The number of internal direct summands of a finitely generated abelian group

Question

Let $G$ be a group. A normal subgroup $H$ of $G$ is called an internal direct summand of $G$ if there exists a normal subgroup $K$ of $G$ so that $G=HK$ and $H\cap K=\{ 1\}$.

My question is that:

Is the number of internal direct summands of a finitely generated abelian group finite?

Answer 1

It's not true even for $G=\Bbb Z^2$. One can take $H=\left<(1,a)\right>$ for any integer $a$ (with $K=\left<(0,1)\right>$).