On the finite subgroups of projective linear groups

Any element of 1-dimensional complex projective line [x:y] and any element of the extended complex plane x/y corresponds. We denote complex numbers by C. The general linear group of 2×2 complex matrices GL(2,C) acts on the complex plane by matrix multiplication and we see that this act on 1-dimensional complex projective line is the same act on the extended complex plane by linear fractional transformation using usual relation between 1-dimensional complex projective line and the extended complex plane.Linear fractional transformations form a group under composition. This group is the quotient group GL(2,C)/Z(GL(2,C)) where Z(GL(2,C)) is the center of GL(2,C) and we have GL(2,C)/Z(GL(2,C)):=PGL(2,C). In other words, the group of linear fractional transformations on the extended complex plane which is called Möbiüs group is isomorphic to PGL(2,C). On the other hand, PGL(2,C) is the automorphism group of the Riemann sphere.Regular convex polyhedras can be inscribed in the sphere in 3-dimensional real space such that their vertices are on the sphere. These regular convex polyhedras are Platonic solids. In Euclidean space, the isometries preserving a Platonic solid P is a finite subgroup of SO(3) and these groups act on the sphere. If we send sphere to the extended complex plane under stereographic projection, we see that the same group acts on 1-dimensional complex projective line. In other words, a group of symmetries of Platonic solids are finite Möbiüs groups. By this aim, in this work it is explained that the Möbiüs group of linear fractional transformations with complex coefficients, any subgroup of PGL(2,C), is isomorphic to a finite group of the symmetries of the sphere, in other words these finite subgorups are n dimensional cyclic group, n dimensional dihedral group, 4. dimensional alternating group, 4. dimensional symmetric group and 5. dimensional alternating group.Moreover, since the linear fractional transformations can also be expressed by their fixed points, the fixed points of any linear fractional transformation are found, then these points are interpreted geometrically.Finally, we consider that any symmetry of a Platonic solid in 3-dimensional real space is a direct isometry of 3-dimensional real space that leaves it invariant. It is said that all symmetries of this Platonic solid form a group and the Möbiüs groups of all Platonic solids are given.Keywords: Projective linear group, Linear fractional transformation, Platonic solids,Riemann sphere.