# Properties of Cyclic Groups

__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

while for

When (say)

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 .

__Proof__**:**

We know that

Therefore the order of is .

Also if , because does not divide , nor does it divide , therefore it does not divide .

Now let,

and

Thus, is a generator of the group.