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

Solutions 123
of symmetries of the tetrahedron if and only if they belong to the same
class.
In the group of rotations of the tetrahedron classes 4 and 5 are absent,
whereas class 2 splits into two subclasses: 2a) all the clockwise rotations
by 120° about the altitudes (looking on the base of the tetrahedron from
the vertex from which the altitude is drawn); 2b) all rotations by 240°.
Solution. Let the elements of the group of symmetries of the tetrahedron
be divided into classes as explained above. Such classes are characterized
by the following properties: all elements of class 2 have order
3 and preserve the orientation of the tetrahedron; all elements of class
3 have order 2 and preserve the orientation; all elements of class 4 have
order 2 and change the orientation; all elements of the class 5 have order
4 and change the orientation. Since an internal automorphism is an isomorphism
(see 94) two elements of different order cannot be transformed
one into the other (see 49). Moreover, and either both change
the orientation or both preserve it (it suffices to consider two cases: when
changes and when preserves the orientation). Consequently two elements
of distinct classes cannot be transformed one into the other by an
internal automorphism.
Let and be two rotations by 180° about two axes through the
middle points of two opposite edges and let be a rotation sending the
first axis onto the other. Thus the rotation sends the second
axis into itself without reversing it. Moreover, (otherwise
Hence coincides with Therefore any two
elements of class 3 can be transformed one into the other by an internal
automorphism in the group of rotations (and therefore in the group of
symmetries) of the tetrahedron.
Let and be two reflections of the tetrahedron with respect to
two planes of symmetry and let be a rotation sending the first plane
onto the second one. Thus as before we have
If and then and
Hence It follows that if can be transformed
either into or into then and can be transformed one
into the other. Therefore it suffices to show that any element of a given
class can be sent into all the other elements of the same class.
Let be an element of the class 5 and let
be the rotations such that