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I am curious if there is any structure result for an infinitely generated abelian group $G$?

In particular, is the following naive guess true?

$$G\cong \bigoplus_{i\in A}\mathbb{Z}\oplus \bigoplus_{i\in B }\mathbb{Z}_{q_i}$$

where $A$, $B$ are some infinite sets, and $\mathbb{Z}_{qi}$ are primary cyclic groups.

Or perhaps it is not a direct sum, i.e.

$$G\cong \prod_{i\in A}\mathbb{Z}\times \prod_{i\in B }\mathbb{Z}_{q_i}$$?

Thanks a lot.

yoyostein
  • 19,608
  • I think it's not true if you use the implied addition on the right hand side of your equation. I don't think you can write a group isomorphism between the additive group $(\mathbb{Q},+)$ and a direct sum of copies of $\mathbb{Z}$. – desiigner Jul 16 '19 at 03:30
  • This has been asked multiple times. See here, and here. – Arturo Magidin Jul 16 '19 at 04:25
  • @desiigner You are correct: every quotient of $\mathbb{Q}$ is finite, so we would need $\mathbb{Q}\cong X\times Y$ for some finite group $X$. However, every element of $\mathbb{Q}$ has infinite order so we cannot embed any finite group $X$ into $\mathbb{Q}$, a contradiction. – user1729 Jul 16 '19 at 14:20

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