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

110 Problems of Chapter 1
2) any integer can be written in the form where
and is some integer. Thus (see 27)
(see 28) =
where
31. Hint. The generator is a rotation of order
32. Answer. In the group of rotations of the triangle the generators
are: rotation by 120° and rotation by 240°; in the group of rotations
of the square they are: rotation by 90° and rotation by 270°.
33. Let where Thus (see solution 30-2))
But if and only if (see 30-1)) Therefore if
and only if
34. Hence (see 33) the order of the
element must divide the number Since is prime the statement
follows.
35. Since the numbers and are non-zero integers,
If is an integer such that
then (see 33) is divisible by and is divisible by Since the
numbers and are relatively prime is divisible by Hence
the smallest non-zero integer such that is
36. Let be a rotation counterclockwise by an angle equal to
The elements of the group considered are thus The element
in order to be a generator, must have its order equal to 12, and
therefore the numbers and 12 (see 35) must be relatively prime. Hence
will be a generator whenever Answer. The generators
are the rotations by angles
37. Let and Thus and
contradicting the hypothesis that is an element of infinite order.
38. Hint. If one considers the group under addition, then by one
indicates the sum and by one indicates
The generators are 1 and –1.