover of PSL(2,C). Since SL(2,C) is simply-connected, it is the universal cover of the Möbius group, and the fundamental group of the Möbius group is Z2. Specifying a transformation by three points Given a set of three distinct points on the Riemann sphere and a second set of distinct points , there exists precisely one Möbius transformation with for . (In other words: the action of the Möbius group on the Riemann sphere is sharply 3-transitive.) There are several ways to determine from the given sets of points. Mapping first to 0, 1, It is easy to check that the Möbius transformation with matrix maps to respectively. If one of the is , then the proper formula for is obtained from the above one by first dividing all entries by and then taking the limit . If is similarly defined to map to then the matrix which maps to becomes The stabilizer of (as an unordered set) is a subgroup known as the anharmonic group. Explicit determinant formula The equation is equivalent to the equation of a standard hyperbola in the -plane. The problem of constructing a Möbius transformation mapping a triple to another triple is thus equivalent to finding the coefficients of the hyperbola passing through the points . An explicit equation can be found by evaluating the determinant by means of a Laplace expansion along the first row. This results in the determinant formulae for the coefficients of the representing matrix . The constructed matrix has determinant equal to which does not vanish if the resp. are pairwise different thus the Möbius transformation is well-defined. If one of the points or is , then we first divide all four determinants by this variable and then take the limit as the