# Kernel of Homomorphism

__Definition__

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 .

__Theorem__**:**

Let and be any two groups and let and be their respective identities. If is a homomorphism of into , then

**(i)**

**(ii)** for all

**(iii)** is a normal subgroup of .

__Proof__**:**

**(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.