I have learnt the following three isomorphisms for a while but without true understanding:
A group homomorphism $\phi:G\to G'$ can be decomposed into \begin{equation}G\xrightarrow{\text{quotient}}G/\operatorname{ker}(\phi)\simeq \operatorname{Im}(\phi)\hookrightarrow G'. \end{equation}
and
$H$ is a normal subgroup of $G$ and $K$ is another subgroup. Then $H\cap K$ is normal in $K$, $HK$ is a subgroup inside which $H$ is normal, and \begin{equation}\frac{K}{H\cap K}\simeq \frac{HK}{H}. \end{equation}
and
$H$ is a subgroup $G$ and $K\supset H$ is another subgroup. Then $K/H$ is normal in $G/H$ if and only if $K$ is normal in $G$. If $K$ is normal then \begin{equation}\frac{G}{K}\simeq \frac{G/H}{K/H}. \end{equation}
The proofs for these three theorems are rather straightforward, and after teaching myself some category theory I am more comfortable with the first one. But I do not feel them. (Like in this post by Gowers he explains Orbit-Stablizer by moving a cube and with this picture you get the feeling that such a theorem has to be right.)
I wonder whether someone can share similar insights on the three isomorphisms maybe by using intuitive-but-nontrivial examples like Gowers.
Thanks!