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

Groups 33
FIGURE 8
127. Prove that all rotations of the cube sending each tetrahedron
onto itself form: a) a subgroup; b) a normal subgroup of the group of
rotations of the cube.
128. Prove that the commutant of the group of rotations of the cube
is isomorphic to the group of rotations of the tetrahedron.
We now prove three properties of the commutant which will be of use
later on.
129. Prove that the quotient group of an arbitrary group G by its
commutant is commutative.
130. Let N be a normal subgroup of a group G and let the quotient
group G/N be commutative. Prove that N contains the commutant of
the group G.
131. Let N be a normal subgroup of a group G and K(N) the commutant
of a subgroup N. Prove that K(N) is a normal subgroup of G
(compare with Example 11, §1.10).
1.13 Homomorphisms
Let G and F be two groups. A mapping such that
for all elements and of the group G (here the product
is taken in G and in F) is called a homomorphism of G into
F. Homomorphisms are distinguishable from isomorphisms because the
homomorphisms are not necessarily bijective.