The Kernel of a Homomorphism

Question

" 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.

Answer 1

The point is that $f(a)=f(b)$ if and only if $ab^{-1}\in \ker f$. So, if you have checked beforehand that $\ker f=\{e\}$, then $f(a)=f(b)$ automatically implies that $a=b$, which means that $f$ is injective. And it works backwards too, if $f$ is injective, then $\ker f=\{e\}$, because $f(a)=e =f(e)$ implies $a=e$.

Answer 2

If $c$ belongs to the image, then there is a $g_0\in G$ such that $f(g_0)=c$. But then$$\{g\in G\mid f(g)=c\}=\{g_0h\mid h\in\ker f\}.$$