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

Solutions 119
hypothesis. The group of rotations of the tetrahedron therefore does not
contain any subgroup of order 6.
89. Answer. a) The left and right partitions coincide:
b) left partition: right partition:
90. Answer. a) The two partitions coincide:
b) left partition: right partition:
91. Answer. The two partitions coincide and contain three cosets: 1)
all numbers of type all numbers of type
all numbers of type
92. Answer. a) Two groups: and
b) two groups: and the group of symmetries of the equilateral
triangle;
c) five groups: the group of symmetries of
the square, the group of quaternions with elements and
the following table of multiplication (Table 11).
Solution. a) Let be the elements of the initial group. The
elements can thus be of order 2 or 4 (see 83). Let us consider some
cases.
1) Amongst there is an element of order 4. Hence the given
group is the cyclic group