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

126 Problems of Chapter 1
such that Since N is a normal subgroup
Hence there exists in N an element such that Therefore
Since the element belongs to N
and belong to the same coset
107. Let and be the elements arbitrarily chosen in and
respectively. By the definition of multiplication of cosets‚
and are the cosets containing the elements and respectively.
Since
108. Hint. Take as a representant of class E.
109. Hint. Let be an arbitrary element of class T. Take as the class
the coset containing the element
110. It is easy to verify (see table 2‚ §1.11) that
Hence this quotient group is isomorphic to the group of symmetries of
the rhombus.
111. We will show only the normal subgroups different from and
the whole group:
a) see 58 (1)‚ 95‚ 96‚ 102. Answer. The normal subgroup is the
group of rotations of the triangle; the corresponding quotient group is
isomorphic to
b) see 99‚ 74‚ 75. Let be the given group. Answer.
The normal subgroups are:
The corresponding quotients groups are isomorphic
to
c) for the notations see examples 3‚ 4 (§1.1). If a normal subgroup
of the group of symmetries of the square contains the element or the
element then it contains every subgroup of the group of rotations of
the square. We obtain in this case the normal subgroup (see
102)‚ and the quotient group
We have and Thus if one of the elements
belongs to the normal subgroup the other one also does. Since in
this case the element also belongs to the normal subgroup. We obtain
the normal subgroup (see 102)‚ and the quotient group
Since and we obtain‚ as before‚ the
normal group and the quotient group
If‚ on the contrary‚ the normal subgroup does not contain the elements
then it coincides with the normal subgroup and the
corresponding quotient group is isomorphic to (see 100‚ 110).