A group is called a cyclic group if, for some , every element is of the form , where is some integer. The element is called generator of .
There may be more than one generator of a cyclic group. If is a cyclic group generated by , then we shall write or . The elements of will be of the form . Of course, they are not necessarily all distinct.
(1) The multiplicative group is cyclic. The generators are and .
(2) The multiplicative group is cyclic. We can write . The generators are and .
(3) The multiplicative group of , nth roots of unity are cyclic, a generator being