I took symmetry course and was wondering about unfaithful group represntations.
I know that representation is unfaithful if group homomorphism is not injective. I also know that if kernel is nontrivial more than one element is mapped to identity. That break injectiveness so representation is unfaithful. If I can find elements that map to same element hence break injectiveness that are not in kernel that means I found an unfaithful representation with trivial kernel. That is impossible according to Wikipedia.
Why is that so?
I also read that elements not in the kernel can map to the same element if they are in the same coset. Is that true? Why?