Isomorphism Theorem
For a homomorphism , the quotient by its kernel satisfies .
Theorem 7.7 (Isomorphism Theorem)
If is a homomorphism, then
Proof
Since , the quotient is a group. Define by .
If , then for some , so . The map is a homomorphism because is a homomorphism. If , then , so and ; hence the map is injective. Every element of has the form , so the map is surjective.
Examples
The homomorphism given by has image and kernel , so . The homomorphism given by has image and kernel , so .
Related
Stated in
- Theorem 7.7 (Isomorphism Theorem)§7.3 The Isomorphism Theorem