" The kernel is important because it controls the entire homomorphism. It tells us not only which elements of G are mapped to the identity in G', but also which pairs of elements have the same image in G'. " (Taken from *Algebra* by Artin.)

How does the kernel tell us which pairs of elements have the same image in G'? 

What I tried:

let $f$ be a homomorphism such that $f(a) = f(b) = c$. $c$ is in $G'$, so it has an inverse $c^{-1}$. So then, $c^{-1}f(a) = c^{-1}f(b) = cc^{-1} = 1$. 

But, this is tedious to compute, and I didn't use the kernel to do it.