If $K$ is a subgroup of $G$ and $N$ a normal subgroup of $G$, prove that $K/K\cap N \cong KN/N$.
The above is the second isomorphism theorem but I would like to prove it as an exercise.
Clearly, $N \triangleleft G$ so $\forall g \in G: gN=Ng$
$K\cap N\triangleleft K$ so $\forall x \in K\cap N: xK=Kx$.
I want to show an isomorphism exists to $KN/N$. Should I be looking towards the first isomorphism theorem?
Any hints are appreciated. Thanks in advance.
Hint: Consider the projection $K\to KN/N$ defined by $$ k\mapsto kN. $$ Prove that this is a surjection with kernel $K\cap N$. Then use the first isomorphism theorem.
The idea is that you want to show that $K/K\cap N\cong KN/N$. One way to do this is to notice that if you could find a homomorphism $K\to KN/N$ with kernel $K\cap N$, then the first isomorphism theorem dictates that you will have the desired isomorphism. Thus, you construct the most natural choice of homomorphism and try to show its kernel is what you want it to be.