A cyclic group is a group in which there is an element x
such that each element of the group may be written as 
for some
integer k. In additive notation, this translates to 
.
We say that x is a generator of the cyclic group or that
the group is generated by x.
As an example, the integers under addition is a cyclic group. The
number 1 is a generator. This is because for any n in the integers
we have 
. Note that -1 is also a generator.
Another example is provided by the set of complex numbers
under multiplication of complex numbers. A generator
is i since 
, 
, 
and 
. Note that
-i is also a generator.
For a finite cyclic group G having n elements, any element of
order n is a generator. If x is a generator having order n
then the order of is 
.
It follows that a cyclic group is an abelian group although not every abelian group is a cyclic group. For example, the rational numbers under addition is not cyclic but is abelian.