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

Groups 35
137. Prove that ker is a subgroup of group G.
138. Prove that ker is a normal subgroup of group G.
Consider the partition of G by the kernel ker
139. Prove that and belong to the same coset if and only if
THEOREM 3. Let be a surjective homomorphism of a
group G onto a group F. The mapping sending each
coset to the image by of a certain element of the coset (and thus of an
arbitrary element (see 139)), is an isomorphism.
The proof of this theorem is contained in the solutions of the following
problems.
140. Prove that is surjective.
141. Prove that is bijective.
142. Prove that is an isomorphism.
We will consider some applications of this theorem.
EXAMPLE 13. Problem 110 asked whether the quotient group of the
group of symmetries of the square by the normal subgroup generated by
the central symmetry is isomorphic either to the group of rotations of the
square or to the group of symmetries of the rhombus. To each element of
the group of symmetries of the square there corresponds some permutation
of the axes of symmetry (Figure 9). This permutation can
just exchange between each other the diagonals and as well as the
axes and
FIGURE 9 FIGURE 10
We thus obtain a mapping of the group of symmetries of the square
into a group of permutations of four elements and This mapping
is a homomorphism surjective onto the group of those permutations