Abel's theorem in problems and solutions - School of Mathematics

Solvability of Equations 239
The group of conformal mappings contains the group of index 2
(or of index 1) consisting of the homographic mappings. Hence for the
almost soluble group of conformal mappings an analogous proposition
holds.
LEMMA. A group of conformal mappings of the sphere is almost soluble
if and only if it satisfies at least one of these conditions:
1)
2)
3)
the group has only one invariant point;
the group has an invariant set consisting of two points;
the group is finite.
This lemma follows from the preceding propositions because the set of
invariant points for a normal divisor is invariant under the action of the
group. It is well known that a finite group of homographic mappings
of the sphere is sent by a homographic transformation of coordinates to a
group of rotations.
It is not difficult to prove that if the product of two inversions with
respect to two different circles corresponds under the stereographic projection
to a rotation of the sphere, then these circles correspond to great
circles. Hence every finite group of conformal mappings generated by
the inversions with respect to some circles is sent by a homographic transformation
of coordinates to a group of motions of the sphere, generated
by reflections.
All the finite groups of the motions of the sphere generated by reflections
are well known. They are exactly the symmetry groups of the
following objects:
1) the regular pyramid with a regular as basis;
2) the i.e., the solid made from two regular pyramids
joining their bases;
3) the tetrahedron;
4) the cube or the octahedron;
5) the dodecahedron or the icosahedron.