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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?.

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Yes, I did. Now can anyone help me? – Muniain Dec 16 '12 at 15:22
I think the answer to this question depends on what $\oplus$ and $\times$ mean to you (and to your teacher, and to Wolf Holzmann). Please could you explain? – user108903 Dec 16 '12 at 16:10

In the case at hand, the notations do indeed mean precisely the same. Usually, one uses $\times$ when considering the product of arbitrary groups and $\oplus$ when considering abelian groups (as these are $\mathbb{Z}$-modules), and the direct product ($\times$) and direct sum ($\oplus$) of $\mathbb{Z}$-modules coincide as long as we only take a finite number of modules.

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