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

Groups 29
102. Let be the order of a group G, the order of a subgroup H
and Prove that H is a normal subgroup of the group G.
103. Prove that the intersection (see footnote to Problem 63) of an
arbitrary number of normal subgroups of a group G is a normal subgroup
of the group G.
DEFINITION. The set of elements of a group G which commute with
all elements of the group is called the centre of group G.
104. Prove that the centre of a group G is a subgroup and, moreover,
a normal subgroup of the group G.
105. Let and be two normal subgroups of two groups and
respectively. Prove that is a normal subgroup of the group
The following example shows that a normal subgroup of a subgroup
of a group G can be a non-normal subgroup of the group G.
EXAMPLE 11. Consider the subgroup of the group of symmetries
of the square, generated by the reflections with respect to the diagonals
and to the centre (see Examples 3,4 §1.1, the subgroup This
subgroup contains one half of the elements of the group of symmetries of
the square, and it is therefore a normal subgroup (see 102). The subgroup
generated by the reflection with respect to one of the diagonals,
contains one half of the elements of the subgroup and it is
therefore a normal subgroup of this subgroup. But the subgroup is
not a normal subgroup of the group of symmetries of the square, because
is sent by an internal automorphism to the reflection with respect to
the other diagonal:
1.11 Quotient groups
Let us start with an example. Consider the partition of the group of
symmetries of the square by the normal subgroup generated by the
central symmetry (see Example 3,4 §1.1). It is easy to see that the
partition of our group into four cosets has the form shown in Table 2.