$H,K$ be subgroups of an infinite abelian group $G$ such that $K \subseteq H$ and $G \cong K$ , then is it true that $G\cong H$ ?

Question

Let $H,K$ be subgroups of an infinite abelian group $G$ such that $K \subseteq H$ and $G \cong K$ , then is it true that $G\cong H$ ?

( see related $C$ be a subring of $B$ which is again a subring of $A$ , let $A,B,C$ be Noetherian and $A \cong C$ , then is $A \cong B$? )

Answer 1

I don't think so. How about $\bigoplus\limits_{i=1}^\infty \mathbb Q \subset \bigoplus_{i=1}^\infty \mathbb Q \oplus \mathbb Z \subseteq \bigoplus\limits_{i=1}^\infty \mathbb Q\oplus\mathbb Q$?

Answer 2

No. Let $G=\bigoplus_{i\in\mathbb{N}}(\mathbb{Q}, +)$, let $K=\{g\in G: g(0)=g(2)=g(4)=...=0\}$ (that is, $K$ is "half" of $G$), and let $$H=\{g\in G: g(2)=g(4)=g(6)=...=0\mbox{ and }g(0)\in\mathbb{Z}\}.$$ That is, $K\cong G$ and $H\cong G\oplus\mathbb{Z}$.