Let be a Subgroup of Group . Then is normal in if and only if there is a group and a Group homomorphism such that the kernel of is .
Proof
"Normal" implies "is a kernel"
Let be normal, so it is closed under conjugation. Then we may define the Quotient group , whose elements are the left cosets of in , and where the operation is that . This group is well-defined (proof).
Now there is a homomorphism given by . This is indeed a homomorphism, by definition of the group operation .
The kernel of this homomorphism is precisely ; that is simply :
- Certainly (because for all , since is closed as a subgroup of );
- We have because if then in particular (where is the group identity) so .
"Is a kernel" implies "normal"
Let have kernel , so if and only if . We claim that is closed under conjugation by members of .
Indeed, since . But that is , so .
That is, if then , so is normal.