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

Solutions 115
to all and therefore also to H. In virtue of the result of Problem 57
H is a subgroup of the group G.
64. Answer. a) Yes; b) yes; c) no; d) no.
65. The vertex A can be sent onto an arbitrary vertex, B onto any one
of the remaining vertices, C onto any one of the remaining two vertices.
Answer. 4 · 3 · 2 = 24.
b)
66. Answer. a) All symmetries fixing vertex D;
67. Let us formulate the definition of orientation in a more symmetric
way. We have defined the orientation by means of the vertex D, but if
the position of the triangle ABC with respect to the vertex D is given,
then the position of any triangle with respect to the fourth vertex is also
uniquely defined. Hence a transformation preserving the orientation of
the tetrahedron preserves the position of every triangle with respect to
the fourth vertex, whereas a transformation changing the orientation of
the tetrahedron changes the position of every triangle with respect to
the fourth vertex. It is clear that the product of two transformations
preserving the orientation of the tetrahedron, as well as the product of
two transformations which do not preserve it, preserves the orientation.
Conversely, if a transformation preserves the orientation and another permutation
changes it, their product changes the orientation. The identity
obviously preserves the orientation and, since for every transformation
if preserves the orientation also preserves the orientation.
It follows that (see 57) all symmetries of the tetrahedron preserving the
orientation form a subgroup in the group of all symmetries of the tetrahedron.
Since the vertex D can be sent onto an arbitrary vertex, and then
the triangle ABC can take one of three positions, this group contains
4 · 3 = 12 elements.
68. Answer. a) The subgroup of rotations about an axis through the
middle points of two opposite edges; b) the subgroup of rotations about
the axis through D perpendicular to the plane of triangle ABC.
69. Using the associativity in G and in H we shall prove the associativity
in G × H. We have