Theorem 1: Every cyclic group is abelian.
Proof: Let be a generator of a cyclic group and let for any then
(Because for )
Thus the operation is commutative and hence the cyclic group is abelian.
Note: For the addition composition the above proof could have been written as (addition of integer is commutative)
Thus the operation + is commutative in .
Theorem 2: The order of a cyclic group is the same as the order of its generator.
Proof: Let the order of a generator of a cyclic group be , then
We observe that
Thus there are exactly elements in the group by , where . Therefore there are and only distinct elements in the cyclic group, i.e. the order of the group is .
Theorem 3: The generators of a cyclic group of order are all the elements , being prime to and .
We know that
Therefore the order of is .
Also if , because does not divide , nor does it divide , therefore it does not divide .
Thus, is a generator of the group.