**Example 2**.

*Proof*. This is just an obvious result of Example 1 and Theorem 1.

**Example 3**. Let be a cyclic group. If then and if then .

*Proof*. The first part is obvious by Example 1. So suppose that and let be a generator of and Let Then, since we can choose anything we like. Now define by See that is an onto ring homomorphism and (Note that )

**Example 4**. Let be cyclic groups of order respectively. Then as abelian groups.

*Proof*. Let be generators of respectively. Let be defined by See that if and only if which is equivalent to So there are possibility for Let be a generator of and define by and see that is a group isomorphism.

**Example 5**. Let be a prime number and and be cyclic groups of orders and respectively. Then

*Proof.* By Theorem 1:

By Examplse 3, and Also, by Example 4

So