An isomorphism is a one to one mapping
from one group
onto 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 if
then
3. Prove thatis onto, so that if
there exists
such that
4. Prove that the mappingpreserves the group structure so that- this is called the morphism property.
Example:
is an isomorphism between the groups
and
1. Defineas above.
2.
is strictly increasing function, so that if
sois one to one.
3. If
then we need to find
satisfying
Such a solutionalways exists for
sois 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, since
but there is no solution
to the equation
does not preserve the group structure, since