I am studying Finitely Generated Abelian Groups. Now I find a material of Wolf Holzmann abelian.pdf
I have a question in this material: Can I replace all notation $\oplus$ by $\times$?. More precisely, Can I replace $K\cong d_1 \mathbb{Z}\oplus \ldots \oplus d_r \mathbb{Z}$ by $K\cong d_1 \mathbb{Z}\times \ldots \times d_r \mathbb{Z}$, and $G\cong \mathbb{Z}/d_1 \mathbb{Z}\oplus \ldots \oplus \mathbb{Z}/d_r \mathbb{Z}\oplus \mathbb{Z}\oplus\ldots\oplus\mathbb{Z}$ by $G\cong \mathbb{Z}/d_1 \mathbb{Z}\times \ldots \times \mathbb{Z}/d_r \mathbb{Z}\times \mathbb{Z}\times\ldots\times\mathbb{Z}$.
I am very confused when I have a seminar about this topic, my presentation almost based on this material of Wolf Holzmann, but my teacher said that the fact $K\cong d_1 \mathbb{Z}\times \ldots \times d_r \mathbb{Z}$ is not true.
Can anyone explain the fact above true or wrong?.