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

114 Problems of Chapter 1
58. Answer. (For notations see Examples 1–4). 1) The subgroup of
rotations three subgroups generated by the three reflections with
respect to the altitudes: two trivial subgroups:
and the whole group; 2) the subgroup of rotations the subgroup
of generated by the central symmetry four subgroups generated
by the four reflections with respect to the symmetry axes:
the two subgroups: and and the two
trivial subgroups: and the whole group.
59. Answer. Suppose that the elements of the given groups are
Their subgroups are thus: a)
b) c)
(see 60).
60. Let the subgroup H be different from and let be the
element with minimum positive among all the elements of the subgroup
H. Thus H also contains all the elements of the form for every
integer Let be an arbitrary element of the subgroup H. Divide
by with remainder where Thus H contains
element If we obtain a contradiction of the
hypothesis that is the minimum. It follows that and that is
divisible by Since the element belongs to H, divides Hence
subgroup H has the requested properties.
61. The solution is the same of that of Problem 60.
62. If an element of the group has infinite order, then the cyclic
infinite subgroup generated by it contains an infinite number of subgroups
(see 61), which are also subgroups of the initial group. If the orders of
all elements are finite we consider the cyclic subgroups generated by the
following elements: at first an arbitrary element then an element
which does not belong to the subgroup generated by then an element
which does not belong to the subgroups generated neither by nor
by etc.. This procedure does not end because each of the subgroups
so obtained contains a finite number of elements.
63. Let be the subgroups of a group G and H their
intersection. Thus (see 57): 1) if and belong to H, then both and
belong to all Hence also belongs to all and thus it belongs
to H; 2) belongs to all subgroups therefore also to H; 3) if is an
arbitrary element of H, then belongs to all thus also belongs