# Isomorphism of Cyclic Groups

__Theorem 1__**:**

Cyclic groups of the same order are isomorphic.

__Proof__**: **Let and be two cyclic groups of order , which are generated by and respectively. Then

and

The mapping , defined by , is isomorphism.

For,

Therefore the groups are isomorphic.

__Theorem 2__**: **

An infinite cyclic group is isomorphic to the additive group of integers.

__Proof__**:** Let be an infinite cyclic group, generated by , then

The mapping , defined by is an isomorphism, for it is one-one onto, and further,

It follows that is isomorphic to .

__Theorem 3__**:**

A cyclic group of order is isomorphic to the additive group of residue classes modulo .

__Proof__**:** Let be an infinite cyclic group, generated by , then

Let be the additive group or residue classes , i.e.

The mapping , defined by , is isomorphism, for it is one-one onto, and further,

It follows that is isomorphic to .

__Theorem 4__**:**

A subgroup of the infinite cyclic group is isomorphic to the additive group of integral multiples of an integer.

__Proof__**:** Let and let be a subgroup of , given by,

Then is isomorphic to the additive group , given by

The mapping , defined by , is isomorphism, for it is one-one onto, and if , then

It will be observed that is itself an infinite cyclic group, and as such it is isomorphic to . Thus a subgroup of an infinite cyclic group is isomorphic to the group itself.