# Group Isomorphism

__Definition__

Let and be any two groups with binary operation and , respectively. If there exists a one-one onto mapping such that

In this case, the group is said to be isomorphic to the group , and the mapping is said to be an isomorphism. If is isomorphic to , we write or .

In other words, a group is isomorphic to the group if there exists a one-one onto mapping of and such that the image of the product of two elements is the product of the images of the elements with respect to the composition in the respective group.

The last condition may also be stated as follows:

If where and then where .