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

24 Chapter 1
72. Find the subgroups of G × H isomorphic to the groups G and H.
73. Let G and H be two commutative groups. Prove that the group
G × H is also commutative.
74. Let be a subgroup of a group G and a subgroup of a group
H. Prove that is a subgroup of the group G × H.
75. Let G and H be two arbitrary groups. Is it true that every
subgroup of the group G × H can be represented in the form
where is a subgroup of the group G and a subgroup of the group
H?
76. Prove that the group of symmetries of the rhombus is isomorphic
to the group
77. Is it true that: 1) 2)
78. Prove that if and only if the numbers and
are relatively prime.
1.8 Cosets. Lagrange’s theorem
For every subgroup H of a group G there exists a partition of the set
of the elements of G into subsets. For each element of G consider the
set of all elements of the form where runs over all elements of a
subgroup H. The set so obtained, denoted by is called the left coset
of H (or left lateral class of H) in G, generated by the element
79. Find all left cosets of the following subgroups of the group of
symmetries of the equilateral triangle: a) the subgroup of rotations of the
triangle; b) the group generated by the reflection with respect to a single
axis (see Examples 1 and 2, §1.1).
80. Prove that given a subgroup H of a group G each element of G
belongs to one left coset of H in G.
81. Suppose that an element belongs to the left coset of H generated
by an element Prove that the left cosets of H generated by elements
and coincide.
82. Suppose that the left cosets of H, generated by elements and
have a common element. Prove that these left cosets coincide.