Find a certain group $G$ with $H,K\unlhd G$, $H\cong K$ and $(G/H)\ncong(G/K)$

Question

I'm trying to solve a problem from my group theory text book. It says:

Find a certain group $G$ with $H,K\unlhd G$, that verifies $H\cong K$ but $(G/H)\ncong(G/K)$.

I don't know what group to consider. At first I thought about the quaternion $Q_8$, but I didn't find a solution (in my book, this kind of exercises use to use the $Q_8$). Then I considered the diedric groups $D_n$ and the symmetric $Sn$ but also got nothing. What is a possible solution to this problem? Any help will be appreciated, thanks in advance.

Answer 1

So you can just consider $G = \mathbb Z, H= 2 \mathbb Z, K = 3 \mathbb Z$. Then $H \cong K$ but $\lvert G / H \rvert = 2$ and $\lvert G / K \rvert = 3$, hence $G / H \not\cong G / K$.