An isomorphism is a one to one mappingfrom one grouponto another(onto in the sense that every element ofis the image of some element of) that preserves the structure of the group, or the group operation, that is, for all
If such a mappingexists, we sayandare isomorphic and write
There are four steps in proving that a group G is isomorphic to a group G.
1. Define a mappingfromto
2. Prove thatis one to one, so that ifthen
3. Prove thatis onto, so that ifthere existssuch that
4. Prove that the mappingpreserves the group structure so that- this is called the morphism property.
Example:is an isomorphism between the groupsand
1. Defineas above.
2.is strictly increasing function, so that ifsois one to one.
3. Ifthen we need to findsatisfyingSuch a solutionalways exists forsois onto.
4.so the group structure is preserved by
Examples:is not an isomorphism between the groupsand
is not one to one since
is not onto, sincebut there is no solutionto the equation
does not preserve the group structure, since