Hw11 Problem 10

Let $H, K \unlhd G$. Give an example showing we may have $H \simeq K$ while $G / H$ is not isomorphic to $G / K$.
Let $G = \mathbb{Z}_2 \times \mathbb{Z}_4$, $H = \langle (1, 0) \rangle$, and $K = \langle (0, 2) \rangle$ where $H \simeq K$. Then $G / H \simeq \mathbb{Z}_{4}$ but $G / K \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$.