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

Groups 37
We now observe what happens to subgroups, to normal subgroups,
and to commutants under the action of a homomorphism. Let
be a homomorphism. Chose in G a subset M. The set of the elements of
F having at least a pre-image in M is called the image of the set M by the
homomorphism (denoted by Conversely, let P be a subset of F;
the set of all elements of G having an image in P is called the pre-image
of P (denoted by Note that the symbol has no meaning
outside P: a homomorphism, in general, has no inverse mapping. Note
also that if then is contained in M, but it does not
necessarily coincide with M (see Figure 11).
FIGURE 11
150. Prove that the image of a subgroup H of a group G under a
homomorphism is a subgroup of the group F.
151. Let H be a subgroup of F and a homomorphism.
Prove that is a subgroup of G.
152. Let N be a normal subgroup of a group F and a
homomorphism. Prove that is a normal subgroup of the group
G.
153. Let be a homomorphism, and the commutants
of G and F. Prove that is contained in and that is contained
in
154. Let N be a normal subgroup of a group G and a
homomorphism surjective of group G onto a group F. Prove that
is a normal subgroup of F.