Let $ψ:A\to B$ is a group homomorphism , then can i say ψ is injective iff kerψ={e1}

Question

I don't know my assumption is true or not.But one part is easy i.e. when ψ is injective then kerψ contain the identity of A . But another part i am stuck. Even i can't find any counter example Here e1 and e2 are identity element of A and B respectively.