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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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My advice: first thing do something really radical with that accept rate of yours. Many people won't feel like trying to help you if they think you don't appreciate their efforts to help you. –  DonAntonio Dec 16 '12 at 15:00
    
Yes, I did. Now can anyone help me? –  Firmino 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
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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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