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?

This is close, but not quite right. For example, if $G^\prime$ is a proper subgroup of $H$ then $\phi$ defines a homomorphism
$$
\bar\phi:G\to H
$$
with $\ker\bar\phi=\ker\phi$. Your result would then imply
$$
\lvert G^\prime\rvert=\frac{\lvert G\rvert}{\lvert\ker \phi\rvert}=\frac{\lvert G\rvert}{\lvert\ker\bar\phi\rvert}=\lvert H\rvert
$$
which is absurd since $\lvert G^\prime\rvert\neq\lvert H\rvert$ in general. 
To remedy the situation we must replace $G^\prime$ in your equation with $\DeclareMathOperator{Im}{Im}\Im\phi$. The formula should be
$$
\frac{\lvert G\rvert}{\lvert\ker\phi\rvert}=\lvert\Im \phi\rvert
$$
which is a consequence of the first isomorphism theorem for groups.
