Understanding homomorphism and kernels Let $G$ be a group and $\phi$ a Homomorphism
$$
\phi:G\to G'
$$
Now I know that the size of the kernel tells you how many elements in $G$ map to the same element in $G'$
I couldn't find this in my book, but I have concluded the following. 
$$
\frac{|G|}{| \:\text{ker} \: \phi \:|} = |G'|
$$
Is that true?

Let's assume all groups are finite. You observe correctly that $|\ker (\phi)|=|\phi^{-1}(g)|$, for any $g\in G'$, as long as $g$ is in the image of $\phi$, otherwise, of course $|\phi^{-1}(g)|=0$. So, the conclusion is: $|G'|=|{\rm Im}(\phi)|+|G'-{\rm Im}(\phi)|=\frac{|G|}{|\ker(\phi)|}+|G'-\Im (\phi)|$. Of course, if $\phi$ is surjective, then the second summand is $0$, and you get the formula you wrote. As noted in other answers, the same result follows easily by invoking the (first) isomorphism theorem, though (perhaps) that is a bit of an over-kill. 
