Let G be a group. Use the FHT (Fundamental Homomorphism Theorem) to prove that the quotient group G/{e} is isomorphic to G.

Question

Ok so the Fundamental Homomorphism Theorem (or First Group Isomorphism Theorem) states that if θ : G → H and ker (θ) = K, then the quotient group G/K is isomorphic to H.

I know that θ : G → G has to be a surjective homomorphism with kernel {e}, but I'm not sure how to prove this or if it is automatically assumed. Help?

Answer 1

Hint: Is $x\mapsto x$ a homomorphism?