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

Groups 39
FIGURE 12
vertices; 4) rotations around the axes through the middle points of opposite
edges.
158. How many elements are contained in each class (without counting
the identity transformation in classes 2–4)?.
159. Let N be an arbitrary normal subgroup of the group of rotations
of the dodecahedron and suppose that N contains at least one element
of one among classes 1–4. Prove that N contains the entire class of this
element.
As a consequence each one of classes 1–4 either belongs entirely to N
or has no elements in common with N.
160. Prove that in the group of rotations of the dodecahedron there
are no other normal subgroups except and the whole group.
161. Suppose that a group G is not commutative and that it has no
normal subgroups other than and G. Prove that G is not soluble.
From problems 160 and 161 it follows that the group of rotations of
the dodecahedron is not soluble.
We shall consider some more problems whose results will be of use
later on.
162. Prove that every subgroup of a soluble group is soluble.
163. Let a homomorphism surjective of a group G onto a
group F and suppose that group G is soluble. Prove that the group F is
also soluble.