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

Groups 23
tude); (rotation by 180° around the axis through
the middle points of the edges AD and BC); (re-
flection with respect to the plane containing edge AD and the middle
point of edge BC); (cyclic permutation of the
vertices)?
All symmetries of the regular tetrahedron obviously form a group,
which is called the group of symmetries of the tetrahedron.
65. How many elements does the group of symmetries of tetrahedron
contain?
66. In the group of symmetries of the tetrahedron find the subgroups
isomorphic to: a) the group of symmetries of the equilateral triangle; b)
the cyclic group
67. Prove that all symmetries of the tetrahedron preserving its orientation
form a subgroup. How many elements does it contain?
The group of symmetries of the tetrahedron preserving its orientation
is called the group of rotations of the tetrahedron.
68. Find in the group of rotations of the tetrahedron the subgroups
isomorphic to: a) b)
1.7 Direct product
Starting from two groups one may define a third group.
DEFINITION. The direct product G × H of groups G and H is the set of
all the ordered pairs where is any element of G and any element
of H, with the following binary operation:
where the product is taken in the group G, and in the group
H.
69. Prove that G × H is a group.
70. Suppose that a group G has elements, and that a group H has
elements. How many elements does the group G × H contain?
71. Prove that the groups G × H and H × G are isomorphic.