First Isomorphism Theorem

Let be Group with a homomorphism.
Then the Kernel of is Normal and

Proof

Firstly, , so nonempty.
Let .
Then so ,
so is a Subgroup of .
Now note

so for all .
Also so .
Let , given by .
Note is surjective homomorphism (need to check well defined).
Can also check injective, so is isomorphism - done.