## Isomorphism

An isomorphism is a one to one mapping from one group onto another (onto in the sense that every element of is the image of some element of ) that preserves the structure of the group, or the group operation, that is, for all If such a mapping exists, we say and are isomorphic and write There are four steps in proving that a group G is isomorphic to a group G.

1. Define a mapping from to 2. Prove that is one to one, so that if then 3. Prove that is onto, so that if there exists such that 4. Prove that the mapping preserves the group structure so that - this is called the morphism property.

Example: is an isomorphism between the groups and 1. Define as above.

2. is strictly increasing function, so that if so is one to one.

3. If then we need to find satisfying Such a solution always exists for so is onto.

4. so the group structure is preserved by Examples: is not an isomorphism between the groups and  is not one to one since  is not onto, since but there is no solution to the equation  does not preserve the group structure, since  