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

18 Chapter 1
In this way and for every integer indicate the same
element, which we will denote by Moreover, for every element
27. Prove that for any integers and
28. Prove that for any integers and
1.4 Cyclic groups
The simplest groups are the cyclic groups. They are, however, very important.
DEFINITION. Let be an element of a group G. The smallest integer
such that the element is called the order of the element If
such an integer does not exist one says that is an element of infinite
order.
29. Find the order of all elements of the groups of symmetries of the
equilateral triangle, of the square and of the rhombus (see 3,5,6).
30. Let the order of an element be equal to Prove that: 1) elements
are all distinct; 2) for every integer the element
coincides with one of the elements listed above.
DEFINITION. If an element has order and in a group G there are
no other elements but the group G is called the cyclic
group of order generated by the element and the element is called
a generator of the group.
EXAMPLE 8. Consider a regular (polygon with sides) and all
rotations of the plane that transform the into itself.
31. Prove that these rotations form a cyclic group of order
32. Find all generators in the group of rotations of the equilateral
triangle and in the group of rotations of the square (see Examples 1 and
3 in §1.1).
33. Let the order of an element be equal to Prove that
if and only if where is any integer.
34. Suppose that the order of an element is equal to a prime number
and that is an arbitrary integer. Prove that either or has
order