If is a homomorphism of a group into a , then the set of all those elements of which is mapped by onto the identity of is called the kernel of the homomorphism .
Let and be any two groups and let and be their respective identities. If is a homomorphism of into , then
(ii) for all
(iii) is a normal subgroup of .
(i) We know that for , .
, and therefore by using left cancellation law we have or
(ii) Since for any , , we get
Similarly , gives
Hence by the definition of in we obtain the result
(iii) Since , , this shows that , now let , , ,
This establishes that is a subgroup of .
Now, to show that it is also normal we prove the following:
Therefore, , hence the result.