$H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then are $H,K$ equal or atleast isomorphic?

Question

Let $H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then is it true that $H=K$ ? Or atleast $H \cong K$ ? ( If $G$ were finite then it would be trivially true , so I am asking for infinite $G$ ) I know the standard examples here Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$? , but there the condition $H \subseteq K$ is not imposed . Please help . Thnaks in advance

Answer 1

Let $G = \bigoplus_{n \geq 0} \mathbb{Z}$ and consider the subgroups $K = \mathbb{Z} \oplus \mathbb{Z} \oplus \bigoplus_{n \geq 2} 0$ and $H = \mathbb{Z} \oplus \bigoplus_{n \geq 1} 0$. Then $H \subseteq K \subseteq G$ and $G/H \cong G \cong G/K$, but $H \cong \mathbb{Z} \ncong \mathbb{Z}^2 \cong K$.