# Properties of Isomorphism

__Theorem 1__**:**

If isomorphism exists between two groups, then the identities correspond, i.e. if is an isomorphism and are respectively the identities in , then .

__Theorem 2__**: **

If isomorphism exists between two groups, then the identities correspond, i.e. if is an isomorphism and , where then .

__Theorem 3__**:**

In an isomorphism the order of an element is preserved, i.e. if is an isomorphism, and the order of is , then the order of is also .

__Proof__**:
**As , then we have and in general we can write it as .

But , by using the statement of Theorem 1,

therefore . Also for , i.e. .

It follows that the order of an element of , if finite, is equal to the order of its image in . If the order of is infinite, we can similarly show that the order of cannot be finite.

__Theorem 4__**:**

The relation of isomorphism in the set of groups is an equivalence relation.