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

128 Problems of Chapter 1
is isomorphic to
As in the case (a) one obtains that the quotient group
113. See 97. A set of rotations is a normal subgroup of the group of
rotations of the tetrahedron if and only if it consists of some one of the
classes given in the solution of Problem 97 (for the group of rotations)
and it is a subgroup. If a normal subgroup contains one rotation (by 120°
or by 240°) about one altitude of the tetrahedron‚ then it also contains the
other rotation about the same altitude and hence it contains all rotations
about all altitudes of the tetrahedron. If and
are two rotations about the altitudes from the
points A and B‚ respectively‚ then is a rotation
by 180° about the axis through the middle points of the edges AD and
BC. Hence in this case the normal subgroup contains all rotations by 180°
about the axes through the middle points of opposite edges‚ and therefore
it coincides with the entire group of rotations of the tetrahedron.
In this way in the group of rotations of the tetrahedron there is only
one normal subgroup (non-trivial): it consists in the identity and in the
three rotations by 180° about the axes through the middle points of the
opposite edges. The quotient group by this normal subgroup contains 3
elements‚ and is thus isomorphic to (see 50).
114. Let be an arbitrary element of and an arbitrary
element of Thus the element
(see solution 69) belongs to
Hence is a normal subgroup of
Let be an arbitrary element of the group We look
for which of the cosets of the normal subgroup contains this
element. If we multiply the element by all elements of the normal
subgroup (for example‚ on the right) we obtain all elements of
the type where runs over all the elements of the group We
denote this coset by In this way the cosets of the normal subgroup
are the cosets of type where runs over all the elements
of the group Since and belong to classes
and respectively‚ and we have that then
The bijective mapping of the group in the quotient
group such that for every in is an isomorphism‚ because