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

Groups 31
111. Find all normal subgroups and the corresponding quotient groups
of the following groups 8 : a) the group of symmetries of the triangle; b)
c) the group of symmetries of the square; d) the group of quaternions
(see solution of Problem 92).
112. Describe all normal subgroups and the corresponding quotient
groups of the following groups: a) b)
113. Find all normal subgroups and the corresponding quotient groups
of the group of rotations of the tetrahedron.
114. Consider the subgroup in the direct product
Prove that it is a normal subgroup and that the corresponding quotient
group is
1.12 Commutant
Recall that two elements and of a group G are said to be commuting if
The degree of non-commutativity of two elements of a group can
be measured by the product which is equal to the unit element
if and only if and commute.
DEFINITION. The element is called the commutator of the
elements and The set of all possible products of a finite number of
commutators of a group G is called the commutant of the group G and it
is denoted by K(G).
115. Prove that the commutant is a subgroup.
116. Prove that the commutant is a normal subgroup.
117. Prove that the commutant coincides with the unit element
if and only if the group is commutative.
118. Find the commutant in the following groups: a) of symmetries
of the triangle; b) of symmetries of the square; c) the group of quaternions
(see solution of Problem 92).
119. Prove that the commutant in the group of symmetries of the
regular is isomorphic to the group if is odd and to the group
if is even.
8 In the sequel finding the quotient group will mean showing a group, among those
already considered, to be isomorphic to the group requested.